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Transcript

Crystallography Made
Crystal Clear
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Crystallography Made
Crystal Clear
A Guide for Users of Macromolecular Models
Third Edition
Gale Rhodes
Chemistry Department
University of Southern Maine
Portland, Maine
CMCC Home Page:
www.usm.maine.edu/∼rhodes/CMCC
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Rhodes, Gale.
Crystallography made crystal clear / Gale Rhodes.– 3rd ed.
p. cm. – (Complementary science series)
Includes bibliographical references and index.
ISBN 0-12-587073-6 (alk. paper)
1. X-ray crystallography. 2. Macromolecules–Structure. 3. Proteins–Structure. I.
Title. II. Series.
QP519.9.X72R48 2006
547′.7–dc22
2005057239
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A catalogue record for this book is available from the British Library.
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ISBN 10: 0-12-587073-6
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06 07 08 09 10 8 7 6 5 4 3 2 1
Like everything, for Pam.

Contents
Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
1 Model and Molecule 1
2 An Overview of Protein Crystallography 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Obtaining an image of a microscopic object . . . . . . . . . . . . . . . . . . 8
2.1.2 Obtaining images of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 A thumbnail sketch of protein crystallography . . . . . . . . . . . . . . . . 9
2.2 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 The nature of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Growing crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Collecting X-ray data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Simple objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.2 Arrays of simple objects: Real and reciprocal lattices . . . . . . . . . . . 16
2.4.3 Intensities of reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.4 Arrays of complex objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.5 Three-dimensional arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Coordinate systems in crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 The mathematics of crystallography: A brief description . . . . . . . . . . . . . . . 20
2.6.1 Wave equations: Periodic functions . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6.2 Complicated periodic functions: Fourier series and sums . . . . . . . . 23
2.6.3 Structure factors: Wave descriptions of X-ray reflections . . . . . . . . 24
2.6.4 Electron-density maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.5 Electron density from structure factors . . . . . . . . . . . . . . . . . . . . . . 27
2.6.6 Electron density from measured reflections . . . . . . . . . . . . . . . . . . . 28
2.6.7 Obtaining a model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
ix x Contents
3 Protein Crystals 31
3.1 Properties of protein crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Size, structural integrity, and mosaicity . . . . . . . . . . . . . . . . . . . . . . 31
3.1.3 Multiple crystalline forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.4 Water content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Evidence that solution and crystal structures are similar . . . . . . . . . . . . . . . . 35
3.2.1 Proteins retain their function in the crystal . . . . . . . . . . . . . . . . . . . 35
3.2.2 X-ray structures are compatible with other structural evidence . . . 36
3.2.3 Other evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Growing protein crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.2 Growing crystals: Basic procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.3 Growing derivative crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.4 Finding optimal conditions for crystal growth . . . . . . . . . . . . . . . . . 41
3.4 Judging crystal quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.5 Mounting crystals for data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Collecting Diffraction Data 49
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Geometric principles of diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1 The generalized unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2 Indices of the atomic planes in a crystal . . . . . . . . . . . . . . . . . . . . . 50
4.2.3 Conditions that produce diffraction: Bragg’s law . . . . . . . . . . . . . . 55
4.2.4 The reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.5 Bragg’s law in reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.6 Number of measurable reflections . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.7 Unit-cell dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.8 Unit-cell symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3 Collecting X-ray diffraction data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 X-ray sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.3 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.4 Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.5 Scaling and postrefinement of intensity data . . . . . . . . . . . . . . . . . . 85
4.3.6 Determining unit-cell dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3.7 Symmetry and the strategy of collecting data . . . . . . . . . . . . . . . . . 88
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 From Diffraction Data to Electron Density 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Fourier sums and the Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 One-dimensional waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.2 Three-dimensional waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Contents xi
5.2.3 The Fourier transform: General features . . . . . . . . . . . . . . . . . . . . . 96
5.2.4 Fourier this and Fourier that: Review . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3 Fourier mathematics and diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Structure factor as a Fourier sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Electron density as a Fourier sum . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3.3 Computing electron density from data . . . . . . . . . . . . . . . . . . . . . . . 100
5.3.4 The phase problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Meaning of the Fourier equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4.1 Reflections as terms in a Fourier sum: Eq. (5.18) . . . . . . . . . . . . . . 101
5.4.2 Computing structure factors from a model: Eq. (5.15)
and Eq. (5.16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.3 Systematic absences in the diffraction pattern: Eq. (5.15) . . . . . . . 105
5.5 Summary: From data to density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 Obtaining Phases 109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Two-dimensional representation of structure factors . . . . . . . . . . . . . . . . . . . 112
6.2.1 Complex numbers in two dimensions . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.2 Structure factors as complex vectors . . . . . . . . . . . . . . . . . . . . . . . . 112
6.2.3 Electron density as a function of intensities and phases . . . . . . . . . 115
6.3 Isomorphous replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.1 Preparing heavy-atom derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.2 Obtaining phases from heavy-atom data . . . . . . . . . . . . . . . . . . . . . 119
6.3.3 Locating heavy atoms in the unit cell . . . . . . . . . . . . . . . . . . . . . . . . 124
6.4 Anomalous scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.2 Measurable effects of anomalous scattering . . . . . . . . . . . . . . . . . . 128
6.4.3 Extracting phases from anomalous scattering data . . . . . . . . . . . . . 130
6.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.4.5 Multiwavelength anomalous diffraction phasing . . . . . . . . . . . . . . . 133
6.4.6 Anomalous scattering and the hand problem . . . . . . . . . . . . . . . . . . 135
6.4.7 Direct phasing: Application of methods from small-molecule
crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Molecular replacement: Related proteins as phasing models . . . . . . . . . . . . 136
6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.5.2 Isomorphous phasing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5.3 Nonisomorphous phasing models . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.5.4 Separate searches for orientation and location . . . . . . . . . . . . . . . . . 139
6.5.5 Monitoring the search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.5.6 Summary of molecular replacement . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.6 lterative improvement of phases (preview of Chapter 7) . . . . . . . . . . . . . . . . 143
7 Obtaining and Judging the Molecular Model 145
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 lterative improvement of maps and models—overview . . . . . . . . . . . . . . . . 146
xii Contents
7.3 First maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3.1 Resources for the first map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3.2 Displaying and examining the map . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3.3 Improving the map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.4 The Model becomes molecular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4.1 New phases from the molecular model . . . . . . . . . . . . . . . . . . . . . . 153
7.4.2 Minimizing bias from the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4.3 Map fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.5 Structure refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.5.1 Least-squares methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.5.2 Crystallographic refinement by least squares . . . . . . . . . . . . . . . . . . 160
7.5.3 Additional refinement parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.5.4 Local minima and radius of convergence . . . . . . . . . . . . . . . . . . . . 162
7.5.5 Molecular energy and motion in refinement . . . . . . . . . . . . . . . . . . 163
7.5.6 Bayesian methods: Ensembles of models . . . . . . . . . . . . . . . . . . . . 164
7.6 Convergence to a final model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.6.1 Producing the final map and model . . . . . . . . . . . . . . . . . . . . . . . . . 168
7.6.2 Guides to convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.7 Sharing the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8 A User’s Guide to Crystallographic Models 179
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.2 Judging the quality and usefulness of the refined model . . . . . . . . . . . . . . . . 181
8.2.1 Structural parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2.2 Resolution and precision of atomic positions . . . . . . . . . . . . . . . . . 183
8.2.3 Vibration and disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.2.4 Other limitations of crystallographic models . . . . . . . . . . . . . . . . . . 187
8.2.5 Online validation tools: Do it yourself! . . . . . . . . . . . . . . . . . . . . . . 189
8.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.3 Reading a crystallography paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.3.2 Annotated excerpts of the preliminary (8/91) paper . . . . . . . . . . . . 193
8.3.3 Annotated excerpts from the full structure-determination
(4/92) paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
9 Other Diffraction Methods 211
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.2 Fiber diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
9.3 Diffraction by amorphous materials (scattering) . . . . . . . . . . . . . . . . . . . . . . 219
9.4 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9.5 Electron diffraction and cryo-electron microscopy . . . . . . . . . . . . . . . . . . . . 227
9.6 Laue diffraction and time-resolved crystallography . . . . . . . . . . . . . . . . . . . . 231
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Contents xiii
10 Other Kinds of Macromolecular Models 237
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
10.2 NMR models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10.2.2 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
10.2.3 Assigning resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.2.4 Determining conformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
10.2.5 PDB files for NMR models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.2.6 Judging model quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.3 Homology models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
10.3.2 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
10.3.3 Databases of homology models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
10.3.4 Judging model quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
10.4 Other theoretical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
11 Tools for Studying Macromolecules 269
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
11.2 Computer models of molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
11.2.1 Two-dimensional images from coordinates . . . . . . . . . . . . . . . . . . . 269
11.2.2 Into three dimensions: Basic modeling operations . . . . . . . . . . . . . 270
11.2.3 Three-dimensional display and perception . . . . . . . . . . . . . . . . . . . 272
11.2.4 Types of graphical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
11.3 Touring a molecular modeling program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
11.3.1 Importing and exporting coordinate files . . . . . . . . . . . . . . . . . . . . . 276
11.3.2 Loading and saving models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11.3.3 Viewing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11.3.4 Editing and labeling the display . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
11.3.5 Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
11.3.6 Measuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
11.3.7 Exploring structural change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
11.3.8 Exploring the molecular surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
11.3.9 Exploring intermolecular interactions: Multiple models . . . . . . . . . 286
11.3.10 Displaying crystal packing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
11.3.11 Building models from scratch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
11.3.12 Scripts and macros: Automating routine structure analysis . . . . . . . 287
11.4 Other tools for studying structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
11.4.1 Tools for structure analysis and validation . . . . . . . . . . . . . . . . . . . . 288
11.4.2 Tools for modeling protein action . . . . . . . . . . . . . . . . . . . . . . . . . . 290
11.5 Final note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Appendix Viewing Stereo Images 293
Index 295

Preface to the Third Edition
Three prefaces make quite a moat to dig around this little castle of crystallography,
so if you are tempted to get inside more quickly by skipping the introductions, at
least take a quick look at the Preface to the First Edition, which still stands as the
best guide to my aims in writing this book, and to your most efficient use of it.
The second and this third preface are sort of like release notes for new versions of
software. They are mostly about changes from previous editions. In brief, in the first
edition, I taught myself the basics of crystallography by writing about it, drawing
on a year or two of sabbatical experience in the field, preceded by quite a few
years of enthusiastic sideline observation. In the second edition, I added material
on other diffraction methods (neutron and electron diffraction, for instance) and
other kinds of models (NMR and homology), and updated the crystallography
only superficially. This time, the main subject, macromolecular crystallography,
got almost all of my attention, and I hope the result is clearer, more accurate,
and more up-to-date. One thing for sure, it’s more colorful. Modern publishing
methods have made color more affordable, and I found it very liberating to use
color wherever I thought it would make illustrations easier to understand.
Just before writing this edition, I took a course, “X-Ray Methods in Structural
Biology,” at Cold Spring Harbor Laboratory. Professor David Richardson of Duke
University, one of many accomplished crystallographers who contributed to this
excellent course, seemed surprised to find me among the students there. When
I told him I was looking for help in updating my book, he quickly offered this
advice: “After taking this course, you will be tempted to complicate your book.
Don’t.” I tried to keep David’s words in mind as I worked on this edition. My
main goal was to make the crystallography chapters more timely, accurate, and
clear, by weeding out withered ideas and methods, replanting with descriptions of
important new developments, culling out errors that readers of previous editions
kindly took the trouble to point out, and adding only those new ideas and details
xv xvi Preface to the Third Edition
that will truly help you to get a feeling for how crystallography produces models
of macromolecules.
So what is new in crystallography since the last edition? First of all, it is faster
than ever. Three developments—more powerful multiwavelength X-ray sources,
low-temperature crystallography, and fast molecular biology methods for producing just about any protein and abundant variants of it—have set off an explosion
of new structures. Automation has reached into every nook and cranny of the field,
to the point that “high-throughput” crystallography is giving us models of proteins faster than we can figure out their functions. Just now I searched the Protein
Data Bank (PDB), the world’s primary repository for macromolecular models, for
entries in which the protein function is listed as “unknown.” I found almost 800
entries, many also marked “structural genomics.” This means that the structures
were determined as part of sweeping efforts, a prominent current one called the
Protein Structure Initiative, to determine the structure of every protein in sight.
Well, not quite; a research group participating in this effort usually focuses on
a specific organism, like the tuberculosis bacterium, and works to determine the
structures of proteins from every open reading frame (ORF) in its genome. For
the first five years of this initiative, participating groups emphasized developing
the technology to automate every step of structure determination: expressing and
purifying the proteins, crystallizing them, collecting X-ray data, solving the structures, and refining the models. As I write these words, they are just beginning
to turn their attention to cranking out new structures, although there are debates
about whether the technology is ready for mass production. The goal of the Protein
Structure Initiative is 10,000 structures by 2010, and even if the initiative falls a
few thousand short, high-throughput crystallography is here to stay. You might
not need to determine the structure of that protein whose action you just detected
for the first time. It may already be in the Protein Data Bank, marked “structural
genomics, unknown function.”
Second, if the structure of that new protein of yours is not already lurking in
the PDB, you might be able to determine it yourself. Methods of crystallization,
data collection, and structure determination are more transparent and user-friendly
than ever. Some of my fellow students at Cold Spring Harbor had already determined protein structures before they arrived, guided by modern crystallization
screens, automated data collection, fast new software, and usually, a post-doctoral
colleague with some crystallography experience. Now they wanted to know what
goes on under the hood, in case a future venture stalls and requires an expert
mechanic to make a few adjustments. If the next steps in your research would
profit from your knowing the structure of a new protein, consider adding crystallography to your research skills. It’s no longer necessary to make it your whole
career.
Third, even if you never do crystallography, you are in a better position than
ever to use models wisely. Powerful software and online tools allow you to make
sound decisions about whether a model will support the conclusions you would
like to draw from it, and with greater ease and clarity than ever. Today’s validation
Preface to the Third Edition xvii
tools can tell you a great deal about model quality, even if the original model
publication is very sketchy on experimental methods and results.
Although the pace of crystallography is quickening toward mass production, I
still wrote this edition about crystallography the old-fashioned way, one model
at a time, with attention to the details of every step. Why? Because these details
are essential to understanding crystallography and to assessing the strengths and
limitations of each model. If you know the whole story, from purified protein to
refined model, then you have a better understanding of the model and all that it
might tell you. And if you try crystallography yourself, you will know something
about the decisions the software is making for you, and when to ask if there are
alternative routes to a model, perhaps better ones.
Many people helped me with this edition. At the top of the list are the instructors at the Cold Spring Harbor course who, for sixteen years, have offered what
many crystallographers tout as the best classroom and hands-on diffraction training
session on the planet—2.5 weeks, 9 AM to 9 PM, packed with labs, lectures, and
computer tutorials, with homework for your spare time. The course gave me great
confidence in choosing what to keep, what to revise, and what to throw out. The
four organizers and main instructors, Bill Furey, Gary Gilliland, Alex McPherson,
and Jim Pflugrath, were patient, helpful, and brimming with good ideas about how
to do and teach crystallography. My fifteen CSH classmates (the oldest among
them about half my age) were friendly, helpful, and inspirational. It was sad to
realize that my presence in the course had displaced another one like them.
Thanks also to readers who pointed out errors in the first two editions, and
to reviewers for their careful readings and helpful suggestions. Thanks to USM
colleagues for granting me a sabbatical leave for this project (again!). Thanks
to my wife, Pam, for proofreading, editing, and helpful suggestions on text and
figures. I had to pay her, but she finally read my book. Thanks to the staff at
Elsevier/Academic Press for guiding my words and pictures through the international maze of operations needed to get a book to market, and especially to Jeremy
Hayhurst for talking me into doing this again, and to Jeff Freeland for overseeing
production. Finally, thanks to all my students for constantly reminding me that
teachers, whether they teach by lecturing, writing books, or building web pages,
have more fun than people.
Gale Rhodes
Portland, Maine
May 2005

Preface to the Second Edition
The first edition of this book was hardly off the press before I was kicking myself
for missing some good bets on how to make the book more helpful to more people.
I am thankful that heartening acceptance and wide use of the first edition gave me
another crack at it, even before much of the material started to show its age. In
this new edition, I have updated the first eight chapters in a few spots and cleaned
up a few mistakes, but otherwise those chapters, the soul of this book’s argument,
are little changed. I have expanded and modernized the last chapter, on viewing
and studying models with computers, bringing it up-to-date (but only fleetingly,
I am sure) with the cyberworld to which most users of macromolecular models
now turn to pursue their interests, and with today’s desktop computers—sleek,
friendly, cheap, and eminently worthy successors to the five-figure workstations
of the eighties.
My main goal, as outlined in the Preface to the First Edition, which appears
herein, is the same as before: to help you see the logical thread that connects those
mysterious diffraction patterns to the lovely molecular models you can display
and play with on your personal computer. An equally important aim is to inform
you that not all crystallographic models are perfect and that cartoon models do not
exhaust the usefulness of crystallographic analysis. Often there is both less and
more than meets the eye in a crystallographic model.
So what is new here? Two chapters are entirely new. The first one is “Other
Diffraction Methods.” In this chapter (the one I should have thought of the first
time), I use your new-found understanding of X-ray crystallography to build an
overview of other techniques in which diffraction gives structural clues. These
methods include scattering of light, X-rays, and neutrons by powders and solutions; diffraction by fibers; crystallography using neutrons and electrons; and
time-resolved crystallography using many X-ray wavelengths at the same time.
These methods sound forbidding, but their underlying principles are precisely the
same as those that make the foundation of single-crystal X-ray crystallography.
xix
xx Preface to the Second Edition
The need for the second new chapter, “Other Types of Models,” was much less
obvious in 1992, when crystallography still produced most of the new macromolecular models. This chapter acknowledges the proliferation of such models
from methods other than diffraction, particularly NMR spectroscopy and homology modeling. Databases of homology models now dwarf the Protein Data Bank,
where all publicly available crystallographic and NMR models are housed. Nuclear
magnetic resonance has been applied to larger molecules each year, with further
expansion just a matter of time. Users must judge the quality of all macromolecular
models, and that task is very different for different kinds of models. By analogies
with similar aids for crystallographic models, I provide guidance in quality control,
with the hope of making you a prudent user of models from all sources.
Neither of the new chapters contains full or rigorous treatments of these “other”
methods. My aim is simply to give you a useful feeling for these methods, for the
relationship between data and structures, and for the pitfalls inherent in taking any
model too literally.
By the way, some crystallographers and NMR spectroscopists have argued for
using the term structure to refer to the results of experimental methods, such
as X-ray crystallography and NMR, and the term model for theoretical models
such as homology models. To me, molecular structure is a book forever closed to
our direct view, and thus never completely knowable. Consequently, I am much
more comfortable with the term model for all of the results of attempts to know
molecular structure. I sometimes refer loosely to a model as a structure and to
the process of constructing and refining models as structure determination, but in
the end, no matter what the method, we are trying to construct models that agree
with, and explain, what we know from experiments that are quite different from
actually looking at structure. So in my view, models, experimental or theoretical
(an imprecise distinction itself), represent the best we can do in our diverse efforts
to know molecular structure.
Many thanks to Nicolas Guex for giving to me and to the world a glorious
free tool for studying proteins—Swiss-PdbViewer, since renamed DeepView—
along with plenty of support and encouragement for bringing macromolecular
modeling to my undergraduate biochemistry students; for his efforts to educate me
about homology modeling; for thoughtfully reviewing the sections on homology
modeling; and for the occasional box of liqueur-loaded Swiss chocolates (whoa!).
Thanks to Kevin Cowtan, who allowed me to adapt some of the clever ideas from
his Book of Fourier to my own uses, and who patiently computed image after
image as I slowly iterated toward the final product. Thanks to Angela Gronenbom,
Duncan McRee, and John Ricci for thorough, thoughtful, and helpful reviews of
the manuscript. Thanks to Jonathan Cooper and Martha Teeter, who found and
reported subtle and interesting errors lurking within figures in the first edition.
Thanks to all those who provided figures—you are acknowledged alongside the
fruits of your labors. Thanks to Emelyn Eldredge at Academic Press for inducing
me to tiptoe once more through the minefields of Microsoft Word to update this
Preface to the Second Edition xxi
little volume, and to Joanna Dinsmore for a smooth trip through production. Last
and most, thanks to Pam for generous support, unflagging encouragement, and
amused tolerance for over a third of a century. Time certainly does fly when we’re
having fun.
Gale Rhodes
Portland, Maine
March 1999
xxiv Preface to the First Edition
and limitations of crystallographic models will enable you to use them wisely and
effectively.
If you are part of my intended audience, I do not believe you need to know, or
are likely to care about, all the gory details of crystallographic methods and all
the esoterica of crystallographic theory. I present just enough about methods to
give you a feeling for the experiments that produce crystallographic data. I present
somewhat more theory, because it underpins an understanding of the nature of a
crystallographic model. I want to help you follow a logical thread that begins with
diffraction data and ends with a colorful picture of a protein model on the screen
of a graphics computer. The novice crystallographer, or the student pondering a
career in crystallography, may find this book a good place to start, a means of
seeing if the subject remains interesting under closer scrutiny. But these readers
will need to consult more extensive works for fine details of theory and method.
I hope that reading this book makes those texts more accessible. I assume that you
are familiar with protein structure, at least at the level presented in an introductory
biochemistry text.
I wish I could teach you about crystallography without using mathematics, simply because so many readers are apt to throw in the towel upon turning the page
and finding themselves confronted with equations. Alas (or hurrah, depending on
your mathematical bent), the real beauty of crystallography lies in the mathematical and geometric relationships between diffraction data and molecular images.
I attempt to resolve this dilemma by presenting no more math than is essential
and taking the time to explain in words what the equations imply. Where possible,
I emphasize geometric explanations over equations.
If you turn casually to the middle of this book, you will see some forbidding
mathematical formulas. Let me assure you that I move to those bushy statements
step-by-step from nearby clearings, making minimum assumptions about your
facility and experience with math. For example, when I introduce periodic functions, I tell you how the simplest of such functions (sines and cosines) “work,” and
then I move slowly from that clear trailhead into the thicker forest of complicated
wave equations that describe X-rays and the molecules that diffract them. When
I first use complex numbers, I define them and illustrate their simplest uses and
representations, sort of like breaking out camping gear in the dry safety of a garage.
Then I move out into real weather and set up a working camp, showing how the
geometry of complex numbers reveals essential information otherwise hidden in
the data. My goal is to help you see the relationships implied by the mathematics,
not to make you a calculating athlete. My ultimate aim is to prove to you that the
structure of molecules really does lie lurking in the crystallographic data—that, in
fact, the information in the diffraction pattern implies a unique structure. I hope
thereby to remove the mystery about how structures are coaxed from data.
If, in spite of these efforts, you find yourself flagging in the most technical
chapters (4 and 7), please do not quit. I believe you can follow the arguments
of these chapters, and thus be ready for the take-home lessons of Chapters 8
and 11, even if the equations do not speak clearly to you. Jacob Bronowski once
described the verbal argument in mathematical writing as analogous to melody

Preface to the First Edition
Most texts that treat biochemistry or proteins contain a brief section or chapter on
protein crystallography. Even the best of such sections are usually mystifying—
far too abbreviated to give any real understanding. In a few pages, the writer
can accomplish little more than telling you to have faith in the method. At the
other extreme are many useful treatises for the would-be, novice, or experienced
crystallographer. Such accounts contain all the theoretical and experimental details
that practitioners must master, and for this reason, they are quite intimidating to
the noncrystallographer. This book lies in the vast and heretofore empty region
between brief textbook sections on crystallography and complete treatments of
the method aimed at the professional crystallographer. I hope there is just enough
here to help the noncrystallographer understand where crystallographic models
come from, how to judge their quality, and how to glean additional information
that is not depicted in the model but is available from the crystallographic study
that produced the model.
This book should be useful to protein researchers in all areas; to students of
biochemistry in general and of macromolecules in particular; to teachers as an auxiliary text for courses in biochemistry, biophysical methods, and macromolecules;
and to anyone who wants an intellectually satisfying understanding of how crystallographers obtain models of protein structure. This understanding is essential for
intelligent use of crystallographic models, whether that use is studying molecular
action and interaction, trying to unlock the secrets of protein folding, exploring
the possibilities of engineering new protein functions, or interpreting the results
of chemical, kinetic, thermodynamic, or spectroscopic experiments on proteins.
Indeed, if you use protein models without knowing how they were obtained, you
may be treading on hazardous ground. For instance, you may fail to use available
information that would give you greater insight into the molecule and its action.
Or worse, you may devise and publish a detailed molecular explanation based on
a structural feature that is quite uncertain. Fuller understanding of the strengths
xxiii
Preface to the First Edition xxv
in music, and thus a source of satisfaction in itself. He likened the equations
to musical accompaniment that becomes more satisfying with repeated listening.
If you follow and retain the melody of arguments and illustrations in Chapters
4 through 7, then the last chapters and their take-home lessons should be useful
to you.
I aim further to enable you to read primary journal articles that announce and
present new protein structures, including the arcane sections on experimental
methods. In most scientific papers, experimental sections are directed primarily
toward those who might use the same methods. In crystallographic papers, however, methods sections contain information from which the quality of the model
can be roughly judged. This judgment should affect your decision about whether
to obtain the model and use it, and whether it is good enough to serve as a guide in
drawing the kinds of conclusions you hope to draw. In Chapter 8, to review many
concepts, as well as to exercise your new skills, I look at and interpret experimental
details in literature reports of a recent structure determination.
Finally, I hope you read this book for pleasure—the sheer pleasure of turning the
formerly incomprehensible into the familiar. In a sense, I am attempting to share
with you my own pleasure of the past ten years, after my mid-career decision to set
aside other interests and finally see how crystallographers produce the molecular
models that have been the greatest delight of my teaching. Among those I should
thank for opening their labs and giving their time to an old dog trying to learn
new tricks are Professors Leonard J. Banaszak, Jens Birktoft, Jeffry Bolin, John
Johnson, and Michael Rossman.
I would never have completed this book without the patience of my wife, Pam,
who allowed me to turn part of our home into a miniature publishing company,
nor without the generosity of my faculty colleagues, who allowed me a sabbatical
leave during times of great economic stress at the University of Southern Maine.
Many thanks to Lorraine Lica, my Acquisitions Editor at Academic Press, who
grasped the spirit of this little project from the very beginning and then held me
and a full corps of editors, designers, and production workers accountable to that
spirit throughout.
Gale Rhodes
Portland, Maine
August 1992
Phase
These still days after frost have let down
the maple leaves in a straight compression
to the grass, a slight wobble from circular to
the east, as if sometime, probably at night, the
wind’s moved that way—surely, nothing else
could have done it, really eliminating the as
if, although the as if can nearly stay since
the wind may have been a big, slow
one, imperceptible, but still angling
off the perpendicular the leaves’ fall:
anyway, there was the green-ribbed, yellow,
flat-open reduction: I just now bagged it up.
A. R. Ammons1
1“Phase,” from The Selected Poems, Expanded Edition by A. R. Ammons. Copyright @ 1987, 1977,
1975, 1974, 1972, 1971, 1970, 1966, 1965, 1964, 1955 by A. R. Ammons. Reprinted by permission
of W. W. Norton & Company, Inc.
 Chapter 1
Model and Molecule
Proteins perform many functions in living organisms. For example, some proteins
regulate the expression of genes. One class of gene-regulating proteins contains
structures known as zinc fingers, which bind directly to DNA. Figure 1.1a shows
a complex composed of a double-stranded DNA molecule and three zinc fingers
from the mouse protein Zif268 (PDB 1zaa).
The protein backbone is shown as a yellow ribbon. The two DNA strands are red
and blue. Zinc atoms, which are complexed to side chains in the protein, are green.
The green dotted lines near the top center indicate two hydrogen bonds in which
nitrogen atoms of arginine-18 (in the protein) share hydrogen atoms with nitrogen
and oxygen atoms of guanine-10 (in the DNA), an interaction that holds the sharing
atoms about 2.8 Å apart. Studying this complex with modern graphics software,
you could zoom in, as in Fig. 1.1b, measure the hydrogen-bond lengths, and find
them to be 2.79 and 2.67 Å. From a closer study, you would also learn that all
of the protein–DNA interactions are between protein side chains and DNA bases;
the protein backbone does not come in contact with the DNA. You could go on to
discover all the specific interactions between side chains of Zif268 and base pairs
of DNA. You could enumerate the additional hydrogen bonds and other contacts
that stabilize this complex and cause Zif268 to recognize a specific sequence of
bases in DNA. You might gain some testable insights into how the protein finds
the correct DNA sequence amid the vast amount of DNA in the nucleus of a cell.
The structure might also lead you to speculate on how alterations in the sequence
of amino acids in the protein might result in affinity for different DNA sequences,
and thus start you thinking about how to design other DNA-binding proteins.
Now look again at the preceding paragraph and examine its language rather than
its content. The language is typical of that in common use to describe molecular
structure and interactions as revealed by various experimental methods, including
single-crystal X-ray crystallography, the primary subject of this book. In fact, this
1
2 Chapter 1 Model and Molecule
Figure 1.1  (a) Divergent stereo image of Zif268/DNA complex (N. P. Pavletich
and C. O. Pabo, Science 252, 809, 1991). (b) Detail showing hydrogen bonding between
arginine-18 of the protein and guanine-10 of the DNA.Atomic coordinates for preparing this
display were obtained from the Protein Data Bank (PDB), which is described in Chapter 7.
The PDB file code is 1zaa. To allow easy access to all models shown in this book, I provide
file codes in this format: PDB 1zaa. Image created by DeepView (formerly called SwissPdbViewer), rendered by POV-Ray. To obtain these programs, see the CMCC home page
at http://www.usm.maine.edu/∼rhodes/CMCC/index.html. For help with viewing stereo
images, see Appendix, page 293.
language is shorthand for more precise but cumbersome statements of what we
learn from structural studies.
First, Fig. 1.1, of course, shows not molecules, but models of molecules, in
which structures and interactions are depicted, not shown. Second, in this specific case, the models are of molecules not in solution, but in the crystalline state,
because the models are derived from analysis of X-ray diffraction by crystals
of the Zif268/DNA complex. As such, these models depict the average structure of somewhere between 1013 and 1015 complexes throughout the crystals that
Chapter 1 Model and Molecule 3
were studied. In addition, the structures are averaged over the time of the X-ray
experiment, which may range from minutes to days.
To draw the conclusions found in the first paragraph requires bringing additional
knowledge to bear upon the graphics image, including a more precise knowledge
of exactly what we learn from X-ray analysis. The same could be said for structural models derived from spectroscopic data or any other method. In short, the
graphics image itself is incomplete. It does not reveal things we may know about
the complex from other types of experiments, and it does not even reveal all that
we learn from X-ray crystallography.
For example, how accurately are the relative positions of atoms known? Are
the hydrogen bonds precisely 2.79 and 2.67 Å long, or is there some tolerance
in those figures? Is the tolerance large enough to jeopardize the conclusion that
hydrogen bonds join these atoms? Further, do we know anything about how rigid
this complex is? Do parts of these molecules vibrate, or do they move with respect
to each other? Still further, in the aqueous medium of the cell, does this complex
have the same structure as in the crystal, which is a solid? As we examine this
model, are we really gaining insight into cellular processes? Two final questions
may surprise you: First, does the model fully account for the chemical composition
of the crystal? In other words, are any of the known contents of the crystal missing
from the model? Second, does the crystallographic data suggest additional crystal
contents that have not been identified, and thus are not shown in the model?
The answers to these questions are not revealed in the graphics image, which
is more akin to a cartoon than to a molecule. Actually, the answers vary from one
model to the next, and from one region of a model to another region, but they
are usually available to the user of crystallographic models. Some of the answers
come from X-ray crystallography itself, so the crystallographer does not miss or
overlook them. They are simply less accessible to the noncrystallographer than is
the graphics image.
Molecular models obtained from crystallography are in wide use as tools for
revealing molecular details of life processes. Scientists use models to learn how
molecules “work”: how enzymes catalyze metabolic reactions, how transport proteins load and unload their molecular cargo, how antibodies bind and destroy
foreign substances, and how proteins bind to DNA, perhaps turning genes on and
off. It is easy for the user of crystallographic models, being anxious to turn otherwise puzzling information into a mechanism of action, to treat models as everyday
objects seen as we see clouds, birds, and trees. But the informed user of models
sees more than the graphics image, recognizing it as a static depiction of dynamic
objects, as the average of many similar structures, as perhaps lacking parts that are
present in the crystal but not revealed by the X-ray analysis, as perhaps failing to
show as-yet unidentified crystal contents, and finally, as a fallible interpretation of
data. The informed user knows that the crystallographic model is richer than the
cartoon.
In the following chapters, I offer you the opportunity to become an informed
user of crystallographic models. Knowing the richness and limitations of
models requires an understanding of the relationship between data and structure.
4 Chapter 1 Model and Molecule
In Chapter 2, I give an overview of this relationship. In Chapters 3 through 7, the
heart of the crystallography in this book, I simply expand Chapter 2 in enough detail
to produce an intact chain of logic stretching from diffraction data to final model.
Topics come in roughly the same order as the tasks that face a crystallographer
pursuing an important structure.
As a practical matter, informed use of a model requires evaluating its quality,
which may entail using online model validation tools to assess model quality, as
well as reading the crystallographic papers and data files that report the new structure, in order to extract from them criteria of model quality. In Chapter 8, I discuss
these criteria and provide guided exercises in extracting them from model files
themselves and from the literature. Chapter 8 includes an annotated version of a
published structure determination and its supporting data, as well as an introduction to online validation tools. Equipped with the background of previous chapters
and experienced with the real-world exercises of using validation tools and taking
a guided tour through a recent publication, you should be able to read new structure publications in the scientific literature, understand how the structures were
obtained, and be aware of just what is known—and what is still unknown—about
the molecules under study. Then you should be better equipped to use models
wisely.
Chapter 9, “Other Kinds of Macromolecular Methods,” builds on your understanding of X-ray crystallography to help you understand other methods in which
diffraction provides insights into the structure of large molecules. These methods include fiber diffraction, neutron diffraction, electron diffraction, and various
forms of X-ray spectroscopy. These methods often seem very obscure, but their
underlying principles are similar to those of X-ray crystallography.
In Chapter 10, “Other Kinds of Macromolecular Models,” I discuss alternative
methods of structure determination: NMR spectroscopy and various forms of theoretical modeling. Just like crystallographic models, NMR and theoretical models
are sometimes more, sometimes less, than meets the eye. A brief description of
how these models are obtained, along with some analogies among criteria of quality for various types of models, can help make you a wiser user of all types of
models.
For new or would-be users of models, I present in Chapter 11 an introduction to
molecular modeling, demonstrating how modern graphics programs allow users
to display and manipulate models and to perform powerful structure analysis, as
well as model validation, on desktop computers. I also provide information on
how to use the World Wide Web to obtain graphics programs and learn how to
use them. Finally, I introduce you to the Protein Data Bank (PDB), a World Wide
Web resource from which you can obtain most of the available macromolecular
models.
There is an additional chapter that does not lie between the covers of this book.
It is the Crystallography Made Crystal Clear (CMCC) home page on the World
Wide Web at www.usm.maine.edu/∼rhodes/CMCC. This web site is devoted to
making sure that you can find all the Internet resources mentioned here. Because
even major Internet resources and addresses may change (the Protein Data Bank
Chapter 1 Model and Molecule 5
moved while I was writing the second edition of this book), I include only one web
address in this book. For all web resources that I describe, I refer you to the CMCC
home page. At that web address, I maintain links to all resources mentioned here
or, if they disappear or change markedly, to new ones that serve the same or similar
functions. For easy reference, the address of the CMCC home page is shown on
the cover and title page of this book.
Today’s scientific textbooks and journals are filled with stories about the molecular processes of life. The central character in these stories is often a protein
or nucleic acid molecule, a thing never seen in action, never perceived directly.
We see models of molecules in books and on computer screens, and we tend to treat
them as everyday objects accessible to our normal perceptions. In fact, models are
hard-won products of technically difficult data collection and powerful but subtle
data analysis. And they are richer and more informative than any single image, or
even a rotating computer image, can convey. This book is concerned with where
our models of structure come from and how to use them wisely.
 Chapter 2
An Overview of Protein
Crystallography
2.1 Introduction
The most common experimental means of obtaining a detailed model of a large
molecule, allowing the resolution of individual atoms, is to interpret the diffraction of X-rays from many identical molecules in an ordered array like a crystal.
This method is called single-crystal X-ray crystallography. As of January 2005,
the Protein Data Bank (PDB), the world’s largest repository of macromolecular
models obtained from experimental data (called experimental models), contains
roughly 25,000 protein and nucleic-acid models determined by X-ray crystallography. In addition, the PDB holds roughly 4500 models, mostly proteins of fewer
than 200 residues, that have been solved by nuclear magnetic resonance (NMR)
spectroscopy, which provides a model of the molecule in solution, rather than
in the crystalline state. (Because many proteins appear in multiple forms—for
example, wild types and mutants, or solo and also as part of protein-ligand or
multiprotein complexes—the number of unique proteins represented in the PDB
is only a fraction of the almost 30,000 models.) Finally, there are theoretical
models, either built by analogy with the structures of known proteins having similar sequence, or based on simulations of protein folding. (Theoretical models are
available from databases other than the PDB.) All methods of obtaining models
have their strengths and weaknesses, and they coexist happily as complementary
methods. One of the goals of this book is to make users of crystallographic models
aware of the strengths and weaknesses of X-ray crystallography, so that users’
expectations of the resulting models are in keeping with the limitations of crystallographic methods. Chapter 10 provides, in brief, complementary information
about other types of models.
7
8 Chapter 2 An Overview of Protein Crystallography
In this chapter, I provide a simplified overview of how researchers use the technique of X-ray crystallography to obtain models of macromolecules. Chapters 3
through 8 are simply expansions of the material in this chapter. In order to keep
the language simple, I will speak primarily of proteins, but the concepts I describe
apply to all macromolecules and macromolecular assemblies that possess ordered
structure, including carbohydrates, nucleic acids, and nucleoprotein complexes
like ribosomes and whole viruses.
2.1.1 Obtaining an image of a microscopic object
When we see an object, light rays bounce off (are diffracted by) the object and enter
the eye through the lens, which reconstructs an image of the object and focuses it
on the retina. In a simple microscope, an illuminated object is placed just beyond
one focal point of a lens, which is called the objective lens. The lens collects light
diffracted from the object and reconstructs an image beyond the focal point on the
opposite side of the lens, as shown in Fig. 2.1.
For a simple lens, the relationship of object position to image position in Fig. 2.1
is (OF)(IF ′) = (FL)(F ′L). Because the distances FL and F ′L are constants
(but not necessarily equal) for a fixed lens, the distance OF is inversely proportional to the distance IF ′. Placing the object just beyond the focal point F results
in a magnified image produced at a considerable distance from F ′ on the other
side of the the lens, which is convenient for viewing. In a compound microscope,
Figure 2.1  Action of a simple lens. Rays parallel to the lens axis strike the lens and
are refracted into paths passing through a focus (F or F ′). Rays passing through a focus
strike the lens and are refracted into paths parallel to the lens axis. As a result, the lens
produces an image at I of an object at O such that (OF)(IF ′) = (FL)(F ′L).
Section 2.1 Introduction 9
the most common type, an additional lens, the eyepiece, is added to magnify the
image produced by the objective lens.
2.1.2 Obtaining images of molecules
In order for the object to diffract light and thus be visible under magnification, the
wavelength (λ) of the light must be, roughly speaking, no larger than the object.
Visible light, which is electromagnetic radiation with wavelengths of 400–700 nm
(nm= 10−9 m), cannot produce an image of individual atoms in protein molecules,
in which bonded atoms are only about 0.15 nm or 1.5 angstroms (Å = 10−10 m)
apart. Electromagnetic radiation of this wavelength falls into the X-ray range, so
X-rays are diffracted by even the smallest molecules. X-ray analysis of proteins
seldom resolves the hydrogen atoms, so the protein models described in this book
include elements on only the second and higher rows of the periodic table. The
positions of all hydrogen atoms can be deduced on the assumption that bond
lengths, bond angles, and conformational angles in proteins are just like those in
small organic molecules.
Even though individual atoms diffract X-rays, it is still not possible to produce a focused image of a single molecule, for two reasons. First, X-rays cannot
be focused by lenses. Crystallographers sidestep this problem by measuring the
directions and strengths (intensities) of the diffracted X-rays and then using a
computer to simulate an image-reconstructing lens. In short, the computer acts
as the lens, computing the image of the object and then displaying it on a screen
(Fig. 2.2).
Second, a single molecule is a very weak scatterer of X-rays. Most of the X-rays
will pass through a single molecule without being diffracted, so the diffracted
beams are too weak to be detected. Analyzing diffraction from crystals, rather than
individual molecules, solves this problem. A crystal of a protein contains many
ordered molecules in identical orientations, so each molecule diffracts identically,
and the diffracted beams for all molecules augment each other to produce strong,
detectable X-ray beams.
2.1.3 A thumbnail sketch of protein crystallography
In brief, determining the structure of a protein by X-ray crystallography entails
growing high-quality crystals of the purified protein, measuring the directions
and intensities of X-ray beams diffracted from the crystals, and using a computer
to simulate the effects of an objective lens and thus produce an image of the
crystal’s contents, like the small section of a molecular image shown in Fig. 2.3a.
Finally, the crystallographer must interpret that image, which entails displaying it
by computer graphics and building a molecular model that is consistent with the
image (Fig. 2.3b).
The resulting model is often the only product of crystallography that the user
sees. It is therefore easy to think of the model as a real entity that has been directly
observed. In fact, our “view” of the molecule is quite indirect. Understanding just
how the crystallographer obtains models of protein molecules from diffraction
measurements is essential to fully understanding how to use models properly.
10 Chapter 2 An Overview of Protein Crystallography
Figure 2.2  Crystallographic analogy of lens action. X-rays diffracted from the object
are received and measured by a detector. The measurements are fed to a computer, which
simulates the action of a lens to produce a graphics image of the object. Compare Fig. 2.2
with Fig. 2.1 and you will see that to magnify molecules, you merely have to replace the
light bulb with a synchrotron X-ray source (175 feet in diameter), replace the glass lens with
the equivalent of a 5- to 10-megapixel camera, and connect the camera output to a computer
running some of the world’s most complex and sophisticated software. Oh, yes, and you
will need to spend somewhere between a few days and the rest of your life getting your
favorite protein to form satisfactory crystals. No, it’s not quite as simple as microscopy.
2.2 Crystals
2.2.1 The nature of crystals
Under certain circumstances, many molecular substances, including proteins,
solidify to form crystals. In entering the crystalline state from solution, individual
molecules of the substance adopt one or a few identical orientations. The resulting crystal is an orderly three-dimensional array of molecules, held together by
noncovalent interactions. Figure 2.4 depicts such a crystalline array of molecules.
The lines in the figure divide the crystal into identical unit cells. The array of
points at the corners or vertices of unit cells is called the lattice. The unit cell is
the smallest and simplest volume element that is completely representative of the
whole crystal. If we know the exact contents of the unit cell, we can imagine the
whole crystal as an efficiently packed array of many unit cells stacked beside and
on top of each other, more or less like identical boxes in a warehouse.
Section 2.2 Crystals 11
Figure 2.3  (a) Small section of molecular image displayed on a computer.
(b) Image (a) is interpreted by building a molecular model to fit within the image. Computer graphics programs allow the crystallographer to add parts to the model and adjust
their positions and conformations to fit the image. The protein shown here is adipocyte
lipid binding protein (ALBP, PDB 1alb).
From crystallography, we obtain an image of the electron clouds that surround
the molecules in the average unit cell in the crystal. We hope this image will allow
us to locate all atoms in the unit cell. The location of an atom is usually given by a
set of three-dimensional Cartesian coordinates, x, y, and z. One of the vertices (a
lattice point or any other convenient point) is used as the origin of the unit cell’s
coordinate system and is assigned the coordinates x = 0, y = 0, and z = 0,
usually written (0, 0, 0) (Fig. 2.5).
2.2.2 Growing crystals
Crystallographers grow crystals of proteins by slow, controlled precipitation from
aqueous solution under conditions that do not denature the protein. A number of
substances cause proteins to precipitate. Ionic compounds (salts) precipitate proteins by a process called “salting out.” Organic solvents also cause precipitation,
but they often interact with hydrophobic portions of proteins and thereby denature them. The water-soluble polymer polyethylene glycol (PEG) is widely used
12 Chapter 2 An Overview of Protein Crystallography
Figure 2.4  Six unit cells in a crystalline lattice. Each unit cell contains two
molecules of alanine (hydrogen atoms not shown) in different orientations.
Figure 2.5  One unit cell from Fig. 2.4. The position of an atom in the unit cell can
be specified by a set of spatial coordinates x, y, z.
because it is a powerful precipitant and a weak denaturant. It is available in preparations of different average molecular masses, such as PEG 400, with average
molecular mass of 400 daltons.
One simple means of causing slow precipitation is to add denaturant to an aqueous solution of protein until the denaturant concentration is just below that required
to precipitate the protein. Then water is allowed to evaporate slowly, which gently
Section 2.3 Collecting X-ray data 13
raises the concentration of both protein and denaturant until precipitation occurs.
Whether the protein forms crystals or instead forms a useless amorphous solid
depends on many properties of the solution, including protein concentration, temperature, pH, and ionic strength. Finding the exact conditions to produce good
crystals of a specific protein often requires many careful trials, and is perhaps
more art than science. I will examine crystallization methods in Chapter 3.
2.3 Collecting X-ray data
Figure 2.6 depicts the collection of X-ray diffraction data. A crystal is mounted
between an X-ray source and an X-ray detector. The crystal lies in the path of a
narrow beam of X-rays coming from the source. The simplest source is an X-ray
tube, and the simplest detector is X-ray film, which when developed exhibits dark
spots where X-ray beams have impinged. These spots are called reflections because
they emerge from the crystal as if reflected from planes of atoms.
Figure 2.7 shows the complex diffraction pattern of X-ray reflections produced
on a detector by a protein crystal. Notice that the crystal diffracts the source beam
into many discrete beams, each of which produces a distinct reflection on the film.
The greater the intensity of the X-ray beam that reaches a particular position, the
darker the reflection.
Figure 2.6  Crystallographic data collection. The crystal diffracts the source beam
into many discrete beams, each of which produces a distinct spot (reflection) on the film.
The positions and intensities of these reflections contain the information needed to determine
molecular structures.
14 Chapter 2 An Overview of Protein Crystallography
Figure 2.7  Diffraction pattern from a crystal of the MoFe (molybdenum-iron) protein of the enzyme nitrogenase from Clostridium pasteurianum, recorded on film. Notice
that the reflections lie in a regular pattern, but their intensities (darkness of spots) are highly
variable. The hole in the middle of the pattern results from a small metal disk (beam stop)
used to prevent the direct X-ray beam, most of which passes straight through the crystal,
from destroying the center of the film. Photo courtesy of Professor Jeffrey Bolin.
An optical scanner precisely measures the position and the intensity of each
reflection and transmits this information in digital form to a computer for analysis.
The position of a reflection can be used to obtain the direction in which that particular beam was diffracted by the crystal. The intensity of a reflection is obtained by
measuring the optical absorbance of the spot on the film, giving a measure of the
strength of the diffracted beam that produced the spot. The computer program that
reconstructs an image of the molecules in the unit cell requires these two parameters, the relative intensity and direction, for each diffracted beam that produces
a reflection at the detector. The intensity is simply a number that tells how dark
the reflection is in comparison to the others. The beam direction, as I will describe
shortly, is specified by a set of three-dimensional coordinates h, k, and l for each
reflection.
Section 2.4 Diffraction 15
Although film for data collection has almost completely been replaced by devices
that feed diffraction data (positions and intensities of each reflection) directly into
computers, I will continue, in this overview, to speak of the data as if collected
on film because of the simplicity of that format, and because diffraction patterns
are usually published in a form identical to their appearance on film. I will discuss
modern methods of collecting data in Chapter 4.
2.4 Diffraction
2.4.1 Simple objects
You can develop some visual intuition for the information available from X-ray
diffraction by examining the diffraction patterns of simple objects like spheres or
arrays of spheres (Figs. 2.8–2.11). Figure 2.8 depicts diffraction by a single sphere,
shown in cross section on the left. The diffraction pattern, on the right, exhibits
high intensity at the center, and smoothly decreasing intensity as the diffraction
angle increases.1
For now, just accept the observation that diffraction by a sphere produces this
pattern, and think of it as the diffraction signature of a sphere. In a sense, you are
already equipped to do very simple structure determination; that is, you can now
recognize a simple sphere by its diffraction pattern.
Figure 2.8  Sphere (cross-section, on left) and its diffraction pattern (right). Images
for Figs. 2.8–2.11 were generously provided by Dr. Kevin Cowtan.
1The images shown in Figs. 2.8–2.11 are computed, rather than experimental, diffraction patterns.
Computation of these patterns involves use of the Fourier transform (Section 2.6.5).
16 Chapter 2 An Overview of Protein Crystallography
2.4.2 Arrays of simple objects: Real and reciprocal lattices
Figure 2.9 depicts diffraction by spheres in a crystalline array, with a cross section
of the crystalline lattice on the left, and its diffraction pattern on the right.
The diffraction pattern, like that produced by crystalline nitrogenase (Fig. 2.7),
consists of reflections (spots) in an orderly array on the film. The spacing of
the reflections varies with the spacing of the spheres in their array. Specifically,
observe that although the lattice spacing of the crystal is smaller vertically, the
diffraction spacing is smaller horizontally. In fact, there is a simple inverse relationship between the spacing of unit cells in the crystalline lattice, called the real
lattice, and the spacing of reflections in the lattice on the film, which, because of
its inverse relationship to the real lattice, is called the reciprocal lattice.
Because the real lattice spacing is inversely proportional to the spacing of reflections, crystallographers can calculate the dimensions, in angstroms, of the unit cell
of the crystalline material from the spacings of the reciprocal lattice on the X-ray
film (Chapter 4). The simplicity of this relationship is a dramatic example of
how the macroscopic dimensions of the diffraction pattern are connected to the
submicroscopic dimensions of the crystal.
2.4.3 Intensities of reflections
Now look carefully at the intensities of the reflections in Fig. 2.9. Some are intense
(“bright”), whereas others are weak or perhaps missing from the otherwise evenly
spaced pattern. These variations in intensity contain important information. If you
blur your eyes slightly while looking at the diffraction pattern, so that you cannot
see individual spots, you will see the intensity pattern characteristic of diffraction
by a sphere, with lower intensities farther from the center, as in Fig. 2.8. (You just
Figure 2.9  Lattice of spheres (left) and its diffraction pattern (right). If you look at
the pattern and blur your eyes, you will see the diffraction pattern of a sphere. The pattern is
that of the average sphere in the real lattice, but it is sampled at the reciprocal lattice points.
Section 2.4 Diffraction 17
determined your first crystallographic structure.) The diffraction pattern of spheres
in a lattice is simply the diffraction pattern of the average sphere in the lattice, but
this pattern is incomplete. The pattern is sampled at points whose spacings vary
inversely with real-lattice spacings. The pattern of varied intensities is that of the
average sphere because all the spheres contribute to the observed pattern. To put
it another way, the observed pattern of intensities is actually a superposition of the
many identical diffraction patterns of all the spheres.
2.4.4 Arrays of complex objects
This relationship between (1) diffraction by a single object and (2) diffraction by
many identical objects in a lattice holds true for complex objects also. Figure 2.10
depicts diffraction by six spheres that form a planar hexagon, like the six carbon
atoms in benzene. Notice the starlike six-fold symmetry of the diffraction pattern.
Again, just accept this pattern as the diffraction signature of a hexagon of spheres.
(Now you know enough to recognize two simple objects by their diffraction patterns.) Figure 2.11 depicts diffraction by these hexagonal objects in a lattice of the
same dimensions as that in Fig. 2.9.
As before, the spacing of reflections varies reciprocally with lattice spacing, but
if you blur your eyes slightly, or compare Figs. 2.10 and 2.11 carefully, you will see
that the starlike signature of a single hexagonal cluster is present in Fig. 2.11. From
these simple examples, you can see that the reciprocal-lattice spacing (the spacing
of reflections in the diffraction pattern) is characteristic of (inversely related to)
the spacing of identical objects in the crystal, whereas the reflection intensities are
characteristic of the shape of the individual objects. From the reciprocal-lattice
spacing in a diffraction pattern, we can compute the dimensions of the unit cell.
From the intensities of the reflections, we can learn the shape of the individual
molecules that compose the crystal. It is actually advantageous that the object’s
Figure 2.10  A planar hexagon of spheres (left) and its diffraction pattern (right).
18 Chapter 2 An Overview of Protein Crystallography
Figure 2.11  Lattice of hexagons (left) and its diffraction pattern (right). If you look
at the pattern and blur your eyes, you will see the diffraction pattern of a hexagon. The
pattern is that of the average hexagon in the real lattice, but it is sampled at the reciprocal
lattice points.
diffraction pattern is sampled at reciprocal-lattice positions. This sampling reduces
the number of intensity measurements we must take from the film and makes it
easier to program a computer to locate and measure the intensities.
2.4.5 Three-dimensional arrays
Unlike the two-dimensional arrays in these examples, a crystal is a threedimensional array of objects. If we rotate the crystal in the X-ray beam, a different
cross section of objects will lie perpendicular to the beam, and we will see a different diffraction pattern. In fact, just as the two-dimensional arrays of objects I
have discussed are cross sections of objects in the three-dimensional crystal, each
two-dimensional array of reflections (each diffraction pattern recorded on film) is
a cross section of a three-dimensional lattice of reflections. Figure 2.12 shows a
hypothetical three-dimensional diffraction pattern, with the reflections that would
be produced by all possible orientations of a crystal in the X-ray beam.
Notice that only one plane of the three-dimensional diffraction pattern is superimposed on the film. With the crystal in the orientation shown, reflections shown
in the plane of the film (solid spots) are the only reflections that produce spots
on the film. In order to measure the directions and intensities of all additional
reflections (shown as hollow spots), the crystallographer must collect diffraction
patterns from all unique orientations of the crystal with respect to the X-ray beam.
The direct result of crystallographic data collection is a list of intensities for each
point in the three-dimensional reciprocal lattice. This set of data is the raw material
for determining the structures of molecules in the crystal.
(Note: The spatial relationship involving beam, crystal, film, and reflections is
more complex than shown here. I will discuss the actual relationship in Chapter 4.)
Section 2.5 Coordinate systems in crystallography 19
Figure 2.12  Crystallographic data collection, showing reflections measured at one
particular crystal orientation (solid, on the film) and those that could be measured at other
orientations (hollow, within the sphere but not on the film). Each reflection is located
by its three-dimensional coordinates h, k, and l. The relationship between measured and
unmeasured reflections is more complex than shown here (see Chapter 4).
2.5 Coordinate systems in crystallography
Each reflection can be assigned three coordinates or indices in the imaginary threedimensional space of the diffraction pattern. This space, the strange land where
the reflections live, is called reciprocal space. Crystallographers usually use h, k,
and l to designate the position of an individual reflection in the reciprocal space
of the diffraction pattern. The central reflection (the round solid spot at the center
of the film in Fig. 2.12) is taken as the origin in reciprocal space and assigned the
coordinates (h, k, l) = (0, 0, 0), usually written hkl = 000. (The 000 reflection is
not measurable because it is always obscured by X-rays that pass straight through
the crystal, and are blocked by the beam stop.) The other reflections are assigned
whole-number coordinates counted from this origin, so the indices h, k, and l are
integers. Thus the parameters we can measure and analyze in the X-ray diffraction
pattern are (1) the position hkl and (2) the intensity Ihkl of each reflection. The
position of a reflection is related to the angle by which the diffracted beam diverges
20 Chapter 2 An Overview of Protein Crystallography
from the source beam. For a unit cell of known dimensions, the angle of divergence
uniquely specifies the indices of a reflection, as I will show in Chapter 4.
Alternatively, actual distances, rather than reflection indices, can be measured
in reciprocal space. Because the dimensions of reciprocal space are the inverse
of dimensions in the real space of the crystal, distances in reciprocal space are
expressed in the units Å−1 (called reciprocal angstroms). Roughly speaking, the
inverse of the reciprocal-space distance from the origin out to the most distant
measurable reflections gives the potential resolution of the model that we can
obtain from the data. So a crystal that gives measurable reflections out to a distance
of 1/(3 Å) from the origin is said to yield a model with a resolution of 3 Å.
Crystallographers work back and forth between two different coordinate systems. I will review them briefly. The first system (see Fig. 2.5, p. 12) is the unit
cell (real space), where an atom’s position is described by its coordinates x, y, z.
A vertex of the unit cell, or any other convenient position, is taken as the origin,
with coordinates x, y, z = (0, 0, 0). Coordinates in real space designate real spatial positions within the unit cell. Real-space coordinates are usually given in
angstroms or nanometers, or in fractions of unit cell dimensions. The second system (see Fig. 2.12, p. 19) is the three-dimensional diffraction pattern (reciprocal
space), where a reflection’s position is described by its indices hkl. The central
reflection is taken as the origin with the index hkl = 000 (round black dot at
center of sphere). The position of a reflection is designated by counting reflections from 000, so the indices h, k, and l are integers. Distances in reciprocal
space, expressed in reciprocal angstroms (Å−1) or reciprocal nanometers (nm−1),
are used to judge the potential resolution of the model that the diffraction data
can yield.
Like Alice’s looking-glass world, reciprocal space may seem strange to you
at first (Fig. 2.13). We will see, however, that some aspects of crystallography
are actually easier to understand, and some calculations are more convenient, in
reciprocal space than in real space (Chapter 4).
2.6 The mathematics of crystallography:
A brief description
The problem of determining the structure of objects in a crystalline array from
their diffraction pattern is, in essence, a matter of converting the experimentally
accessible information in the reciprocal space of the diffraction pattern to otherwise
inaccessible information about the real space inside the unit cell. Remember that
a computer program that makes this conversion is acting as a simulated lens to
reconstruct an image from diffracted radiation. Each reflection is produced by a
beam of electromagnetic radiation (X-rays), so the computations entail treating
the reflections as waves and recombining these waves to produce an image of the
molecules in the unit cell.
Section 2.6 The mathematics of crystallography: A brief description 21
Figure 2.13  Fun in reciprocal space. © The New Yorker Collection, 1991. John
O’Brien, from cartoonbank.com. All rights reserved.
2.6.1 Wave equations: Periodic functions
Each reflection is the result of diffraction from complicated objects, the molecules
in the unit cell, so the resulting wave is complicated also. Before considering
how the computer represents such an intricate wave, I will consider mathematical
descriptions of the simplest waves (Fig. 2.14).
Asimple wave, like that of visible light or X-rays, can be described by a periodic
function, for instance, an equation of the form
f (x) = F cos 2π(hx + α) (2.1)
or f (x) = F sin 2π(hx + α). (2.2)
In these functions, f (x) specifies the vertical height of the wave at any horizontal position x along the wave. The variable x and the constant α are angles
expressed in fractions of the wavelength; that is, x = 1 implies a position of one
full wavelength (2π radians or 360◦) from the origin. The constant F specifies the
amplitude of the wave (the height of crests from the horizontal wave axis). For
example, the crests of the wave f (x) = 3 cos 2πx are three times as high and the
troughs are three times as deep as those of the wave f (x) = cos 2πx (compare b
with a in Fig. 2.14).
22 Chapter 2 An Overview of Protein Crystallography
Figure 2.14  Graphs of four simple wave equations f (x) = F cos 2π(hx + α).
(a) F = 1, h = 1, α = 0: f (x) = cos 2π(x). (b) F = 3, h = 1, α = 0: f (x) =
3 cos 2π(x). Increasing F increases the amplitude of the wave. (c) F = 1, h = 3, α = 0:
f (x) = cos 2π(3x). Increasing h increases the frequency (or decreases the wavelength λ)
of the wave. (d) F = 1, h = 1, α = 1/4: f (x) = cos 2π(x + 1/4). Changing α changes
the phase (position) of the wave.
The constant h in a simple wave equation specifies the frequency or wavelength
of the wave. For example, the wave f (x) = cos 2π(3x) has three times the
frequency (or one-third the wavelength) of the wave f (x) = cos 2πx (compare c
with a in Fig. 2.14). (In the wave equations used in this book, h takes on integral
values only.)
Finally, the constant α specifies the phase of the wave, that is, the position of
the wave with respect to the origin of the coordinate system on which the wave is
Section 2.6 The mathematics of crystallography: A brief description 23
Figure 2.15  Visualizing a one-dimensional function, the average daily high temperature, requires two dimensions. The height of the curve f (x) (black) represents the average
temperature on day x. To illustrate the notion of phase difference, note that the phase α
of this wave is shifted with respect to a plot of day length, with maximum value around
June 20 and minimum around December 20 (green curve).
plotted. For example, the position of the wave f (x) = cos 2π(x + 1/4) is shifted
by one-quarter of 2π radians (or one-quarter of a wavelength, or 90◦) from the
position of the wave f (x) = cos 2πx (compare d with a in Fig. 2.14). Because
the wave is repetitive, with a repeat distance of one wavelength or 2π radians, a
phase of 1/4 is the same as a phase of 1 14 , or 2
1
4 , or 3
1
4 , and so on. In radians, a
phase of 0 is the same as a phase of 2π , or 4π , or 6π , and so on. (This use of the
term phase is different from common parlance, in which, for example, the new
moon is called a phase of the lunar month, or autumn is thought of as a phase or
time of the year. Mathematically, phase gives the position of the entire wave with
respect to a specified origin, not merely a location on that wave; location is given
by x.)
These equations describe one-dimensional waves, in which a property (in this
case, the height of the wave) varies in one direction. Visualizing a one-dimensional
function f (x) requires a two-dimensional graph, with the second dimension used
to represent the numerical value off (x) (Fig. 2.15). For example, iff (x) describes
the average daily high temperature over a year’s time, the x-axis represents time in
days, and the height of the curve f (x) on day x represents the average temperature
on that day. The temperature f (x) is in no real sense perpendicular to the time x,
but it is convenient to use the perpendicular direction to show the numerical value
of the temperature. In general, visualizing a function in n dimensions requires
n+ 1 dimensions.
2.6.2 Complicated periodic functions: Fourier series and sums
As discussed in Sec. 2.6.1, p. 21, any simple sine or cosine wave can be described
by three constants—the amplitude F , the frequency h, and the phase α. It is less
obvious that far more complicated waves can also be described with this same
simplicity. The French mathematician Jean Baptiste Joseph Fourier (1768–1830)
showed that even the most intricate periodic functions can be described as the sum
of simple sine and cosine functions whose wavelengths are integral fractions of
24 Chapter 2 An Overview of Protein Crystallography
the wavelength of the complicated function. Such a sum is called a Fourier series
and each simple sine or cosine function in the sum is called a Fourier term.
Figure 2.16 shows a periodic function, called a step function, and the beginning
of a Fourier series that describes it. A method called Fourier synthesis is used to
compute the sine and cosine terms that describe a complex wave, which I will call
the “target” of the synthesis. I will discuss the results of Fourier synthesis, but
not the method itself. In the example of Fig. 2.16, the first four terms produced
by Fourier synthesis are shown individually (f0 through f3, on the left), and each
is added sequentially to the Fourier sum (on the right). Notice that the first term
in the series, f0 = 1, simply displaces the sums upward so that they have only
positive values like the target function. (Sine and cosine functions themselves have
both positive and negative values, with average values of zero.) The second term
f1 = cos 2πx, has the same wavelength as the step function, and wavelengths
of subsequent terms are simple fractions of that wavelength. (It is equivalent to
say, and it is plain in the equations, that the frequencies h are simple multiples
of the frequency of the step function.) Notice that the sum of only the first few
Fourier terms merely approximates the target. If additional terms of shorter wavelength are computed and added, the fit of the approximated wave to the target
improves, as shown by the sum of the first six terms. Indeed, using the tenets of
Fourier theory, it can be proved that such approximations can be made as similar as desired to the target waveform, simply by including enough terms in the
series.
Look again at the components of this Fourier series, functions f0 through f3.
The low-frequency terms like f1 approximate the gross features of the target
wave. Higher-frequency terms like f3 improve the approximation by filling in
finer details, for example, making the approximation better in the sharp corners of
the target function. We would need to extend this series infinitely to reproduce the
target perfectly.
2.6.3 Structure factors: Wave descriptions of X-ray reflections
Each diffracted X-ray that arrives at the film to produce a recorded reflection
can also be described as the sum of the contributions of all scatterers in the unit
cell. The sum that describes a diffracted ray is called a structure-factor equation.
The computed sum for the reflection hkl is called the structure factor Fhkl . As
I will show in Chapter 4, the structure-factor equation can be written in several
different ways. For example, one useful form is a sum in which each term describes
diffraction by one atom in the unit cell, and thus the sum contains the same number
of terms as the number of atoms.
If diffraction by atom A in Fig. 2.17 is represented by fA, then one diffracted
ray (producing one reflection) from the unit cell of Fig. 2.17 is described by a
structure-factor equation of this form:
Fhkl = fA + fB + · · · + fA′ + fB ′ + · · · + fF ′ . (2.3)
Section 2.6 The mathematics of crystallography: A brief description 25
Figure 2.16  Beginning of a Fourier series to approximate a target function, in this
case, a step function or square wave. f0 = 1; f1 = cos 2π(x); f2 = (−1/3) cos 2π(3x);
f3 = (1/5) cos 2π(5x). In the left column are the target and terms f1 through f3. In the
right column are f0 and the succeeding sums as each term is added to f0. Notice that the
approximation improves (that is, each successive sum looks more like the target) as the
number of Fourier terms in the sum increases. In the last graph, terms f4, f5, and f6 are
added (but not shown separately) to show further improvement in the approximation.
26 Chapter 2 An Overview of Protein Crystallography
Figure 2.17  Every atom contributes to every reflection in the diffraction pattern,
as described for this unit cell by Eq. 2.3.
The structure-factor equation implies, and correctly so, that each reflection on
the film is the result of diffractive contributions from all atoms in the unit cell.
That is, every atom in the unit cell contributes to every reflection in the diffraction
pattern. The structure factor Fhkl is a wave created by the superposition of many
individual waves fj , each resulting from diffraction by an individual atom. So the
structure factor is the sum of many wave equations, one for diffraction by each
atom. In that sense, the structure factor equation is a Fourier sum (sometimes called
a Fourier summation, but I prefer one syllable to three), but not a Fourier series.
In a Fourier series, each succeeding term can be generated from the previous one
by some repetitive formula, as in Fig. 2.16.
2.6.4 Electron-density maps
To be more precise about diffraction, when we direct an X-ray beam toward a
crystal, the actual diffractors of the X rays are the clouds of electrons in the
molecules of the crystal. Diffraction should therefore reveal the distribution of
electrons, or the electron density, of the molecules. Electron density, of course,
reflects the molecule’s shape; in fact, you can think of the molecule’s boundary
as a van der Waals surface, the surface of a cloud of electrons that surrounds the
molecule. Because, as noted earlier, protein molecules are ordered, and because,
in a crystal, the molecules are in an ordered array, the electron density in a crystal
can be described mathematically by a periodic function.
If we could walk through the crystal depicted in Fig. 2.4, p. 12, along a linear
path parallel to a cell edge, and carry with us a device for measuring electron
density, our device would show us that the electron density varies along our path
in a complicated periodic manner, rising as we pass through molecules, falling in
the space between molecules, and repeating its variation identically as we pass
through each unit cell. Because this statement is true for linear paths parallel to
all three cell edges, the electron density, which describes the surface features and
overall shapes of all molecules in the unit cell, is a three-dimensional periodic
Section 2.6 The mathematics of crystallography: A brief description 27
Figure 2.18  Small volume element m within the unit cell, one of many elements
formed by subdividing the unit cell with planes parallel to the cell edges. The average electron density within m is ρm(x, y, z). Every volume element contributes to every reflection
in the diffraction pattern, as described by Eq. 2.4.
function. I will refer to this function as ρ(x, y, z), implying that it specifies a
value ρ for electron density at every position x, y, z in the unit cell. A graph of
the function is an image of the electron clouds that surround the molecules in the
unit cell. The most readily interpretable graph is a contour map—a drawing of a
surface along which there is constant electron density (refer to Fig. 2.3, p. 11).
The graph is called an electron-density map. The map is, in essence, a fuzzy image
of the molecules in the unit cell. The goal of crystallography is to obtain the
mathematical function whose graph is the desired electron-density map.
2.6.5 Electron density from structure factors
Because the electron density we seek is a complicated periodic function, it can
be described as a Fourier sum. Do the many structure-factor equations, each a
sum of wave equations describing one reflection in the diffraction pattern, have
any connection with the Fourier function that describes the electron density? As
mentioned earlier, each structure-factor equation can be written as a sum in which
each term describes diffraction from one atom in the unit cell. But this is only
one of many ways to write a structure-factor equation. Another way is to imagine
dividing the electron density in the unit cell into many small volume elements by
inserting planes parallel to the cell edges (Fig. 2.18).
These volume elements can be as small and numerous as desired. Now because
the true diffractors are the clouds of electrons, each structure-factor equation can be
written as a Fourier sum in which each term describes diffraction by the electrons in
one volume element. In this sum, each term contains the average numerical value
of the desired electron density function ρ(x, y, z) within one volume element.
If the cell is divided into n elements, and the average electron density in volume
element m is ρm, then one diffracted ray from the unit cell of Fig. 2.18 is described
28 Chapter 2 An Overview of Protein Crystallography
by a structure-factor equation, another Fourier sum, of this form:
Fhkl = f (ρ1)+ f (ρ2)+ · · · + f (ρm)+ · · · + f (ρn). (2.4)
Each reflection is described by an equation like this one, giving us a large number
of equations describing reflections in terms of the electron density. Is there any
way to solve these equations for the function ρ(x, y, z) in terms of the measured
reflections? After all, structure factors like Eq. 2.4 describe the reflections in terms
of ρ(x, y, z), which is precisely the function the crystallographer is trying to learn.
I will show in Chapter 5 that a mathematical operation called the Fourier transform
solves the structure-factor equations for the desired function ρ(x, y, z), just as if
they were a set of simultaneous equations describing ρ(x, y, z) in terms of the
amplitudes, frequencies, and phases of the reflections.
The Fourier transform describes precisely the mathematical relationship
between an object and its diffraction pattern. In Figs. 2.8–2.11 (pp. 15–18),
the diffraction patterns are the Fourier transforms of the corresponding objects
or arrays of objects. To put it another way, the Fourier transform is the lenssimulating operation that a computer performs to produce an image of molecules
(or more precisely, of electron clouds) in the crystal. This view of ρ(x, y, z) as
the Fourier transform of the structure factors implies that if we can measure three
parameters—amplitude, frequency, and phase—of each reflection, then we can
add them together to obtain the function ρ(x, y, z), graph the function, and “see”
a fuzzy image of the molecules in the unit cell.
2.6.6 Electron density from measured reflections
Are all three of these parameters accessible in the data that reaches our detectors?
I will show in Chapter 5 that the measurable intensity Ihkl of one reflection gives
the amplitude of one Fourier term in the series that describes ρ(x, y, z), and that
the position hkl specifies the frequency for that term. But the phase α of each
reflection is not recorded on any kind of detector. In Chapter 6, I will show how
to obtain the phase of each reflection, completing the information we need to
calculate ρ(x, y, z).
A final note: Even though we cannot measure phases by simply collecting diffraction patterns, we can compute them from a known structure, and we can depict
them by adding color to images like those of Figs. 2.8–2.11. In his innovative
World Wide Web Book of Fourier, Kevin Cowtan illustrates phases in diffraction
patterns in this clever manner. For example, Fig. 2.19 shows a simple group of
atoms, like the carbon atoms in ethylbenzene. Figure 2.19b is the computed Fourier
transform of (a). Image (c) depicts a lattice of the objects in (a), and (d) is the
corresponding Fourier transform.
Because patterns (b) and (d) were computed from objects of known structure,
rather than measured experimentally from real objects, the phases are included in
the calculated results, and thus are known. The phase of each reflection is depicted
by its color, according to the color wheel ( f ). The phase can be expressed as an
Section 2.6 The mathematics of crystallography: A brief description 29
Figure 2.19  Simple asymmetric object, alone (a) and in a lattice (c), and the
computed Fourier transforms of each (b and d). Phases in b and d are depicted by color.
Darkness of color indicates the intensity of a reflection. The phase angle of a region in b
or a reflection in d corresponds to the angle of its color on the color wheel ( f ). Experimental diffraction patterns do not contain phase information, as in (e). Images computed
and generously provided by Dr. Kevin Cowtan. For additional vivid illustrations of Fourier
transforms as they apply to crystallography, direct your web browser to the CMCC home
page and select Kevin Cowtan’s Book of Fourier.
30 Chapter 2 An Overview of Protein Crystallography
angle between 0◦ and 360◦ [this is the angle α in Eqs. (2.1) or (2.2)]. In Fig. 2.19,
the phase angle of each region (in b) or reflection (in d) is the angle that corresponds
to the angle of its color on the color wheel ( f ). For example, red corresponds to a
phase angle of 0◦, and green to an angle of about 135◦. So a dark red reflection has
a high intensity (dark color) and a phase angle of 0◦(red). A pale green reflection
has a low intensity (faint color) and a phase angle of about 135◦(green). With the
addition of color, these Fourier transforms give a full description of each reflection,
including the phase angle that we do not learn from diffraction experiments, which
would give us only the intensities, as shown in (e). In a sense then, Figs. 2.8–2.11
show diffraction patterns, whereas Fig. 2.19b and d show structure-factor patterns,
which depict the structure factors fully. Note again that (d) is a sampling of (b) at
points corresponding to the reciprocal lattice of the lattice in (c). In other words,
the diffraction pattern (d) still contains the diffraction signature, including both
intensities and phases, of the object in (a).
In these terms, I will restate the central problem of crystallography: In order
to determine a structure, we need a full-color version of the diffraction pattern—
that is, a full description of the structure factors, including amplitude, frequency,
and phase. But diffraction experiments give us only the black-and-white version
(e in Fig. 2.19), the positions and intensities of the reflections, but no information
about their phases. We must learn the phase angles from further experimentation,
as described fully in Chapter 6.
2.6.7 Obtaining a model
Once we obtain ρ(x, y, z), we graph the function to produce an electron-density
map, an image of the molecules in the unit cell. Finally, we interpret the map by
building a model that fits it (Fig. 2.3b, p. 11). In interpreting the molecular image
and building the model, a crystallographer takes advantage of all current knowledge about the protein under investigation, as well as knowledge about protein
structure in general. The most important element of this prior knowledge is the
sequence of amino acids in the protein. In a few rare instances, the amino-acid
sequence has been learned from the crystallographic structure. But in almost all
cases, crystallographers know the sequence to start with, from the work of chemists
or molecular biologists, and use it to help them interpret the image obtained from
crystallography. In effect, the crystallographer starts with knowledge of the chemical structure, but without knowledge of the conformation. Interpreting the image
amounts to finding a chemically realistic conformation that fits the image precisely.
A crystallographer interprets a map by displaying it on a graphics computer and
building a graphics model within it. The final model must be (1) consistent with
the image and (2) chemically realistic; that is, it must possess bond lengths, bond
angles, conformational angles, and distances between neighboring groups that are
all in keeping with established principles of molecular structure and stereochemistry. With such a model in hand, the crystallographer can begin to explore the
model for clues about its function.
In Chapters 3–7, I will take up in more detail the principles introduced in this
chapter.
 Chapter 3
Protein Crystals
3.1 Properties of protein crystals
3.1.1 Introduction
As the term X-ray crystallography implies, the sample being examined is in the
crystalline state. Crystals of many proteins and other biomolecules have been
obtained and analyzed in the X-ray beam. A few macromolecular crystals are
shown in Fig. 3.1.
In these photographs, the crystals appear much like inorganic materials such
as sodium chloride. But there are several important differences between protein
crystals and ionic solids.
3.1.2 Size, structural integrity, and mosaicity
Whereas inorganic crystals can often be grown to dimensions of several centimeters
or larger, it is frequently impossible to grow protein crystals as large as 1 mm in
their shortest dimension. In addition, larger crystals are often twinned (two or more
crystals grown into each other at different orientations) or otherwise imperfect and
not usable. Roughly speaking, protein crystallography requires a crystal of at least
0.2 mm in its shortest dimension, although modern methods of data collection can
sometimes succeed with smaller crystals, and modern software can sometimes
decipher data from twinned crystals.
Inorganic crystals derive their structural integrity from the electrostatic attraction of fully charged ions. On the other hand, protein crystals are held together
by weaker forces, primarily hydrogen bonds between hydrated protein surfaces.
In other words, proteins in the crystal stick to each other primarily by hydrogen
bonds through intervening water molecules. Protein crystals are thus much more
fragile than inorganic crystals; gentle pressure with a needle is enough to crush the
hardiest protein crystal. Growing, handling, and mounting crystals for analysis
31
32 Chapter 3 Protein Crystals
Figure 3.1  Some protein crystals grown by a variety of techniques and using a
number of different precipitating agents. They are (a) deer catalase, (b) trigonal form of
fructose-1,6-diphosphatase from chicken liver, (c) cortisol binding protein from guinea pig
sera, (d) concanavalin B from jack beans, (e) beef liver catalase, (f ) an unknown protein
from pineapples, (g) orthorhombic form of the elongation factor Tu from Escherichia coli,
(h) hexagonal and cubic crystals of yeast phenylalanine tRNA, (i), monoclinic laths of
the gene 5 DNA unwinding protein from bacteriophage fd, ( j) chicken muscle glycerol3-phosphate dehydrogenase, and (k) orthorhombic crystals of canavalin from jack beans.
From A. McPherson, in Methods in Enzymology 114, H. W. Wyckoff, C. H. W. Hirs, and
S. N. Timasheff, eds., Academic Press, Orlando, Florida, 1985, p. 114. Photo generously
provided by the author; photo and caption reprinted with permission.
Section 3.1 Properties of protein crystals 33
Figure 3.2  Crystals are not perfectly ordered. They consist of many small arrays in
rough alignment with each other. As a result, reflections are not points, but are spherical or
ovoid, and must be measured over a small angular range.
thus require very gentle techniques. If possible, protein crystals are often harvested, examined, and mounted for crystallography within their mother liquor,
the solution in which they formed.
The textbook image of a crystal is that of a perfect array of unit cells stretching
throughout. Real macroscopic crystals are actually mosaics of many submicroscopic arrays in rough alignment with each other, as illustrated in Fig. 3.2. The
result of mosaicity is that an X-ray reflection actually emerges from the crystal
as a narrow cone rather than a perfectly linear beam. Thus the reflection must be
measured over a very small range of angles, rather than at a single, well-defined
angle. In protein crystals, composed as they are of relatively flexible molecules
held together by weak forces, this mosaicity is more pronounced than in crystals
of rigid organic or inorganic molecules, and the reflections from protein crystals
therefore suffer greater mosaic spread than do those from more ordered crystals.
3.1.3 Multiple crystalline forms
In efforts to obtain crystals, or to find optimal conditions for crystal growth, crystallographers sometimes obtain a protein or other macromolecule in more than one
crystalline form. Compare, for instance, Figs. 3.1a and e, which show crystals of
the enzyme catalase from two different species. Although these enzymes are almost
identical in molecular structure, they crystallize in different forms. In Fig. 3.1h,
you can see that highly purified yeast phenylalanyl tRNA (transfer ribonucleic
acid) crystallizes in two different forms. Often, the various crystal forms differ
in quality of diffraction, in ease and reproducibility of growth, and perhaps in
other properties. The crystallographer must ultimately choose the best form with
which to work. Quality of diffraction is the most important criterion, because it
determines the ultimate quality of the crystallographic model. Among forms that
diffract equally well, more symmetrical forms are usually preferred because they
require less data collection (see Chapter 4).
34 Chapter 3 Protein Crystals
3.1.4 Water content
Early protein crystallographers, proceeding by analogy with studies of other
crystalline substances, examined dried protein crystals and obtained no diffraction patterns. Thus X-ray diffraction did not appear to be a promising tool for
analyzing proteins. In 1934, J. D. Bernal and Dorothy Crowfoot (later Hodgkin)
measured diffraction from pepsin crystals still in the mother liquor. Bernal and
Crowfoot recorded sharp diffraction patterns, with reflections out to distances in
reciprocal space that correspond in real space to the distances between atoms. The
announcement of their success was the birth announcement of protein crystallography.
Careful analysis of electron-density maps usually reveals many ordered water
molecules on the surface of crystalline proteins (Fig. 3.3). Additional disordered
water is presumed to occupy regions of low density between the ordered particles.
Ordered water refers to water molecules that occupy the same site on every protein
molecule in every unit cell (or a high percentage of them) and thus show up clearly
in electron-density maps. Disordered water refers to bulk water molecules that
occupy the spaces between protein molecules, are in different arrangements in
each unit cell, and thus show up only as uniform regions of low electron density.
The quantity of water varies among proteins and even among different crystal
forms of the same protein. The number of detectable ordered water molecules
averages about one per amino-acid residue in the protein. Both the ordered and
disordered water are essential to crystal integrity, so drying destroys the crystal
structure. For this reason, protein crystals are subjected to X-ray analysis in a very
humid atmosphere or in a solution that will not dissolve them, such as the mother
liquor or a protective harvest buffer.
Figure 3.3  Model (stereo) of one molecule of crystalline adipocyte lipid-binding
protein (ALBP, PDB 1alb), showing ordered water molecules on the surface and within a
molecular cavity where lipids are usually bound. Protein is shown as a ball-and-stick model
with carbon dark gray, oxygen red, and nitrogen blue. Ordered water molecules, displayed
as space-filling oxygen atoms, are green. Image: DeepView/POV-Ray.
Section 3.2 Evidence that solution and crystal structures are similar 35
NMR analysis of protein structure suggests that the ordered water molecules
seen by X-ray diffraction on protein surfaces have very short residence times in
solution. Thus most of these molecules may be of little importance to an understanding of protein function. However, ordered water is of great importance to
the crystallographer. As the structure determination progresses, ordered water
becomes visible in the electron-density map. For example, in Fig. 2.3, p. 11, water
molecules are implied by small regions of disconnected density. Positions of these
molecules are indicated by red crosses. Assignment of water molecules to these
isolated areas of electron density improves the overall accuracy of the model, and
for reasons I will discuss in Chapter 7, improvements in accuracy in one area of
the model give accompanying improvements in all other regions.
3.2 Evidence that solution and crystal
structures are similar
Knowing that crystallographers study proteins in the crystalline state, you may
be wondering if these molecules are altered when they crystallize, and whether
the structure revealed by X-rays is pertinent to the molecule’s action in solution.
Crystallographers worry about this problem also, and with a few proteins, it has
been found that crystal structures are in conflict with chemical or spectroscopic
evidence about the protein in solution. These cases are rare, however, and the
large majority of crystal structures appear to be identical to the solution structure.
Because of the slight possibility that crystallization will alter molecular structure,
an essential part of any structure determination project is an effort to show that the
crystallized protein is not significantly altered.
3.2.1 Proteins retain their function in the crystal
Probably the most convincing evidence that crystalline structures can safely be
used to draw conclusions about molecular function is the observation that many
macromolecules are still functional in the crystalline state. For example, substrates
added to suspensions of crystalline enzymes are converted to product, albeit at
reduced rates, suggesting that the enzyme’s catalytic and binding sites are intact.
The lower rates of catalysis can be accounted for by the reduced accessibility of
active sites within the crystal, in comparison to solution.
In a dramatic demonstration of the persistence of protein function in the crystalline state, crystals of deoxyhemoglobin shatter in the presence of oxygen.
Hemoglobin molecules are known to undergo a substantial conformational change
when they bind oxygen. The conformation of oxyhemoglobin is apparently incompatible with the constraints on deoxyhemoglobin in crystalline form, and so
oxygenation disrupts the crystal.
It makes sense, therefore, after obtaining crystals of a protein and before
embarking on the strenuous process of obtaining a structure, to determine whether
36 Chapter 3 Protein Crystals
the protein retains its function in the crystalline state. If the crystalline form is
functional, the crystallographer can be confident that the model will show the
molecule in its functional form.
3.2.2 X-ray structures are compatible with other
structural evidence
Further evidence for the similarity of solution and crystal structures is the compatibility of crystallographic models with the results of chemical studies on proteins.
For instance, two reactive groups in a protein might be linked by a cross-linking
reagent, demonstrating their nearness. The groups shown to be near each other by
such studies are practically always found near each other in the crystallographic
model.
In a growing number of cases, both NMR and X-ray methods have been used to
determine the structure of the same molecule. Figure 3.4 shows the alpha-carbon
backbones of two models of the protein thioredoxin. The blue model was obtained
by X-ray crystallography and the red model by NMR. Clearly the two methods
produce similar models. The models are most alike in the pleated-sheet core and
the alpha helices. The greatest discrepancies, even though they are not large, lie
in the surface loops at the top and bottom of the models. This and other NMRderived models confirm that protein molecules are very similar in crystals and in
solution. In some cases, small differences are seen and can usually be attributed to
crystal packing. Often these packing effects are detectable in the crystallographic
Figure 3.4  Models (stereo) of the protein thioredoxin (human, reduced form) as
obtained from X-ray crystallography (blue, PDB 1ert) and NMR (red, PDB 3trx). Only
backbone alpha carbons are shown. The models were superimposed by least-squares minimization of the distances between corresponding alpha carbons, using DeepView. Image:
DeepView/POV-Ray.
Section 3.3 Growing protein crystals 37
model itself. For instance, in the crystallographic model of cytoplasmic malate
dehydrogenase (PDB file 4mdh), whose functional form is a symmetrical dimer,
an external loop has different conformations in the two molecules of one dimer.
On examination of the dimer in the context of neighboring dimers, it can be seen
that one molecule of each pair lies very close to a molecule of a neighboring pair.
It was thus inferred that the observed difference between the oligomers in a dimer
is due to crystal packing, and further, that the unaffected molecule of each pair is
probably more like the enzyme in solution.
3.2.3 Other evidence
In a few cases, the structure of a protein has been obtained from more than one
type of crystal. The resulting models were identical, suggesting that the molecular
structure was not altered by crystallization.
Recall that stable protein crystals contain a large amount of both ordered and
disordered water molecules. As a result, the proteins in the crystal are still in the
aqueous state, subject to the same solvent effects that stabilize the structure in
solution. Viewed in this light, it is less surprising that proteins retain their solution
structure in the crystal.
3.3 Growing protein crystals
3.3.1 Introduction
Crystals suffer damage in the X-ray beam, primarily due to free radicals generated
by X-rays. For this reason and others discussed later, a full structure determination project usually consumes many crystals. I will now consider the problem
of developing a reliable, reproducible source of protein crystals. This entails not
only growing good crystals of the pure protein, but also obtaining derivatives, or
crystals of the protein in complex with various nonprotein components (loosely
called ligands). For example, in addition to pursuing the structures of proteins
themselves, crystallographers also seek structures of proteins in complexes with
ligands such as cofactors, substrate analogs, inhibitors, and allosteric effectors.
Structure determination then reveals the details of protein-ligand interactions,
giving insight into protein function.
Another vital type of ligand is a heavy-metal atom or ion. Crystals of protein/
heavy-metal complexes, often called heavy-atom derivatives, are usually needed
in order to solve the phase problem mentioned in Sec. 2.6.6, p. 28. I will show
in Chapter 6 that, for the purpose of obtaining phases, it is crucial that crystals
of heavy-atom derivatives be isomorphic with crystals of the pure protein. This
means that derivatives must possess the same unit-cell dimensions and symmetry,
and the same protein conformation, as the pure protein, which in discussions of
derivatives are called native crystals. So in most structure projects, the crystallographer must produce both native and derivative crystals under the same or very
38 Chapter 3 Protein Crystals
similar circumstances. Modern methods of obtaining phases can often succeed
with proteins in which residues of the amino acid methionine are replaced by
selenomethionine, in which selenium replaces the usual sulfur of methionine.
This substitution provides selenium as built-in heavy atoms that usually do not
alter protein conformation or unit-cell structure. I will discuss the production of
crystals of heavy-atom and so-called selenomet derivatives after describing general
procedures for crystallization.
3.3.2 Growing crystals: Basic procedure
Crystals of an inorganic substance can often be grown by preparing a hot, saturated
solution of the substance and then slowly cooling it. Polar organic compounds can
sometimes be crystallized by similar procedures or by slow precipitation from
aqueous solutions by addition of organic solvents. If you work with proteins, just
the mention of these conditions probably makes you cringe. Proteins, of course,
are usually denatured by heat or exposure to organic solvents, so techniques used
for small molecules are not appropriate. In the most common methods of growing protein crystals, purified protein is dissolved in an aqueous buffer containing
a precipitant, such as ammonium sulfate or polyethylene glycol, at a concentration, [precipitant], just below that necessary to precipitate the protein. Then water
is removed by controlled evaporation to raise both [protein] and [precipitant],
resulting in precipitation. Slow precipitation is more likely to produce larger crystals, whereas rapid precipitation may produce many small crystals, or worse, an
amorphous solid.
In theory, precipitation should occur when the combination of [protein] and
[precipitant] exceeds threshold values, as shown in the phase diagram of Fig. 3.5a.
Crystal formation occurs in two stages, nucleation, and growth. Nucleation, the
initial formation of molecular clusters from which crystals grow, requires protein
and/or precipitant concentrations higher than those optimal for slow precipitation
(Fig 3.5a, blue region). In addition, nucleation conditions, if they persist, result
in the formation of many nuclei, and as a result, either an amorphous precipitate
or many small crystals instead of a few larger ones. An ideal strategy (Fig. 3.5b)
would be to start with conditions corresponding to the blue region of the phase
diagram, and then, when nuclei form, move into the green region, where growth,
but not additional nucleation, can occur.
One widely used crystallization technique is vapor diffusion, in which the
protein/precipitant solution is allowed to equilibrate in a closed container with
a larger aqueous reservoir whose precipitant concentration is optimal for producing crystals. One of many examples of this technique is the hanging-drop method
(Fig. 3.6).
Less than 25 µL of the solution of purified protein is mixed with an equal
amount of the reservoir solution, giving precipitant concentration about 50% of
that required for protein crystallization (conditions represented by the red circle
in Fig. 3.5b). This solution is suspended as a droplet underneath a cover slip,
which is sealed onto the top of the reservoir with grease. Because the precipitant
Section 3.3 Growing protein crystals 39
Figure 3.5  (a) Phase diagram for crystallization mediated by a precipitant. The
red region represents concentrations of protein and precipitant at which the solution is not
saturated with protein, so neither nucleation nor growth occurs. The green and blue regions
represent unstable solutions that are supersaturated with protein. Conditions in the blue
region support both nucleation and growth, while conditions in the green support growth
only. (b) An ideal strategy for growing large crystals is to allow nucleation to occur under
conditions in the blue region, then to move to conditions in the green region until crystal
growth ceases.
Figure 3.6  Growing crystals by the hanging-drop method. The droplet hanging
under the cover slip contains buffer, precipitant, protein, and, if all goes well, protein
crystals.
40 Chapter 3 Protein Crystals
is the major solute present, vapor diffusion (evaporation and condensation) in
this closed system results in net transfer of water from the protein solution in
the drop to the reservoir, until the precipitant concentration is the same in both
solutions. Because the reservoir is much larger than the protein solution, the final
concentration of the precipitant in the drop is nearly equal to that in the reservoir.
When the system comes to equilibrium, net transfer of water ceases, and the
protein solution is maintained at constant precipitant concentration. At this point,
drop shrinkage has increased both [precipitant] and [protein], moving conditions
diagonally into the nucleation region (blue circle in Fig. 3.5b). In this way, the
precipitant concentration in the protein solution rises to the level required for
nucleation and remains there without overshooting because, at equilibrium, the
vapor pressure in the closed system equals the inherent vapor pressure of both
protein solution and reservoir. As nuclei form, the protein concentration decreases,
moving the conditions vertically into the growth region (green circle in Fig. 3.5b).
Frequently the crystallographer obtains many small crystals instead of a few
that are large enough for diffraction measurements. If many crystals grow at once,
the supply of dissolved protein will be depleted before crystals are large enough
to be useful. Small crystals of good quality can be used as seeds to grow larger
crystals. The experimental setup is the same as before, except that each hanging
droplet is seeded with a few small crystals. Seed crystals are sometimes etched
before use by brief soaking in buffer with precipitant concentration lower than that
of the mother liquor. This soak dissolves outer layers of the seed crystal, exposing
fresh surface on which crystallization can proceed. Seeds may also be obtained
by crushing small crystals or by stroking a crystal with a hair and passing the hair
through the crystallization droplet (it is reported that animal whiskers are best—
really). Whatever the seeding method, crystals may grow from seeds up to ten
times faster than they grow anew, so most of the dissolved protein goes into only
a few crystals.
3.3.3 Growing derivative crystals
Crystallographers obtain the derivatives needed for phase determination and
for studying protein-ligand interactions by two methods: cocrystallizing protein
and ligand, and soaking preformed protein crystals in mother-liquor solutions
containing ligand.
It is sometimes possible to obtain crystals of protein-ligand complexes by
crystallizing protein and ligand together, a process called cocrystallization.
For example, a number of NAD+-dependent dehydrogenase enzymes readily crystallize as NAD+ or NADH complexes from solutions containing these
cofactors. Cocrystallization is the only method for producing crystals of proteins
in complexes with large ligands, such as nucleic acids or other proteins.
A second means of obtaining crystals of protein-ligand complexes is to soak
protein crystals in mother liquor that contains ligand.As mentioned earlier, proteins
retain their binding and catalytic functions in the crystalline state, and ligands can
diffuse to active sites and binding sites through channels of water in the crystal.
Soaking is usually preferred over cocrystallization when the crystallographer plans
Section 3.3 Growing protein crystals 41
to compare the structure of a pure protein with that of a protein-ligand complex.
Soaking preformed protein crystals with ligands is more likely to produce crystals
of the same form and unit-cell dimensions as those of pure protein, so this method
is recommended for first attempts to make isomorphic heavy-atom derivatives.
Making selenomet derivatives requires taking advantage of modern methods
of molecular biology, in which the gene encoding a desired protein is introduced
(for example, on a plasmid) into a specially designed strain of bacterium or other
microbe, which is called an expression vector. The microbe, in turn, expresses the
gene, which means that it produces messenger RNA from the gene and synthesizes
the desired protein. To produce a selenomet derivative, the gene for the desired
protein is expressed in a mutant microbe that cannot make its own methionine,
and thus can live only in a growth medium that provides methionine. If the
growth medium provides selenomethionine instead of methionine, the microbe
usually grows normally, and expression results in incorporation of selenomethionine wherever methionine would normally appear. Purification and crystallization
of the selenomet derivative usually follow the same procedures as for the native
protein.
Finally, some proteins naturally contain metal ions that can serve the same purpose in phasing as introduced heavy-atom compounds. For example, hemoglobin
contains iron (II) ions that can be used to obtain phase information. For such
proteins, there is often no need to produce heavy-atom or selenomet derivatives.
3.3.4 Finding optimal conditions for crystal growth
The two most important keys to success of a crystallographic project are purity
and quantity of the macromolecule under study. Impure samples will not make
suitable crystals, and even for proteins of the highest purity, repeated trials will be
necessary before good crystals result.
Many variables influence the formation of macromolecular crystals. These
include obvious ones like protein purity, concentrations of protein and precipitant, pH, and temperature, as well as more subtle ones like cleanliness, vibration
and sound, convection, source and age of the protein, and the presence of ligands.
Clearly, the problem of developing a reliable source of crystals entails controlling
and testing a large number of parameters. The difficulty and importance of obtaining good crystals has prompted the invention of crystallization robots that can be
programmed to set up many trials under systematically varied conditions.
The complexity of this problem is illustrated in Fig. 3.7, which shows the
effects of varying just two parameters, the concentrations of protein (in this case,
the enzyme lysozyme) and precipitant (NaCl). Notice the effect of slight changes
in concentration of either protein or precipitant on the rate of crystallization, as
well as the size and quality of the resulting crystals.
A sample scheme for finding optimum crystallization conditions is to determine
the effect of pH on precipitation with a given precipitant, repeat this determination
at various temperatures, and then repeat these experiments with different precipitating agents. Notice in Fig. 3.7 that the region of [protein] versus [precipitant]
that gives best crystals is in the shape of an arc, like the arc-shaped growth region
42 Chapter 3 Protein Crystals
Figure 3.7  Schematic map of crystallization kinetics as a function of lysozyme and
NaCI concentration obtained from a matrix of dishes. Inserts show photographs of dishes
obtained one month after preparation of solutions. From G. Feher and X. Kam, in Methods
in Enzymology 114, H. W. Wyckoff, C. H. W. Hirs, and S. N. Timasheff, eds., Academic
Press, Orlando, Florida, 1985, p. 90. Photo and caption reprinted with permission.
of Fig. 3.5a. It turns out that if these same data are plotted as [protein] versus
([protein] × [precipitant]), this arc-shaped region becomes a rectangle, which
makes it easier to survey the region systematically. For such surveys of crystallization conditions, multiple batches of crystals can be grown conveniently by
the hanging-drop or other methods in crystallization plates of 24, 48, or 96 wells
(Fig. 3.8), each with its own cover. This apparatus has the advantage that the growing crystals can be observed through the cover slips with a dissecting microscope.
Then, once the ideal conditions are found, many small batches of crystals can be
grown at once, and each batch can be harvested without disturbing the others.
Crystallographers have developed sophisticated schemes for finding and optimizing conditions for crystal growth. One approach, called a response-surface
procedure, begins with the establishment of a scoring scheme for results, such
as giving higher scores for lower ratios of the shortest to the longest crystal
dimension. This method gives low scores for needles and higher scores for cubes.
Section 3.3 Growing protein crystals 43
Figure 3.8  Well-plate, in which 24 sitting-drop crystallization trials can be carried
out. Each well contains a pedestal with a concave top, in which the drop sits. Vapor diffusion
occurs between drop and reservoir in the bottom of the well.
Then crystallization trials are carried out, varying several parameters, including
pH, temperature, and concentrations of protein, precipitant, and other additives.
The results are scored, and the relationships between parameters and scores are analyzed. These relationships are fitted to mathematical functions (like polynomials),
which describe a complicated multidimensional surface (one dimension for each
variable or for certain revealing combinations of variables) over which the score
varies. The crystallographer wants to know the location of the “peaks” on this
surface, where scores are highest. Such peaks may lie at sets of crystallization
conditions that were not tried in the trials and may suggest new and more effective
conditions for obtaining crystals. Finding peaks on such surfaces is just like finding
the maximum or minimum in any mathematical function. You take the derivative
of the function, set it equal to zero, and solve for the values of the parameters. The
sets of values obtained correspond to conditions that lie at the top of mountains on
the surface of crystal scores.
An example of this approach is illustrated in Fig. 3.9. The graph in the center
is a two-dimensional slice of a four-dimensional surface over which [protein],
([protein] × [precipitant]), pH, and temperature were varied, in attempts to find
optimal crystallization conditions for the enzyme tryptophanyl-tRNA synthetase.
Note that this surface samples the rectangular region [protein] versus ([protein] ×
[precipitant]), mentioned earlier. The height of the surface is the score for the
crystallization. Surrounding the graph are photos of typical crystals obtained in
multiple trials of each set of conditions. None of the trial conditions were near the
peak of the surface. The photos labeled Opt1 and Opt2 are of crystals obtained
from conditions defined by the surface peak. In this instance, the response-surface
approach predicted conditions that produced better crystals than any from the trials
that pointed to these conditions.
44 Chapter 3 Protein Crystals
Figure 3.9  Optimization of conditions for crystallization of tryptophanyl-tRNA
synthetase. Photo insets show crystals obtained from various conditions represented by
points on the surface. Coordinates of the surface are protein concentration (PROTEIN),
product of protein concentration and precipitant concentration (PRO_PPNT), and the shape
of the crystal as reflected by the ratio of its two smallest dimensions, width and length
(WL_RATIO). From C. W. Carter, in Methods in Enzymology 276, C. W. Carter and R. M.
Sweet, eds., Academic Press, New York, 1997, p. 75. Reprinted with permission.
So if you decide to try to grow some of your own crystals, how should you
proceed? Theoretical studies like those described above, as well as the recorded
experience of myriad crystallization successes and failures, have led to development of commercial screening kits that can often streamline the pursuit of crystals.
Typical kits are sets of 24, 48, or 96 solutions containing various buffers, salts,
and precipitants, representing a wide variety of potential crystallization conditions.
After establishing appropriate protein concentration for screening (there is a kit
for that, too), you would set up one trial with each of the screen solutions in cells
of crystallization plates like the one shown in Fig. 3.8 ($ome kit$ even come with
prefilled well plate$). If a particular screen solution produces promising crystals,
you can then try to optimize the conditions by varying pH, [salt] or [precipitant]
around the values of the screen solution.
Another way to tap accumulated wisdom about crystallization is through online
databases. For example, at the combined Biological Macromolecule Crystallization Database and NASA Archive for Protein Crystal Growth Data (see CMCC
home page), you can search for successful crystallization conditions for thousands
Section 3.3 Growing protein crystals 45
of macromolecules. You can search by many criteria, including molecule name,
source species, prosthetic groups, molecular weight, space groups, as well as specific precipitants, methods, or conditions. Conditions that have succeeded with
proteins similar to your target may be good starting points.
When varying the more conventional parameters fails to produce good crystals,
the crystallographer may take more drastic measures. Sometimes limited digestion
of the protein by a proteolytic enzyme removes a disordered surface loop, resulting in a more rigid, hydrophilic, or compact molecule that forms better crystals.
A related measure is adding a ligand, such as a cofactor, that is known to bind
tightly to the protein. The protein-ligand complex may be more likely to crystallize than the free protein, either because the complex is more rigid than the free
protein or because the cofactor induces a conformational change that makes the
protein more amenable to crystallizing. Desperation has even prompted addition
of coffee (usually readily at hand in research labs) to precipitant mixtures, but I
am aware of no successes from this measure.
Many membrane-associated proteins will not dissolve in aqueous buffers and
tend to form amorphous precipitates instead of crystals. The intractability of such
proteins often results from hydrophobic domains or surface regions that are normally associated with the interior of membranes. Such proteins have sometimes
been crystallized in the presence of detergents, which coat the hydrophobic portion and decorate it with ionic groups, thus rendering it more soluble in water.
A small number of proteins have been diffused into crystalline phases of lipid
to produce ordered arrays that diffracted well and yielded structures. In some
cases, limited proteolysis of membrane-associated proteins has removed exposed
hydrophobic portions, leaving crystallizable fragments that are more like a typical water-soluble protein. Membrane proteins are greatly under-represented in the
Protein Data Bank, due to their resistance to crystallization. The search for widely
applicable conditions for crystallizing membrane proteins is one of crystallography’s holy grails. The announcement of a model of a new membrane protein is
usually greeted with much attention, and the first question is usually, “How did
they crystallize it?”
The effects of modifications of the target protein, as well as the potential
crystallizability of a newly purified protein, can be tentatively assessed before
crystallization trials begin, through analysis of laser light scattering by solutions
of the macromolecule. Simple, rapid light-scattering experiments (see Sec. 9.3,
p. 219) can reveal much about the nature of the substance in solutions of varied
composition, pH, and temperature, including estimates of average molecular mass
of the particles, radius of gyration (dependent on shape of particles), rates of diffusion through the solution, and range and distribution of particle sizes (degree
of polydispersity). Some of the measured properties correlate well with crystallizability. In particular, monodisperse preparations—those containing particles of
uniform size—are more promising candidates for crystallization than those in
which the protein is polydisperse. In many cases, polydispersity arises from nonspecific interactions among the particles, which at higher concentrations is likely
to result in random aggregation rather than orderly crystallization.
46 Chapter 3 Protein Crystals
When drastic measures like proteolysis are required to yield good crystals, the
crystallographer is faced with the question of whether the resulting fragment is
worthy of the arduous effort to determine its structure. This question is similar to
the basic issue of whether a protein has the same structure in crystal and in solution,
and the question must be answered in the same way. Specifically, it may be possible
to demonstrate that the fragment maintains at least part of the biological function of
the intact molecule, and further, that this function is retained after crystallization.
3.4 Judging crystal quality
The acid test of a crystal’s suitability for structure determination is, of course, its
capacity to give sharp diffraction patterns with clear reflections at large angles from
the X-ray beam. Using equipment typical of today’s crystallography laboratories,
researchers can collect preliminary diffraction data quickly and decide whether
to obtain a full data set. However, a brief inspection of crystals under a lowpower light microscope can also provide some insight into quality and can help
the crystallographer pick out the most promising crystals.
Desirable visible characteristics of crystals include optical clarity, smooth faces,
and sharp edges. Broken or twinned crystals sometimes exhibit dark cleavage
planes within an otherwise clear interior. Depending on the lattice type (Chapter 4)
and the direction of viewing relative to unit-cell axes, some crystals strongly rotate
plane-polarized light. This property is easily observed by examining the crystal
between two polarizers, one fixed and one rotatable, under a microscope. Upon
rotation of the movable polarizer, a good-quality crystal will usually brighten and
darken sharply.
Once the crystallographer has a reliable source of suitable crystals, data
collection can begin.
3.5 Mounting crystals for data collection
The classical method of mounting crystals is to transfer them into a fine glass
capillary along with a droplet of the mother liquor. The capillary is then sealed at
both ends and mounted onto a goniometer head (see Fig. 4.25, p. 81, and Sec. 4.3.4,
p. 80), a device that allows control of the crystal’s orientation in the X-ray beam.
The droplet of mother liquor keeps the crystal hydrated.
For many years, crystallographers have been aware of the advantages of collecting X-ray data on crystals at very low temperatures, such as that of liquid nitrogen
(boiling point −196◦C). In theory, lowering the temperature should increase
molecular order in the crystal and improve diffraction. In practice, however, early
Section 3.5 Mounting crystals for data collection 47
attempts to freeze crystals resulted in damage due to formation of ice crystals. Then
crystallographers developed techniques for flash freezing crystals in the presence
of agents like glycerol, which prevent ice from forming. Crystallography at low
temperatures is called cryocrystallography and the ice-preventing agents are called
cryoprotectants. Other cryoprotectants include xylitol or sugars such as glucose.
Some precipitants, for example, polyethylene glycol, also act as cryoprotectants,
and often it is only necessary to increase their concentration in order to achieve
protection from ice formation.
If the crystal was not grown in cryoprotectant, preparation for cryocrystallography typically entails placing it in a cryoprotected mother liquor for 5–15 seconds
to wash off the old mother liquor (this liquid is sometimes called a harvest buffer).
If sudden exposure to cryoprotectant damages the crystal, it might be serially
transferred through several solutions of gradually increasing cryoprotectant concentration. After transfer into protectant, the crystal is picked up in a small (<1 mm)
circular loop of glass wool or synthetic fiber, where it remains suspended in a thin
film of solvent, sort of like the soap film in a plastic loop for blowing soap bubbles.
The crystal is then flash frozen by dipping the loop into liquid nitrogen. If flashfreezing is successful, the liquid film in the loop freezes into a glass and remains
clear (if it is frosty, crystalline water has formed, usually destroying the crystal
in the process). For data collection, the loop is mounted onto the goniometer (see
Fig. 4.25b, p. 81), where it is held in a stream of cold nitrogen gas coming from
a reservoir of liquid nitrogen. A temperature of −100◦C can be maintained in this
manner.
In addition to better diffraction, other benefits of cryocrystallography include
reduction of radiation damage to the crystal and hence the possibility of collecting
more data—perhaps an entire data set—from a single crystal; reduction of X-ray
scattering from water (resulting in cleaner backgrounds in diffraction patterns)
because the amount of water surrounding the crystal is far less than that in a droplet
of mother liquor in a capillary; and the possibility of safe storage, transport, and
reuse of crystals. Crystallographers can take or ship loop-mounted flash-frozen
crystals, in liquid-nitrogen-filled insulated containers, to sites of data collection,
minimizing handling of crystals at the collection site. With all these benefits, it is
not surprising that cryocrystallography is now common practice.
 Chapter 4
Collecting Diffraction Data
4.1 Introduction
In this chapter, I will discuss the geometric principles of diffraction, revealing, in
both the real space of the crystal’s interior and in reciprocal space, the conditions
that produce reflections. I will show how these conditions allow the crystallographer to determine the dimensions of the unit cell and the symmetry of its contents
and how these factors determine the strategy of data collection. Finally, I will look
at the devices used to produce and detect X-rays and to measure precisely the
intensities and positions of reflections.
4.2 Geometric principles of diffraction
W. L. Bragg showed that the angles at which diffracted beams emerge from a
crystal can be computed by treating diffraction as if it were reflection from sets
of equivalent, parallel planes of atoms in a crystal. (This is why each spot in the
diffraction pattern is called a reflection.) I will first describe how crystallographers
denote the planes that contribute to the diffraction pattern.
4.2.1 The generalized unit cell
The dimensions of a unit cell are designated by six numbers: the lengths of three
unique edges a, b, and c; and three unique angles α, β, and γ (Fig. 4.1, p. 50).
Notice the use of bold type in naming the unit cell edges or the axes that correspond
to them. I will use bold letters (a, b, c) to signify the edges or axes themselves,
and letters in italics (a, b, c) to specify their length. Thus a is the length of unit
cell edge a, and so forth.
49
50 Chapter 4 Collecting Diffraction Data
Figure 4.1  General (triclinic) unit cell, with edges a, b, c and angles α, β, γ .
A cell in which a = b = c and α = β = γ , as in Fig. 4.1, is called triclinic,
the simplest crystal system. If a = b = c, α = γ = 90◦, and β > 90◦, the cell
is monoclinic. If a = b, α = β = 90◦ and γ = 120◦, the cell is hexagonal. For
cells in which all three cell angles are 90◦, if a = b = c, the cell is cubic; if
a = b = c, the cell is tetragonal; and if a = b = c, the cell is orthorhombic. The
possible crystal systems are shown in Fig. 4.2. The crystal systems form the basis
for thirteen unique lattice types, which I will describe later in this chapter.
The most convenient coordinate systems for crystallography adopt coordinate
axes based on the directions of unit-cell edges. For cells in which at least one cell
angle is not 90◦, the coordinate axes are not the familiar orthogonal (mutually
perpendicular) x, y, and z. In this book, for clarity, I will emphasize unit cells
and coordinate systems with orthogonal axes (α = β = γ = 90◦), and I will use
orthorhombic systems most often, making it possible to distinguish the three cell
edges by their lengths. In such systems, the a edges of the cell are parallel to the
x-axis of an orthogonal coordinate system, edges b are parallel to y, and edges c
are parallel to z. Bear in mind, however, that the principles discussed here can be
generalized to all unit cells.
4.2.2 Indices of the atomic planes in a crystal
The most readily apparent sets of planes in a crystalline lattice are those determined
by the faces of the unit cells. These and all other regularly spaced planes that
can be drawn through lattice points can be thought of as sources of diffraction
and can be designated by a set of three numbers called lattice indices or Miller
indices. Three indices hkl identify a particular set of equivalent, parallel planes.
The index h gives the number of planes in the set per unit cell in the x direction
or, equivalently, the number of parts into which the set of planes cut the a edge
Section 4.2 Geometric principles of diffraction 51
Cubic
a= b= c,
α=β= γ=90°
Tetragonal
a= b≠ c,
α=β= γ=90°
Orthorhombic
a≠b≠ c,
α=β= γ=90°
Hexagonal
a= b= c,
α=β=90°
γ = 120°
Rhombohedral
a= b= c,
α=β= γ≠90°
Triclinic
a≠ b≠ c,
α≠β≠ γ≠90°
Monoclinic
a≠ b≠ c,
α=γ=90°
β≠90°
Figure 4.2  Crystal systems beginning with the most symmetric (cubic, upper left),
and ending with the least symmetric (triclinic, lower right).
of each cell. The indices k and l specify how many such planes exist per unit cell
in the y and z directions. An equivalent way to determine the indices of a set of
planes is to start at any lattice point and move out into the unit cell away from
the plane cutting that lattice point. If the first plane encountered cuts the a edge
at some fraction 1/na of its length, and the same plane cuts the b edge at some
fraction 1/nb of its length, then the h index is na and the k index is nb (examples
are given later). Indices are written in parentheses when referring to the set of
planes; hence, the planes having indices hkl are the (hkl) planes.
In Fig. 4.3, each face of an orthorhombic unit cell is labeled with the indices
of the set of planes that includes that face. (The crossed arrows lie on the labeled
face, and parallel faces have the same indices.)
52 Chapter 4 Collecting Diffraction Data
Figure 4.3  Indices of faces in an orthorhombic unit cell.
The set of planes including and parallel to the bc face, and hence normal to the
x-axis, is designated (100) because there is one such plane per lattice point in the
x direction. In like manner, the planes parallel to and including the ac face are
called (010) planes (one plane per lattice point along y). Finally, the ab faces of the
cell determine the (001) planes. (To recognize these planes easily, notice that if you
think of the index as (abc), the zeros tell you the location of the plane: the zeros in
(010) occupy the a and c positions, so the plane corresponds to the ac face.) In the
Bragg model of diffraction as reflection from parallel sets of planes, any of these
sets of planes can be the source of one diffracted X-ray beam. (Remember that
an entire set of parallel planes, not just one plane, acts as a single diffractor and
produces one reflection.) But if these three sets of planes were the only diffractors,
the number of diffracted beams would be small, and the information obtainable
from diffraction would be very limited.
In Fig. 4.4, an additional set of planes, and thus an additional source of diffraction, is indicated. The lattice (solid lines) is shown in section parallel to the ab
faces or the xy plane. The dashed lines represent the intersection of a set of equivalent, parallel planes that are perpendicular to the xy plane of the paper. Note that
the planes cut each a edge of each unit cell into two parts and each b edge into
one part, so these planes have indices 210. Because all (210) planes are parallel
to the z axis (which is perpendicular to the plane of the paper), the l index is zero.
Or equivalently, because the planes are infinite in extent, and are coincident with
c edges, and thus do not cut edges parallel to the z axis, there are zero (210) planes
per unit cell in the z direction. As another example, for any plane in the set shown
in Fig. 4.5, the first plane encountered from any lattice point cuts that unit cell at
a/2 and b/3, so the indices are 230.
All planes perpendicular to the xy plane have indices hk0. Planes perpendicular
to the xz plane have indices h0l, and so forth. Many additional sets of planes are
not perpendicular to x, y, or z. For example, the (234) planes cut the unit cell
edges a into two parts, b into three parts, and c into four parts (Fig. 4.6).
Section 4.2 Geometric principles of diffraction 53
Figure 4.4  (210) planes in a two-dimensional section of lattice.
Figure 4.5  (230) planes in a two-dimensional section of lattice.
54 Chapter 4 Collecting Diffraction Data
Figure 4.6  The intersection of three (234) planes with a unit cell. Note that the (234)
planes cut the unit-cell edges a into two parts, b into three parts, and c into four parts.
Finally, Miller indices can be negative as well as positive. Sets of planes in
which all indices have opposite signs are identical. For example, the (210) planes
are the same as (−2 −1 0), which is commonly written (210). The (210) or (210)
planes are identical (all signs opposite), but tilt in the opposite direction from the
(210) planes (Fig. 4.7). To determine the sign of indices the h and k indices for
a set of planes that cut the xy plane, look at the direction of a line perpendicular
to the planes (green arrows in Fig. 4.7), and imagine the line passing through the
origin of an xy coordinate system. If this perpendicular line points into the (++)
and (−−) quadrants of the xy plane, then the indices h and k are both positive or
both negative. If the perpendicular points into the (+−) and (−+) quadrants, then
h and k have opposite signs. This mnemonic works in three dimensions as well:
a perpendicular to the planes in Fig. 4.6 points into the (+ + +) and (− − −)
octants of an xyz coordinate system. You will see why this mnemonic works when
I show how to construct the reciprocal lattice.
In Bragg’s way of looking at diffraction as reflection from sets of planes in the
crystal, each set of parallel planes described here (as well as each additional set of
planes interleaved between these sets) is treated as an independent diffractor and
produces a single reflection. This model is useful for determining the geometry
of data collection. Later, when I discuss structure determination, I will consider
another model in which each atom or each small volume element of electron density
is treated as an independent diffractor, represented by one term in a Fourier sum
that describes each reflection. What does the Fourier sum model add to the Bragg
model? Bragg’s model tells us where to look for the data. The Fourier sum model
tells us what the data has to say about the molecular structure, that is, about where
the atoms are located in the unit cell.
Section 4.2 Geometric principles of diffraction 55
Figure 4.7  The (210) and (210) planes are identical. They tilt in the opposite
direction from (210) and (210) planes.
4.2.3 Conditions that produce diffraction: Bragg’s law
Notice that the different sets of equivalent parallel planes in the preceding figures
have different interplanar spacing d. Among sets of planes (hkl), interplanar spacing decreases as any index increases (more planes per unit cell means more closely
spaced planes). W. L. Bragg showed that a set of parallel planes with index hkl and
interplanar spacing dhkl produces a diffracted beam when X-rays of wavelength λ
impinge upon the planes at an angle θ and are reflected at the same angle, only if
θ meets the condition
2dhkl sin θ = nλ, (4.1)
where n is an integer. The geometric construction in Fig. 4.8 demonstrates the
conditions necessary for producing a strong diffracted ray. The red dots represent
two parallel planes of lattice points with interplanar spacing dhkl . Two rays R1 and
R2 are reflected from them at angle θ .
Lines AC are drawn from the point of reflection A of R1 perpendicular to
the ray R2. If ray R2 is reflected at B, then the diagram shows that R2 travels
the same distance as R1 plus an added distance 2BC. Because AB in the small
triangle ABC is perpendicular to the atomic plane, and AC is perpendicular to the
56 Chapter 4 Collecting Diffraction Data
Figure 4.8  Conditions that produce strong diffracted rays. If the additional distance
traveled by the more deeply penetrating ray R2 is an integral multiple of λ, then rays R1
and R2 interfere constructively.
incident ray, the angle CAB equals θ , the angle of incidence (two angles are equal
if corresponding sides are perpendicular). Because ABC is a right triangle, the sine
of angle θ is BC/AB or BC/dhkl . Thus BC equals dhkl sin θ , and the additional
distance 2BC traveled by ray R2 is 2dhkl sin θ .
If this difference in path length for rays reflected from successive planes
(2dhkl sin θ ) is equal to an integral number of wavelengths (nλ) of the impinging X rays (that is, if 2dhkl sin θ = nλ), then the rays reflected from successive
planes emerge from the crystal in phase with each other, interfering constructively to produce a strong diffracted beam. For other angles of incidence θ ′ (where
2dhkl sin θ ′ does not equal an integral multiple of λ), waves emerging from successive planes are out of phase, so they interfere destructively, and no beam emerges
at that angle. Think of it this way: If X-rays impinge at an angle θ ′ that does not
satisfy the Bragg conditions, then for every reflecting plane p, there will exist, at
some depth in the crystal, another parallel plane p′ producing a wave precisely
180◦ out of phase with that from p, and thus precisely cancelling the wave from p.
So all such waves will be cancelled by destructive interference, and no diffracted
ray will emerge at the angle θ ′. Diffracted rays reflect from (hkl) planes of spacing
dhkl only at angles θ for which 2dhkl sin θ = nλ. Notice that what I am calling
the diffraction angle θ is the angle of incidence and the angle of reflection. So
the actual angle by which this reflection diverges from the incident X-ray beam
is 2θ . I should also add that the intensity of this diffracted ray will depend on how
many atoms, or much electron density, lies on this set of planes in the unit cell. If
electron density on this set of planes is high, the ray will be strong (high intensity).
Section 4.2 Geometric principles of diffraction 57
If there is little electron density of this set of planes, this ray, although allowed by
Bragg’s law, will be weak or undetectable.
Notice that the angle of diffraction θ is inversely related to the interplanar spacing
dhkl (sin θ is proportional to 1/dhkl). This implies that large unit cells, with large
spacings, give small angles of diffraction and hence produce many reflections that
fall within a convenient angle from the incident beam. On the other hand, small unit
cells give large angles of diffraction, producing fewer measurable reflections. In a
sense, the number of measurable reflections depends on how much information is
present in the unit cell. Large cells contain many atoms and thus more information,
and they produce more information in a diffraction pattern of the same size. Small
unit cells contain fewer atoms, and diffraction from them contains less information.
It is not coincidental that I use the variable names h, k, and l for both the indices
of planes in the crystal and the indices of reflections in the diffraction pattern
(Sec. 2.5, p. 19). I will show later that in fact the electron density on (or parallel to)
the set of planes (hkl) produces the reflection hkl of the diffraction pattern. In the
terms I used in Chapter 2, each set of parallel planes in the crystal produces one
reflection, or one term in the Fourier sum that describes the electron density within
the unit cell. The intensity of that reflection depends on the electron distribution
and density along the planes that produce the reflection.
4.2.4 The reciprocal lattice
Now let us consider the Bragg conditions from another point of view: in reciprocal
space—the space occupied by the reflections. Before looking at diffraction from
this vantage point, I will define and tell how to construct a new lattice, the reciprocal
lattice, in what will at first seem an arbitrary manner. But I will then show that the
points in this reciprocal lattice are the locations of all the Bragg reflections, and
thus they are guides that tell the crystallographer the angles at which all reflections
will occur.
Figure 4.9a shows an ab section of lattice with an arbitrary lattice point O
chosen as the origin of the reciprocal lattice I am about to define. This point is thus
the origin for both the real and reciprocal lattices. Each red + in the figure is a real
lattice point.
Through a neighboring lattice point N , draw one plane from each of the sets
(110), (120), (130), and so forth. These planes intersect the ab section in lines
labeled (110), (120), and (130) in Fig. 4.9a. From the origin, draw a line normal to
the (110) plane. Make the length of this line 1/d110, the inverse of the interplanar
spacing d110. Define the reciprocal lattice point 110 as the point at the end of this
line (green dot). Now repeat the procedure for the (120) plane, drawing a line from
O normal to the (120) plane, and of length 1/d120. Because d120 is smaller than
d110 (recall that d decreases as indices increase), this second line is longer than
the first. The end of this line defines a second reciprocal lattice point, with indices
120 (green dot). Repeat for the planes (130), (140), and so forth. Notice that the
reciprocal lattice points lie on a straight line.
Now continue this operation for planes (210), (310), (410), and so on, defining
reciprocal lattice points 210, 310, 410, and so on (Fig. 4.9b). Note that the points
58 Chapter 4 Collecting Diffraction Data
Figure 4.9  (a) Construction of reciprocal lattice. Real-lattice points are red+ signs,
and reciprocal lattice points are green dots. Notice the real cell edges b and a reciprocal cell
edge b*. (b) Continuation of (a). Notice the real cell edges a and a reciprocal cell edge a*.
Section 4.2 Geometric principles of diffraction 59
Figure 4.10  Reciprocal unit cells of large and small real cells.
defined by continuing these operations form a lattice, with the arbitrarily chosen
real lattice point as the origin (indices 000). This new lattice (green dots) is the
reciprocal lattice. The planes hk0, h0l, and 0kl correspond, respectively, to the xy,
xz, and yz planes. They intersect at the origin and are called the zero-level planes
in this lattice. Other planes of reciprocal-lattice points parallel to the zero-level
planes are called upper-level planes.
We can also speak of the reciprocal unit cell in such a lattice (Fig. 4.10). If the
angles α, β, and λ in the real cell (red) are 90◦, the reciprocal unit cell (green)
has axes a* lying along (colinear with) real unit cell edge a, b* lying along b, and
c* along c. The lengths of edges a*, b*, and c* are reciprocals of the lengths of
corresponding real cell edges a, b, and c: a* = 1/a, and so forth, so small real
cells have large reciprocal cells and vice versa. If axial lengths are expressed in
angstroms, then reciprocal-lattice spacings are in the unit 1/Å or Å−1 (reciprocal
angstroms).
For real unit cells with nonorthogonal axes, the spatial relationships between the
real and reciprocal unit-cell edges are more complicated. Examples of monoclinic
real and reciprocal unit cells are shown in Fig. 4.11 with a brief explanation. I will
make no further use of nonorthogonal unit cells in this book (much to your relief,
I expect).
Now envision this lattice of imaginary points surrounding the crystal in space.
For a small real unit cell, interplanar spacings dhkl are small, and hence the lines
from the origin to the reciprocal lattice points are long. Therefore, the reciprocal
unit cell is large, and lattice points are widely spaced. On the other hand, if the real
unit cell is large, the reciprocal unit cell is small, and reciprocal space is densely
populated with reciprocal lattice points.
60 Chapter 4 Collecting Diffraction Data
Figure 4.11  Example of real and reciprocal cells in the monoclinic system (stereo).
As in Fig. 4.10, the real monoclinic unit cell is red, and its reciprocal unit cell is green.
Even when unit cell angles are not 90◦, the following relationship always holds: a* is
perpendicular to the real-space plane bc, b* is perpendicular to ac, and c* is perpendicular
to ab. In this monoclinic cell, b* and b are colinear (both perpendicular to ac), but a* and
c* are not colinear with corresponding real axes a and c.
The reciprocal lattice is spatially linked to the crystal because of the way the
lattice points are defined, so if we rotate the crystal, the reciprocal lattice rotates
with it. So now when you think of a crystal, and imagine the many identical unit
cells stretching out in all directions (real space), imagine also a lattice of points
in reciprocal space, points whose lattice spacing is inversely proportional to the
interplanar spacings within the crystal.
You are now in a position to understand the logic behind the mnemonic device for
signs of Miller indices (p. 54). Recall that reciprocal lattice points are constructed
on normals or perpendiculars to Miller planes. As a result, if the real and reciprocal
lattices are superimposed as they were during the construction of Fig. 4.9, p. 58, a
normal to a Miller plane points toward the corresponding reciprocal-lattice point.
If the normal points toward the (++) quadrant of a two-dimensional coordinate
system, then the corresponding reciprocal-lattice point lies in that (++) quadrant
of reciprocal space, and thus its corresponding plane is assigned positive signs for
both of its indices. The same normal also points toward the (−−) quadrant, where
the corresponding (−−) reciprocal-lattice point lies, so the same set of planes can
also be assigned negative signs for both indices.
4.2.5 Bragg’s law in reciprocal space
Now I will look at diffraction from within reciprocal space. I will show that the
reciprocal-lattice points give the crystallographer a convenient way to compute
the direction of diffracted beams from all sets of parallel planes in the crystalline
lattice (real space). This demonstration entails showing how each reciprocal-lattice
point must be arranged with respect to the X-ray beam in order to satisfy Bragg’s
Section 4.2 Geometric principles of diffraction 61
law and produce a reflection from the crystal. It will also show how to predict the
direction of the diffracted ray.
Figure 4.12a shows an a*b* plane of reciprocal lattice. Assume that an X-ray
beam (arrow XO) impinges upon the crystal along this plane. Point O is arbitrarily
chosen as the origin of the reciprocal lattice. (Remember that O is also a reallattice point in the crystal.) Imagine the X-ray beam passing through O along
the line XO (arrow). Draw a circle of radius 1/λ having its center C on XO and
passing throughO. This circle represents the wavelength of the X-rays in reciprocal
space. (If the wavelength is λ in real space, it is 1/λ in reciprocal space.) Rotating
the crystal about O rotates the reciprocal lattice about O, successively bringing
reciprocal lattice points like P and P ′ into contact with the circle. In Fig. 4.12a, P
(whose indices are hkl) is in contact with the circle, and the lines OP and BP are
drawn. The angle PBO is θ . Because the triangle PBO is inscribed in a semicircle,
it is a right triangle and
sin θ = OP
OB
= OP
2/λ
. (4.2)
Rearranging Eq. 4.2 gives
2
1
OP
sin θ = λ. (4.3)
Because P is a reciprocal lattice point, the length of the line OP is 1/dhkl , where
h, k, and l are the indices of the set of planes represented by P . (Recall from
the construction of the reciprocal lattice that the length of a line from O to a
reciprocal-lattice point hkl is 1/dhkl .) So 1/OP = dhkl and
2dhkl sin θ = λ, (4.4)
which is Bragg’s law with n = 1. So Bragg’s law is satisfied, and reflection occurs,
when a reciprocal-lattice point touches this circle.
In Fig. 4.12b, the crystal, and hence the reciprocal lattice, has been rotated
clockwise about origin O until P ′, with indices h′k′l′, touches the circle. The
same construction as in Fig. 4.12a now shows that
2dh′k′l′ sin θ = λ. (4.5)
We can conclude that whenever the crystal is rotated about origin O so that a
reciprocal-lattice point comes in contact with this circle of radius 1/λ, Bragg’s
law is satisfied and a reflection occurs.
What direction does the diffracted beam take? Recall (from construction of the
reciprocal lattice) that the line defining a reciprocal-lattice point is normal to the
set of planes having the same indices as the point. So BP, which is perpendicular
to OP, is parallel to the planes that are producing reflection P in Fig. 4.12a.
62 Chapter 4 Collecting Diffraction Data
Figure 4.12  Diffraction in reciprocal space. (a) Ray R emerges from the crystal
when reciprocal lattice point P intersects the circle. (b) As the crystal rotates clockwise
around origin O, point P ′ intersects the circle, producing ray R′.
Section 4.2 Geometric principles of diffraction 63
If we draw a line (red) parallel to BP and passing through C, the center of the
circle, this line (or any other line parallel to it and separated from it by an integral
multiple of dhkl) represents a plane in the set that reflects the X-ray beam under
these conditions. The beam impinges upon this plane at the angle θ , is reflected
at the same angle, and so diverges from the beam at C by the angle 2θ , which
takes it precisely through the point P . So CP gives the direction of the reflected
ray R in Fig. 4.12a. In Fig. 4.12b, the reflected ray R′ follows a different path, the
line CP′.
The conclusion that reflection occurs in the direction CP when reciprocal latticepoint P comes in contact with this circle also holds for all points on all circles
produced by rotating the circle of radius 1/λ about the X-ray beam. The figure that
results, called the sphere of reflection, or the Ewald sphere, is shown in Fig. 4.13
intersecting the reciprocal-lattice planes h0l and h1l. In the crystal orientation
shown, reciprocal-lattice point 012 is in contact with the sphere, so a diffracted
ray R is diverging from the source beam in the direction defined by C and point
012. This ray would be detected as the 012 reflection.
As the crystal is rotated in the X-ray beam, various reciprocal-lattice points come
into contact with this sphere, each producing a beam in the direction of a line from
the center of the sphere of reflection through the reciprocal-lattice point that is in
contact with the sphere. The reflection produced when reciprocal-lattice point Phkl
contacts the sphere is called the hkl reflection and, according to Bragg’s model, is
caused by reflection from the set of equivalent, parallel, real-space planes (hkl).
This model of diffraction implies that the directions of reflection, as well as the
number of reflections, depend only upon unit-cell dimensions and not on the contents of the unit cell. As stated earlier, the intensity of reflection hkl depends upon
the amount of electron density, or the average value of ρ(x, y, z), on planes (hkl).
Figure 4.13  Sphere of reflection. When reciprocal lattice point 012 intersects the
sphere, ray R emerges from the crystal as reflection 012.
64 Chapter 4 Collecting Diffraction Data
I will show (Chapter 5) that the intensities of the reflections give us the structural
information we seek.
4.2.6 Number of measurable reflections
If the sphere of reflection has a radius of 1/λ, and the crystal is rotated about
origin 000 on the surface of this sphere, then any reciprocal-lattice point within a
distance 2/λ of the origin can be rotated into contact with the sphere of reflection
(Fig. 4.14). This distance defines the limiting sphere. The number of reciprocal
lattice points within the limiting sphere is equal to the number of reflections that can
be produced by rotating the crystal through all possible orientations in the X-ray
beam. This demonstrates that the unit-cell dimensions and the wavelength of the
X-rays determine the number of measurable reflections. Shorter wavelengths make
a larger sphere of reflection, bringing more reflections into the measurable realm.
Larger unit cells mean smaller reciprocal unit cells, which populate the limiting
sphere more densely, also increasing the number of measurable reflections.
Because there is one lattice point per reciprocal unit cell (one-eighth of each lattice point lies within each of the eight unit-cell vertices), the number of reflections
within the limiting sphere is approximately the number of reciprocal unit cells
within this sphere. So the number N of possible reflections equals the volume of
the limiting sphere divided by the volume Vrecip of one reciprocal cell. The volume
of a sphere of radius r is (4π/3)r3, and r for the limiting sphere is 2/λ, so
N = (4π/3) · (2/λ)
3
Vrecip
. (4.6)
Figure 4.14  Limiting sphere. All reciprocal-lattice points within the limiting sphere
of radius 2/λ can be rotated through the sphere of reflection.
Section 4.2 Geometric principles of diffraction 65
The volume V of the real unit cell is V −1recip, so
N = 33.5 · V
λ3
. (4.7)
Equation 4.7 shows that the number of available reflections depends only on
V and λ, the unit-cell volume and the wavelength of the X radiation. For a
modest-size protein unit cell of dimensions 40× 60× 80 Å, 1.54-Å radiation can
produce 1.76× 106 reflections, an overwhelming amount of data. Fortunately,
because of cell and reciprocal-lattice symmetry, not all of these reflections are
unique (Sec. 4.3.7, p. 88). Still, getting most of the available information from the
diffraction experiment with protein crystals usually requires measuring somewhere
between one thousand and one million reflections.
It can be further shown that the limit of resolution in an image derived from
diffraction information is roughly equal to 0.707 times dmin, the minimum interplanar spacing that gives a measurable reflection at the wavelength of the X-rays.
For instance, with 1.54-Å radiation, the resolution attainable from all the available data is 0.8 Å, which is more than needed to resolve atoms. A resolution of
1.5 Å, which barely resolves adjacent atoms, can be obtained from about half the
available data. Interpretable electron-density maps can usually be obtained with
data only out to 2.5 or 3 Å. The number of reflections out to 2.5 Å is roughly the
volume of a limiting reciprocal sphere of radius 1/(2.5Å) multiplied by the volume
of the real unit cell. For the unit cell in the preceding example, this gives about
50,000 reflections. (For a sample calculation, see Chapter 8.)
4.2.7 Unit-cell dimensions
Because reciprocal-lattice spacings determine the angles of reflection, the spacings
of reflections on the detector are related to reciprocal-lattice spacings. (The exact
relationship depends on the geometry of recording the reflections, as discussed
later.) Reciprocal-lattice spacings, in turn, are simply the inverse of real-lattice
spacings. So the distances between reflections on the detector and the dimensions
of the unit cell are closely connected, making it possible to measure unit-cell
dimensions from reflection spacings. I will discuss the exact geometric relationship
in Sec. 4.3.6, p. 86, in the context of data-collection devices, whose geometry
determines the method of computing unit-cell size.
4.2.8 Unit-cell symmetry
If the unit-cell contents are symmetric, then the reciprocal lattice is also symmetric
and certain sets of reflections are equivalent. In theory, only one member of each
set of equivalent reflections need be measured, so awareness of unit-cell symmetry
can greatly reduce the magnitude of data collection. In practice, redundancy of
measurements improves accuracy, so when more than one equivalent reflection
is observed (measured), or when the same reflection is observed more than once,
the average of these multiple observations is considered more accurate than any
single observation.
66 Chapter 4 Collecting Diffraction Data
In this section, I will discuss some of the simplest aspects of unit-cell symmetry.
Crystallography in practice requires detailed understanding of these matters, but
users of crystallographic models need only understand their general importance.
As I will show later (Sec. 4.3.7, p. 88, and Chapter 5), the crystallographer can
determine the unit-cell symmetry from a limited amount of X-ray data and thus can
devise a strategy for data collection that will control the redundancy of observations
of equivalent reflections. While today’s data-collection software can often make
such decisions automatically, sometimes it can do no more than reduce the decision
to several alternatives. The crystallographer must be able to check the decisions
of the software, and know how to make the correct choice if the software offers
alternatives.
When it comes to data collection, the internal symmetry of the unit cell is
fundamental, and the equalities of Fig. 4.2, p. 51 are actually identities. That is,
when Fig. 4.2 says a=b, it means not only that the axis lengths are the same,
but that the contents of the unit cell must be identical along those axes. So the
equalities must reflect symmetry within the unit cell. For example, a cell in which
edges a, b, and c are equal, and cell angles α, β, and λ are 90◦ to within the
tolerance of experimental measurement, would appear to be cubic. But it might
actually be triclinic if it possesses no internal symmetry.
The symmetry of a unit cell and its contents is described by its space group,
which describes the cell’s internal symmetry elements. Space group is designated
by a cryptic symbol (like P 212121), in which a capital letter indicates the lattice
type and the other symbols represent symmetry operations (defined below) that can
be carried out on the unit cell without changing its appearance. Mathematicians
in the late 1800s showed that there are exactly 230 possible space groups. The
unit cells of a few lattice types in cubic crystal systems are shown in Fig. 4.15.
P designates a primitive lattice, containing one lattice point at each comer or vertex
of the cell. Because each lattice point is shared among eight neighboring unit cells,
a primitive lattice contains eight times one-eighth or one lattice point per unit
cell. Symbol I designates a body-centered or internal lattice, with an additional
lattice point in the center of the cell, and thus two lattice points per unit cell.
Figure 4.15  Primitive (P ), body-centered or internal (I ), and face-centered (F ) unit
cells.
Section 4.2 Geometric principles of diffraction 67
Figure 4.16  Of the several possible ways to divide a lattice into unit cells, the
preferred choice is the unit cell that is most symmetrical. Centered unit cells are chosen
when they are more symmetrical than any of the possible primitive cells.
SymbolF designates a face-centered lattice, with additional lattice points (beyond
the primitive ones) on the centers of some or all faces.
There are usually several ways to choose the unit cell in any lattice (Fig 4.16).
By convention, the choice is the unit cell that is most symmetrical. Body-centered
and face-centered unit cells are chosen when they have higher symmetry than any
of the primitive unit-cell choices. In Fig. 4.16, the first three choices are primitive,
but are not as symmetrical as the centered cell on the right. With the addition of
body- and face-centered lattices to the primitive lattices of Fig. 4.2, p. 51, there
are just 13 allowable lattices, known as the Bravais lattices.
After defining the lattice type, the second part of identifying the space group
is to describe the internal symmetry of the unit cell, using symmetry operations.
To illustrate with a familiar object, one end of rectangular table looks just like
a mirror reflection of the other end (Fig. 4.17a). We say that the table possesses
a mirror plane of symmetry, cutting the table perpendicularly across its center
(actually, it has two mirror planes; where is the other one?). In addition, if the
rectangular table is rotated 180◦ about an axis perpendicular to and centered on
the tabletop, the table looks just the same as it did before rotation (ignoring imperfections such as coffee stains). We say that the table also possesses a twofold
rotation axis because, in rotating the table one full circle about this axis, we find
two positions that are equivalent: 0◦ and 180◦. The mirrors and the twofold axis
are examples of a symmetry elements.
Protein molecules are inherently asymmetric, being composed of chiral aminoacid residues coiled into larger chiral structures such as right-handed helices
or twisted beta sheets. If only one protein molecule occupies a unit cell, then
the cell itself is chiral, and there are no symmetry elements, as in Fig. 4.17b,
when one chiral object is placed anywhere on the table, destroying all symmetry
68 Chapter 4 Collecting Diffraction Data
Figure 4.17  (a) Table with two symmetry elements, mirror plane and twofold
rotation axis. (b) Placing a chiral object anywhere on the table destroys all symmetry,
but (c) if two identical chiral objects are properly placed, they restore twofold rotational
symmetry. No placement of them will restore mirror symmetry.
Section 4.2 Geometric principles of diffraction 69
in the figure. This situation is rare; in most cases, the unit cell contains several
identical molecules or oligomeric complexes in an arrangement that produces symmetry elements. In the unit cell, the largest aggregate of molecules that possesses
no symmetry elements, but can be juxtaposed on other identical entities by symmetry operations, is called the asymmetric unit. In the simplest case for proteins,
the asymmetric unit is a single protein molecule.
The simplest symmetry operations and elements needed to describe unit-cell
symmetry are translation, rotation (element: rotation axis), and reflection (element:
mirror plane). Combinations of these elements produce more complex symmetry
elements, including centers of symmetry, screw axes, and glide planes (discussed
later). Because proteins are inherently asymmetric, mirror planes and more complex elements involving them are not found in unit cells of proteins. For example,
notice in Fig. 4.17c that proper placement of two identical chiral objects restores
twofold rotational symmetry, but no placement of the two objects will restore
mirror symmetry. Because of protein chirality, symmetry elements in protein crystals include only translations, rotations, and screw axes, which are rotations and
translations combined. This limitation on symmetry of unit cells containing asymmetrical objects reduces the number of space groups for chiral molecules from
230 to 65.
Now let us examine some specific symmetry operations. Translation simply
means movement by a specified distance. For example, by the definition of unit
cell, movement of its contents along one of the unit-cell axes by a distance equal
to the length of that axis superimposes the atoms of the cell on corresponding
atoms in the neighboring cell. This translation by one axial length is called a unit
translation. Unit cells often exhibit symmetry elements that entail translations by
a simple fraction of axial length, such as a/4.
In the space-group symbols, rotation axes such as the twofold axis of the table
in Fig. 4.17a or c are represented in general by the symbol n and specifically by a
number. For example, 4 means a fourfold rotation axis. If the unit cell possesses
this symmetry element, then it has the same appearance after each 90◦ rotation
around the axis.
The screw axis results from a combination of rotation and translation. The
symbol nm represents an n-fold screw axis with a translation of m/n of the unit
translation. For example, Fig. 4.18 shows models of the amino acid alanine on
a 31 screw axis in a hypothetical unit cell. On the screw axis, each successive
molecule is rotated by 120◦ (360◦/3) with respect to the previous one, and translated
one-third of the axial length in the direction of the rotation axis.
Figure 4.19 shows alanine in hypothetical unit cells of two space groups. A
triclinic unit cell (Fig. 4.19a) is designated P 1, being a primitive lattice with
only a one-fold axis of symmetry (that is, with no symmetry). P 21 (Fig. 4.19b)
describes a primitive unit cell possessing a twofold screw axis parallel to c, which
points toward you as you view the figure. Notice that along any 21 screw axis,
successive alanines are rotated 180◦ and translated one-half the axis length. A cell
in space group P 212121 possesses three perpendicular twofold screw axes.
70 Chapter 4 Collecting Diffraction Data
Figure 4.18  Three alanine molecules on a 31 screw axis in a hypothetical unit cell
(stereo).
Because crystallographers deal with unit cell contents by specifying their x, y,
and z coordinates, one of the most useful ways to describe unit-cell symmetry is
by equivalent positions, locations in the unit cell that are superimposed on each
other by the symmetry operations. In a P 21 cell with the screw axis on or parallel
to cell axis b, for an atom located at (x, y, z), an identical atom can always be
found at (−x, y + 1/2,−z) (more commonly written (x, y + 1/2, z), because the
operation of a 21 screw axis interchanges these positions. So a P 21 cell has the
equivalent positions (x, y, z) and (x, y+1/2, z). (The 1/2 means one-half of a unit
translation along b, or a distance b/2 along the y-axis.) To see an example, look
again at Fig. 4.19b, and focus on the carboxyl carbon atom of the alanine at bottom
front left of the figure. If the b-axis is perpendicular to the page, with a horizontal
and c vertical, you can see that moving this atom from position (x, y, z) to position
(−x, y + 1/2,−z) will superimpose it on the next alanine carboxy carbon along
the b-axis.
Lists of equivalent positions for the 230 space groups can be found in International Tables for X-ray Crystallography, a reference series that contains an
enormous amount of practical information that crystallographers need in their
daily work. So the easiest way to see how asymmetric units are arranged in a cell
of complex symmetry is to look up the space group in International Tables. Each
entry contains a list of equivalent positions for that space group, and several types
of diagrams of the unit cell. The entry for space group P 21 is shown in Fig. 4.20.
Section 4.2 Geometric principles of diffraction 71
Figure 4.19  Alanine molecules in P 1 (a) and P 21 (b) unit cells (stereo).
Certain symmetry elements in the unit cell announce themselves in the diffraction pattern by causing specific reflections to be missing (intensity of zero). For
example, a twofold screw axis (21) along the b edge causes all 0k0 reflections
having odd values of k to be missing. Notice in Fig. 4.20 that “Reflection conditions” in a P 21 cell includes the condition that, along the (0k0) axis, k = 2n,
meaning that only the even-numbered reflections are present along the k-axis.
So the missing reflections include 010, 030, 050, and so forth. As another example, body-centered (I ) lattices show missing reflections for all values of hkl where
the sum of h, k, and l is odd. International Tables will list among “Reflection
conditions” the following: 0kl: k = l = 2n. This means that half the reflections
in the (0kl) plane are missing, including reflections 010, 001, 030, 003, and so
72 Chapter 4 Collecting Diffraction Data
Figure 4.20  Entry for space group P 21 in International Tables for Crystallography, Brief Teaching Edition of Volume A, Space-Group Symmetry, Theo Hanh, ed., Kluwer
Academic Publishers, Norwell, MA, 5th, revised edition 2002, pp. 91–92. Reprinted
with kind permission from Kluwer Academic Publishers and the International Union of
Crystallography.
Section 4.3 Collecting X-ray diffraction data 73
forth, giving the reflections in zero-level plane a diamond-shaped pattern. These
patterns of missing reflections are called systematic absences, and they allow the
crystallographer to determine the space group by looking at a few crucial planes
of reflections. I will show later in this chapter how symmetry guides the strategy
of data collection. In Chapter 5, I will show why symmetry causes systematic
absences.
4.3 Collecting X-ray diffraction data
4.3.1 Introduction
Simply stated, the goal of data collection is to determine the indices and record
the intensities of as many reflections as possible, as rapidly and efficiently as possible. One cause for urgency is that crystals, especially those of macromolecules,
deteriorate in the beam because X-rays generate heat and reactive free radicals in
the crystal. Thus the crystallographer would like to capture as many reflections
as possible during every moment of irradiation. Often the diffracting power of
the crystal limits the number of available reflections. Protein crystals that produce
measurable reflections from interplanar spacings down to about 3 Å or less are
usually suitable for structure determination.
In the following sections, I will discuss briefly a few of the major instruments
employed in data collection. These include the X-ray sources, which produce an
intense, narrow beam of radiation; detectors, which allow quantitative measurement of reflection intensities; and cameras, which control the orientation of the
crystal in the X-ray beam, and thus direct reflections having known indices to
detectors.
4.3.2 X-ray sources
X-rays are electromagnetic radiation of wavelengths 0.1–100 Å. X-rays in the
useful range for crystallography can be produced by bombarding a metal target
(most commonly copper, molybdenum, or chromium) with electrons produced
by a heated filament and accelerated by an electric field. A high-energy electron
collides with and displaces an electron from a low-lying orbital in a target metal
atom. Then an electron from a higher orbital drops into the resulting vacancy,
emitting its excess energy as an X-ray photon.
The metal in the target exhibits narrow characteristic lines (specific wavelengths) of emission resulting from the characteristic energy-level spacing of that
element. The wavelengths of emission lines are longer for elements of lower atomic
numberZ. Electronic shells of atoms are designated, starting from the lowest level,
as K, L, M, . . .. Electrons dropping from the L shell of copper (Z = 29) to replace
displaced K electrons (L −→ K or Kα transition) emit X-rays of λ = 1.54 Å. The
M −→ K transition produces a nearby emission band (Kβ ) at 1.39 Å (Fig. 4.21a,
solid curve). For molybdenum (Z = 42), λ(Kα) = 0.71 Å, and λ(Kβ) = 0.63 Å.
74 Chapter 4 Collecting Diffraction Data
Figure 4.21  (a) Emission (solid red line) and absorption (dashed green line) spectra
of copper. (b) Emission spectrum of copper (solid red) and absorption spectrum of nickel
(dashed green). Notice that Ni absorbs copper Kβ more strongly than Kα .
A monochromatic (single-wavelength) source of X-rays is desirable for crystallography because the radius of the sphere of reflection is 1/λ. A source producing
two distinct wavelengths of radiation gives two spheres of reflection and two
interspersed sets of reflections, making indexing difficult or impossible because of
overlapping reflections. Elements like copper and molybdenum make good X-ray
sources if the weaker Kβ radiation can be removed.
At wavelengths away from the characteristic emission lines, each element
absorbs X-rays. The magnitude of absorption increases with increasing X-ray
wavelength and then drops sharply just at the wavelength of Kβ . The green curve
in Fig. 4.21a shows the absorption spectrum for copper. The wavelength of this
absorption edge, or sharp drop in absorption, like that of characteristic emission
lines, increases as Z decreases such that the absorption edge for element Z−1 lies
slightly above the Kβ emission line of element Z. This makes element Z − 1 an
effective Kβ filter for elementZ, leaving almost pure monochromatic Kα radiation.
For example, a nickel filter 0.015 mm in thickness reduces Cu–Kβ radiation to
about 0.01 times the intensity of Cu–Kα . Figure 4.21b shows the copper emission
spectrum (red) and the nickel absorption spectrum (green). Notice that Ni absorbs
strongly at the wavelength of Cu–Kβ radiation, but transmits Cu–Kα .
There are three common X-ray sources, X-ray tubes (actually a cathode ray
tube sort of like a television tube), rotating anode tubes, and particle storage
rings, which produce synchrotron radiation in the X-ray region. In the X-ray tube,
electrons from a hot filament (cathode) are accelerated by electrically charged
Section 4.3 Collecting X-ray diffraction data 75
Figure 4.22  (a) X-ray tube. (b) Rotating anode source.
plates and collide with a water-cooled anode made of the target metal (Fig. 4.22a).
X-rays are produced at low angles from the anode, and emerge from the tube
through windows of beryllium.
Output from X-ray tubes is limited by the amount of heat that can be dissipated from the anode by circulating water. Higher X-ray output can be obtained
from rotating anode sources, in which the target is a rapidly rotating metal disk
(Fig. 4.22b). This arrangement improves heat dissipation by spreading the powerful electron bombardment over a much larger piece of metal. Rotating anode
sources are more than ten times as powerful as tubes with fixed anodes.
Particle storage rings, which are associated with the particle accelerators used
by physicists to study subatomic particles, are the most powerful X-ray sources.
In these giant rings, electrons or positrons circulate at velocities near the speed of
light, driven by energy from radio-frequency transmitters and maintained in circular motion by powerful magnets. A charged body like an electron emits energy
(synchrotron radiation) when forced into curved motion, and in accelerators, the
energy is emitted as X-rays. Accessory devices called wigglers cause additional
bending of the beam, thus increasing the intensity of radiation. Systems of focusing mirrors and monochromators tangential to the storage ring provide powerful
monochromatic X-rays at selectable wavelengths.
A particle-storage ring designed expressly for producing X-rays, part of the
National Synchrotron Light Source (NSLS) at Brookhaven National Laboratory
on Long Island in New York, is shown in Fig. 4.23. In the interior floor plan
76 Chapter 4 Collecting Diffraction Data
Figure 4.23  National Synchrotron Light Source, Brookhaven National Laboratory,
Brookhaven NY. (a) Aerial view of exterior. (b) Interior floor plan.
(Fig. 4.18b), paths of particle-storage rings are shown as dark circles. The largest
of the three rings in the building is the X-ray ring, which is 54 meters (177 feet)
in diameter. The dark lines tangent to rings represent beam lines for work in
experimental stations (green blocks). NSLS began operations in 1984. Crystallographers can apply to NSLS and other synchrotron sources for grants of time for
Section 4.3 Collecting X-ray diffraction data 77
data collection. For a detailed virtual tour of NSLS and other synchrotron sources,
see the CMCC home page.
Although synchrotron sources are available only at storage rings and require the
crystallographer to collect data away from the usual site of work, there are many
advantages that compensate for the inconvenience. X-ray data that requires several
hours of exposure to a rotating anode source can often be obtained in seconds or
minutes at a synchrotron source like NSLS. In a day or two at a synchrotron
source, a crystallographer can collect data that might take weeks to acquire with
conventional sources. Another advantage, as I will show in Chapter 6, is that
X-rays of selectable wavelength can be helpful in solving the phase problem.
Whatever the source of X-rays, the beam is directed through a collimator, a
narrow metal tube that selects and reflects the X-rays into parallel paths, producing
a narrow beam. After collimation, beam diameter can be further reduced with
systems of metal plates called focusing mirrors. In the ideal arrangement of source,
collimators, and crystal, all points on the crystal can “see” through the collimator
and mirrors to all line-of-sight points on the X-ray source.
SAFETY NOTE: X-ray sources pose dangers that the crystallographer must
consider in daily work. X-ray tubes require high-voltage power supplies containing large condensers that can produce a dangerous shock even after equipment
is shut off. The X-rays themselves are relatively nonpenetrating, but can cause
serious damage to surface tissues. Even brief exposure to weak X-rays can damage eyes, so protective goggles are standard attire in the vicinity of X-ray sources.
The direct beam is especially powerful, and sources are electronically interlocked
so that the beam entrance shutter cannot be opened while the user is working with
the equipment. The beam intensity is always reduced to a minimum during alignment of collimating mirrors or cameras. During data collection, the direct beam
is blocked just beyond the crystal by a piece of metal called a beam stop, which
also has the beneficial effect of preventing excessive radiation from reaching the
center of the detector, thus obscuring low-angle reflections. In addition, the entire
source, camera, and detector are usually surrounded by Plexiglas to block scattered radiation from the beam stop or collimators but to allow observation of the
equipment. As a check on the efficacy of measures to prevent X-ray exposure, the
prudent crystallographer wears a dosage-measuring ring or badge during all work
with X-ray equipment. These devices are periodically sent to radiation-safety labs
for measurement of the X-ray dose received by the worker.
4.3.3 Detectors
Reflection intensities can be measured by scintillation counters, which in essence
count X-ray photons and thus give quite accurate intensities over a wide range.
Scintillation counters contain a phosphorescent material that produces a flash of
light (a scintillation) when it absorbs an X-ray photon. A photocell counts the
flashes. With simple scintillation counters, each reflection must be measured separately, an arrangement that was the basis of diffractometry, which is now used
only for small-molecule crystallography, where the number of reflections per data
78 Chapter 4 Collecting Diffraction Data
set is very small. Macromolecular crystallography requires means to collect many
reflections at once. Such devices are called area detectors.
The simplest X-ray area detector, and for years the workhorse of detectors, is
X-ray-sensitive film (for example, see Fig. 2.7, p. 14), but film has been replaced
by image plates and CCD detectors, which I will describe below. Various types
of cameras (next section) can direct reflections to area detectors in useful arrangements, allowing precise determination of indices and intensities for thousands of
reflections from a single image.
Image plate detectors are somewhat like reusable films that can store diffraction
images reversibly and have a very wide dynamic range, or capacity to record reflections of widely varying intensity. Image plates are plastic sheets with a coating of
small crystals of a phosphor, such as BaF:Eu2+. The crystals can be stimulated by
X-rays into a stable excited state in which Eu2+ loses an electron to the F− layer,
which contains electron vacancies introduced by the manufacturing process. Further stimulation by visible light causes the electrons to drop back to the Eu layer,
producing visible light in proportion to the intensity of the previously absorbed
X-rays. After X-ray exposure, data are read from the plate by a scanner in which
a fine laser beam induces luminescence from a very small area of the plate, and a
photocell records the intensity of emitted light. The intensities are fed to a computer, which can then reconstruct an image of the diffraction pattern. Image plates
can be erased by exposure to bright visible light and reused indefinitely.
Multiwire area detectors were the first type to combine the accuracy and wide
dynamic range of scintillation counting; the simultaneous measurement of many
reflections, as with image plates; and the advantage of direct collection of data
by computer, without a separate scanning step. In these detectors, some of which
are still in use, two oppositely charged, perpendicular sets of parallel wires in
an inert gas detect and accurately locate ionization of the gas induced by single
X-ray photons, relaying to a computer the positions of X-ray photons almost
instantaneously as they strike the detector. The computer records events and their
locations to build up an image of the reflections that reach the detector. Multiwire
systems are insensitive to X-rays for a short time after recording each event, a
period known as dead time, which sets a limit on their rate of X-ray photon
counting.
The latest designs in area detectors employ charge-coupled devices (CCDs) as
detectors (Fig 4.24). CCDs first found use in astronomy as light collectors of great
sensitivity, and have since replaced other kinds of light detectors in many common
devices, most notably digital cameras. In effect, CCDs are photon counters, solidstate devices that accumulate charge (electrons) in direct proportion to the amount
of light that strikes them. An incident photon raises an electron to a higher energy
level, allowing it to move into, and become trapped within, a positively charged
region at the center of the pixel (Fig. 4.24a). Each pixel in the array can accurately
accumulate electrons from several hundred thousand absorption events before
reaching its capacity (becoming saturated). Before saturation, its contents must
be read out to a computer. At the end of a collection cycle, which produces one
frame of data, the charges are read out by a process in which rows of pixel charge
Section 4.3 Collecting X-ray diffraction data 79
Figure 4.24  (a) Schematic diagram of CCD (8 pixels), showing path of data readout.
(b) Diagram of CCD X-ray detector.
are transferred sequentially, by flipping the voltage applied at the edges, into a
serial readout row at one edge of the CCD. The charges in the readout row are
transferred serially to an amplifier at the end of the row, and then the next row of
pixel charges is transferred into the readout row. Because all data are read out at
the end of data collection, a CCD, unlike a multiwire area detector, has no dead
time, and thus no practical limit on its rate of photon counting.
CCDs are sensitive to visible light, not X-rays, so phosphors that produce visible
light in response to X-ray absorption must be interposed between the diffracted
X-rays and the CCD. In addition, the individual pixels of commercially available
CCD arrays are smaller and more densely packed than is best for typical X-ray
detection geometry, and in addition, some light is lost by reflection from the CCD
surface. To overcome all these obstacles, CCD arrays are bonded to a tapered
bundle of optical fibers (Fig 4.24b). The large end of the optical taper is coated
with phosphors that emit visible light in response to X-rays, producing, in effect,
a very dense array of scintillation counters. As an example of detector and CCD
dimensions, the CCD area detector of the NSLS beamline X-12, which is designed
80 Chapter 4 Collecting Diffraction Data
especially for macromolecular crystallography, consists of four 1150×1150-pixel
CCD arrays in a 2 × 2 array, giving a 5.3 megapixel array. The optical taper is
about 188 mm (roughly 7.5 inches) on a side at the phosphor-coated face of the
detector, and tapers down to about 50 mm (less than 2 inches) at the face of the
CCD array. Each pixel can count 450,000 to 500,000 photons between readouts,
and readouts take from 1 to 10 seconds, depending on the desired signal-to-noise
ratio in the output. These numbers are typical in 2005, but such technology changes
rapidly.
4.3.4 Cameras
Between the X-ray source and the detector lies a mounted crystal, in the grip of
an X-ray camera. The term is a misnomer. The word camera is from Latin for
chamber or vault, referring in common photographic cameras to the darkened
chamber inside that prevents stray light from reaching the film (or CCD array).
X-ray cameras have no darkened chamber; they are simply mechanical devices,
usually a combination of a crystal-holding head on a system of movable circular
mounts for orienting and moving crystals with great precision (to within 10−5
degrees). The goal in data collection is to use carefully controlled movement of
the crystal by the camera to direct all unique reflections to a detector like one of
those described in the previous section. In this section, I will describe cameras
that can rotate a crystal through a series of known orientations, causing specified
reciprocal lattice points to pass through the sphere of reflection and thus produce
diffracted X-ray beams.
In all forms of data collection, the crystal is mounted on a goniometer head, a
device that allows the the crystal orientation to be set and changed precisely. The
most complex goniometer heads (Fig. 4.25a) consist of a holder for a capillary
tube or cryoloop containing the crystal; two arcs (marked by angle scales), which
permit rotation of the crystal by up to 40◦ in each of two perpendicular planes; and
two dovetailed sledges, which permit small translations of the arcs for centering the
crystal on the rotation axis of the head. Simpler heads (Fig. 4.25b) contain sledges
only. Protein crystals, either sealed in capillary tubes with mother liquor (as in a)
or flash-frozen in a fiber loop (as in b) are mounted on the goniometer head, which
is adjusted to center the crystal in the X-ray beam and to allow rotation of the
crystal while maintaining centering. Flash-frozen crystals are held in a stream of
cold nitrogen gas emerging from a reservoir of liquid nitrogen, and the goniometer
head is heated to prevent condensation from forming ice on it.
The goniometer head and crystal are mounted in a system of movable circles
called a goniostat, which allows automated, highly precise movement of the crystal into almost any orientation with respect to the X-ray beam and the detector.
The crystal orientation is specified by a system of angles, whose nomeclature
is a vestige of diffractometry, in which individual reflections are directed to a
scintillation counter for intensity measurement (diffractometry is still in wide
use for crystallography of small molecules). Figure 4.26 shows this system of
angles. A complete diffractometer consists of a fixed X-ray source, the goniostat,
Section 4.3 Collecting X-ray diffraction data 81
Figure 4.25  (a) Full goniometer head, with capillary tube holder at top. The tool
(right) is an Allen wrench for adjusting arcs and sledges. Photo courtesy of Charles Supper
Company. (b) Simple goniometer head, having only sledges. The holder on top is a magnetic
disk that accepts a cryoloop holder. The wiring is for heating the head, to prevent ice
formation from the nitrogen stream that keeps the crystal at low temperatures. Images
courtesy of Hampton Research.
and a movable scintillation-counter detector. The system of circles (Fig. 4.26)
allows rotation of the goniometer head (angle φ), movement of the head around a
circle centered on the X-ray beam (angle χ ), and rotation of the χ circle around
an axis perpendicular to the beam (angle ω). Furthermore, the detector moves on
a circle coplanar with the beam. The axis of this circle coincides with the ω-axis.
The position of the detector with respect to the beam is denoted by the angle 2θ .
(Why? Think about it, then look at Fig. 4.12, p. 62.) With this arrangement, the
crystal can be moved to bring any reciprocal lattice point that lies within the limiting sphere into the plane of the detector and into contact with the sphere of
reflection, producing diffracted rays in the detector plane. The detector can be
moved into proper position to receive, and measure the intensity of, the resulting
diffracted beam.
82 Chapter 4 Collecting Diffraction Data
Figure 4.26  System of angles in diffractometry. The crystal in the center is mounted
on a goniometer head.
Diffractometry gives highly accurate intensity measurement but is slow in comparison with methods that record many reflections at once. In addition, the total
irradiation time is long, so crystals may deteriorate and have to be replaced. While
one reflection is being recorded, there are usually other unmeasured reflections
present, so a considerable amount of diffracted radiation is wasted. Diffractometers teamed up with area detectors give substantial increases in the efficiency
of data collection, but are no match for more modern experimental arrangement,
which can direct hundreds of reflections to area detectors simultaneously.
Modern, high-speed data collection relies on the rotation/oscillation method
(Fig. 4.27). An oscillation camera is far simpler than a diffractometer, providing,
at the minimum, means to rotate the crystal about an axis perpendicular to the
beam (φ-axis), as well as to oscillate it back and forth by a few degrees about the
same axis. Figure 4.27 shows a diffractometer, X-ray source, and area detector
set up for rotation or oscillation photography. The camera provides means for
movement through all the diffractometer angles, as shown in the figure labels.
In the arrangement shown, the detector and X-ray source (a rotating anode, not
shown) are colinear, and rotations about φ and χ are used for data collection.
Rotation of the crystal through a small angle casts large numbers of reflections in
a complex pattern onto the area detector, as illustrated in Fig. 4.28.
Figure 4.28a shows how rotation of a crystal tips many planes of the reciprocal
lattice through the sphere of reflection, producing many reflections. The crystal
(dark red) lies in the middle of the Ewald sphere (sphere of reflection, light brown
surface). The smaller sphere of gray dots represents the reciprocal lattice. Purple
lines show reflections being produced at the current orientation of the crystal, from
Section 4.3 Collecting X-ray diffraction data 83
Figure 4.27  X-ray instrumentation used in the 2004 course, X-ray Methods in Structural Biology, Cold Spring Harbor Laboratory. The instrument is actually a four-circle
diffractometer using only the φ-axis for rotation photography. X-rays come from the left
from a rotating anode source (not shown), and emerge through a metal collimator to strike the
crystal. A CCD detector (right) captures reflections. The crystal is kept frozen by a stream
of cold nitrogen gas coming from a liquid nitrogen supply. The slight cloudiness just to the
right of the goniometer head is condensation of moisture in the air as it is cooled by the
nitrogen stream. Rotation of the crystal about the φ- and χ axes directs many reflections to
the detector. In the arrangement shown, axes φ, 2θ , and ω are coincident.
reciprocal-lattice points in contact with the Ewald sphere. The other reflections on
the detector were produced as the crystal was rotated through a small angle to its
current position.
Figure 4.28b illustrates the actual geometry of an oscillation image (called a
frame of data). To produce this figure, a computer program calculated the indices
of reflections expected when a crystal is oscillated a few degrees about its c-axis.
At the expected position of each reflection, the program plotted the indices of
that reflection. Only the l index of each reflection is shown here, revealing that
reflections from many levels of reciprocal space are recorded at once. Although
frames of oscillation data are very complex, modern software can index them.
As the crystal oscillates about a fixed starting position, many reciprocal-lattice
points pass back and forth through the sphere of reflection, and their intensities
are recorded. The amount of data from a single oscillation is limited only by
overlap of reflections. The strategy is to collect one frame by oscillating the crystal through a small angle about a starting position of rotation, recording all the
84 Chapter 4 Collecting Diffraction Data
Figure 4.28  (a) (Stereo) rotation or oscillation of a crystal by a few degrees tips
many reciprocal-lattice planes through the sphere of reflection, sending many reflections to
the detector. Image created with the free program XRayView, which allows the user to study
diffraction geometry interactively. To obtain the program, see the CMCC home page. Image
used with permission of Professor George N. Phillips, Jr. (b) Diagram showing expected
positions of reflections in a frame of oscillation data. Diagram courtesy of Professor Michael
Rossmann.
Section 4.3 Collecting X-ray diffraction data 85
resulting reflections, and then rotating the crystal to a new starting point such that
the new oscillating range overlaps the previous one slightly. From this new position, oscillation produces additional reflections in a second frame. This process is
continued until all unique reflections have been recorded.
In classical crystallography, the goniometer head was viewed through a microscope to orient the mounted crystal precisely, using crystal faces as guides. This
allowed the crystallographer to make photographs of the reciprocal lattice in specific orientations, such as to record the zero-level planes for measuring unit-cell
lengths and angles. Well-formed crystals show distinct faces that are parallel to
unit-cell edges, and first attempts to obtain a diffraction pattern were made by
placing a crystal face perpendicular to the X-ray beam. Preliminary photos of the
diffraction pattern allowed the initial setting to be refined. Photographing the very
revealing zero-level planes required a complex movement of crystal and detector,
called precession photography. Images of zero-level planes are still sometimes
called precession photographs. (Figure 2.7, p. 14 is a true precession photograph,
taken on film around 1985.) Examination of the zero-level planes of diffraction
allowed determination of unit-cell dimensions and space group. The crystallographer then devised a data-collection strategy that would record all unique reflections
from a minimum number of crystal orientations, as described later.
None of this is necessary today, because data-collection software can quickly
determine the crystal orientation from frames taken over a small range of rotation, making initial orientation of the crystal unimportant (which is good, because
orientation is hard to control in cryoloops). The usual procedure is to use the
goniometer head sledges to position the randomly oriented crystal so that it stays
centered in the beam through all rotations, collect reflections over a few degrees
of rotation, and then let the software determine the crystal orientation, index the
reflections, determine unit-cell parameters and space group, and devise the collection strategy. The procedure is sometimes called the American method—shoot
first, ask questions afterward. The software cannot always make the determination
unequivocally, and so sometimes offers choices. The crystallographer’s knowledge of crystal systems and symmetry guide the final decision. After full data
collection, the software provides means to view and analyze specific sections of
the data, such as the zero-level planes.
4.3.5 Scaling and postrefinement of intensity data
The goal of data collection is a set of consistently measured, indexed intensities
for as many of the reflections as possible. After data collection, the raw intensities
must be processed to improve their consistency and to maximize the number of
measurements that are sufficiently accurate to be used.
A complete set of measured intensities often includes many frames, as well as
distinct blocks of data obtained from several (or many) crystals in different orientations. Because of variability in the diffracting power of crystals, the difference
in the length of the X-ray path through the crystal in different orientations, and the
intensity of the X-ray beam, the crystallographer cannot assume that the absolute
intensities are consistent from one frame or block of data to the next. An obvious
86 Chapter 4 Collecting Diffraction Data
way to obtain this consistency is to compare reflections of the same index that were
measured from more than one crystal or in more than one frame and to rescale the
intensities of the two blocks of data so that identical reflections are given identical
intensities. This process is called scaling. Scaling is often preliminary to a more
complex process, postrefinement, which recovers usable data from reflections that
were only partially measured.
Primarily because real crystals are mosaics of submicroscopic crystals (Fig. 3.2,
p. 33), a reciprocal-lattice point acts as a small three-dimensional entity (sphere
or ovoid) rather than as an infinitesimal point. As a reciprocal-lattice point moves
through the sphere, diffraction is weak at first, peaks when the center of the point
lies precisely on the sphere, and then weakens again before it is extinguished.
Accurate measurement of intensity thus entails recording the X-ray output during
the entire passage of the point through the sphere. Any range of oscillation will
record some reflections only partially, but these may be recorded fully at another
rotation angle, allowing partial reflections to be discarded from the data. The problem of partial reflections is serious for large unit cells, where smaller oscillation
angles are employed to minimize overlap of reflections. In such cases, if partial
reflections are discarded, then a great deal of data is lost. Data from partial reflections can be interpreted accurately through postrefinement of the intensity data.
This process produces an estimate of the partiality of each reflection. Partiality
is a fraction p (0 ≥ p ≥ 1) that can be used as a correction factor to convert
the measured intensity of a partial reflection to an estimate of that reflection’s full
intensity.
Scaling and postrefinement are the final stages in producing a list of internally
consistent intensities for as many of the available reflections as possible.
4.3.6 Determining unit-cell dimensions
The unit-cell dimensions determine the reciprocal-lattice dimensions, which in
turn tell us where we must look for the data. Methods like oscillation photography require that we (or our software) know precisely which reflections will fall
completely and partially within a given oscillation angle so that we can collect as
many reflections as possible without overlap. So we need the unit-cell dimensions
in order to devise a strategy of data collection that will give us as many identifiable
(by index), measurable reflections as possible.
Modern software can search the reflections, measure their precise positions, and
subsequently compute unit-cell parameters. This search entails complexities we
need not encounter here. Instead, I will illustrate the simplest method for determining unit-cell dimensions: measuring reflection spacings from an orthorhombic
crystal on an image of a zero-level plane. Because reciprocal-lattice spacings are
the inverse of real-lattice spacings, the unit-cell dimensions are inversely proportional to the spacing of reflections on planes in reciprocal space, and determining
unit-cell dimensions from reciprocal-lattice spacings is a remarkably simple geometric problem. Figure 4.29 shows the geometric relationship between reflection
spacings on the film and actual reciprocal-lattice spacings.
Section 4.3 Collecting X-ray diffraction data 87
Figure 4.29  Reflection spacings on the film are directly proportional to reciprocallattice spacings, and so they are inversely proportional to unit-cell dimensions.
The crystal at C is precessing about its c*-axis, and therefore recording hk0
reflections on the detector, with the h00 axis horizontal and the 0k0 axis vertical.
Point P is the reciprocal-lattice point 100, in contact with the sphere of reflection,
and O is the origin. Point F is the origin on the detector and R is the recording
of reflection 100 on the detector. The distance OP is the reciprocal of the distance
d100, which is the length of unit-cell edge a. Because CRF and CPO are similar
triangles (all corresponding angles equal), and because the radius of the sphere of
reflection is 1/λ,
RF
CF
= PO
CO
= PO
1/λ
= PO · λ. (4.8)
Therefore,
PO = RF
CF · λ. (4.9)
Because d100 = 1/PO,
d100 = CF · λ
RF
. (4.10)
In other words, the axial length a (length of unit-cell edge a) can be determined
by dividing the crystal-to-detector distance (CF) by the distance from the detector
88 Chapter 4 Collecting Diffraction Data
origin to the 100 reflection (RF) and multiplying the quotient by the wavelength
of X-rays used in taking the photograph.
In like manner, the vertical reflection spacing along 0k0 or parallel axes gives
1/d010, and from it, the length of unit-cell axis b. Because we are considering
an orthorhombic crystal, which has unit-cell angles of 90◦, a second zero-level
image, taken after rotating this orthorhombic crystal by 90◦ about its vertical axis,
would record the 00l axis horizontally, giving 1/d001, and the length of c.
Of course, the distance from the detector origin to the 100 reflection on a zerolevel image is the same as the distance between any two reflections along this or
other horizontal lines, so one zero-level image allows many measurements to determine accurately the average spacing of reciprocal-lattice points along two different
axes. From accurate average values, unit-cell-axis lengths can be determined with
sufficient accuracy to guide a data-collection strategy.
Except perhaps for fun or curiosity, no one today works out this little geometry
problem to determine unit-cell dimensions, and I was tempted to omit the topic
during this revision. But nothing in crystallography shows more dramatically how
simple is the relationship between the submicroscopic dimensions of the unit cell
and the macroscopic dimensions of spacing between reflections at the detector.
I once sat down with an X-ray photo, used a small ruler under a magnifier to measure the distances between the centers of reflections, and used a pocket calculator
to compute the unimaginably small dimensions of unit cells in some of the first
crystals I had ever grown. To make a measurement at the molecular level with an
ordinary ruler was a magical experience.
4.3.7 Symmetry and the strategy of collecting data
Strategy of data collection is guided not only by the unit cell’s dimensions but also
by its internal symmetry. If the cell and its contents are highly symmetric, then
certain sets of crystal orientations produce exactly the same reflections, reducing
the number of crystal orientations needed in order to obtain all of the distinct or
unique reflections.
As mentioned earlier, the unit-cell space group can be determined from systematic absences in the diffraction pattern. With the space group in hand, the
crystallographer can determine the space group of the reciprocal lattice, and thus
know which orientations of the crystal will give identical data. All reciprocal lattices possess a symmetry element called a center of symmetry or point of inversion
at the origin. That is, the intensity of each reflection hkl is identical to the intensity of reflection hkl. To see why, recall from our discussion of lattice indices
(Sec. 4.2.2, p. 50) that the the index of the (230) planes can also be expressed as
(230). In fact, the 230 and the 230 reflections come from opposite sides of the
same set of planes, and the reflection intensities are identical. (The equivalence
of Ihkl and Ihkl is called Friedel’s law, but there are exceptions, as I will show in
Sec. 6.4, p. 128) This means that half of the reflections in the reciprocal lattice are
redundant, and data collection that covers 180◦ about any reciprocal-lattice axis
will capture all unique reflections.
Section 4.4 Summary 89
Additional symmetry elements in the reciprocal lattice allow further reduction
in the total angle of data collection. It can be shown that the reciprocal lattice
possesses the same symmetry elements as the unit cell, plus the additional point of
inversion at the origin. The 230 possible space groups reduce to only 11 different
groups, called Laue groups, when a center of symmetry is added. For each Laue
group, and thus for all reciprocal lattices, it is possible to compute the fraction of
reflections that are unique. For monoclinic systems, such as P 2, the center of symmetry is the only element added in the reciprocal lattice and the fraction of unique
reflections is 1/4. At the other extreme, for the cubic space group P 432, which
possesses four-, three-, and twofold rotation axes, only 1/48 of the reflections are
unique. Determination of the crystal symmetry can greatly reduce the number of
reflections that must be measured. It also guides the crystallographer in choosing
the best axis about which to rotate the crystal during data collection. In practice,
crystallographers collect several times as many reflections as the minimum number of unique reflections. They use the redundancy to improve the signal-to-noise
ratio by averaging the multiple determinations of equivalent reflections. They also
use redundancy to correct for X-ray absorption, which varies with the length of
the X-ray path through the crystal.
4.4 Summary
The result of X-ray data collection is a list of intensities, each assigned an index
hkl corresponding to its position in the reciprocal lattice. The intensity assigned to
reflection hkl is therefore a measure of the relative strength of the reflection from
the set of lattice planes having indices hkl. Recall that indices are counted from
the origin (indices 000), which lies in the direct path of the X-ray beam. In an
undistorted image of the reciprocal lattice, such as an image of a zero-level plane,
reflections having low indices lie near the origin, and those with high indices
lie farther away. Also recall that as indices increase, there is a corresponding
decrease in the spacing dhkl of the real-space planes represented by the indices.
This means that the reflections near the origin come from sets of widely spaced
planes, and thus carry information about larger features of the molecules in the
unit cell. On the other hand, the reflections far from the origin come from closely
spaced lattice planes in the crystal, and thus they carry information about the fine
details of structure.
In this chapter, I have shown how the dimensions and symmetry of the unit
cell determine the dimensions and symmetry of the diffraction pattern. Next I will
show how the molecular contents of the unit cell determine the contents (that is
the reflection intensities) of the diffraction pattern. In the next three chapters, I
will examine the relationship between the intensities of the reflections and the
molecular structures we seek, and thus show how the crystallographer extracts
structural information from the list of intensities.
 Chapter 5
From Diffraction Data to
Electron Density
5.1 Introduction
In producing an image of molecules from crystallographic data, the computer
simulates the action of a lens, computing the electron density within the unit
cell from the list of indexed intensities obtained by the methods described in
Chapter 4. In this chapter, I will discuss the mathematical relationships between
the crystallographic data and the electron density.
As stated in Chapter 2, computation of the Fourier transform is the lenssimulating operation that a computer performs to produce an image of molecules
in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to
convert a Fourier-sum description of the reflections to a Fourier-sum description
of the electron density. A reflection can be described by a structure-factor equation,
containing one term for each atom, or for each volume element, in the unit cell.
In turn, the electron density is described by a Fourier sum in which each term is
a structure factor. The crystallographer uses the Fourier transform to convert the
structure factors to ρ(x, y, z), the desired electron density equation.
First I will discuss Fourier sums and the Fourier transform in general terms.
I will emphasize the form of these equations and the information they contain,
in the hope of helping you to interpret the equations—that is, to translate the
equations into words and visual images. Then I will present the specific types of
Fourier sums that represent structure factors and electron density and show how
the Fourier transform interconverts them.
91
92 Chapter 5 From Diffraction Data to Electron Density
5.2 Fourier sums and the Fourier transform
5.2.1 One-dimensional waves
Recall from Sec. 2.6.1, p. 21, that waves are described by periodic functions, and
that simple wave equations can be written in the form
f (x) = F cos 2π(hx + α) (5.1)
or f (x) = F sin 2π(hx + α), (5.2)
where f (x) specifies the vertical height of the wave at any horizontal position x
(measured in wavelengths, where x = 1 implies one full wavelength or one full
repeat of the periodic function). In these equations, F specifies the amplitude of
the wave (the distance from from horizontal axis to peak or valley), h specifies its
frequency (number of wavelengths per radian), and α specifies its phase (position
of the wave, in radians, with respect to the origin). These equations are onedimensional in the sense that they represent a numerical value [f (x), the height
of the wave] at all points along one axis, in this case, the x-axis. See Fig. 2.14,
p. 22 for graphs of such equations.
I also stated in Chapter 2 (see Fig. 2.16, p. 25) that any wave, no matter how
complicated, can be described as the sum of simple waves. This sum is called a
Fourier sum and each simple wave equation in the sum is called a Fourier term.
Either Eq. (5.1) or (5.2) could be used as a single Fourier term. For example, we
can write a Fourier sum of n terms using Eq. (5.1) as follows:
f (x) = F0 cos 2π(0x + α0)+ F1 cos 2π(1x + α1)
+ F2 cos 2π(2x + α2)+ · · · + Fn cos 2π(nx + αn), (5.3)
or equivalently,
f (x) =
n∑ h=0
Fh cos 2π(hx + αh). (5.4)
According to Fourier theory, any complicated periodic function can be approximated by such a sum, by putting the proper values of h, Fh, and αh in each term.
Think of the cosine terms as basic wave forms that can be used to build any other
waveform. Also according to Fourier theory, we can use the sine function or, for
that matter, any periodic function in the same way as the basic wave for building
any other periodic function.
A very useful basic waveform is [cos 2π(hx)+ i sin 2π(hx)]. Here, the waveforms, of cosine and sine are combined to make a complex number, whose general
Section 5.2 Fourier sums and the Fourier transform 93
form is a + ib, where i is the imaginary number (−1)1/2. Although the phase α
of this waveform is not shown, it is implicit in the combination of the cosine and
sine functions, and it depends only upon the values of h and x. As I will show in
Chapter 6, expressing a Fourier term in this manner gives a clear geometric means
of representing the phase α and allows us to see how phases are computed. For
now, just accept this convention as a convenient way to write completely general
Fourier terms. In Chapter 6, I will discuss the properties of complex numbers and
show how they are used to represent and compute phases.
With the terms written in this fashion, a general Fourier sum looks like this:
f (x) =
n∑ h=0
Fh[cos 2π(hx)+ i sin 2π(hx)] (5.5)
In words, this sum consists of n simple Fourier terms, one for each integral value
of h beginning with zero and ending with n. Each term is a simple wave with its
own amplitude Fh, its own frequency h, and (implicitly) its own phase α.
Next, we can express the complex number in square brackets as an exponential,
using the following equality from complex number theory:
cos θ + i sin θ = eiθ . (5.6)
In our case, θ = 2π(hx), so the Fourier sum becomes
f (x) =
n∑ h=0
Fhe
2πi(hx) (5.7)
or simply
f (x) =

h Fhe
2πi(hx), (5.8)
in which the sum is taken over all values of h, and the number of terms is
unspecified.
I will write Fourier sums in this form throughout the remainder of the book. This
kind of equation is compact and handy, but quite opaque at first encounter. Take
the time now to look at this equation carefully and think about what it represents.
Whenever you see an equation like this, just remember that it is a Fourier sum,
a sum of sine and cosine wave equations, with the full sum representing some
complicated wave. The hth term in the sum, Fhe2πi(hx), can be expanded to
Fh[cos 2π(hx) + i sin 2π(hx)], making plain that the hth term is a simple wave
of amplitude Fh, frequency h, and implicit phase αh.
94 Chapter 5 From Diffraction Data to Electron Density
5.2.2 Three-dimensional waves
The Fourier sum that the crystallographer seeks isρ(x, y, z), the three-dimensional
electron density of the molecules under study. This function is a wave equation or
periodic function because it repeats itself in every unit cell. The waves described
in the preceding equations are one-dimensional: they represent a numerical value
f (x) that varies in one direction, along the x-axis. How do we write the equations
of two-dimensional and three-dimensional waves? First, what do the graphs of
such waves look like?
When you graph a function, you must use one more dimension than specified
by the function. You use the additional dimension to represent the numerical value
of the function. For example, in graphing f (x), you use the y-axis to show the
numerical value of f (x). In Fig. 2.14, p. 22, for example, the y-axes are used to
represent f (x), the height of each wave at point x. Graphing a two-dimensional
function f (x, y) requires the third dimension to represent the numerical value of
the function.
For example, imagine a weather map with mountains whose height at location
(x, y) represents the temperature at that location. Such a map graphs a twodimensional function t (x, y), which gives the temperature t at all locations (x, y)
on the plane represented by the map. If we must avoid using the third dimension,
for instance in order to print a flat map, the best we can do is to draw a contour map
on the plane map (Fig. 5.1), with continuous lines (contours, in this case called
isotherms) representing locations having the same temperature.
Graphing the three-dimensional function ρ(x, y, z) in the same manner would
require four dimensions, one for each of the spatial dimensions x, y, and z, and
a fourth one for representing the value of ρ. Here a contour map is the only
choice. In three dimensions, contours are continuous surfaces (rather than lines)
on which the function has a constant numerical value. A contour map of the threedimensional wave ρ(x, y, z) exhibits surfaces of constant electron density ρ. You
are already familiar with such contour maps. The common drawings of electronic
orbitals (such as the 1s orbital of a hydrogen atom, often drawn as a simple sphere)
is a contour map of a three-dimensional function. Everywhere on the surface of
this sphere, the electron density is the same. Orbital surfaces are often drawn to
enclose the region that contains 98% (or some specified value) of the total electron
density.
The blue netlike surface in Fig. 2.3, p. 11 is also a contour map of a threedimensional function. It represents a surface on which the electron density
ρ(x, y, z) of adipocyte lipid binding protein (ALBP) is constant. Imagine that
the net encloses some specified value, say 98%, of the protein’s electron density,
and so the net is in essence an image of the protein’s surface. The actual value of
ρ(x, y, z) on the plotted surface is specified as the map’s contour level, usually
given in units of σ , the standard deviation of the overall electron density, from
the mean electron density. For example, in a map contoured at 2σ , the displayed
surface is two standard deviations higher than the mean electron density for the
whole map.
Section 5.2 Fourier sums and the Fourier transform 95
Figure 5.1  Seasonable February morning in Maine. Lines of constant temperature
(isotherms) allow plotting a two-dimensional function without using the third dimension.
This is a contour map of t (x, y), giving the temperature t at all locations (x, y). Along each
contour line lie all points having the same temperature. A planar contour map of a function
of two variables takes the form of contour lines on the plane. In contrast, a contour map
of a function of three variables takes the form of contour surfaces in three dimensions (see
Fig. 2.3, p. 11).
I hope the foregoing helps you to imagine three-dimensional waves. What
do the equations of such waves look like? A three-dimensional wave has three
frequencies, one along each of the x-, y-, and z-axes. So three variables h, k, and l
are needed to specify the frequency in each of the three directions. A general
Fourier sum for the wave f (x, y, z), written in the compact form of Eq. (5.8) is
as follows:
f (x, y, z) =

h
k
l Fhkle
2πi(hx+ky+lz). (5.9)
In words, Eq. (5.9) says that the complicated three-dimensional wave f (x, y, z)
can be represented by a Fourier sum. Each term in the sum is a simple threedimensional wave whose frequency is h in the x-direction, k in the y-direction,
and l in the z-direction. For each possible set of values h, k, and l, the associated
wave has amplitude Fhkl and, implicitly, phase αhkl . The triple sum simply means
to add up terms for all possible sets of integers h, k, and l. The range of values for
h, k, and l depends on how many terms are required to represent the complicated
wave f (x, y, z) to the desired precision.
96 Chapter 5 From Diffraction Data to Electron Density
5.2.3 The Fourier transform: General features
Fourier demonstrated that for any function f (x), there exists another function
F(h) such that
F(h) =
∫ +∞
−∞
f (x)e2πi(hx) dx, (5.10)
where F(h) is called the Fourier transform (FT) of f (x), and the units of the
variable h are reciprocals of the units of x. For example, if x is time in seconds (s),
then h is reciprocal time, or frequency, in reciprocal seconds (s−1). So if f (x) is a
function of time, F(h) is a function of frequency. Taking the FT of time-dependent
functions is a means of decomposing these functions into their component frequencies and is sometimes referred to as Fourier analysis. The FT in this form is used
in infrared (IR) and nuclear magnetic resonance (NMR) spectroscopy to obtain the
frequencies of many spectral lines simultaneously (as I will describe in Chapter
10 on obtaining models from NMR).
On the other hand, if x is a distance or length in Å, h is reciprocal length in
Å−1. You can thus see that this highly general mathematical form is naturally
adapted for relating real and reciprocal space. In fact, as I mentioned earlier,
the Fourier transform is a precise mathematical description of diffraction. The
diffraction patterns in Figs. 2.8–2.11 (pp. 15–18) are Fourier transforms of the
corresponding simple objects and arrays. If these figures give you some intuition
about how an object is related to its diffraction pattern, then they provide the
same perception about the kinship between an object and its Fourier transform.
According to Eq. (5.10), to compute F(h), the Fourier transform of f (x), just
multiply the function by e2πi(hx) and integrate (or better, let a computer integrate)
the combined functions with respect to x. The result is a new function F(h), which
is the FT of f (x). Computer routines for calculating FTs of functions are widely
available, and form one of the vital internal organs of crystallographic software.
The Fourier transform operation is reversible. That is, the same mathematical
operation that gives F(h) from f (x) can be carried out in the opposite direction,
to give f (x) from F(h); specifically,
f (x) =
∫ +∞
−∞
F(h)e−2πi(hx)dh (5.11)
In other words, if F(h) is the transform of f (x), then f (x) is in turn the transform
of F(h). In this situation, f (x) is sometimes called the back-transform of (h),
but this is a loose term that simply refers to the second successive transform that
recreates the original function. Notice that the only difference between Eqs. (5.10)
and (5.11) is the sign of the exponential term. You can think of this sign change
as analogous (very roughly analogous!) to the sign change that makes subtraction
the reverse of addition. Adding 3 to 5 gives 8: 5+ 3 = 8. To reverse the operation
and generate the original 5, you subtract 3 from the previous result: 8 − 3 = 5.
Section 5.2 Fourier sums and the Fourier transform 97
If you think of 8 as a simple transform of 5 made by adding 3, the back-transform
of 8 is 5, produced by subtracting 3.
Returning to the visual transforms of Figs. 2.8–2.11 (pp. 15–18), each object
(the sphere in Fig. 2.8, for instance) is the Fourier transform (the back-transform,
if you wish) of its diffraction pattern. If we build a model that looks like the
diffraction pattern on the right, and then obtain its diffraction pattern, we get an
image of the object on the left.
There is one added complication. The preceeding functions f (x) and F(h) are
one-dimensional. Fortunately, the Fourier transform applies to periodic functions
in any number of dimensions. To restate Fourier’s conclusion in three dimensions,
for any function f (x, y, z) there exists the function F(h, k, l) such that
F(h, k, l) =

x
y
z f (x, y, z)e2πi(hx+ky+lz) dx dy dz. (5.12)
As before, F(h, k, l) is called the Fourier transform of f (x, y, z), and in turn,
f (x, y, z) is the Fourier transform of F(h, k, l) as follows:
f (x, y, z) =

h
k
l F (h, k, l)e−2πi(hx+ky+lz) dh dk dl. (5.13)
5.2.4 Fourier this and Fourier that: Review
I have used Fourier’s name in discussing several types of equations and operations,
and I want to be sure that I have not muddled them in your mind. First, a Fourier
sum is a sum of simple wave equations or periodic functions that describes or
approximates a complicated periodic function. Second, constructing a Fourier
sum—that is, determining the proper F, h, and α values to approximate a specific
function—is called Fourier synthesis. For example, the sum of f0 through f6
in Fig. 2.16, p. 25 is at once a Fourier sum and the result of Fourier synthesis.
Third, decomposing a complicated function into its components is called Fourier
analysis. Fourth, the Fourier transform is an operation that transforms a function
containing variables of one type (say time) into a function whose variables are
reciprocals of the original type [in this case, 1/(time) or frequency]. The function
f (x) is related to its Fourier transform F(h) by Eq. (5.10). The term transform is
commonly used as a noun to refer to the function F(h) and also loosely as a verb
to denote the operation of computing a Fourier transform. Finally, a Fourier series
is an infinite sum based on some iterative formula for generating each term (the
sum in Fig. 2.16, p. 25 is actually the beginning of an infinite series). So a sum of
experimental terms (such as X-ray reflections), or a sum of individual atomic or
electron-density contributions to a reflection is a Fourier sum, not a series. (Last,
a grammar note: the word series is both singular and plural. You must gather from
context whether a writer is talking about one series or many series. But from here
on, I will be talking about sums, not series.)
98 Chapter 5 From Diffraction Data to Electron Density
5.3 Fourier mathematics and diffraction
5.3.1 Structure factor as a Fourier sum
I have stated that both structure factors and electron density can be expressed as
Fourier sums. A structure factor describes one diffracted X-ray, which produces
one reflection received at the detector. A structure factor Fhkl can be written as a
Fourier sum in which each term gives the contribution of one atom to the reflection
hkl [see Fig. 2.17, p. 26 and Eq. (2.3), p. 24]. Here is a single term, called an atomic
structure factor, fhkl , in such a series, representing the contribution of the single
atom j to reflection hkl:
fhkl = fj e2πi(hxj+kyj+lzj ). (5.14)
The term fj is called the scattering factor of atom j , and it is a mathematical
function (called a δ function) that amounts to treating the atom as a simple sphere
of electron density. The function is slightly different for each element, because
each element has a different number of electrons (a different value of Z) to diffract
the X-rays. The exponential term should be familiar to you by now. It represents
a simple three-dimensional periodic function having both cosine and sine components. But the terms in parenthesis now possess added physical meaning: xj ,
yj , and zj are the coordinates of atom j in the unit cell (real space), expressed as
fractions of the unit-cell axis lengths; and h, k, and l, in addition to their role as
frequencies of a wave in the three directions x, y, and z, are also the indices of a
specific reflection in the reciprocal lattice.
As mentioned earlier, the phase of a diffracted ray is implicit in the exponential
formulation of a structure factor and depends only upon the atomic coordinates (xj , yj , zj ) of the atom. In fact, the phase for diffraction by one atom is
2π(hxj + kyj + lyj ), the exponent of e (ignoring the imaginary i) in the structure
factor. For its contribution to the 220 reflection, an atom at (0, 1/2, 0) has phase
2π(hxj + kyj + lzj ) or 2π(2[0] + 2[1/2] + 0[0]) = 2π , which is the same as a
phase of zero. This atom lies on the (220) plane, and all atoms lying on (220) planes
contribute to the 220 reflection with phase of zero. [Try the above calculation for
another atom at (1/2, 0, 0), which is also on a (220) plane.] This conclusion is in
keeping with Bragg’s law, which says that all atoms on a set of equivalent, parallel
lattice planes diffract in phase with each other.
Each diffracted ray is a complicated wave, the sum of diffractive contributions
from all atoms in the unit cell. For a unit cell containing n atoms, the structure
factor Fhkl is the sum of all the atomic fhkl values for individual atoms. Thus, in
parallel with Eq. (2.3), p. 24, we write the structure factor for reflection Fhkl as
follows:
Fhkl =
n∑ j=1
fj e
2πi(hxj+kyj+lzj ). (5.15)
Section 5.3 Fourier mathematics and diffraction 99
In words, the structure factor that describes reflection hkl is a Fourier sum in
which each term is the contribution of one atom, treated as a simple sphere of
electron density. So the contribution of each atom j to Fhkl depends on (1) what
element it is, which determines fj , the amplitude of the contribution, and (2) its
position in the unit cell (xj , yj , zj ), which establishes the phase of its contribution.
Alternatively, Fhkl can be written as the sum of contributions from each volume
element of electron density in the unit cell [see Fig. 2.18, p. 27 and Eq. (2.4), p. 28].
The electron density of a volume element centered at (x, y, z) is, roughly, the average value of ρ(x, y, z) in that region. The smaller we make our volume elements,
the more precisely these averages approach the correct values of ρ(x, y, z) at all
points. We can, in effect, make our volume elements infinitesimally small, and
the average values of ρ(x, y, z) precisely equal to the actual values at every point,
by integrating the function ρ(x, y, z) rather than summing average values. Think
of the resulting integral as the sum of the contributions of an infinite number of
vanishingly small volume elements. Written this way,
Fhkl =

x
y
z ρ(x, y, z)e2πi(hx+ky+lz) dx dy dz, (5.16)
or equivalently,
Fhkl =

V
ρ(x, y, z)e2πi(hx+ky+lz) dV , (5.17)
where the integral over V , the unit-cell volume, is just shorthand for the integral
over all values of x, y, and z in the unit cell. Each volume element contributes
to Fhkl with a phase determined by its coordinates (x, y, z), just as the phase of
atomic contributions depend on atomic coordinates.
You can see by comparing Eq. (5.17) with Eq. (5.10) [or Eq. (5.16) with
Eq. (5.12)] that Fhkl is the Fourier transform of ρ(x, y, z). More precisely, Fhkl is
the transform of ρ(x, y, z) on the set of real-lattice planes (hkl). All of the Fhkls
together compose the transform of ρ(x, y, z) on all sets of equivalent, parallel
planes throughout the unit cell.
5.3.2 Electron density as a Fourier sum
Because the Fourier transform operation is reversible [Eqs. (5.10) and (5.11)], the
electron density is in turn the transform of the structure factors, as follows:
ρ(x, y, z) = 1
V

h
k
l Fhkle
−2πi(hx+ky+lz), (5.18)
where V is the volume of the unit cell.
This transform is a triple sum rather than a triple integral because theFhkls represent a set of discrete entities: the reflections of the diffraction pattern. The transform
100 Chapter 5 From Diffraction Data to Electron Density
of a discrete function, such as the reciprocal lattice of measured intensities, is a
summation of discrete values of the function. The transform of a continuous function, such as ρ(x, y, z), is an integral, which you can think of as a sum also, but a
sum of an infinite number of infinitesimals. Superficially, except for the sign change
(in the exponential term) that accompanies the transform operation, this equation
appears identical to Eq. (5.9), a general three-dimensional Fourier sum. But here,
each Fhkl is not just one of many simple numerical amplitudes for a standard set of
component waves in a Fourier sum. Instead, each Fhkl is a structure factor, itself
a Fourier sum, describing a specific reflection in the diffraction pattern.
“Curiouser and curiouser,” said Alice.
5.3.3 Computing electron density from data
Equation (5.18) tells us, at last, how to obtain ρ(x, y, z). We need merely to
construct a Fourier sum from the structure factors. By now, you might be wondering
about the practical aspects of calculating an electron-density map from this rather
abstract equation. Widely available software can turn archived structure factors into
electron-density maps for viewing with graphics programs, making possible a firsthand look at the most critical evidence in support of a model. But if you think about
the calculation itself, you might wonder about those imaginary terms and what they
describe, or what happens to them in the calculations. What physical meaning could
be ascribed to imaginary terms in a representation of electron density? I told you
earlier that the exponential description (ei(hx)) is computationally more efficient
than its trigonometric equivalent [cos(hx) + i sin(hx)], but when it comes to
calculating a concrete image, what happens to those i terms?
In short, they go away. The Fourier sum is taken over all values of indices h, k,
and l, and for every term containing a positive index h, there is a term containing
the same negative index −h. Under these circumstances, sines and cosines behave
very differently, because cos(hx) = cos(−hx), but sin(hx) = − sin(−hx) [try it
on your hand calculator, with thirty degrees for (hx)]. So when you use Eq. (5.18)
or similar equations to compute anything from structure factors, the cosine terms
for positive and negative indices add together, but the sine terms all cancel out
precisely. So the actual computation of an electron density map makes use only
of the cosine terms, and the imaginaries go away. But that does not mean that the
sine terms serve no purpose. As I have said before, in the structure factor equation,
the [cos(hx)+ i sin(hx)] terms implicitly define the phases of reflections.
A structure factor describes a diffracted ray, and a full description of a diffracted
ray, like any description of a wave, must include three parameters: amplitude,
frequency, and phase. In discussing data collection, however, I have mentioned
only two measurements: the indices of each reflection and its intensity. Looking
again at Eq. (5.18), you see that the indices of a reflection play the role of the three
frequencies in one Fourier term. The only measurable variable remaining in the
equation isFhkl . Does the measured intensity of a reflection, the only measurement
we can make in addition to the indices, completely define Fhkl? Unfortunately, the
answer is “no.”
Section 5.4 Meaning of the Fourier equations 101
5.3.4 The phase problem
Because Fhkl is a periodic function, it possesses amplitude, frequency, and phase.
The amplitude of Fhkl is proportional to the square root of the reflection intensity
Ihkl , so structure-factor amplitudes are directly obtainable from measured reflection intensities. The three frequencies of this three-dimensional wave function are
h, k, and l, the indices of the planes that produce the reflection described by Fhkl .
So the frequency of a structure factor is equal to 1/dhkl , making the wavelength
the same as the spacing of the planes producing the reflection. But the phase of
Fhkl is not directly obtainable from a single measurement of the reflection intensity. In order to compute ρ(x, y, z) from the structure factors, we must obtain,
in addition to the intensity of each reflection, the phase of each diffracted ray.
In Chapter 6, I will present an expression for ρ(x, y, z) as a Fourier series in
which the phases are explicit (finally, huh?), and I will discuss means of obtaining
phases. This is one of the most difficult problems in crystallography. For now, on
the assumption that the phases can be obtained, and thus that complete structure
factors are obtainable, I will consider further the implications of Eq. (5.15) (structure factors F expressed in terms of atoms), Eq. (5.16) [structure factors in terms
of ρ(x, y, z)], and Eq. (5.18) [ρ(x, y, z) in terms of structure factors].
5.4 Meaning of the Fourier equations
5.4.1 Reflections as terms in a Fourier sum: Eq. (5.18)
First consider Eq. (5.18) (ρ in terms of F s). Each term in this Fourier-series
description of ρ(x, y, z) is a structure factor representing a single X-ray reflection.
The indices hkl of the reflection give the three frequencies necessary to describe
the Fourier term as a simple wave in three dimensions. Recall from Sec. 2.6.2,
p. 23 that any periodic function can be approximated by a Fourier sum, and that the
approximation improves as more terms are added to the sum (see Fig. 2.16, p. 25).
The low-frequency terms in Eq. (5.18) determine gross features of the periodic
function ρ(x, y, z), whereas the high-frequency terms improve the approximation
by filling in fine details. You can also see in Eq. (5.18) that the low-frequency
terms in the Fourier sum that describes our desired function ρ(x, y, z) are given by
reflections with low indices, that is, by reflections near the center of the diffraction
pattern (Fig. 5.2). In some crystallographic circles (so to speak), they are called
low-angle reflections.
The high-frequency terms are given by reflections with high indices, reflections
farthest from the center of the pattern (high-angle reflections). Thus you can see the
importance of how well a crystal diffracts. If a crystal does not produce diffracted
rays at large angles from the direct beam (reflections with large indices), the Fourier
sum constructed from all the measurable reflections lacks high-frequency terms,
and the resulting transform is not highly detailed—the resolution of the resulting
102 Chapter 5 From Diffraction Data to Electron Density
Figure 5.2  Structure factors of reflections near the center of the diffraction pattern
are low-frequency terms in the Fourier sum that approximates ρ(x, y, z). Structure factors
of reflections near the outer edge of the pattern are high-frequency terms.
image is poor. The Fourier series of Fig. 2.16, p. 25 is truncated in just this manner
and does not fit the target function in fine details like the sharp corners.
This might sound like a Zen Buddhist question: What is the meaning of one
reflection? What does the FT of one structure factor tell you? In Fig. 5.3, I have
again used a page from Kevin Cowtan’s Book of Fourier to demonstrate what
individual structure factors contribute to the molecular image we are seeking. The
first column (a) shows the full diffraction pattern (top panel, with phases indicated
by color) of a simple molecular model in its unit cell (a, bottom panel). The
second column (b) shows contour plots of the Fourier transforms of individual
structure factors, with reflection indices listed to the left of each transform. The
FT of the 01 reflection is simply a sinusoidal electron-density function showing
electron density on the 01 planes. The density exhibits peaks (red) on the planes
and reaches minima (blue) between the planes. The FT of any single reflection,
like the 10 reflection shown next, will look something like this: merely a repetitive
rise and fall of electron density. After all, the only thing you learn from a single
reflection is the average electron-density along the set of Miller planes that share
their indices with that specific reflection.
But look what happens when you add structure-factor FTs to each other
(column c). When the FTs of the 01 and 10 reflections are added together, we
see interference between them, in the form of high positive density (intense red,
where red crosses red), zero density (white, where red crosses blue), and high negative density (blue crossing blue). The result is a large lobe of positive density that
shows us approximately where the molecule lies in this unit cell. In this simple
example, only two reflections, when transformed, roughly locate the molecule!
Section 5.4 Meaning of the Fourier equations 103
Figure 5.3  (a) Structure-factor pattern (top) calculated from a simple model
(bottom). (b) Fourier transforms of individual reflections from a. Red is positive electron
density, blue is negative. (c) Sums of FTs from b. In each square, the FT of one reflection
is added to the sum above it.
104 Chapter 5 From Diffraction Data to Electron Density
As we continue adding FTs of structure factors for more reflections, the molecular
image becomes more sharply defined, showing what each reflection contributes
to the final image. Notice that FTs of reflections with higher indices have higher
frequencies (more red-blue repeats per unit distance). These FTs add fine details
to the sum. After we have summed only seven reflections, we have located all the
atoms approximately. Interference among the full set of FTs gives a sharp image
of all atoms (a, lower panel), with only very weak negative density, which is due
to the finiteness of our data set. Remember that full data sets for biological macromolecules comprise from thousands to millions of structure factors, each telling
us the average electron density on a specific set of parallel planes.
There is even more than meets the eye in Fig. 5.3, p. 103. We will visit it again
in Chapter 6, when I take up the phase problem. Can’t wait? See p. 111.
5.4.2 Computing structure factors from a model:
Eq. (5.15) and Eq. (5.16)
Equation (5.15) describes one structure factor in terms of diffractive contributions
from all atoms in the unit cell. Equation (5.16) describes one structure factor in
terms of diffractive contributions from all volume elements of electron density in
the unit cell. Notice in both cases that all parts of the structure—every atom or every
scrap of electron density—contribute to every structure factor. These two equations
suggest that we can calculate the full set of structure factors either from an atomic
model of the protein or from an electron density function. In short, if we know
the structure, either as atoms or as electron density, we can calculate the diffraction pattern, including the phases of all reflections. This computation, of course,
appears to go in just the opposite direction that the crystallographer desires. It turns
out, however, that computing structure factors from a model of the unit cell (backtransforming the model) is an essential part of crystallography, for several reasons.
First, this computation is used in obtaining phases. As I will discuss in Chapter 6,
the crystallographer obtains phases by starting from rough estimates of them
and then undertaking an iterative process to improve the estimates. This iteration entails a cycle of three steps. In step 1, an estimated ρ(x, y, z) (that is, a crude
model of the structure) is computed using Eq. (5.18) with observed intensities (Iobs)
and estimated phases (αcalc). In step 2, the crystallographer attempts to improve
the model by viewing the electron-density map [a computer plot of ρ(x, y, z)] and
identifying molecular features such as the molecule-solvent boundaries or specific
groups of atoms (called interpreting the map). Step 3 entails computing new structure factors (Fcalc), using either Eq. (5.16) with the improved ρ(x, y, z) model
from step 2, or Eq. (5.15) with a partial atomic model of the molecule, containing
only those atoms that can be located with some confidence in the electron-density
map. Calculation of new Fcalcs in step 3 produces a new (better, we hope) set of
estimated phases, and the cycle is repeated: a new ρ(x, y, z) is computed from
the original measured intensities and the newest phases, interpretation produces a
more detailed model, and calculation of structure factors from this model produces
improved phases. In each cycle, the crystallographer hopes to obtain an improved
Section 5.4 Meaning of the Fourier equations 105
ρ(x, y, z), which means a more detailed and interpretable electron-density map,
and thus a more complete and accurate model of the desired structure. I will discuss the iterative improvement of phases and electron-density maps in Chapter
7. For now just take note that obtaining the final structure entails both calculating ρ(x, y, z) from structure factors and calculating structure factors from some
preliminary model, either a rough form of ρ(x, y, z) or a partial atomic model.
Note further that when we compute structure factors from a known or assumed
model, the results include the phases for that model. In other words, the computed results give us the information needed for a “full-color” diffraction pattern,
like that shown in Fig. 2.19d, p. 29, whereas experimentally obtained diffraction
patterns lack the phases and are merely black and white, like Fig. 2.19e.
The second use of back-transforms is to assess the progress of structure determination. Equations (5.15) and (5.16) provide means to monitor the iterative process
to see whether it is converging toward improved phases and improved ρ(x, y, z).
The computed structure factors Fcalc include both the desired phases αcalc and a
new set of intensities. I will refer to these calculated intensities as Icalc to distinguish them from the measured reflection intensities Iobs taken from the diffraction
pattern. As the iteration proceeds, the values of Icalc should approach those of
Iobs. So the crystallographer compares the Icalc and Iobs values at each cycle in
order to see whether the iteration is converging. When cycles of computation provide no further improvement in correspondence between calculated and measured
intensities, then the process is complete, and the model can be improved no further.
5.4.3 Systematic absences in the diffraction pattern: Eq. (5.15)
A third application of Eq. (5.15) allows us to understand how systematic absences
in the diffraction pattern reveal symmetry elements in the unit cell, thus guiding the crystallographer in assigning the space group of the crystal. Recall from
Sec. 4.2.8, p. 65, that if the unit cell possesses symmetry elements, then certain
sets of reciprocal-lattice points are equivalent, and so certain reflections in the
diffraction pattern are redundant. The crystallographer must determine the unitcell space group (i.e., determine what symmetry elements are present) in order
to devise an efficient strategy for measuring as many unique reflections as efficiently as possible. I stated without justification in Chapter 4 that certain symmetry
elements announce themselves in the diffraction pattern as systematic absences:
regular patterns of missing reflections. Now I will use Eq. (5.15) to show how a
symmetry element in the unit cell produces systematic absences in the diffraction
pattern.
For example, as indicated by the “Reflection conditions” in Fig. 4.20, p. 72,
if the b-axis of the unit cell is a twofold screw axis, then reflections 010, 030,
050, along with all other 0k0 reflections in which k is an odd number, are missing.
We can see why by using the concept of equivalent positions (Sec. 4.2.8, p. 65).
For a unit cell with a twofold screw axis along edge b, the equivalent positions
are (x, y, z) and (−x, y + 1/2,−z). That is, for every atom j with coordinates
(x, y, z) in the unit cell, there is an identical atom j ′ at (−x, y+ 1/2,−z). Atoms
j and j ′ are called symmetry-related atoms. According to Eq. (5.15), the structure
106 Chapter 5 From Diffraction Data to Electron Density
factor for reflections F0k0 is
F0k0 =

j fj e
2πi(kyj ). (5.19)
The exponential term is greatly simplified in comparison to that in Eq. (5.15)
because h = l = 0 for reflections on the 0k0 axis. Now I will separate the
contributions of atoms j from that of their symmetry-related atoms j ′:
F0k0 =

j fj e
2πi(kyj ) +

j ′
fj ′e
2πi(kyj ′ ). (5.20)
Because atoms j and j ′ are identical, they have the same scattering factor f , and
so I can substitute fj for fj ′ and factor out the f terms:
F0k0 =

j fj
∑
j e2πikyj +

j ′
e 2πikyj ′

 . (5.21)
If the y coordinate of atom j is y, then the y coordinate of atom j ′ is y + 1/2.
Making these substitutions for zj and zj ′ ,
F0k0 =

j fj
∑
j [e2πiky + e2πik(y+1/2)]

 . (5.22)
The fj terms are nonzero, so F0k0 is zero, and the corresponding 010 reflection is
missing, only if all the summed terms in square brackets equal zero. Simplifying
one of these terms,
e2πiky + e2πik(y+1/2) = e2πiky(1 + eπik). (5.23)
This term is zero, and hence F0k0 is zero, if eπik equals −1. Converting this
exponential to its trigonometric form [see Eq. (5.6)],
eπik = cos(πk)+ i sin(πk). (5.24)
The cosine of π radians (180o), or any odd multiple of π radians, is −1. The sine
of π radians is 0. Thus eπik equals −1 for all odd values of k, and F0k0 equals
zero if l is odd.
The preceding shows that F0k0 disappears for odd values of k when the b edge
of a unit cell is a twofold screw axis. But what is going on physically? In short, the
diffracted rays from two atoms at (x, y, z) and (−x, y + 1/2,−z) are identical in
amplitude (fj = fj ′) but precisely opposite in phase. Recall that the phase of an
Section 5.5 Summary: From data to density 107
atom’s contribution to Fhkl is 2π(hx + ky + lz). Consider an atom j lying at the
origin of the unit cell (0, 0, 0), and its symmetry-related atom j ′ at (0, 1/2, 0). The
phase of j ’s contribution to F010 is 2π([0 · 0]+ [1 · 0]+ [0 · 0)] = 0 radians or 0o.
The phase for atom j ′ is 2π([0·0]+[1·(1/2)]+[0·0)] = π radians or 180o, which
is precisely 180o out of phase with j ’s contribution, thus cancelling it to makeF010
a missing reflection. So the symmetry-related pair of atoms contributes nothing to
F0k0 when k is odd, and because every atom has such a symmetry-related partner,
this cancellation occurs no matter where the atoms lie. Putting it another way, if
the unit cell contains a twofold screw axis along edge b, then every atom in the
unit cell is paired with a symmetry-related atom that cancels its contributions to all
odd-numbered 0k0 reflections. (Can you show that two atoms related by a twofold
screw axis along b diffract in phase for the 020 reflection?)
Similar computations have been carried out for all symmetry elements and combinations of elements. Like equivalent positions, systematic absences are tabulated
for all space groups in International Tables, so the crystallographer can use this
reference as an aid to space-group determination. As mentioned above, the International Tables entry for space group P 21 (Fig. 4.20, p. 72), which possess a 21
axis on edge b, indicates that, for reflections 0k0, the “Reflection conditions” are
0k0 : k = 2n. In other words, in this space group, reflections 0k0 are present only
if k is even (2 times any integer n), so they are absent if k is odd, as I proved above.
5.5 Summary: From data to density
When we describe structure factors and electron density as Fourier sums, we find
that they are intimately related. The electron density is the Fourier transform of the
structure factors, which means that we can convert the crystallographic data into
an image of the unit cell and its contents. One necessary piece of information is,
however, missing for each structure factor. We can measure only the intensity Ihkl
of each reflection, not the complete structure factor Fhkl . What is the relationship
between them? It can be shown that the amplitude of structure factor Fhkl is
proportional to (Ihkl)1/2, the square root of the measured intensity. So if we know
Ihkl from diffraction data, we know the amplitude of Fhkl . Unfortunately, we do
not know its phase αhkl . In focusing light reflected from an object, a lens maintains
all phase relationships among the rays, and thus constructs an image accurately.
When we record diffraction intensities, we lose the phase information that the
computer needs in order to simulate an X-ray-focusing lens. In Chapter 6, I will
describe methods for learning the phase of each reflection, and thus obtaining the
complete structure factors needed to calculate the electron density.
 Chapter 6
Obtaining Phases
6.1 Introduction
The molecular image that the crystallographer seeks is a contour map of the electron
density ρ(x, y, z) throughout the unit cell. The electron density, like all periodic
functions, can be represented by a Fourier sum. The representation that connects
ρ(x, y, z) to the diffraction pattern is
ρ(x, y, z) = 1
V

h
k
l Fhkle
−2πi(hx+ky+lz). (5.18)
Equation (5.18) tells us how to calculate ρ(x, y, z): simply construct a Fourier
sum using the structure factors Fhkl . For each term in the sum, h, k, and l are
the indices of reflection hkl, and Fhkl is the structure factor that describes the
reflection. Each structure factor Fhkl is a complete description of a diffracted ray
recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency,
amplitude, and phase. Its three frequency terms h, k, and l are the indices of the
set of parallel planes that produce the reflection. Its amplitude is proportional to
(Ihkl)
1/2, the square root of the measured intensity Ihkl of reflection hkl. Its phase
is unknown and is the only additional information the crystallographer needs in
order to compute ρ(x, y, z) and thus obtain an image of the unit cell contents. In
this chapter, I will discuss some of the common methods of obtaining phases.
Let me emphasize that each reflection has its own phase (see Fig. 2.19d, p. 29),
so the phase problems must be solved for every one of the thousands of reflections used to construct the Fourier sum that approximates ρ(x, y, z). Let me also
emphasize how crucial this phase information is. In his Book of Fourier, Kevin
Cowtan illustrates the relative importance of phases and intensities in solving a
structure, as shown in Fig. 6.1. Images (a) and (b) show two simple models, a duck
109
110 Chapter 6 Obtaining Phases
Figure 6.1  Relative amounts of information contained in reflection intensities and
phases. (a) and (b) Duck and cat, along with their Fourier transforms. (c) Intensity (shading)
of the duck transform, combined with the phases (colors) of the cat transform. (d) Backtransform of (c) produces recognizable image of cat, but not duck. Phases contain more
information than intensities. Figure generously provided by Dr. Kevin Cowtan.
Section 6.1 Introduction 111
and a cat, along with their calculated Fourier transforms. As in Fig. 2.19, p. 29,
phases are shown as colors, while the intensity of color reflects the amplitude of
the structure factor at each location. (Note that these are continuous transforms,
because the model is not in a lattice.) Back-transforming each Fourier transform
would produce an image of the duck or cat. In (c) the colors from the cat transform
are superimposed on the intensities from the duck transform. This gives us a transform in which the intensities come from the duck and the phases come from the
cat. In (d) we see the back-transform of (c). The image of the cat is obvious, but
you cannot find any sign of the duck. Ironically, the diffraction intensities, which
are relatively easy to measure, contain far less information than do the phases,
which are much more difficult to obtain.
Before I begin showing you how to obtain phases, I want to give you one
additional bit of feeling for their physical meaning, and as well, to confess to a
small half-truth I have quietly sustained until now, for simplicity. Take another
look at Fig. 5.3, p. 103. If you look very closely at the FT of individual reflections,
you will see that the peak of repetitive electron density does not pass precisely
through the origin of the unit cell. (The 10 reflection is a good one to check, and
you may have to compare its FT with the lower panel of Fig. 5.3a in order to see
the unit cell clearly.) This observation implies that the electron density producing,
say, the 10 reflection does not peak precisely on the 10 plane. While we commonly
say that the 234 reflection comes from the planes whose Miller indices are 234, the
truth is that only if its phase is zero does the reflection come precisely from those
planes. If its phase is π (180o), then the reflection is coming from halfway between
those planes. Physically, the electron density that repeats with the orientation and
frequency of the 234 planes may not have its peak on the planes; it is more likely
that the peak lies somewhere between them. So to be most accurate, we should say
that the 234 reflection comes from repetitive electron density whose orientation and
frequency corresponds to that of the 234 planes, but whose peak may lie anywhere
between those planes. Its exact position constitutes its phase. Whatever its phase,
if that repetitive electron density is high, the 234 reflection will be strong; if low,
it will be weak or missing. If you look again at the basic Bragg model (Fig. 4.8,
p. 56), you will see that the actual position of the planes is immaterial. Any set of
planes with the same orientation and spacing would produce the reflection shown.
(It appears that no one ever bothers to add this last little bit of precision to the
Bragg model, but there it is. Now I feel better.)
So let us get to phasing. In order to illuminate both the phase problem and its
solution, I will represent structure factors as vectors on a two-dimensional plane
of complex numbers of the form a+ ib, where i is the imaginary number (−1)1/2.
This allows me to show geometrically how to compute phases. I will begin by
introducing complex numbers and their representation as points having coordinates
(a, b) on the complex plane. Then I will show how to represent structure factors
as vectors on the same plane. Because we will now start thinking of the structure
factor as a vector, I will hereafter write it in boldface (Fhkl) instead of the italics
used for simple variables and functions. Finally, I will use the vector representation
of structure factors to explain a few common methods of obtaining phases.
112 Chapter 6 Obtaining Phases
6.2 Two-dimensional representation of
structure factors
6.2.1 Complex numbers in two dimensions
Structure-factor equations like Eq. (5.15), p. 98, present the structure factor as a sum of terms each containing the exponential element e2πi(hx+ky+lz).
Remember that these exponential elements can also be expressed trigonometrically as [cos 2π(hx + ky + lz)+ i sin 2π(hx + ky + lz)]. In this form, each term
in the Fourier sum, and hence the structure factor itself, is a complex number of
the form a+ ib. Complex numbers can be represented as points in two dimensions
(Fig. 6.2). I will use this representation to help you understand the nature of the
phase problem and various ways of solving it. The horizontal axis in the figure
represents the real-number line. Any real number a is a point on this line, which
stretches from a = −∞ to a = +∞. The vertical axis is the imaginary-number
line, on which lie all imaginary numbers ib between b = −i∞ and b = +i∞.
A complex number a+ ib, which possesses both real (a) and imaginary (ib) parts,
is thus a point at position (a, ib) on this plane.
6.2.2 Structure factors as complex vectors
A representation of structure factors on this plane must include the two properties
we need in order to construct ρ(x, y, z): amplitude and phase. Crystallographers
represent each structure factor as a complex vector, that is, a vector (not a point) on
the plane of complex numbers. The length of this vector represents the amplitude
of the structure factor. Thus the length of the vector representing structure factor
Fhkl is proportional to (Ihkl)1/2. The second property, phase, is represented by the
Figure 6.2  The complex number N = a + ib, represented as a point on the plane
of complex numbers.
Section 6.2 Two-dimensional representation of structure factors 113
Figure 6.3  (a) The structure factor F, represented as a vector on the plane of complex
numbers. The length of F is proportional to I1/2, the square root of the measured intensity I .
The angle between F and the positive real axis is the phase α. (b) (Stereo) F can be pictured
as a complex vector spinning around its line of travel. The projection of the path taken by
the head of the vector is the familiar sine wave. The spinning vector reaches the detector
pointing in a specific direction that corresponds to its phase.
angle α that the vector makes with the positive real-number axis when the origin
of the vector is placed at the origin of the complex plane, which is the point 0+ i0
(Fig. 6.3a).
We can represent a structure factor F as a vector A + iB on this plane. The
projection of F on the real axis is its real part A, a vector of length |A| (absolute
value of A) on the real-number line; and the projection of F on the imaginary axis
is its imaginary part iB, a vector of length |B| on the imaginary-number line. The
length or magnitude (or in wave terminology, the amplitude) of a complex vector
is analogous to the absolute value of a real number, so the length of vector Fhkl is
|Fhkl |; therefore, |Fhkl | is proportional to (Ihkl)1/2, and if the intensity is known
from data collection, we can treat |Fhkl | as a known quantity. The angle that Fhkl
makes with the real axis is represented in radians as α(0 ≤ α ≤ 2π), or in cycles
as α′(0 ≤ α′ ≤ 1), and is referred to as the phase angle.
114 Chapter 6 Obtaining Phases
This representation of a structure factor is equivalent to thinking of a wave as
a complex vector spinning around its axis as it travels thorough space (Fig. 6.3b).
If its line of travel is perpendicular to the tail of the vector, then a projection of the
head of the vector along the line of travel is the familiar sinusoidal wave. When
the wave strikes the detector, the vector is pointing in a specific direction that
corresponds to its phase. The phase of a structure factor tells us the direction of
the vector at some arbitrary origin (in this case, the plane of the detector), and to
know the phase of all reflections means to know all their individual phase angles
with respect to a common origin.
In Sec. 4.3.7, p. 88, I mentioned Friedel’s law, that Ihkl = Ihkl . It will be helpful
for later discussions to look at the vector representations of pairs of structure factors
Fhkl and Fhkl , which are called Friedel pairs. Even though Ihkl and Ihkl are equal,
Fhkl and Fhkl are not. The structure factors of Friedel pairs have different phases,
as shown in Fig. 6.4. Specifically, if the phase of Fhkl is α, then the phase of Fhkl
is 360o −α. Another way to put it is that Friedel pairs are reflections of each other
in the real axis; Fhkl is the mirror image of Fhkl with the real axis serving as the
mirror.
The representation of structure factors as vectors in the complex plane (that is,
complex vectors) is useful in several ways. Because the diffractive contributions
of atoms or volume elements to a single reflection are additive, each contribution
can be represented as a complex vector, and the resulting structure factor is the
vector sum of all contributions. For example, in Fig. 6.5, F (green) represents a
structure factor of a three-atom structure, in which f1, f2, and f3 (black) are the
atomic structure factors. The length of each atomic structure factor fn represents
its amplitude, and its angle with the real axis, αn, represents its phase. The vector
sum F = f1 + f2 + f3 is obtained by placing the tail of f1 at the origin, the tail of
Figure 6.4  Structure factors of a Friedel pair. F
hkl
is the reflection of Fhkl in the
real axis.
Section 6.2 Two-dimensional representation of structure factors 115
Figure 6.5  Molecular structure factor F (green) is the vector sum of three atomic
structure factors (black). Vector addition of f1, f2, and f3 gives the amplitude and phase of F.
f2 on the head of f1, and the tail of f3 on the head of f2, all the while maintaining
the phase angle of each vector. The structure factor F is thus a vector with its tail
at the origin and its head on the head of f3. This process sums both amplitudes
and phases, so the resultant length of F represents its amplitude, and the resultant
angle α is its phase angle. (The atomic vectors may be added in any order with the
same result.)
In subsequent sections of this chapter, I will use this simple vector arithmetic
to show how to compute phases from various kinds of data. In the next section,
I will use complex vectors to derive an equation for electron density as a function
of reflection intensities and, at last, phases.
6.2.3 Electron density as a function of intensities and phases
Figure 6.3 shows how to decompose Fhkl into its amplitude |Fhkl |, which is the
length of the vector, and its phase αhkl , which is the angle the vector makes
with the real number line. This allows us to express ρ(x, y, z) as a function of
the measurable amplitude of F (measurable because it can be computed from the
reflection intensity I ) and the unknown phase α. For clarity, I will at times drop
the subscripts on F, I , and α, but remember that these relationships hold for all
reflections.
In Fig. 6.3,
cosα = |A||F| and sin α =
|B|
|F| , (6.1)
116 Chapter 6 Obtaining Phases
and therefore
|A| = |F| · cosα and |B| = |F| · sin α. (6.2)
Expressing F as a complex vector A + iB,
F = |A| + i |B| = |F| · (cosα + i sin α). (6.3)
Expressing the complex term in the parentheses as an exponential [Eq. (5.6)],
F = |F| · eiα . (6.4)
Substituting this expression for Fhkl in Eq. (5.18), the electron-density equation
(remembering that α is the phase αhkl of the specific reflection hkl), gives
ρ(x, y, z) = 1
V

h
k
l |Fhkl | eiαhkl e−2πi(hx+ky+lz). (6.5)
We can combine the exponential terms more simply by expressing the phase angle
as α′, using α = 2πα′:
ρ(x, y, z) = 1
V

h
k
l |Fhkl |e2πiα′hkl e−2πi(hx+ky+lz). (6.6)
This substitution allows us to combine the exponentials by adding their exponents:
ρ(x, y, z) = 1
V

h
k
l |Fhkl |e−2πi(hx+ky+lz−α′hkl ). (6.7)
This equation gives the desired electron density as a function of the known
amplitudes |Fhkl | and the unknown phases α′hkl of each reflection. Recall that this
equation represents ρ(x, y, z) in a now-familiar form, as a Fourier sum, but this
time with the phase of each structure factor expressed explicitly. Each term in the
series is a three-dimensional wave of amplitude |Fhkl |, phase α′hkl , and frequencies
h along the x-axis, k along the y-axis, and l along the z-axis.
The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called
phase problem, that of determining the phase angle αhkl for each reflection.
In the remainder of this chapter, I will discuss some of the common methods
for overcoming this obstacle. These include the heavy-atom method (also called
isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates
of phases, which must be improved before an interpretable electron-density map
Section 6.3 Isomorphous replacement 117
can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include
as many reflections as possible. In Chapter 7, I will discuss methods of phase
improvement and phase extension, which ultimately result in accurate phases and
an interpretable electron-density map.
6.3 Isomorphous replacement
Each atom in the unit cell contributes to every reflection in the diffraction pattern
[Eq. (5.15)]. The contribution of an atom is greatest to the reflections whose indices
correspond to lattice planes that intersect that atom, so a specific atom contributes
to some reflections strongly, and to some weakly or not at all. If we could add one
or a very small number of atoms to identical sites in all unit cells of a crystal, we
would expect to see changes in the diffraction pattern, as the result of the additional
contributions of the added atom. As I will show later, the slight perturbation in the
diffraction pattern caused by an added atom can be used to obtain initial estimates
of phases. In order for these perturbations to be large enough to measure, the added
atom must be a strong diffractor, which means it must be an element of higher
atomic number than most of the other atoms—a so-called heavy atom. In smallmolecule crystallography, a larger atom like sulfur that is already present among
a modest number of smaller atoms like carbon and hydrogen can be used for this
purpose. This approach is properly called the heavy-atom method. For proteins, it
is usually necessary to add a heavy atom, such as mercury, lead, or gold, in order
to see perturbations against the signal of hundreds or thousands of smaller atoms.
Addition of one or more heavy atoms to a protein for phasing is properly called
isomorphous replacement, but is sometimes loosely referred to as the heavy-atom
method.
6.3.1 Preparing heavy-atom derivatives
After obtaining a complete set of X-ray data and determining that these data are
adequate to produce a high-resolution structure, the crystallographer undertakes
to prepare one or more heavy-atom derivatives. In the most common technique,
crystals of the protein are soaked in solutions of heavy ions, for instance ions or
ionic complexes of Hg, Pt, or Au. In many cases, such ions bind to one or a few
specific sites on the protein without perturbing its conformation or crystal packing.
For instance, surface cysteine residues react readily with Hg2+ ions, and cysteine,
histidine, and methionine displace chloride from Pt complexes like PtCl2−4 to form
stable Pt adducts. The conditions that give such specific binding must be found by
simply trying different ionic compounds at various pH values and concentrations.
(The same companies that sell crystal screening kits sell heavy-atom screening
kits.)
118 Chapter 6 Obtaining Phases
Several diffraction criteria define a promising heavy-atom derivative. First, the
derivative crystals must be isomorphous with native crystals. At the molecular
level, this means that the heavy atom must not disturb crystal packing or the
conformation of the protein. Unit-cell dimensions are quite sensitive to such disturbances, so heavy-atom derivatives whose unit-cell dimensions are the same
as native crystals are probably isomorphous. The term isomorphous replacement
comes from this criterion.
The second criterion for useful heavy-atom derivatives is that there must be
measurable changes in at least a modest number of reflection intensities. These
changes are the handle by which phase estimates are pulled from the data, so they
must be clearly detectable, and large enough to measure accurately.
Figure 6.6 shows precession photographs for native and derivative crystals of
the MoFe protein of nitrogenase. Underlined in the figure are pairs of reflections
whose relative intensities are altered by the heavy atom. If examining heavy-atom
data frames by eye, the crystallographer would look for pairs of reflections whose
relative intensities are reversed. This distinguishes real heavy-atom perturbations
from simple differences in overall intensity of two photos. For example, consider
the leftmost underlined pairs in each photograph. In the native photo (a), the
reflection on the right is the darker of the pair, whereas in the derivative photo (b),
the reflection on the left is darker. Several additional differences suggest that
this derivative might produce good phases. In modern practice, software computes intensity differences between native and heavy-atom reflections, and gives
a quantitative measure of the potential phasing power of a heavy-atom derivative.
Figure 6.6  Precession photographs of the hk0 plane in native (a) and heavy-atom
(b) crystals of the MoFe protein from nitrogenase. Corresponding underlined pairs in
the native and heavy-atom patterns show reversed relative intensities. Photos courtesy
of Professor Jeffrey Bolin.
Section 6.3 Isomorphous replacement 119
Finally, the derivative crystal must diffract to reasonably high resolution,
although the resolution of derivative data need not be as high as that of native
data. Methods of phase extension (Chapter 7) can produce phases for higher-angle
reflections from good phases of reflections at lower angles.
Having obtained a suitable derivative, the crystallographer faces data collection
again. Because derivatives must be isomorphous with native crystals, the strategy
is the same as that for collecting native data. You can see that the phase problem
effectively multiplies the magnitude of the crystallographic project by the number
of derivative data sets needed. As I will show, at least two, and often more,
derivatives are required in isomorphous replacement.
6.3.2 Obtaining phases from heavy-atom data
Consider a single reflection of amplitude |FP| (P for protein) in the native data, and
the corresponding reflection of amplitude |FPH| (PH for protein plus heavy atom)
in data from a heavy-atom derivative. Because the diffractive contributions of all
atoms to a reflection are additive, the difference in amplitudes (|FPH| − |FP|) is
the amplitude contribution of the heavy atom alone. If we compute a diffraction
pattern in which the amplitude of each reflection is (|FPH| − |FP|)2, the result is
the diffraction pattern of the heavy atom alone in the protein’s unit cell, as shown
in Fig. 6.7. In effect, we have subtracted away all contributions from the protein
atoms, leaving only the heavy-atom contributions. Now we see the diffraction
pattern of one atom (or only a small number of atoms) rather than the far more
complex pattern of the protein.
In comparison to the protein structure, this “structure”—a sphere (or very few
spheres) in a lattice—is very simple. It is usually easy to “determine” this structure,
that is, to find the location of the heavy atom in the unit cell. Before considering
how to locate the heavy atom, I will show how finding it helps us to solve the
phase problem.
Suppose we are able to locate a heavy atom in the unit cell of derivative crystals.
Recall that Eq. (5.15) gives us the means to calculate the structure factors Fhkl
for a known structure. This calculation gives us not just the amplitudes but the
complete structure factors, including each of their phases. So we can compute the
amplitudes and phases of our simple structure, the heavy atom in the protein unit
cell (Fig. 6.7e). Now consider a single reflection hkl as it appears in the native
and derivative data. Let the structure factor of the native reflection be FP. Let the
structure factor of the corresponding derivative reflection be FPH. Finally, let FH
be the structure factor for the heavy atom itself, which we can compute if we can
locate the heavy atom.
Figure 6.8 shows the relationship among the vectors FP, FPH, and FH on the
complex plane. (Remember that we are considering this relationship for a specific
reflection, but the same relationship holds, and the same kind of equation must
be solved, for all reflections.) Because the diffractive contributions of atoms are
additive vectors,
FPH = FH + FP. (6.8)
120 Chapter 6 Obtaining Phases
Figure 6.7  Heavy-atom method. (a) Protein in unit cell, and its diffraction pattern.
(b) Heavy-atom derivative, and its diffraction pattern. Can you find slight differences in
relative intensities of reflections in (a) and (b)? Taking the difference between diffraction
patterns in (a) and (b) gives (c), the diffraction pattern of the heavy atom alone. (d) Interpretation of (c) by Patterson methods locates the heavy atom (dark gray) in the unit cell of
the protein. FT of (d) gives (e), the structure factors of the heavy atom, with phases. These
FH terms allow solution of Eq. (6.9).
That is, the structure factor for the heavy-atom derivative (red in the figure) is the
vector sum of the structure factors for the protein alone (green) and the heavy atom
alone (blue). For each reflection, we wish to know FP. (We already know that its
length is obtainable from the measured reflection intensity IP, but we want to learn
its phase angle.) According to the previous equation,
FP = FPH − FH. (6.9)
We can solve this vector equation for FP, and thus obtain the phase angle of the
structure factor, by use of a Harker diagram which represents the equation in the
complex plane (Fig. 6.9).
We know |FPH| and |FP| from measuring reflection intensities IPH and IP. So
we know the length of the vectors FPH and FP, but not their directions or phase
angles. We know FH, including its phase angle, from locating the heavy atom and
calculating all its structure factors (Fig. 6.7). To solve Eq. (6.9) for FP and thus
obtain its phase angle, we place the vector −FH at the origin and draw a circle of
radius |FPH| centered on the head of vector −FH (Fig. 6.9a). All points on this
Section 6.3 Isomorphous replacement 121
Figure 6.8  Astructure factor FPH for the heavy-atom derivative is the sum of contributions from the native structure (FP) and the heavy atom (FH). COLOR KEY: In this and
all subsequent diagrams of this type (called Harker diagrams), the structure factor whose
phase we are seeking (in this case, that of the protein) is green. The structure factor of the
heavy atom (or its equivalent in other types of phasing) is blue, and the structure factor of
the heavy atom derivative of the protein is red. Circles depicting possible orientations of
structure factors carry the same colors.
Figure 6.9  Vector solution of Eq. (6.9). (a) All points on the red circle equal the
vector sum |FPH| − FH. (b) Vectors from the origin to intersections of the two circles are
solutions to Eq. (6.9).
122 Chapter 6 Obtaining Phases
circle equal the vector sum |FPH| − FH. In other words, we know that the head of
FPH lies somewhere on this circle of radius |FPH|. Next, we add a circle of radius
|FP| centered at the origin (Fig. 6.9b). We know that the head of the vector FP lies
somewhere on this circle, but we do not know where because we do not know its
phase angle. Equation (6.9) holds only at points where the two circles intersect.
Thus the phase angles of the two vectors FaP and F
b P that terminate at the points of
intersection of the circles are the only possible phases for this reflection.
Our heavy-atom derivative allows us to determine, for each reflection hkl, that
αhkl has one of two values. How do we decide which of the two phases is correct?
In some cases, if the two intersections lie near each other, the average of the
two phase angles will serve as a reasonable estimate. I will show in Chapter 7 that
certain phase improvement methods can sometimes succeed with such phases from
only one derivative, in which case the structure is said to be solved by the method
of single isomorphous replacement (SIR). More commonly, however, a second
heavy-atom derivative must be found and the vector problem outlined previously
must be solved again. Of the two possible phase angles found by using the second
derivative, one should agree better with one of the two solutions from the first
derivative, as shown in Fig. 6.10.
Figure 6.10a shows the phase determination using a second heavy-atom derivative; F′H is the structure factor for the second heavy atom. The radius of the red
circle is |F′PH|, the amplitude of F′PH for the second heavy-atom derivative. For this
derivative, FP = F′PH − F′H. Construction as before shows that the phase angles
of FcP and F
d P are possible phases for this reflection. In Fig. 6.10b, the circles from
Figure 6.10  (a) A second heavy-atom derivative indicates two possible phases, one
of which corresponds to Fa in Fig. 6.9b. (b) FP, which points from the origin to the common
intersection of the three circles, is the solution to Eq. (6.9) for both heavy-atom derivatives.
Thus α is the correct phase for this reflection.
Section 6.3 Isomorphous replacement 123
Figure 6.11  The MIR solution for this structure factor gives phase of high
uncertainty.
Figs. 6.9b and 6.10a are superimposed, showing that FcP is identical to F
a P. This
common solution to the two vector equations is FP, the desired structure factor.
The phase of this reflection is therefore the angle labeled α in the figure, the only
phase compatible with data from both derivatives.
In order to resolve the phase ambiguity from the first heavy-atom derivative, the
second heavy atom must bind at a different site from the first. If two heavy atoms
bind at the same site, the phases of FH will be the same in both cases, and both
phase determinations will provide the same information. This is true because the
phase of an atomic structure factor depends only on the location of the atom in the
unit cell, and not on its identity (Sec. 5.3.1, p. 98). In practice, it sometimes takes
three or more heavy-atom derivatives to produce enough phase estimates to make
the needed initial dent in the phase problem. Obtaining phases with two or more
derivatives is called the method of multiple isomorphous replacement (MIR). For
many years, MIR was one of the most successful methods in macromolecular
crystallography.
To compute a high-resolution structure, we must ultimately know the phases
αhkl for all reflections. High-speed computers can solve large numbers of these
vector problems rapidly, yielding an estimate of each phase along with a measure
of its precision.1 For many phases, the precision of the first phase estimate is so low
that the phase is unusable. For instance, in Fig. 6.11, the circles graze each other
rather than intersecting sharply, so there is a large uncertainty in α. In some cases,
1Computer programs calculate phases for each derivative numerically (rather than geometrically)
by obtaining two solutions to the equation
F2PH = F2P + F2H + 2FPFH cos(αP − αH).
The pairs of solutions for two heavy-atom derivatives should have one solution in common.
124 Chapter 6 Obtaining Phases
because of inevitable experimental errors in measuring intensities, the circles do
not intersect at all. This situation is referred to as lack of closure, and there are
computer algorithms for making phase estimates when it occurs.
Computer programs for calculating phases also compute statistical parameters
representing attempts to judge the quality of phases. Some parameters, usually
called phase probabilities, are measures of the uncertainty of individual phases.
Other parameters, including figure of merit, closure errors, phase differences, and
various R-factors are attempts to assess the quality of groups of phases obtained
by averaging results from several heavy-atom derivatives (or results from other
phasing methods). In most cases, these parameters are numbers between 0 (poor
phases) and 1 (perfect phases). No single one of these statistics is an accurate
measure of the goodness of phases. Crystallographers often use two or more of
these criteria simultaneously in order to cull out questionable phases. In short, until
correct phases are obtained (see Chapter 7), there is no sure way to measure the
quality of estimates. The acid test of phases is whether they give an interpretable
electron-density map. In Chapter 7, I will say more about the most modern methods
of improving phases at all stages in crystallography.
When promising phases are available, the crystallographer carries out Fourier
summation [Eq. (6.7)] to calculate ρ(x, y, z). Each Fourier term is multiplied by
the probability of correctness of the associated phase. This procedure gives greater
weight to terms with more reliable phases. Every phase that defies solution or is
too uncertain (and for that matter every intensity that is too weak to measure accurately) forces the crystallographer to omit one term from the Fourier series when
calculatingρ(x, y, z). Each omitted term lowers the accuracy of the approximation
to ρ(x, y, z), degrading the quality and resolution of the resulting map. In practice, a good pair of heavy-atom derivatives may allow us to estimate only a small
percentage of the phases. We can enlarge our list of precise phases by iterative
processes mentioned briefly in Sec. 5.4.2, p. 104, which I will describe more fully
in Chapter 7. For now, I will complete this discussion of isomorphous replacement
by considering how to find heavy atoms, which is necessary for calculating FH.
6.3.3 Locating heavy atoms in the unit cell
Before we can obtain phase estimates by the method described in the previous
section, we must locate the heavy atoms in the unit cell of derivative crystals.
As I described earlier, this entails extracting the relatively simple diffraction signature of the heavy atom from the far more complicated diffraction pattern of
the heavy-atom derivative, and then solving a simpler “structure,” that of one
heavy atom (or a few) in the unit cell of the protein (Fig. 6.7). The most powerful tool in determining the heavy-atom coordinates is a Fourier sum called the
Patterson function P(u, v,w), a variation on the Fourier sum I have described
for computing ρ(x, y, z) from structure factors. The coordinates (u, v,w) locate
a point in a Patterson map, in the same way that coordinates (x, y, z) locate a
point in an electron-density map. The Patterson function or Patterson synthesis
is a Fourier sum without phases. The amplitude of each term is the square of
one structure factor, which is proportional to the measured reflection intensity.
Section 6.3 Isomorphous replacement 125
Thus we can construct this series from intensity measurements, even though we
have no phase information. Here is the Patterson function in general form:
P(u, v,w) = 1
V

h
k
l |Fhkl |2e−2πi(hu+kv+lw). (6.10)
To obtain the Patterson function solely for the heavy atoms in derivative crystals,
we construct a difference Patterson function, in which the amplitudes are (
F)2 =
(|FPH| − |FP|)2. The difference between the structure-factor amplitudes with and
without the heavy atom reflects the contribution of the heavy atom alone. The
difference Patterson function is
P(u, v,w) = 1
V

h
k
l F2hkle
−2πi(hu+kv+lw). (6.11)
In words, the difference Patterson function is a Fourier sum of simple sine
and cosine terms. (Remember that the exponential term is shorthand for these
trigonometric functions.) Each term in the series is derived from one reflection
hkl in both the native and derivative data sets, and the amplitude of each term
is (|FPH| − |FP|)2, which is the amplitude contribution of the heavy atom to
structure factor FPH. Each term has three frequencies h in the u-direction, k in
the v-direction, and l in the w-direction. Phases of the structure factors are not
included; at this point, they are unknown. (If we knew them, we wouldn’t have to
do all this.)
Because the Patterson function contains no phases, it can be computed from any
raw set of crystallographic data, but what does it tell us?Acontour map ofρ(x, y, z)
displays areas of high density (peaks) at the locations of atoms. In contrast, it can
be proven that a Patterson map, which is a contour map of P(u, v,w), displays
peaks at locations corresponding to vectors between atoms. (This is a strange
idea at first, but the following example will make it clearer.) Of course, there are
more vectors between atoms than there are atoms, so a Patterson map is more
complicated than an electron-density map. But if the structure is simple, like that
of one or a few heavy atoms in the unit cell, the Patterson map may be simple
enough to allow us to locate the atom(s). You can see now that the main reason for
using the difference Patterson function instead of a simple Patterson using FPHs
is to eliminate the enormous number of peaks representing vectors between light
atoms in the protein.
I will show, in a two-dimensional example, how to construct the Patterson map
from a simple crystal structure and then how to use a calculated Patterson map
to deduce a structure (Fig. 6.12). The simple molecular structure in Fig. 6.12a
contains three atoms (red circles) in each unit cell. To construct the Patterson map,
first draw all possible vectors between atoms in one unit cell, including vectors
between the same pair of atoms but in opposite directions. (For example, treat
1 → 2 and 2 → 1 as distinct vectors.) Two of the six vectors (1 → 3 and 3 → 2)
126 Chapter 6 Obtaining Phases
Figure 6.12  Construction and interpretation of a Patterson map. (a) Structure of unit
cell containing three atoms. Two of the six interatomic vectors are shown. (b) Patterson
map is constructed by moving the tails of all interatomic vectors to the origin. Patterson
“atoms” [peaks (purple) in the contour map] occur at the head of each vector. (c) Complete
Patterson map, containing all peaks from (b) in all unit cells. Peak at origin results from
self-vectors. Image of original structure is present (red peaks) amid other peaks. (d) Trial
solution of map (c). If origin and Patterson atoms a and b were the image of the real unit
cell, the interatomic vector a → b would produce a peak in the small green box. Absence
of the peak disproves this trial solution.
are shown in the figure. Then draw empty unit cells around an origin (Fig. 6.12b),
and redraw all vectors with their tails at the origin. The head of each vector is
the location of a peak in the Patterson map, sometimes called a Patterson “atom”
(purple circles in b, c, and d). The coordinates (u, v,w) of a Patterson atom
representing a vector between atom 1 at (x1, y1, z1) and atom 2 at (x2, y2, z2)
are (u, v,w) = (x1 − x2, y1 − y2, z1 − z2). The vectors from Fig. 6.12a are
redrawn in Fig. 6.12b, along with all additional Patterson atoms produced by this
procedure. Finally, in each of the unit cells, duplicate the Patterson atoms from all
four unit cells. The result (Fig. 6.12c) is a complete Patterson map of the structure
in Fig. 6.12a. In this case, there are six Patterson atoms in each unit cell. You can
easily prove to yourself that a real unit cell containing n atoms will give a Patterson
unit cell containing n(n− 1) Patterson atoms.
Now let’s think about how to go from a computed Patterson map to a structure—
that is, how to locate real atoms from Patterson atoms. A computed Patterson map
exhibits a strong peak at the origin because this is the location of all vectors
Section 6.3 Isomorphous replacement 127
between an atom and itself. Notice in Fig. 6.12c that the origin and two of the
Patterson atoms (red circles) reconstruct the original arrangement of atoms. Finding six peaks (ignoring the peak at the origin) in each unit cell of the calculated
Patterson map, we infer that there are three real atoms per unit cell. [Solve the
equation n(n− 1) = 6.] We therefore know that the origin and two peaks reconstruct the relationship among the three real atoms, but we do not know which two
peaks to choose. To solve the problem, we pick a set of peaks—the origin and two
others—as a trial solution, and follow the rules described earlier to generate the
expected Patterson map for this arrangement of atoms. If the trial map has the same
peaks as the calculated map, then the trial arrangement of atoms is correct. By trial
and error, we can determine which pair of Patterson atoms, along with an atom
at the origin, would produce the remaining Patterson atoms. Figure 6.12d shows
an incorrect solution (the origin plus green peaks a and b). The vector a → b is
redrawn at the origin to show that the map does not contain the Patterson atom
a → b, and hence that this solution is incorrect.
You can see that as the number of real atoms increases, the number of Patterson
atoms, and with it the difficulty of this problem, increases rapidly. Computer
programs can search for solutions to such problems and, upon finding a solution,
can refine the atom positions to give the most likely arrangement of heavy atoms.
Unit-cell symmetry can also simplify the search for peaks in a three-dimensional
Patterson map. For instance, in a unit cell with a 21 axis (twofold screw) on edge b,
recall (equivalent positions, Sec. 4.2.8, p. 65) that each atom at (x, y, z) has an
identical counterpart atom at (−x, y + 1/2,−z). The vectors connecting such
symmetry-related atoms will all lie at (u, v,w) = (2x, [1/2], 2z) in the Patterson
map (just subtract one set of coordinates from the other, and realize that the position
v = −1/2 is the same as v = 1/2), which means they all lie in the plane that cuts
the Patterson unit cell atw = 1/2—theu[1/2]w plane. Such planes, which contain
the Patterson vectors for symmetry-related atoms, are called Harker sections or
Harker planes. If heavy atoms bind to the protein at equivalent positions, heavyatom peaks in the Patterson map can be found on the Harker sections. (Certain
symmetry elements give Patterson vectors that all lie upon a line, called a Harker
line, rather than on a plane.)
Given the location of a first heavy-atom Patterson peak on a Harker section,
what is the location of the heavy atom in the real unit cell? In the P 21 unit cell, the
equality u = 2x means that x = u/2, giving the x-coordinate of the atom. In like
manner, z = w/2. The location of the peak on a Harker section at v = 1/2 places
no restrictions on the y-coordinate, so it can be given a convenient arbitrary value
like y = 0. Thus the heavy-atom coordinates in the real unit cell are (u/2, 0, w/2).
Additional heavy-atom sites will have their y-coordinates specified relative to this
arbitrary assignment.
There is an added complication (in crystallography, it seems there always is):
the arrangement of heavy atoms in a protein unit cell is often enantiomeric. For
example, if heavy atoms are found along a threefold screw axis, the screw may
be left- or right-handed. The Patterson map does not distinguish between mirrorimage, or more accurately, inverted-image, arrangements of heavy atoms because
128 Chapter 6 Obtaining Phases
you cannot tell whether a Patterson vector ab is a → b or b → a. But the phases
obtained by calculating structure factors from the inverted model are incorrect
and will not lead to an interpretable map. Crystallographers refer to this difficulty
as the hand problem (although hands are mirror images and the two solutions
described here are inversions). If derivative data are available to high resolution,
the crystallographer simply calculates two electron-density maps, one with phases
from each enantiomer of the heavy-atom structure. With luck, one of these maps
will be distinctly clearer than the other. If derivative data are available only at low
resolution, this method may not determine the hand with certainty. The problem
may require the use of anomalous scattering methods, discussed in Sec. 6.4.6,
p. 135.
Having located the heavy atom(s) in the unit cell, the crystallographer can
compute the structure factors FH for the heavy atoms alone, using Eq. (5.15).
This calculation yields both the amplitudes and the phases of structure factors
FH, giving the vector quantities needed to solve Eq. (6.9) for the phases αhkl of
protein structure factors FP. This completes the information needed to compute
a first electron-density map, using Eq. (6.7), p. 116. This map requires improvement because these first phase estimates contain substantial errors. I will discuss
improvement of phases and maps in Chapter 7.
6.4 Anomalous scattering
6.4.1 Introduction
A second means of obtaining phases from heavy-atom derivatives takes advantage
of the heavy atom’s capacity to absorb X-rays of specified wavelength. As a result
of this absorption, Friedel’s law (Sec. 4.3.7, p. 88) does not hold, and the reflections hkl and hkl are not equal in intensity. This inequality of symmetry-related
reflections is called anomalous scattering or anomalous dispersion.
Recall from Sec. 4.3.2, p. 73 that elements absorb X-rays as well as emit them,
and that this absorption drops sharply at wavelengths just below their characteristic
emission wavelength Kβ (Fig. 4.21, p. 74). This sudden change in absorption as a
function of λ is called an absorption edge. An element exhibits anomalous scattering when the X-ray wavelength is near the element’s absorption edge. Absorption
edges for the light atoms in the unit cell are not near the wavelength of X-rays used
in crystallography, so carbon, nitrogen, and oxygen do not contribute to anomalous scattering. However, absorption edges of heavy atoms are in this range, and
if X-rays of varying wavelength are available, as is often the case at synchrotron
sources, X-ray data can be collected under conditions that maximize anomalous
scattering by the heavy atom.
6.4.2 Measurable effects of anomalous scattering
When the X-ray wavelength is near the heavy-atom absorption edge, a fraction
of the radiation is absorbed by the heavy atom and reemitted with altered phase.
Section 6.4 Anomalous scattering 129
Figure 6.13  Real and imaginary anomalous-scattering contributions alter the magnitude and phase of the structure factor. COLOR KEY: By analogy to colors used in MIR
illustrations, the vector whose phase we want to find, in this case, Fλ1PH, is green; the anomalous scattering contributions, whose role is analogous to that of a heavy atom, are shades
of blue; and Fλ2PH, the “anomalous-dispersion derivative” that takes the role of the heavy
atom derivative in MIR, is red.
The effect of this anomalous scattering on a given structure factor FPH in the
heavy-atom data is depicted in vector diagrams as consisting of two perpendicular
contributions, one real (
Fr) and the other imaginary (
Fi).
In Fig. 6.13, Fλ1PH (green) represents a structure factor for the heavy-atom derivative measured at wavelength λ1, where anomalous scattering does not occur.
Fλ2HP (red) is the same structure factor measured at a second X-ray wavelength
λ2 near the absorption edge of the heavy atom, so anomalous scattering alters the
heavy-atom contribution to this structure factor. The vectors representing anomalous scattering contributions are
Fr (real, blue) and
Fi (imaginary, cyan). From
the diagram, you can see that
Fλ2PH = Fλ1PH +
Fr +
Fi. (6.12)

Figure 6.14 shows the result of anomalous scattering for a Friedel pair of structure
factors, distinguished from each other in the figure by superscripts + and −.
Recall that for Friedel pairs in the absence of anomalous scattering, |Fhkl | =
|Fhkl | and αhkl = −αhkl , so Fλ1−PH is the reflection of Fλ1+PH in the real axis. The
real contributions
F+r and
F−r to the reflections of a Friedel pair are, like the
structure factors themselves, reflections of each other in the real axis. On the other
hand, it can be shown (but I will not prove it here) that the imaginary contribution
to Fλ1−PH is the inverted reflection of that for F
λ1+
PH . That is,
F

i is obtained by
reflecting
F+i in the real axis and then reversing its sign or pointing it in the
opposite direction. Because of this difference between the imaginary contributions
130 Chapter 6 Obtaining Phases
Figure 6.14  Under anomalous scattering, at wavelength λ2, Fhkl is no longer the
mirror image of Fhkl .
to these reflections, under anomalous scattering, the two structure factors are no
longer precisely equal in intensity, nor are they precisely opposite in phase. It is
clear from Fig. 6.14 that Fλ2−PH is not the mirror image of F
λ2+
PH . From this disparity
between Friedel pairs, the crystallographer can extract phase information.
6.4.3 Extracting phases from anomalous scattering data
The magnitude of anomalous scattering contributions
Fr and
Fi for a given
element are constant and roughly independent of reflection angle θ , so these quantities can be looked up in tables of crystallographic information. The phases of
Fr and
Fi depend only upon the position of the heavy atom in the unit cell, so
once the heavy atom is located by Patterson methods, the phases can be computed.
The resulting full knowledge of
Fr and
Fi allows Eq. (6.12) to be solved for the
vector Fλ1PH, thus establishing its phase. Crystallographers obtain solutions by computer, but I will solve the general equation using Harker diagrams (Fig. 6.15), and
thus show that the amount of information is adequate to solve the problem. First
consider the structure factor Fλ1+PH in Fig. 6.14. Applying Eq. (6.12) and solving
for Fλ1+PH gives
Fλ1+PH = Fλ2+PH −
F+r −
F+i . (6.13)

To solve this equation (see Fig. 6.15), draw the vector −
F+r with its tail at the
origin, and draw −
F+i with its tail on the head of −
F+r . With the head of

F+i as center, draw a circle of radius |Fλ2+PH |, representing the amplitude of this
reflection in the anomalous scattering data set. The head of the vector Fλ2+PH lies
somewhere on this circle. We do not know where, because we do not know the
phase of the reflection. Now draw a circle of radius |Fλ1+PH | with its center at the
Section 6.4 Anomalous scattering 131
Figure 6.15  Vector solution of Eq. (6.13).
Fr and
Fi play the same role as FH
in Figs. 6.9, p. 121, and 6.10, p. 122.
origin, representing the structure-factor amplitude of this same reflection in the
nonanomalous scattering data set. The two points of intersection of these circles
satisfy Eq. (6.13), establishing the phase of this reflection as either that of Fa or
Fb. As with the SIR method, we cannot tell which of the two phases is correct.
The Friedel partner of this reflection comes to the rescue. We can obtain a second
vector equation involving Fλ1+PH by reflecting F
λ2−
PH and all its vector components
across the real axis (Fig. 6.16a).
After reflection, Fλ1−PH equals F
λ1+
PH ,
F


r equals
F
+
r , and
F

i equals −
F+i .
The magnitude of Fλ2−PH is unaltered by reflection across the real axis. If we make
these substitutions in Eq. (6.13), we obtain
Fλ1+PH =
∣∣Fλ2−PH ∣∣−
F+r − (−
F+i ). (6.14)

We can solve this equation in the same manner that we solved Eq. (6.13), by
placing the vectors−
F+r and
F+i head-to-tail at the origin, and drawing a circle
of radius |Fλ2PH| centered on the head of
F+i (Fig. 6.16b). The circles intersect at
the two solutions to Eq. (6.14). Although the circles graze each other and give two
phases with considerable uncertainty, one of the possible solutions corresponds to
Fa in Fig. 6.15, and neither of them is close to the phase of Fb.
So the disparity between intensities of Friedel pairs in the anomalous scattering
data set establishes their phases in the nonanomalous scattering data set. The
reflection whose phase has been established here corresponds to the vector FPH in
Eq. (6.9). Thus the amplitudes and phases of two of the three vectors in the Eq. (6.9)
are known: (1) FPH is known from the anomalous scattering computation just
132 Chapter 6 Obtaining Phases
Figure 6.16  Reflection of F− components across the real axis gives a second
vector equation involving the desired structure factor. (a) All reflected components are
labeled with their equivalent contributions from F+. (b) Vector solution of Eq. (6.13).
These solutions are compatible only with Fa in Fig. 6.15.
shown, and (2) FH is known from calculating the heavy-atom structure factors
after locating the heavy atom by Patterson methods. The vector FP, then, is simply
the vector difference FPH − FH, establishing the phase of this reflection in the
native data.
6.4.4 Summary
Under anomalous scattering, the members of a Friedel pair can be used to establish
the phase of a reflection in the heavy atom derivative data, thus establishing the
phase of the corresponding reflection in the native data. Let me briefly review
the entire project of obtaining the initial structure factors by SIR with anomalous
scattering (called SIRAS). First, we collect a complete data set with native crystals,
giving us the amplitudes |FP| for each of the native reflections. Then we find a
heavy-atom derivative and collect a second data set at the same wavelength, giving
amplitudes |FPH| for each of the reflections in the heavy-atom data. Next we collect
a third data set at a different X-ray wavelength, chosen to maximize anomalous
scattering by the heavy atom. We use the nonequivalence of Friedel pairs in the
anomalous scattering data to establish phases of reflections in the heavy-atom
Section 6.4 Anomalous scattering 133
data, and we use the phased heavy-atom derivative structure factors to establish
the native phases. (Puff-puff!)
In practice, several of the most commonly used heavy atoms (including uranium,
mercury, and platinum) give strong anomalous scattering with Cu–Kα radiation.
In such cases, crystallographers can measure intensities of Friedel pairs in the
heavy-atom data set. In phase determination (refer to Figs. 6.14–6.16), the average
of |Fhkl | and |Fhkl | serves as both |Fλ1+PH | and |Fλ1−PH |, while |Fhkl | and |Fhkl | separately serve as |Fλ2+PH | and |Fλ2−PH |, so only one heavy atom data set is required. This
method is called single isomorphous replacement with anomalous scattering, or
SIRAS.
Like phases from the MIR method, each anomalous scattering phase can only
serve as an initial estimate and must be weighted with some measure of phase probability. The intensity differences between Friedel pairs are very small, so measured
intensities must be very accurate if any usable phase information is to be derived.
To improve accuracy, crystallographers collect intensities of Friedel partners from
the same crystal, and under very similar conditions. In rotation/oscillation photography, crystallographers can alter the sequence of frame collection, so that frames
of Friedel pairs are measured in succession, thus minimizing artifactual differences between Friedel pairs due to crystal deterioration or changes in the X-ray
beam. For example, if we call a frame of data Fn and the frame of matching
Friedel pairs F′n, then minimizing artifactual differences between Friedel pairs
entails collecting F1, rotating the crystal 180o and collecting frame F′1 of matching Friedel pairs, then collecting F′2 starting at the end of the rotation for F′1, and
finally, rotating back 180o to collect F2.
6.4.5 Multiwavelength anomalous diffraction phasing
Three developments—variable-wavelength synchrotron X rays, cryocrystallography, and the production of proteins containing selenomethionine instead of the
normal sulfur-containing methionine—have recently allowed rapid progress in
maximizing the information obtainable from anomalous dispersion. For proteins
that naturally contain a heavy atom, such as the iron in a globin or cytochrome, the
native heavy atom provides the source of anomalous dispersion. Proteins lacking
functional heavy atoms can be expressed in Escherichia coli containing exclusively
selenomethionine. The selenium atoms then serve as heavy atoms in a protein that
is essentially identical to the “native” form. Isomorphism is, of course, not a problem with these proteins, because the same protein serves as both the native and
derivative forms.
The power of multiwavelength radiation is that data sets from a heavy-atom
derivative at different wavelengths are in many respects like those from distinct
heavy-atom derivatives. Especially in the neighborhood of the absorption maximum of the heavy atom [see, for example, the absorption spectra of copper
and nickel (dotted lines) in Fig. 4.21, p. 74], the real and imaginary anomalous scattering factors
Fr and
Fi vary greatly with X-ray wavelength. At the
absorption maximum,
Fi reaches its maximum value, whereas at the ascending
134 Chapter 6 Obtaining Phases
inflection point or edge,
Fr reaches a minimum and then increases farther from
the absorption peak. So data sets taken at the heavy-atom absorption maximum,
the edge, and at wavelengths distance from the maximum all have distinct values
for the real and imaginary contributions of anomalous dispersion. Thus each measurement of a Freidel pair at a specific wavelength provides the components of
distinct sets of phasing equations like those solved in Figs. 6.15 and 6.16. In
addition to wavelength-dependent differences between Friedel pairs, individual
reflection intensities vary slightly with wavelength (called dispersive differences),
and these differences also contain phase information, which can be extracted by
solving equations much like those for isomorphous replacement. All told, data sets
at different wavelengths from a single crystal can contain sufficient phasing information to solve a structure if the molecule under study contains one or more atoms
that give anomalous dispersion. This method is called multiwavelength anomalous dispersion, or MAD, phasing, and it has become one of the most widely used
phasing methods.
The principles upon which MAD phasing are based have been known for years.
But the method had to await the availability of variable-wavelength synchrotron
X-ray sources. In addition, the intensity differences the crystallographer must
measure are small, and until recently, these small signals were difficult to measure with sufficient precision. To maximize accuracy in the measured differences
between Friedel pairs, corresponding pairs must be measured on the same crystal
and at nearly the same time so that crystal condition and instrument parameters
do not change between measurements. And even more demanding, crystal condition and instrument parameters should be sufficiently constant to allow complete
data sets to be taken at several different wavelengths. These technical demands
are met by cryocrystallography and synchrotron sources. Flash-freezing preserves
the crystal in essentially unchanged condition through extensive data collection.
Synchrotron sources provide X-rays of precisely controllable wavelength and also
of high intensity, which shortens collection time.
The first successes of MAD phasing were small proteins that contained functional heavy atoms. Production of selenomethionine proteins opened the door
to MAD phasing for nonmetalloproteins. But larger proteins may contain many
methionines. Can there be too many heavy atoms? Recall that to solve SIR
and anomalous-dispersion phase equations, we must know the position(s) of the
heavy atom(s) in the unit cell. For MAD phasing, heavy atoms can be located
by Patterson methods (Sec. 6.3.3, p. 124), which entails trial-and-error comparisons of the Patterson map with calculated Pattersons for various proposed models
of heavy-atom locations. But Patterson maps of proteins containing many heavy
atoms may require so many trials that they resist solution. On the other hand, locating a relatively large number (say, tens) of heavy atoms is similar in complexity
to determine the structure of a “small” molecule. And indeed, as I will describe in
Sec. 6.4.7, p. 135, the direct-phasing methods used in small-molecule crystallography have come to the rescue in locating numerous Se atoms in selenomethione
proteins, allowing application of MAD phasing to larger and larger proteins, with
no end in sight.
Section 6.4 Anomalous scattering 135
6.4.6 Anomalous scattering and the hand problem
As I discussed in Sec. 6.3.3, p. 124, Patterson methods do not allow us to distinguish between enantiomeric arrangements of heavy atoms, and phases derived
from heavy-atom positions of the wrong hand are incorrect. When high-resolution
data are available for the heavy-atom derivative, phases and electron-density maps
can be calculated for both enantiomeric possibilities. The map calculated with
phases from the correct enantiomer will sometimes be demonstrably sharper and
more interpretable. If not, and if anomalous scattering data are available, SIR and
anomalous scattering phases can be computed for both hands, and maps can be
prepared from the two sets of phases. The added phase information from anomalous scattering sometimes makes hand selection possible when SIR phases alone
do not.
The availability of two heavy-atom derivatives, one with anomalous scattering,
allows a powerful technique for establishing the hand, even at quite low resolution.
We locate heavy atoms in the first derivative by Patterson methods, and choose one
of the possible hands to use in computing SIR phases. Then, using the same hand
assumption, we compute anomalous-scattering phases. For the second heavy-atom
derivative, instead of using Patterson methods, we compute a difference Fourier
between the native data and the second derivative data, using the SIR phases
from the first derivative. Then we compute a second difference Fourier, adding
the phases from anomalous scattering. Finally, we compute a third difference
Fourier, just like the second except that the signs of all anomalous-scattering
contributions are reversed, which is like assuming the opposite hand. The first
Fourier should exhibit electron-density peaks at the positions of the second heavy
atom. If the initial hand assumption was correct, heavy-atom peaks should be
stronger in the second Fourier. If it was incorrect, heavy-atom peaks should be
stronger in the third Fourier.
6.4.7 Direct phasing: Application of methods from small-molecule
crystallography
Methods involving heavy atoms apply almost exclusively to large molecules (500
or more atoms, not counting hydrogens). For small molecules (up to 200 atoms),
phases can be determined by what are commonly called direct methods. One form
of direct phasing relies on the existence of mathematical relationships among
certain combinations of phases. From these relationships, a sufficient number of
initial phase estimates can be obtained to begin converging toward a complete set
of phases. One such relationship, called a triplet relationship, relates the phases
and indices of three reflections as follows:
αhkl + αh′k′l′ α(h+h′)(k+k′)(l+l′) (6.15)
In words, if the indices of the reflections on the left sum to the indices of
the reflection on the right, then their phases sum approximately. As a specific
example, the phase of reflection 120 is always approximately the sum of the
136 Chapter 6 Obtaining Phases
phases of reflections 110 and 210. If the number of reflections is not too large,
so that a large percentage of these relationships can be examined simultaneously,
they might put enough constraints on phases to produce some initial estimates. For
macromolecules, the number of reflections is far too large to make this method
useful.
Another form of direct phasing, executed by a program called Shake-and-Bake,
in essence tries out random arrangements of atoms, simulates the diffraction patterns they would produce, and compares the simulated patterns with those obtained
from the crystals. Even though the trial arrangements are limited to those that are
physically possible (for example, having no two atoms closer than bonding or
van der Waals forces allow), the number of trial arrangements can be too large for
computation if the number of atoms is large. But this “try-everything” method is
enormously powerful for any number of atoms that computation can handle, and
of course, this number grows with the rapidly growing capacity of computers.
Direct methods work if the molecules, and thus the unit cells and numbers of
reflections, are relatively small. Isomorphous replacement works if the molecules
are large enough that a heavy atom does not disturb their structures significantly.
The most difficult structures for crystallographers are those that are too large for
direct methods and too small to remain isomorphous despite the intrusion of a
heavy atom. If a medium-size protein naturally contains a heavy atom, like iron or
zinc, or if a selenomethionine derivative can be produced, the structure can often
be solved by MAD phasing (Sec. 6.4.5, p. 133). [NMR methods (see Chapter 10)
are also of great power for small and medium-size molecules.]
The Shake-and-Bake style of direct phasing apparently has the potential to solve
the structures of proteins of over 100 residues if they diffract exceptionally well (to
around 1.0 Å). Fewer than 10% of large molecules diffract well enough to qualify.
But in a combined process that shows great promise, direct phasing has been combined with MAD phasing to solve large structures. Recall that larger proteins may
contain too many methionines to allow Patterson or least-squares location of all
seleniums in the selenomethionine derivative. Solving the anomalous-diffraction
phase equations requires knowing the locations of all the heavy atoms. Shake-andBake can solve this problem, even if there are 50 or more seleniums in the protein.
One early success of this combined method was a protein of molecular mass over
250,000 containing 65 selenomethionines.
Our last phasing method applies to all molecules, regardless of size, but it
requires knowledge that the desired structure is similar to a known structure.
6.5 Molecular replacement: Related
proteins as phasing models
6.5.1 Introduction
The crystallographer can sometimes use the phases from structure factors of a
known protein as initial estimates of phases for a new protein. If this method is
Section 6.5 Molecular replacement: Related proteins as phasing models 137
feasible, then the crystallographer may be able to determine the structure of the
new protein from a single native data set. The known protein in this case is referred
to as a phasing model, and the method, which entails calculating initial phases by
placing a model of the known protein in the unit cell of the new protein, is called
molecular replacement.
For instance, the mammalian serine proteases—trypsin, chymotrypsin, and
elastase—are very similar in structure and conformation. If a new mammalian
serine protease is discovered, and sequence homology with known proteases suggests that this new protease is similar in structure to known ones, then one of the
known proteases might be used as a phasing model for determining the structure
of the new protein.
Similarly, having learned the crystallographic structure of a protein, we may
want to study the conformational changes that occur when the protein binds to
a small ligand and to learn the molecular details of protein-ligand binding. We
might be able to crystallize the protein and ligand together or introduce the ligand into protein crystals by soaking. We expect that the protein-ligand complex
is similar in structure to the free protein. If this expectation is realized, we do
not have to work completely from scratch to determine the structure of the complex. We can use the ligand-free protein as a phasing model for the protein-ligand
complex.
In Fig. 6.1, p. 110, I showed that phases contain more information than intensities. How, then, can the phases from a different protein help us find an unknown
structure? In his Book of Fourier, Kevin Cowtan uses computed transforms to
illustrate this concept, as shown in Fig. 6.17. First we see, posing this time as
an unknown structure, the cat (a), with its Fourier transform shown in black and
white. The colorless transform is analogous to an experimental diffraction pattern
because we do not observe phases in experimental data. Next we see a Manx
(tailless) cat (b) (along with its transform) posing as a solved structure that we
also know (because of, say, sequence homology) to be similar in structure to the
cat (a). If we know that the unknown structure, the cat, is similar to a known
structure, the Manx cat, are the intensities of the cat powerful enough to reveal the
differences between the unknown structure and the phasing model—in this case,
the tail? In (c) the phases (colors) of the Manx cat transform are superimposed
on the intensities of the unknown cat transform. In (d) we see the back-transform
of (c), and although the image is weak, the cat’s tail is apparent. The intensities of
cat diffraction do indeed provide enough information to show how the cat differs
from the Manx cat. In like manner, measured intensities from a protein of unknown
structure do indeed have the power to show us how it differs from a similar, known
structure used as a phasing model.
6.5.2 Isomorphous phasing models
If the phasing model and the new protein (the target) are isomorphous, as may be
the case when a small ligand is soaked into protein crystals, then the phases from
the free protein can be used directly to compute ρ(x, y, z) from native intensities
138 Chapter 6 Obtaining Phases
Figure 6.17  Structure determination by molecular replacement. (a) Unknown
structure, cat, and its diffraction pattern (not colored, because phases are unknown).
(b) Known structure and phasing model, Manx cat, and transform computed from the model
(colored, because calculation of transform from a model tells us phases). (c) Manx-cat
phases combined with unknown-cat intensities. (d) Back-transform of (c). Intensities contain enough information to reveal differences (the tail) between phasing model and unknown
structure.
Section 6.5 Molecular replacement: Related proteins as phasing models 139
of the new protein [Eq. (6.16)]:
ρ(x, y, z) = 1
V

h
k
l ∣∣∣Ftargethkl
∣∣∣ e−2πi
(
hx+ky+lz−α′modelhkl
)
(6.16)
In this Fourier synthesis, the amplitudes |Ftargethkl | are obtained from the native
intensities of the new protein, and the phases α′model are those of the phasing
model. During the iterative process of phase improvement (Chapter 7), the phases
should change from those of the model to those of the new protein or complex,
revealing the desired structure. In Fig. 6.17, we not only knew that our phasing
model (the Manx cat) was similar to the unknown (cat with tail), but we had
the added advantage of knowing that its orientation was the same. Otherwise, its
phases would not have revealed the unknown structure.
6.5.3 Nonisomorphous phasing models
If the phasing model is not isomorphous with the target structure, the problem
is more difficult. The phases of atomic structure factors, and hence of molecular
structure factors, depend upon the location of atoms in the unit cell. In order to
use a known protein as a phasing model, we must superimpose the structure of
the model on the structure of the target protein in its unit cell and then calculate
phases for the properly oriented model. In other words, we must find, in the new
unit cell, the position and orientation of the phasing model that superimposes it on
the target protein, and hence, that would give phases most like those of the target.
Then we can calculate the structure factors of the properly positioned model and
use the phases of these computed structure factors as initial estimates of the desired
phases.
Without knowing the structure of the target protein, how can we copy the model
into the unit cell with the proper orientation and position? From native data on
the new protein, we can determine its unit-cell dimensions and symmetry. Clearly
the phasing model must be placed in the unit cell with the same symmetry as the
new protein. This places some constraints upon where to locate the model, but
not enough to give useful estimates of phases. In theory, it should be possible to
conduct a computer search of all orientations and positions of the model in the new
unit cell. For each trial position and orientation, we would calculate the structure
factors (called Fcalc) of the model [Eq. (5.15)], and compare their amplitudes
|Fcalc| with the measured amplitudes |Fobs| obtained from diffraction intensities
of the new protein. Finding the position and orientation that gives the best match,
we would take the computed phases (αcalc) as the starting phases for structure
determination of the new protein.
6.5.4 Separate searches for orientation and location
In practice, the number of trial orientations and positions for the phasing model
is enormous, so a brute-force search is impractical, even on the fastest computers
(as of this writing, of course). The procedure is greatly simplified by separating
140 Chapter 6 Obtaining Phases
the search for the best orientation from the search for the best position. Further,
it is possible to search for the best orientation independently of location by using
the Patterson function.
If you consider the procedure for drawing a Patterson map from a known structure (Sec. 6.3.3, p. 124), you will see that the final map is independent of the
position of the structure in the unit cell. No matter where you draw the “molecule,”
as long as you do not change its orientation (that is, as long as you do not rotate it
within the unit cell), the Patterson map looks the same. On the other hand, if you
rotate the structure in the unit cell, the Patterson map rotates around the origin,
altering the arrangement of Patterson atoms in a single Patterson unit cell. This
suggests that the Patterson map might provide a means of determining the best
orientation of the model in the unit cell of the new protein.
If the model and the target protein are indeed similar, and if they are oriented
in the same way in unit cells of the same dimensions and symmetry, they should
give very similar Patterson maps. We might imagine a trial-and-error method in
which we compute Patterson maps for various model orientations and compare
them with the Patterson map of the desired protein. In this manner, we could
find the best orientation of the model, and then use that single orientation in
our search for the best position of the model, using the structure-factor approach
outlined earlier. A systematic search for the best orientation entails superimposing
the origins of the two three-dimensional Patterson maps, and then rotating the
map of the phasing model through three angles, as shown in Fig. 6.18. First, let
us imagine an orthogonal system of coordinates (x, y, z) established with a fixed
relationship to the Patterson unit cell of the desired protein (Fig. 6.18a). Then
imagine a system of spherical polar angles φ, ϕ, and χ defined with respect to
the orthogonal system such that φ and ϕ give angles of rotation of a directing
axis (blue), and χ gives the angle of rotation around that axis. The Patterson cell
of the phasing model (b, transparent cell containing colored Patterson peaks) is
then rotated with respect to the target Patterson cell (black framework with black
Patterson peaks) through appropriately small intervals of the angles φ, ϕ, and
χ , (b, c, d) while the correlation between Patterson maps for the two models is
monitored (next section). We are searching for values of φ, ϕ, and χ that will
superimpose the Patterson peaks in the two models (e).
How much computing do we actually save by searching for orientation and
location separately? The orientation of the model can be specified by three angles
of rotation about orthogonal axes x, y, and z with their origins at the center
of the model. Specifying location also requires three numbers, the x, y, and z
coordinates of the molecular center with respect to the origin of the unit cell.
For the sake of argument, let us say that we must try 100 different values for
each of the six parameters. (In real situations, the number of trial values is much
larger.) The number of combinations of six parameters, each with 100 possible values is 1006, or 1012. Finding the orientation as a separate search requires
first trying 100 different values for each of three angles, which is 1003 or 106
combinations. After finding the orientation, finding the location requires trying
100 different values of each of three coordinates, again 1003 or 106 combinations.
Section 6.5 Molecular replacement: Related proteins as phasing models 141
Figure 6.18  Rotation search (b–e) finds the values of angles φ, ϕ, and χ (a) that
superimpose the Patterson map of the phasing model (transparent cell with colored Patterson
peaks) on that of the target (black framework with black Patterson peaks). The orientation (e)
that gives the highest correlation between the two Patterson maps gives the best orientation
of the phasing model in the unit cell of the target.
The total number of trials for separate orientation and location searches is 106+106
or 2 × 106. The magnitude of the saving is 1012/2 × 106 or 500,000. In this
case, the problem of finding the orientation and location separately is smaller by
half a million times than the problem of searching for orientation and location
simultaneously.
6.5.5 Monitoring the search
Finally, what mathematical criteria are used in these searches? In other words,
as the computer goes through sets of trial values (angles or coordinates) for
142 Chapter 6 Obtaining Phases
the model, how does it compare results and determine optimum values of the
parameters?
Comparison of orientations is usually done by computing a rotation function,
which evaluates the correlation between Patterson maps for the target protein
and for the phasing model in various orientations. For this orientation search
(often called a rotation search), the computer is looking for large values of the
model Patterson function Pmodel(u, v,w) at locations corresponding to peaks in
the Patterson map of the desired protein. One method is to integrate the product
of the two Patterson maps over the Patterson unit cell for each orientation of the
model Patterson with respect to the target Patterson. Where either Patterson has a
peak and the other does not, the product is zero. Where the two Pattersons have
coincident peaks, the product is large. So the integral of the product will be very
large if there are many coincident peaks in the two maps, and a maximum value at
the relative orientation of maximum overlap. For this type of search, the rotation
function can be expressed as
R(φ, ϕ, χ) =

u,v,w
P target(u, v,w)Pmodel{(u, v,w)× [φ, ϕ, χ]}du dv dw
(6.17)
In words, at each set of rotation angles φ, ϕ, and χ , the value of the rotation
function R is the integral of the product of two Patterson functions: (1) that of
the target molecule [P target(u, v,w)], and (2) that of the model [Pmodel(u, v,w)]
with its coordinates (u, v,w) operated on by rotation matrix2 [φ, ϕ, χ] to produce
a specific orientation relative to the target Patterson. This function will exhibit
maxima where the two Pattersons have many coincident peaks. This maximum
should tell us the best orientation for placing the phasing model in the unit cell
of the desired protein (Fig. 6.18e). Near the maximum, the rotation search can be
repeated at smaller angular intervals to refine the orientation.
For the location or translation search, the criterion is the correspondence between
the expected structure-factor amplitudes from the model in a given trial location
and the actual amplitudes derived from the native data on the desired protein. This
criterion can be expressed as the R-factor, a parameter we will encounter later
as a criterion of improvement of phases in final structure determination. The Rfactor compares overall agreement between the amplitudes of two sets of structure
factors, as follows:
R =
∑ ||Fobs| − |Fcalc||∑ |Fobs| . (6.18)
2The operation of a rotation matrix on a set of coordinates produces a set of simultaneous equations whose solution is the new set of coordinates. For an example of such a set of equations, see
Eq. (11.1), p. 271.
Section 6.6 Iterative improvement of phases (preview of Chapter 7) 143
In words, for each reflection, we compute the difference between the observed
structure-factor amplitude from the native data set |Fobs| and the calculated amplitude from the model in its current trial location |Fcalc| and take the absolute value,
giving the magnitude of the difference. We add these magnitudes for all reflections. Then we divide by the sum of the observed structure-factor amplitudes (the
reflection intensities).
If, on the whole, the observed and calculated intensities agree with each other,
the differences in the numerator are small, and the sum of the differences is small
compared to the sum of the intensities themselves, so R is small. For perfect
agreement, all the differences equal zero, and R equals zero. No single difference
is likely to be larger than the corresponding |Fobs|, so the maximum value of R is
one. For proteins, R-values of 0.3 to 0.4 for the best placement of a phasing model
have often provided adequate initial estimates of phases.
6.5.6 Summary of molecular replacement
If we know that the structure of a new protein is similar to that of a known protein,
we can use the known protein as a phasing model, and thus solve the phase problem
without heavy atom derivatives. If the new crystals and those of the model are
isomorphous, the model phases can be used directly as estimates of the desired
phases. If not, we must somehow superimpose the known protein upon the new
protein to create the best phasing model. We can do this without knowledge of
the structure of the new protein by using Patterson-map comparisons to find the
best orientation of the model protein and then using structure-factor comparisons
to find the best location of the model protein.
6.6 Iterative improvement of phases
(preview of Chapter 7)
The phase problem greatly increases the effort required to obtain an interpretable
electron-density map. In this chapter, I have discussed several methods of obtaining
phases. In all cases, the phases obtained are estimates, and often the set of estimates
is incomplete. Electron-density maps calculated from Eq. (6.7), p. 116, using
measured amplitudes and first phase estimates, are often difficult or impossible to
interpret. In Chapter 7, I will discuss improvement of phase estimates and extension
of phase assignments to as many reflections as possible. As phase improvement and
extension proceed, electron-density maps become clearer and easier to interpret
as an image of a molecular model. The iterative process of structure refinement
eventually leads to a structure that is in good agreement with the original data.
 Chapter 7
Obtaining and Judging the
Molecular Model
7.1 Introduction
In this chapter, I will discuss the final stages of structure determination: obtaining
and improving the electron-density map that is based on the first phase estimates,
interpreting the map to produce an atomic model of the unit-cell contents, and
refining the model to optimize its agreement with the original native reflection
intensities. The criteria by which the crystallographer judges the progress of the
work overlap with criteria for assessing the quality of the final model. These
subjects form the bridge from Chapter 7 to Chapter 8, where I will review many
of the concepts of this book by guiding you through the experimental descriptions
from the description of a structure determination as published in a scientific journal.
With each passing year, crystallography becomes more highly automated, and
the methods of this chapter have probably been the most affected by automation.
In routine structure determination, many of the judgments and choices that I will
describe are now made by software, running sophisticated algorithms that reflect
the collective experience and cleverness of the talented men and women who
propel this challenging field. In many cases, it is possible to go from a first map
based on MIR phases to a model that is more than 90% complete in just a few
minutes of computing that alternates between fitting a model to the map and then
computationally improving the map. But just as it helps to know a little about the
innards of that car that refuses to start, it helps software users to know what is going
on under the hoods of those quiet, sleek, crystallography programs. I will speak
of the decisions inherent in structure refinement as if they were made by a prudent
and discerning human being. Many of the routine decisions can indeed be handed
over to software, and lots of drudgery thus evaded. But there are difficult cases and
145
146 Chapter 7 Obtaining and Judging the Molecular Model
crucial mop-up work that software cannot handle, requiring the crystallographer
to know what the program is trying to do, how it does it, and what the options are
when progress stalls.
7.2 Iterative improvement of maps and
models—overview
In brief, obtaining a detailed molecular model of the unit-cell contents entails
calculating ρ(x, y, z) from Eq. (6.7),
ρ(x, y, z) = 1
V

h
k
l |Fhkl |e−2πi(hx+ky+lz−α′hkl ). (6.7)
using amplitudes (|Fhkl |) computed from measured intensities in the native data
set and phases (α′hkl) computed from heavy-atom data, anomalous scattering, or
molecular replacement. Because the phases are rough estimates, the first map
may be uninformative and disappointing. Crystallographers improve the map by
an iterative process sometimes called bootstrapping. The basic principle of this
iteration is easy to state but demands care, judgment, and much labor to execute.
The principle is the following: any features that can be reliably discerned in, or
inferred from, the map become part of a phasing model for subsequent maps.
Without the input of new information, the map will not improve.
Whatever crude model of unit cell contents that can be discerned in the map is
cast in the form of a simple electron-density function and used to calculate new
structure factors by Eq. (5.16):
Fhkl =

x
y
z ρ(x, y, z)e2πi(hx+ky+lz) dx dy dz, (5.16)
The phases of these structure factors are used, along with the original native intensities, to add more terms to Eq. (6.7), the Fourier-sum description of ρ(x, y, z), in
hopes of producing a clearer map. When the map becomes clear enough to allow
location of atoms, these are added to the model, and structure factors are computed
from this model using Eq. (5.15),
Fhkl =
n∑ j=1
fj e
2πi(hxj+kyj+lzj ). (5.15)
which contains atomic structure factors rather than electron density. As the model
becomes more detailed, the phases computed from it improve, and the model,
computed from the original native structure-factor amplitudes and the latest phases,
Section 7.2 Iterative improvement of maps and models—overview 147
becomes even more detailed. The crystallographer thus tries to bootstrap from the
initial rough phase estimates to phases of high accuracy, and from them, a clear,
interpretable map and a model that fits the map well.
I should emphasize that the crystallographer cannot get any new phase information without modifying the model in some way. The possible modifications
include solvent flattening, noncrystallographic symmetry averaging, or introducing a partial atomic model, all of which are discussed further in this chapter. It is
considered best practice to make sure that initial phases are good enough to make
the map interpretable. If it is not, then the crystallographer needs to find additional
derivatives and collect better data.
The model can be improved in another way: by refinement of the atomic coordinates. This method entails adjusting the atomic coordinates to improve the
agreement between amplitudes calculated from the current model and the original
measured amplitudes in the native data set. In the latter stages of structure determination, the crystallographer alternates between map interpretation and refinement
of the coordinates. At its heart, refinement is an attempt to minimize the differences
between (a) measured diffraction intensities and (b) intensities predicted by the
current model, which, in intermediate stages, is incomplete and harbors errors that
will eventually be removed. Classical refinement algorithms employ the statistical method of least-squares (Sec. 7.5.1, p. 159), but newer methods, including
energy refinement (Sec. 7.5.5, p. 163) and methods based on Bayesian statistics
(Sec. 7.5.6, p. 164), are allowing greater automation, as well as more effective
extraction of structural information in difficult cases.
The block diagram of Fig. 7.1 shows how these various methods ultimately
produce a molecular model that agrees with the native data. The vertical dotted
line in Fig. 7.1 divides the operations into two categories. To the right of the line are
real-space methods, which entail attempts to improve the electron-density map, by
adding information to the map or removing noise from it, or to improve the model,
using the map as a guide. To the left of the line are reciprocal-space methods,
which entail attempts to improve phases or to improve the agreement between
reflection intensities computed from the model and the original measured reflection
intensities. In real-space methods, the criteria for improvement or removal of errors
are found in electron-density maps, in the fit of model to map, or in the adherence
of the model to prior knowledge, such as expected bond lengths and angles (all
real-space criteria); in reciprocal-space methods, the criteria for improvement or
removal of errors involve reliability of phases and agreement of calculated structure
factors with measured intensities (all reciprocal-space criteria). The link between
real and reciprocal space is, of course, the Fourier transform (FT).
I will return to this diagram near the end of the chapter, particularly to amplify the
meaning of error removal, which is indicated by dashed horizontal lines in Fig. 7.1.
For now, I will illustrate the bootstrapping technique for improving phases, map,
and model with an analogy: the method of successive approximations for solving
a complicated algebraic equation. Most mathematics education emphasizes equations that can be solved analytically for specific variables. Many realistic problems
defy such analytic solutions but are amenable to numerical methods. The method
148 Chapter 7 Obtaining and Judging the Molecular Model
Figure 7.1  Block diagram of crystallographic structure determination. Operations
on the left are based on reciprocal space criteria (improving phases), while those on the
right are based in real space (improving the accuracy of atomic coordinates).
of successive approximations has much in common with the iterative process that
extracts a protein model from diffraction data.
Consider the problem of solving the following equation for the variable y:
(
1 + 1
y2 )
(y − 1) = 1. (7.1)
Attempts to simplify the equation produce a cubic equation in y, giving no
straightforward means to a numerical solution. You can, however, easily obtain a
numerical solution for y with a hand calculator. Start by solving for y in terms of
Section 7.3 First maps 149
y2 as follows:
y = 1(
1 + 1
y2 ) + 1. (7.2)
Then make an arbitrary initial estimate of y, say y = 1. (This is analogous
to starting with the MIR phases as initial estimates of the correct phases.) Plug
this estimate into the right-hand y2 term, and calculate y [analogous to computing
a crude structure from measured structure-factor amplitudes (|Fobs|) and phase
estimates]. The result is 1.5. Now take this computed result as the next estimate
(analogous to computing new structure factors from the crude structure), plug it
into the y2 term, and compute y again (analogous to computing a new structure
from better phase estimates). The result is 1.6923. Repeating this process produces
these answers in succession: 1.7412, 1.752, 1.7543, 1.7547, 1.7549, 1.7549, and
so on. After a few iterations, the process converges to a solution; that is, the output
value of y is the same as the input. This value is a solution to the original equation.
With Eq. (7.2), any first estimate above 1.0 (even one million) produces the
result shown. In contrast, for many other equations, the method of successive
approximations works only if the initial estimate is close to a correct solution.
Otherwise, the successive answers do not converge; instead, they may oscillate
among several values (the iteration “hangs up” instead of converging), or they
may continually become larger in magnitude (the iteration “blows up”). In order
for the far more complex crystallographic iteration to converge to a protein model
that is consistent with the diffraction data, initial estimates of many phases must
be close to the correct values. Attempts to start from random phases in hopes of
convergence to correct ones appear doomed to failure because of the large number
of incorrect solutions to which the process can converge.
The following sections describe the crystallographic bootstrapping process in
more detail.
7.3 First maps
7.3.1 Resources for the first map
Entering the final stages of structure determination, the crystallographer is armed
with several sets of data with which to calculate ρ(x, y, z) as a Fourier sum of
structure factors using Eq. (6.7). First is the original native data set, which usually contains the most accurate and complete (highest-resolution) set of measured
intensities. These data will support the most critical tests of the final molecular
model. Next are data sets from heavy-atom derivatives, which are often limited to
lower resolution. Several sets of phases may be available, calculated from heavyatom derivatives and perhaps anomalous dispersion. Because each phase must
be calculated from a heavy-atom reflection, phase estimates are not available for
150 Chapter 7 Obtaining and Judging the Molecular Model
native reflections at higher resolution than that of the best heavy-atom derivative.
Finally, for each set of phases, there is usually some criterion of precision. These
criteria will be used as weighting factors, numbers between 0 and 1, for Fourier
terms containing the phases. A Fourier term containing a phase estimate of low
reliability (see Fig. 6.11) will be multiplied by a low weighting factor in the Fouriersum computation of ρ(x, y, z). In other words, such a term will be multiplied by a
number less than 1.0 to reduce its contribution to the Fourier sum, and thus reduce
bias from a reflection whose phase is questionable. Conversely, a term containing
a phase of high reliability will be given full weight (weighting factor of 1.0) in
the sum.
Here is the Fourier sum that gives the first electron-density map:
ρ(x, y, z) = 1
V

h
k
l whkl |Fobs| e−2πi(hx+ky+lz−α′calc). (7.3)
In words, the desired electron-density function is a Fourier sum in which term
hkl has amplitude |Fobs|, which equals (Ihkl)1/2, the square root of the measured
intensity Ihkl from the native data set. The phase α′hkl of the same term is calculated from heavy-atom, anomalous dispersion, or molecular replacement data, as
described in Chapter 6. The term is weighted by the factor whkl , which will be
near 1.0 if α′hkl is among the most highly reliable phases, or smaller if the phase is
questionable. This Fourier sum is called an Fobs or Fo synthesis (and the map an
Fo map) because the amplitude of each term hkl is |Fobs| for reflection hkl.
The first term in this Fourier series, the F000 term, should contain (I000)1/2,
where I000 is the intensity of reflection 000, which lies at the origin of the reciprocal
lattice. Recall that this reflection is never measured because it is obscured by
the direct beam. Examination of Eq. (7.3) reveals that F000 is a real constant
(as opposed to a complex or imaginary number). The phase α′000 of this term is
assigned a value of zero, with the result that all other phases will be computed
relative to this assignment. Then because h = k = l = 0 for reflection 000, the
exponent of e is zero and the entire exponential term is 1.0. Thus F000 is a constant,
just like f0 in Fig. 2.16, p. 25.
All other terms in the sum are simple trigonometric functions with average
values of zero, so it is clear that the value assigned to F000 will determine the
overall amplitude of the electron-density map. (In the same manner, the f0 term in
Fig. 2.16 displaces all the Fourier sums upward, making the sums positive for all
values of x, like the target function.) The sensible assignment for F000 is therefore
the total number of electrons in the unit cell, making the sum of ρ(x, y, z) over
the whole unit cell equal to the total electron density. In practice, this term can be
omitted from the calculation, and the overall map amplitude can be set by means
described in Sec. 7.3.3, p. 151.
7.3.2 Displaying and examining the map
Until the middle to late 1980s, the contour map of the first calculated electron
density was displayed by printing sections of the unit cell onto Plexiglas or clear
Section 7.3 First maps 151
plastic sheets and stacking them to produce a three-dimensional model, called a
minimap. Today’s computers can display the equivalent of a minimap and allow
much more informative first glimpses of the electron density. With computer displays, the contour level of the map (see p. 94) can be adjusted for maximum detail.
These first glimpses of the molecular image are often attended with great excitement and expectation. If the phase estimates are sufficiently good, the map will
show some of the gross features of unit cell contents. In the rare best cases, with
good phases from molecular replacement, and perhaps with enhancement from
noncrystallographic averaging (explained further in Sec. 7.3.3, p. 151), first maps
are easily interpretable, clearly showing continuous chains of electron density and
features like alpha helices—perhaps even allowing some amino-acid side chains
to be identified. At the worst, the first map is singularly uninformative, signaling
the need for additional phasing information, perhaps from another heavy-atom
derivative. Usually the minimum result that promises a structure from the existing
data is that protein be distinguishable from bulk water, and that dense features like
alpha helices can be recognized. If the boundary of each molecule, the molecular envelope, shows some evidence of recognizable protein structure, then a full
structure is likely to come forth.
I will consider the latter case, in which the first map defines a molecular envelope, with perhaps a little additional detail. If more detail can be discerned, the
crystallographer can jump ahead to later stages of the map-improvement process
I am about to describe. If the molecular envelope cannot be discerned, then the
crystallographer must collect more data.
7.3.3 Improving the map
The crude molecular image seen in the F0 map, which is obtained from the original
indexed intensity data (|Fobs|) and the first phase estimates (α′calc), serves now as
a model of the desired structure. A crude electron density function is devised to
describe the unit-cell contents as well as they can be observed in the first map. Then
the function is modified to make it more realistic in the light of known properties
of proteins and water in crystals. This process is called, depending on the exact
details of procedure, density modification, solvent leveling, solvent flattening, or
solvent flipping.
The electron density function devised by density modification may be no more
than a fixed, high value of ρ(x, y, z) for all regions that appear to be within a
protein molecule, and a fixed, low value of ρ for all surrounding areas of bulk
solvent. One automated method first defines the molecular envelope by dividing
the unit cell into a grid of regularly spaced points. At each point, the value of
ρ(x, y, z) in the F0 map is evaluated. At each grid point, if ρ is negative, it is
reassigned a value of zero; if ρ is positive, it is assigned a value equal to the
average value of ρ within a defined distance of the gridpoint. This procedure
smooths the map (eliminates many small, random fluctuations in density) and
essentially divides the map into two types of regions: those of relatively high (protein) and relatively low (solvent) density. Numerous variations of this method are
in use. In solvent flipping, the density corresponding to solvent is inverted rather
152 Chapter 7 Obtaining and Judging the Molecular Model
than flattened, to enhance contrast between solvent and protein, and to counteract
bias in the model. Some software includes a graphical display that allows the user
to draw a mask to define this boundary. Next, the overall amplitude of the map
is increased until the ratio of high density to low density agrees with the ratio of
protein to solvent in the crystal, usually assuming that the crystal is about half
water. This contrived function ρ(x, y, z) is now used to compute structure factors,
using Eq. (5.16). From this computation, we learn what the amplitudes and phases
of all reflections would be if this very crude new model were correct. We use the
phases from this computation, which constitute a new set of α′hkls, along with
the |Fobs|s derived from the original measured intensities, to calculate ρ(x, y, z)
again, using Eq. (7.3).
We do not throw out old phases immediately, but continue to weight each Fourier
term with some measure of phase quality (sometimes called a figure of merit). In
this manner, we continue to let the data speak for itself as much as possible, rather
than allowing the current model to bias the results. If the new phase estimates
are better, then the new ρ(x, y, z) will be improved, and the electron-density map
will be more detailed. The new map serves to define the molecular boundary more
precisely, and the cycle is repeated. (Refer again to the block diagram in Fig. 7.1,
p. 148.) If we continue to use good judgment in incorporating new phases and new
terms into Eq. (7.3), successive Fourier-series computations of ρ(x, y, z) include
more terms, and successive contour maps become clearer and more interpretable.
In other words, the iterative process of incorporating phases from successively
better and more complete models converges toward a structure that fits the native
data better. The phase estimates “converge” in the sense that the output phases
computed from the current model [Eq. (5.16)] agree better with the input phases
that went into computation of the model [Eq. (7.3)].
As this process continues, and the model becomes more detailed, we begin to get
estimates for the phases of structure factors at resolution beyond that of the heavyatom derivatives. In a process called phase extension, we gradually increase the
number of terms in the Fourier sum of Eq. (7.3), adding terms that contain native
intensities (as |Fobs|) at slightly higher resolution with phases from the current
model. This must be done gradually and judiciously, so as not to let incorrect areas
of the current model bias the calculations excessively. If the new phase estimates
are good, the resulting map has slightly higher resolution, and structure factors
computed from Eq. (5.16) give useful phase estimates at still higher resolution.
In this manner, low-resolution phases are improved, and phase assignments are
extended to higher resolution.
If phase extension seems like getting something from nothing, realize that by
using general knowledge about protein and solvent density, we impose justifiable
restrictions (sometimes referred to as prior knowledge) on the model, giving it
realistic properties that are not visible in the map. In effect, we are using known
crystal properties to increase the resolution of the model. Thus it is not surprising
that the phases calculated from the modified model are good to higher resolution
than those calculated from an electron-density function that does little more than
describe what can be seen in the map.
Section 7.4 The Model becomes molecular 153
Another means of improving the map at this stage depends upon the presence of
noncrystallographic symmetry elements in the unit cell. Recall that the intensity
of reflections results from many molecules in identical orientations diffracting
identically. In a sense, the diffraction pattern is the sum of diffraction patterns
from all individual molecules. This is equivalent to taking a large number of weak,
noisy signals (each the diffraction from one molecule) and adding them together to
produce a strong signal. The noise in the individual signals, which might include
the background intensity of the film or the weak signal of stray X-rays, is random,
and when many weak signals are added, this random noise cancels out.
In some cases, the strength of this signal can be increased further by averaging
the signals from molecules that are identical but have different orientations in the
unit cell, such that no two orientations of the crystal during data collection gives
the same orientation of these molecules in the X-ray beam. These molecules may
be related by symmetry elements that are not aligned with symmetry elements of
the entire unit cell. Thus the diffractive contributions of these identical molecules
are never added together. In such cases, the unit cell is said to exhibit noncrystallographic symmetry (NCS). By knowing the arrangement of molecules in the unit
cell—that is, by knowing the location and type of noncrystallographic symmetry elements—the crystallographer can use a computer to simulate the movement
of these sets of molecules into identical orientations and thus add their signals
together. The result is improved signal-to-noise ratio and, in the end, a clearer
image of the molecules. This method, called symmetry averaging, is spectacularly
successful in systems with high symmetry, such as viruses. Many virus coat proteins are icosahedral, possessing two-, three-, and five-fold rotation axes. Often
one or more two- and threefold axes are noncrystallographic, and fivefold axes are
always noncrystallographic, because no unit cell exhibits fivefold symmetry.
Finding noncrystallographic symmetry elements is another application of rotation searches (Sec. 6.5.4, p. 139, and Fig. 6.18, p. 141) and rotation functions
(Sec. 6.5.5, p. 141, and Eq. (6.17), p. 142) using Patterson maps. In this case, the
target and the model Pattersons are the same, and the rotation function is called a
self-rotation function. In carrying out this rotation search, we are asking whether
the Patterson map of the unit cell is superimposable on itself in a different orientation. The two or more orientations that superimpose the Patterson on itself reveal
the orientations of symmetry axes that relate unit-cell contents, but that do not
apply to the unit cell as a whole. These axes then become the basis of symmetry
averaging.
7.4 The Model becomes molecular
7.4.1 New phases from the molecular model
At some critical point in the iterative improvement of phases, the map becomes
clear enough that we can trace the protein chain through it. In the worst
154 Chapter 7 Obtaining and Judging the Molecular Model
circumstances at this stage, we may only be able to see some continuous tubes of
density. Various aids may be used at this point to help the viewer trace the protein
chain through the map. One is to skeletonize the map, which means to draw line
segments along lines of maximum density. These so-called ridge lines show the
viewer rough lines along which the molecular chain is likely to lie. They can help
to locate both the main chain and the branch points of side chains.
In a clearer map, we may be able to recognize alpha helices, one of the densest
features of a protein, or sheets of beta structure. Now we can construct a partial
molecular model (as opposed to an electron-density model) of the protein, using
computer graphics to build and manipulate a stick model of the known sequence
within small sections of the map. This technique is called map fitting, and is
discussed later. From the resulting model, which may harbor many errors and
undefined regions, we again calculate structure factors, this time using Eq. (5.15),
which treats each atom in the current model as an independent scatterer. In other
words, we calculate new structure factors from our current, usually crude, molecular model rather than from an approximation of ρ(x, y, z). Additional iterations
may improve the map further, allowing more features to be constructed therein.
Here again, as in density modification, we are using prior knowledge—known
properties of proteins—to improve the model beyond what we can actually see in
the map. Thus we are in effect improving the resolution of the model by making
it structurally realistic: giving it local electron densities corresponding to the light
atoms we know are present and connecting atoms at bond lengths and angles that
we know must be correct. So again, our successive models give us phases for
reflections at higher and higher resolution. Electron-density maps computed from
these phases and, as always, the original native amplitudes |Fobs| become more and
more detailed. As the map becomes clearer, even more specific prior knowledge
of protein structure, can be applied. For example, in pleated sheets, successive
carbonyl C=O bonds point in opposite directions. Side chains in pleated sheets
also alternate directions, and are roughly perpendicular to the carbonyls. Near the
ends of pleated sheet strands, if electron density is weakening, this knowledge
of pleated-sheet geometry can often guide the placement and orientation of extra
residues that are not as well-defined by the current map.
7.4.2 Minimizing bias from the model
Conversion to a molecular model greatly increases the hazard of introducing excessive bias from the model into ρ(x, y, z). At this point, bias can be decreased by one
of several alternative Fourier computations of the electron-density map. As phases
from the model begin to be the most reliable, they begin to dominate the Fourier
sum. In the extreme, the series would contain amplitudes purely from the intensity
data and phases purely from the model. In order to compensate for the increased
influence of model phases, and to continue letting the intensity data influence
improvement of the model, the crystallographer calculates electron-density maps
using various difference Fourier syntheses, in which the amplitude of each term is
of the form (|n |Fobs| − |Fcalc||), which reduces overall model influence by subtracting the calculated structure-factor amplitudes (|Fcalc|) from some multiple of
Section 7.4 The Model becomes molecular 155
the observed amplitudes (|Fobs|) within each Fourier term. For n = 1, the Fourier
series is called an Fo − Fc synthesis:
ρ(x, y, z) = 1
V

h
k
l (|Fo| − |Fc|)e−2πi(hx+ky+lz−α′calc). (7.4)
A contour map of this Fourier series is called an Fo − Fc map. How is this map
interpreted? Depending on which of Fo or Fc is larger, Fourier terms can be either
positive or negative. The resulting electron-density map contains both positive
and negative “density.” Positive density in a region of the map implies that the
contribution of the observed intensities (Fos) to ρ are larger than the contribution
of the model (Fcs), and thus that the unit cell (represented by Fos) contains more
electron density in this region than implied by the model (represented by Fcs). In
other words, the map is telling us that the model should be adjusted to increase the
electron density in this region, by moving atoms toward the region. On the other
hand, a region of negative density indicates that the model implies more electron
density in the region than the unit cell actually contains. The region of negative
density is telling us to move atoms away from this region. As an example, if an
amino-acid side chain in the model is in the wrong conformation, the Fo −Fc map
may exhibit a negative peak coincident with the erroneous model side chain and a
nearby positive peak signifying the correct position.
The Fo − Fc map emphasizes errors in the current model. In effect, it removes
the influence of the current model, so that the original data can tell us where the
model is wrong. But the Fo−Fc map lacks the familiar appearance of the molecular
surface found in an Fo map. In addition, if the model still contains many errors,
the Fo − Fc map is “noisy,” full of small positive and negative peaks that are
difficult to interpret. The Fo − Fc map is most useful near the end of the structure
determination, when most of the model errors have been eliminated. The Fo − Fc
map is a great aid in detecting subtle errors after most of the serious errors are
corrected.
A more easily interpreted and intuitively satisfying difference map, but one that
still allows undue influence by the model to be detected, is the 2Fo − Fc map,
calculated as follows
ρ(x, y, z) = 1
V

h
k
l (2 |Fo| − |Fc|)e−2πi(hx+ky+lz−α′calc). (7.5)
In this map, the model influence is reduced, but not as severely as with Fo − Fc.
Unless the model contains extremely serious errors, this map is everywhere positive, and contours at carefully chosen electron densities resemble a molecular
surface. With experience, the crystallographer can often see the bias of an incorrect area of the model superimposed upon the the signal of the correct structure
as implied by the original intensity data. For instance, in a well-refined map (see
Sec. 7.5, p. 145), backbone carbonyl oxygens are found under a distinct rounded
156 Chapter 7 Obtaining and Judging the Molecular Model
bulge in the backbone electron density. If a carbonyl oxygen in the model is pointing 180o away from the actual position in the molecule, the bulge in the map may
be weaker than usual, or misshapen (sometimes cylindrical) and a weak bulge may
be visible on the opposite side of the carbonyl carbon, at the true oxygen position.
Correcting the oxygen orientation in the model, and then recalculating structure
factors, results in loss of the weak, incorrect bulge in the map and intensification
of the bulge in the correct location. This may sound like a serious correction of
the model, requiring the movement of many atoms, but the entire peptide bond
can be flipped 180o around the backbone axis with only slight changes in the positions of neighboring atoms. Such peptide-flip errors are common in early atomic
models.
Various other Fourier syntheses are used during these stages in order to improve
the model. Some crystallographers prefer a 3Fo−2Fc map, a compromise between
Fo−Fc and 2Fo−Fc, for the final interpretation. In areas where the maps continue
to be ambiguous, it is often helpful to examine the original MIR or molecular
replacement maps for insight into how model building in this area might be started
off on a different foot. Another measure is to eliminate the atoms in the questionable
region and calculate structure factors from Eq. (5.15), so that the possible errors
in the region contribute nothing to the phases, and hence do not bias the resulting
map, which is called an omit map.
Rigorous analysis of sources and distributions of errors in atomic coordinates,
intensities, and phases, and of how these introduce bias into maps, has led to the
wide use of figure-of-merit weighted maps, often called sigma-A weighted maps.
The synthesis equivalent to Fo − Fc in this case is mFo − DFc, where m is a
figure of merit for each model phase, and D is an overall estimate of atomiccoordinate errors in the current model. In some treatments of this synthesis, D is
replaced by a the termσA (hence the term sigma-Aweighted), which isD corrected
for completeness of the current model. Comparative tests of σA-weighted maps
versus other types indicate that they are quite powerful at revealing molecular
features that are in conflict with the current model, and such maps have come into
wide use.
Another important type of difference Fourier synthesis, which is used to compare
similar protein structures, is discussed in Sec. 8.3.3, p. 198.
7.4.3 Map fitting
Conversion to a molecular model is usually done as soon as the map reveals recognizable structural features. This procedure, called map fitting or model building,
entails interpreting the electron-density map by building a molecular model that
fits realistically into the molecular surface implied by the map. Map fitting is done
by interactive computer graphics, although automation has eliminated much of
the manual labor of this process. A computer program produces a realistic threedimensional display of small sections of one or more electron-density maps, and
allows the user to construct and manipulate molecular models to fit the map. As
mentioned earlier, such programs can draw ridge lines to help the viewer trace the
chain through areas of weak density. Rapid and impressive advances have been
Section 7.4 The Model becomes molecular 157
made in automated model building, starting from the first electron density maps
that show sufficient detail. As before, I will describe this process as if under manual
control in order to give insights into the types of problems the software is solving.
Some of these decisions are still necessary in areas of the map where the software
is unable to make unequivocal decisions.
As the model is built, the viewer sees the model within the map, as shown in
Fig. 2.3b, p. 11. As the model is constructed or adjusted, the program stores current
atom locations in the form of three-dimensional coordinates. The crystallographer,
while building a model interactively on the computer screen, is actually building
a list of atoms, each with a set of coordinates (x, y, z) to specify its location.
Software updates the atomic coordinates whenever the model is adjusted. This list
of coordinates is the output file from the map-fitting program and the input file
for calculation of new structure factors. When the model is correct and complete,
this file becomes the means by which the model is shared with the community
of scientists who study proteins (Sec. 7.7, p. 173). This list of coordinates is the
desired model, in a form that is convenient for further study and analysis.
In addition to routine commands for inserting or changing amino-acid residues,
moving atoms and fragments, and changing conformations, map-fitting programs
contain many sophisticated tools to aid the model builder. Fragments, treated as
rigid assemblies of atoms, can be automatically fitted to the map by the method
of least squares (see Sec. 7.5.1, p. 159). After manual adjustments of the model,
which may result in unrealistic bond lengths and angles, portions of the model
can be regularized, which entails automatic correction of bond lengths and angles
with minimal movement of atoms. In effect, regularization looks for the most
realistic configuration of the model that is very similar to its current configuration.
Where small segments of the known sequence cannot be easily fitted to the map,
some map-fitting programs can search fragment databases or the Protein Data Bank
(see Sec. 7.7, p. 173) for fragments having the same sequence, and then display
these fragments so the user can see whether they fit the map.
Following is a somewhat idealized description of how map fitting may proceed,
illustrated with views from a modern map-fitting program. The maps and models
are from the structure determination of adipocyte lipid binding protein (ALBP),
which I will discuss further in Chapter 8.
When the map has been improved to the point that molecular features are
revealed, the crystallographer attempts to trace the protein through as much continuous density as possible. At this point, the quality of the map will vary from
place to place, being perhaps quite clear in the molecular interior, which is usually
more ordered, and exhibiting broken density in some places, particularly at chain
termini and surface loops. Because we know that amino-acid side chains branch
regularly off alpha carbons in the main chain, we can estimate the positions of
many alpha carbons. These atoms should lie near the center of the main-chain
density next to bulges that represent side chains. In proteins, alpha carbons are
3.8–4.2 Å apart. This knowledge allows the crystallographer to construct an alphacarbon model of the molecule (Fig. 7.2) and to compute structure factors from this
model.
158 Chapter 7 Obtaining and Judging the Molecular Model
Figure 7.2  Alpha-carbon model (stereo) of ALBP built into electron-density map
of Fig. 2.3a, p. 11.
Further improvement of the map with these phases may reveal side chains more
clearly. Now the trick is to identify some specific side chains so that the known
amino-acid sequence of the protein can be aligned with visible features in the map.
As mentioned earlier, chain termini are often ill-defined, so we need a foothold for
alignment of sequence with map where the map is sharp. Many times the key is a
short stretch of sequence containing several bulky hydrophobic residues, like Trp,
Phe, and Tyr. Because they are hydrophobic, they are likely to be in the interior
where the map is clearer. Because they are bulky, their side-chain density is more
likely to be identifiable. From such a foothold, the detailed model building can
begin.
Regions that cannot be aligned with sequence are often built with polyalanine,
reflecting our knowledge that all amino acids contain the same backbone atoms,
and all but one, glycine, have at least a beta carbon (Fig. 7.3). In this manner,
Figure 7.3  Polyalanine model (stereo) of ALBP built into electron-density map of
Fig. 2.3.
Section 7.5 Structure refinement 159
we build as many atoms into the model as possible in the face of our ignorance
about how to align the sequence with the map in certain areas.
In pleated sheets, we know that successive carbonyl oxygens point in opposite
directions. One or two carbonyls whose orientations are clearly revealed by the
map can allow sensible guesses as to the positions of others within the same sheet.
As mentioned previously with respect to map fitting, we use prior knowledge
of protein structure to infer more than the map shows us. If our inferences are
correct, subsequent maps, computed with phases calculated from the model, will
show enhanced evidence for the inferred features and will show additional features
as well, leading to further improvement of the model. Poor inferences degrade the
map, so where electron density conflicts with intuition, we follow the density as
closely as possible.
With each successive map, new molecular features are added as soon as they
can be discerned, and errors in the model, such as side-chain conformations that no
longer fit the electron density, are corrected. As the structure nears completion, the
crystallographer may simultaneously use maps based on various Fourier syntheses
in order to track down the most subtle disagreements between the model and
the data.
7.5 Structure refinement
7.5.1 Least-squares methods
Cycles of map calculation and model building, which are forms of realspace refinement of the model, are interspersed with computerized attempts to
improve the agreement of the model with the original intensity data. (Everything
goes back to those original reflection intensities, which give us our |Fobs| values.)
Because these computations entail comparison of computed with observed structure factor amplitudes (reciprocal space), rather than examination of maps and
models (real space), these methods are referred to as reciprocal space refinement.
The earliest successful refinement technique was a massive version of least-squares
fitting, the same procedure that freshman chemistry students employ to construct
a straight line that fits a scatter graph of data. More recently other versions, energy
refinement and methods based on Bayesian statistics, have come to the fore. I
will discuss least-squares methods first, and then contrast them with more modern
methods.
In the simple least-squares method for functions of one variable, the aim is to find
a function y = f (x) that fits a series of observations (x1, y1), (x2, y2), . . . (xi, yi),
where each observation is a data point—a measured value of the independent
variable x at some selected value y. (For example, y might be the temperature
of a gas, and x might be its measured pressure.) The solution to the problem is
a function f (x) for which the sum of the squares of distances between the data
points and the function itself is as small as possible. In other words, f (x) is the
160 Chapter 7 Obtaining and Judging the Molecular Model
function that minimizes D, the sum of the squared differences between observed
(yi) and calculated f (xi) values, as follows:
D =

i wi [yi − f (xi)]2 . (7.6)
The differences are squared to make them all positive; otherwise, for a large
number of random differences, D would simply equal zero. The term wi is an
optional weighting factor that reflects the reliability of observation i, thus giving
greater influence to the most reliable data. According to principles of statistics,
wi should be 1/(σ i)2, where σ i is the standard deviation computed from multiple
measurements of the same data point (xi, yi). In the simplest case, f (x) is a straight
line, for which the general equation is f (x) = mx+b, where m is the slope of the
line and b is the intercept of the line on the f (x)-axis. Solving this problem entails
finding the proper values of the parameters m and b. If we substitute (mi + b)
for each f (xi) in Eq. (7.6), take the partial derivative of the right-hand side with
respect to m and set it equal to zero, and then take the partial derivative with
respect to b and set it equal to zero, the result is a set of simultaneous equations in
m and b. Because all the squared differences are to be minimized simultaneously,
the number of equations equals the number of observations, and there must be
at least two observations to fix values for the two parameters m and b. With just
two observations (x1, y1) and (x2, y2), m and b are determined precisely, and f (x)
is the equation of the straight line between (x1, y1) and (x2, y2). If there are more
than two observations, the problem is overdetermined and the values of m and
b describe the straight line of best fit to all the observations. So the solution to
this simple least-squares problem is a pair of parameters m and b for which the
function f (x) = mx + b minimizes D.
7.5.2 Crystallographic refinement by least squares
In the crystallographic case, the parameters we seek (analogous to m and b) are,
for all atoms j , the positions (xj , yj , zj ) that best fit the observed structure-factor
amplitudes. Because the positions of atoms in the current model can be used to
calculate structure factors, and hence to compute the expected structure-factor
amplitudes (|Fcalc|) for the current model, we want to find a set of atom positions that give |Fcalc|s, analogous to calculated values f (xi), that are as close as
possible to the |Fobs|s (analogous to observed values yi). In least-squares terminology, we want to select atom positions that minimize the squares of differences
between corresponding |Fcalc|s and |Fobs|s. We define the difference between the
observed amplitude |Fobs| and the measured amplitude |Fcalc| for reflection hkl as
(|Fo| − |Fc|)hkl , and we seek to minimize the function , where
 =

hkl
whkl (|Fo| − |Fc|)2hkl . (7.7)
Section 7.5 Structure refinement 161
In words, the function  is the sum of the squares of differences between
observed and calculated amplitudes. The sum is taken over all reflections hkl currently in use. Each difference is weighted by the term whkl , a number that depends
on the reliability of the corresponding measured intensity. As in the simple example, according to principles of statistics, the weight should be 1/(σhkl)2, where
σ is the standard deviation from multiple measurements of |Fobs|. Because the
data do not usually contain enough measurements of each reflection to determine
its standard deviation, other weighting schemes have been devised. Starting from
a reasonable model, the least-squares refinement method succeeds about equally
well with a variety of weighting systems, so I will not discuss them further.
7.5.3 Additional refinement parameters
We seek a set of parameters that minimize the function. These parameters include
the atom positions, of course, because the atom positions in the model determine
each Fcalc. But other parameters are included as well. One is the temperature
factor Bj , or B-factor, of each atom j , a measure of how much the atom oscillates
around the position specified in the model.Atoms at side-chain termini are expected
to exhibit more freedom of movement than main-chain atoms, and this movement
amounts to spreading each atom over a small region of space. Diffraction is affected
by this variation in atomic position, so it is realistic to assign a temperature factor
to each atom and include the factor among parameters to vary in minimizing .
From the temperature factors computed during refinement, we learn which atoms
in the molecule have the most freedom of movement, and we gain some insight
into the dynamics of our largely static model. In addition, adding the effects of
motion to our model makes it more realistic and hence more likely to fit the data
precisely.
Another parameter included in refinement is the occupancy nj of each atom j ,
a measure of the fraction of molecules in which atom j actually occupies the
position specified in the model. If all molecules in the crystal are precisely identical,
then occupancies for all atoms are 1.00. Occupancy is included among refinement
parameters because occasionally two or more distinct conformations are observed
for a small region like a surface side chain. The model might refine better if
atoms in this region are assigned occupancies equal to the fraction of side chains
in each conformation. For example, if the two conformations occur with equal
frequency, then atoms involved receive occupancies of 0.5 in each of their two
possible positions. By including occupancies among the refinement parameters,
we obtain estimates of the frequency of alternative conformations, giving some
additional information about the dynamics of the protein molecule. The factor |Fc|
in Eq. (7.7) can be expanded to show all the parameters included in refinement, as
follows:
Fc = G ·

j njfj e
2πi(hxj+kyj+lzj ) · e−Bj [(sin θ)/λ]2 . (7.8)
162 Chapter 7 Obtaining and Judging the Molecular Model
Although this equation is rather forbidding, it is actually the familiar Eq. (5.15)
with the new parameters included. Equation (7.8) says that structure factor Fhkl
can be calculated (Fc) as a Fourier sum containing one term for each atom j in the
current model. G is an overall scale factor to put all Fcs on a convenient numerical
scale. In the j th term, which describes the diffractive contribution of atom j to this
particular structure factor, nj is the occupancy of atom j ; fj is its scattering factor,
just as in Eq. (5.15); xj , yj , and zj are its coordinates; and Bj is its temperature
factor. The first exponential term is the familiar Fourier description of a simple
three-dimensional wave with frequencies h, k, and l in the directions x, y, and z.
The second exponential shows that the effect of Bj on the structure factor depends
on the angle of the reflection [(sin θ)/λ].
7.5.4 Local minima and radius of convergence
As you can imagine, finding parameters (atomic coordinates, occupancies, and
temperature factors for all atoms in the model) to minimize the differences between
all the observed and calculated structure factors is a massive computing task. As in
the simple example, one way to solve this problem is to differentiatewith respect
to all the parameters, which gives simultaneous equations with the parameters as
unknowns. The number of equations equals the number of observations, in this case
the number of measured reflection intensities in the native data set. The parameters
are overdetermined only if the number of measured reflections is greater than the
number of parameters to be obtained. The complexity of the equations rules out
analytical solutions and requires iterative (successive-approximations) methods
that we hope will converge from the starting parameters of our current model to a
set of new parameters corresponding to a minimum in . It has been proved that
the atom positions that minimize  are the same as those found from Eq. (7.3), the
Fourier-series description of electron density. So real-space and reciprocal-space
methods converge to the same solution.
The complicated function  undoubtedly exhibits many local minima, corresponding to variations in model conformation that minimize  with respect
to other quite similar (“neighboring”) conformations. A least-squares procedure
will find the minimum that is nearest the starting point, so it is important that
the starting model parameters be near the global minimum, the one conformation that gives best agreement with the native structure factors. Otherwise the
refinement will converge into an incorrect local minimum from which it cannot
extract itself. The greatest distance from the global minimum from which refinement will converge properly is called the radius of convergence. The theoretically
derived radius is dmin/4, where dmin is the lattice-plane spacing corresponding to
the reflection of highest resolution used in the refinement. Inclusion of data from
higher resolution, while potentially giving more information, decreases the radius
of convergence, so the model must be ever closer to its global minimum as more
data are included in refinement. There are a number of approaches to increasing
the radius of convergence, and thus increasing the probability of finding the global
minimum.
Section 7.5 Structure refinement 163
These approaches take the form of additional constraints and restraints on the
model during refinement computations. A constraint is a fixed value for a certain
parameter. For example, in early stages of refinement, we might constrain all
occupancies to a value of 1.0. A restraint is a subsidiary condition imposed upon
the parameters, such as the condition that all bond lengths and bond angles be
within a specified range of values. The function , with additional restraints on
bond lengths and angles, is as follows:
 =

hkl
whkl(|Fo| − |Fc|)2hkl
+
bonds∑
i wi(d
ideal
i − dmodeli )2
+

j wj (φ
ideal
j − φmodelj )2,
(7.9)
where di is the length of bond i, and φj is the bond angle at location j . Ideal values
are average values for bond lengths and angles in small organic molecules, and
model values are taken from the current model. In minimizing this more complicated , we are seeking atom positions, temperature factors, and occupancies that
simultaneously minimize differences between (1) observed and calculated reflection amplitudes, (2) model bond lengths and ideal bond lengths, and (3) model
bond angles and ideal bond angles. In effect, the restraints penalize adjustments
to parameters if the adjustments make the model less realistic.
7.5.5 Molecular energy and motion in refinement
Crystallographers can take advantage of the prodigious power of today’s computers to include knowledge of molecular energy and molecular motion in the
refinement. In energy refinement, least-squares restraints are placed upon the overall energy of the model, including bond, angle, and conformational energies and
the energies of noncovalent interactions such as hydrogen bonds. Adding these
restraints is an attempt to find the structure of lowest energy in the neighborhood
of the current model. In effect, these restraints penalize adjustments to parameters
if the adjustments increase the calculated energy of the model.
Another form of refinement employs molecular dynamics, which is an attempt
to simulate the movement of molecules by solving Newton’s laws of motion for
atoms moving within force fields that represent the effects of covalent and noncovalent bonding. Molecular dynamics can be turned into a tool for crystallographic
refinement by including an energy term that is related to the difference between
the measured reflection intensities and the intensities calculated from the model.
In effect, this approach treats the model as if its energy decreases as its fit to the
native crystallographic data improves. In refinement by simulated annealing, the
model is allowed to move as if at high temperature, in hopes of lifting it out of local
164 Chapter 7 Obtaining and Judging the Molecular Model
energy minima. Then the model is cooled slowly to find its preferred conformation at the temperature of diffraction data collection. All the while, the computer
is searching for the conformation of lowest energy, with the assigned energy partially dependent upon agreement with diffraction data. In some cases, the radius
of convergence is greatly increased by this process, a form of molecular dynamics
refinement.
7.5.6 Bayesian methods: Ensembles of models
Lurking in all the preceding discussions of phasing and refinement has been a tacit
but questionable assumption. It is the belief that, at every stage in phasing, it is
possible to choose the one best model at the moment, and that it will improve
by a linear path through other best-models-at-the-moment, and eventually lead to
the best possible model. We make this assumption when we choose a working
model and its accompanying set of phases, using procedures like isomorphous
replacement, anomalous dispersion, solvent flattening, molecular replacement,
and noncrystallographic symmetry. In fact, at each of these stages, there is uncertainty in the choice of models—in the exact values of phases from anomalous
dispersion, the exact position of solvent masks, or the exact rotation and translation
for a phasing model in molecular replacement. But typically, the crystallographer or the software makes a single choice and proceeds. As you know, the
model can bias the subsequent refinement process, at the worst sending it into
local minima from which it cannot extricate itself. The gradual release of constraints during least-squares refinement is one means of evading local minima.
Weighting structure factors according to the uncertainty in their intensities and
associated phases is a means of weakening the model’s power to bias the refinement. Finally, simulated annealing is a means of repeatedly getting out of such
minima.
A very powerful way to avoid bias is to avoid the best-model-at-the-moment
assumption entirely, and instead of refining the model, refine instead a representative sample of all models that are compatible with the current level of ambiguity
in the data. The most ambitious movement for permeating crystallography with
this kind of thinking is called the Bayesian program, because it is based on
Bayes’s theorem, which is the foundation of a general and well-established statistical method for decision making in the face of incomplete information. In
essence, Bayesian thinking expresses current knowledge about a range of possible hypotheses in the form of prior probabilities. Observation or data collection
leads to a new state of knowledge described by posterior probabilities. In an
iterative process, models with higher prior probability are permuted and confronted with the data to raise prior probabilities further. In its broadest form,
the Bayesian program in crystallography is an attempt to combine the brawn of
direct phasing methods in small-molecule crystallography with the brains of prior
knowledge in macromolecular crystallography. The idea that unifies these two
realms is a common means of assessing the correctness of their models. Bringing these two forms of inference together promises to extend to macromolecular
Section 7.5 Structure refinement 165
crystallography the kind of automation that has routinized much of small-molecule
crystallography.
First, here is a simple example of how Bayesian inference can alter our assessment of possible outcomes. Suppose Dorothy and Max are running for office.
Knowing no reason that voters might strongly prefer either candidate (prior knowledge), we assign a prior probability of 0.5 to the election of either. Then we
conduct a survey, asking ten randomly chosen voters how they will vote. We find
that seven plan to vote for Dorothy. Bayesian inference allows us to compute the
probability of Dorothy’s success in the light of this survey—the posterior probability of Dorothy’s election. The calculated prior probability is almost 0.9 (for
calculation, see “Bayesian Inference” at the CMCC home page). In this case, a
small amount of data greatly alters our assessment of Dorothy’s chances.
Facing the ambiguities typical of many stages of crystallography, Bayesian
thinking about improving the current model entails iteration of the following steps:
1. Generate an ensemble of models that form a representative sample of all
models that are compatible with the current level of ambiguity about the
structure factors. Each model, referred to as an hypothesis (HJ ), is assigned a
prior probability [Pprior(HJ )] based on knowledge from outside the diffraction measurements. For example, at the stage of solvent flattening, such
knowledge might include the percentages of protein and water in the crystal, and the known average densities of protein and bulk solvent. We were
assigning a prior probability when we assumed that Dorothy and Max had
equal chances of winning the election.
2. Compute a probability distribution of observed structure factors expected
from each model [P(F | HJ )], and remove the phases to obtain a probability distribution of diffraction amplitudes expected from each model
[P(|F| | HJ )]. (Read the single vertical line | as “given that” or just “given,”
so that [P(|F| | HJ )] says “probability distribution P of a set of structure
factor amplitudes |F| given hypothesis HJ .”) This entails recognizing that
structure factors are not independent of each other, but exhibit characteristic distributions in their intensities. (A familiar example of a probability
distribution is the “bell curve” or normal distribution of a large set of exam
scores.) The shapes of distributions of structure factor amplitudes constitute a signal that can be used in assessing the probability of a model’s
correctness.
3. Compute, by comparing the probability distribution of amplitudes given
each model with the probability distribution of amplitudes in the actual data
[that is, comparing P(|F| | HJ ) with P(|Fobs|)], the likelihood (HJ ) of
each model given the observed data. Likelihoods are related to probabilities
as follows:
(HJ | |Fobs|) = P(|Fobs| | HJ ). (7.10)
166 Chapter 7 Obtaining and Judging the Molecular Model
4. Use Bayes’s theorem to compute a posterior probability for each model in
the light of the data. Bayes’s theorem formulated for this situation is
Ppost(HJ | |Fobs|) = Pprior(HJ ) ·(HJ | |Fobs|)∑
K Pprior(HK) ·(HK | |Fobs|)
. (7.11)
In words, the posterior probability of hypothesisHJ given the data (observed
structure factors) equals its prior probability times its likelihood given the
data, divided by the sum of all such P ·  terms for all hypotheses. The
denominator assures that the probabilities for all hypotheses add up to 1.00.
The posterior probability of a model tells how it stacks up against the other
models in their ability to fit the observed data.
5. Use the models of highest posterior probability as the basis for generating
a new ensemble of models of higher likelihood, and use these models in
step one to begin the next interation. Interating this process can retrieve, in
the calculation of structure factors from each hypothesis, missing information, such as uncertain phases or even intensities that are missing or poorly
measured.
The founder of the Bayesian program, Gerard Bricogne, has likened this iterative
process to the molecular biologist’s tool of phage display, a method of “evolving” proteins with specific ligand-binding affinities (Fig. 7.4). The method entails
introducing genes into bacteriophages in such a way that expression of the genes
results in display of the gene product on the phage surface. This makes it possible to use ligand-affinity chromatography to separate phages that express (and
contain) the gene from those that do not. To evolve proteins of higher binding
affinity, the molecular biologist generates diversity by producing large numbers
of mutated forms of the desired gene, expresses them in phages to produce populations displaying many mutated forms of the protein, and selects the best binders
by ligand-affinity chromatography. This process can be interated by generating
mutational diversity within populations of the best binders from the previous generation. By analogy, Bricogne refers to the Bayesian method as “phase display,”
which generates a diverse population of hypothetical models that are compatible
with the current state of ambiguity, typically by permuting phases (analogous to
mutations); “expresses” these hypotheses by converting them into probability distributions of observable diffraction intensities, and finally, selects by their ability
to “bind to the data”—that is, to agree with the experimental data accurately.
How do Bayesian crystallographers tell how they are doing? Recall that, in
least-squares refinement (p. 159), the refinement target is minimization of differences between (a) measured intensities and (b) intensities calculated from the
current model. In the searches typical of molecular replacement (rotation and
translation searches) and noncrystallographic symmetry averaging (self-rotation),
the refinement target is maximization of the correlation between Patterson maps.
Section 7.5 Structure refinement 167
Figure 7.4  Phage display analogy of Bayesian approach to crystallography. Both
methods entail generation of diversity, followed by expression of diversity in a form that is
subject to selection of best results.
168 Chapter 7 Obtaining and Judging the Molecular Model
In applications of Bayesian methods, by contrast, the refinement target is maximization of either likelihood (when data are plentiful) or posterior probability
(a more complex task, required when data are more scarce). An example of a
likelihood function is log-likelihood gain, expressed as
LLG(HJ ) = log (HJ )
(H0)
, (7.12)
which is a measure of the likelihood of a model (HJ ) in comparison to the
likelihood of some null hypothesis (H0). A typical null hypothesis might be
a model in which the atoms are distributed in a physically realistic but random
manner within the unit cell. As knowledge of phases improves, the surviving
models are of higher and higher likelihood.
In essence, Bayesian methods provide rigorous organization, bookkeeping, and
quality control for all the models that remain compatible with the current state of
knowledge as that knowledge grows. In this respect, the methods formalize the
many possibilities that prior knowledge keeps alive in the crystallographer’s mind
during structure determination. At the same time, these methods “try everything”
that remains compatible with the current state of knowledge, reflecting common
ground with direct phasing methods.
If least-squares methods are computationally intensive, then certainly refinement methods involving large numbers of hypothetical data sets and coordinate
sets are much more so. But what is computationally intensive today typically
becomes routine in just a few years—computers get faster and programmers get
cleverer. Each year, more of the demands of the Bayesian program can be accomodated by readily available computing. These methods are sure to be lurking in the
black boxes of software that ultimately automate more and more of crystallographic
structure determination.
7.6 Convergence to a final model
7.6.1 Producing the final map and model
In the last stages of structure determination, the crystallographer alternates computed, reciprocal-space refinement with map fitting, or real-space refinement. The
most powerful software packages can start with the first map that shows any sort of
structural detail, and cycle between automated map fitting and refinement cycles
until most of the model is complete. In an approach that borrows from the Shake
and Bake method, one type of program starts from atoms randomly placed in
density, minimizing initial biases in the atomic model. At the end of refinement,
such programs point the user to residues that are still uncertain, allowing manual
intervention to fix the last details. So in the best cases, manual model building
is only needed in a few problem areas. In general, whether the model building is
Section 7.6 Convergence to a final model 169
manual or automated, constraints and restraints are lifted as refinement proceeds,
so that agreement with the original reflection intensities is gradually given highest
priority. When ordered water becomes discernible in the map, water molecules
are added to the model, and occupancies are no longer constrained, to reflect the
possibility that a particular water site may be occupied in only a fraction of unit
cells. Early in refinement, all temperature factors are assigned a starting value.
Later, the value is held the same for all atoms or for groups of similar atoms (like
all backbone atoms as one group and all side-chain atoms as a separate group), but
the overall value is not constrained. Finally, individual atomic temperature factors
are allowed to refine independently. Early in refinement, the whole model is held
rigid to refine its position in the unit cell. Then blocks of the model are held rigid
while their positions refine with respect to each other. In the end, individual atoms
are freed to refine with only stereochemical restraints. This gradual release of the
model to refine against the original data is an attempt to prevent it from getting
stuck in local minima. Choosing when to relax specific constraints and restraints
was once considered to be more art than science, but many of today’s crystallographic software packages can now negotiate this terrain and greatly reduce the
manual labor of refinement.
Near the end of refinement, the Fo − Fc map becomes rather empty except in
problem areas, and sigma-A weighted maps show good agreement with the model
and no empty density. Map fitting becomes a matter of searching for and correcting
errors in the model, which amounts to extricating the model from local minima
in the reciprocal-space refinement. Wherever model atoms lie outside 2Fo − Fc
contours, the Fo − Fc map will often show the atoms within negative contours,
with nearby positive contours pointing to correct locations for these atoms. Many
crystalline proteins possess disordered regions, where the maps do not clear up
and become unambiguously interpretable. Such regions of structural uncertainty
are simply omitted from the model, and this omission is mentioned in published
papers on the structure and in the header information of Protein Data Bank files
(see Sec. 7.7, p. 173).
At the end of successful refinement, the 2Fo − Fc map almost looks like a
space-filling model of the protein. (Refer to Fig. 2.3b, p. 11, which is the final
model built into the same region shown in Fig. 7.2, p. 158 and Fig. 7.3, p. 158.)
The backbone electron density is continuous, and peptide carbonyl oxygens are
clearly marked by bulges in the backbone density. Side-chain density, especially
in the interior, is sharp and fits the model snugly. Branched side chains, like those
of valine, exhibit distinct lobes of density representing the two branches. Rings
of histidine, phenylalanine, tyrosine, and tryptophan are flat, and in models of the
highest resolution, aromatic rings show a clear depression or hole in the density
at their centers. Looking at the final model in the final map, you can easily underestimate the difficulty of interpreting the early maps, in which backbone density
is frequently weak and broken, and side chains are missing or shapeless.
You can get a rough idea of how refinement gradually reveals features of the
molecule by comparing electron-density maps computed at low, medium, and
high resolution, as in Fig. 7.5. Each photo in this set shows a section of the final
170 Chapter 7 Obtaining and Judging the Molecular Model
Figure 7.5  Electron-density maps at increasing resolution (stereo). Maps were calculated using final phases, and Fourier sums were truncated at the the following resolution
limits: (a) 6.0 A; (b) 4.5 A; (c) 3.0 A; (d) 1.6 A.
Section 7.6 Convergence to a final model 171
ALBP model in a map calculated with the final phases, but with |Fobs|s limited
to specified resolution. In (a), only |Fobs|s of reflections at resolution 6 Å or
greater are used. With this limit on the data (which amounts to including in the
2Fo − Fc Fourier sum only those reflections whose indices hkl correspond to sets
of planes with spacing dhkl of 6 Å or greater), the map of this pleated-sheet region
of the protein is no more than a featureless sandwich of electron density. As we
extend the Fourier sum to include reflections out to 4.5 Å, the map (b) shows
distinct, but not always continuous, tubes of density for each chain. Extending
the resolution to 3.0 Å, we see density that defines the final model reasonably
well, including bulges for carbonyl oxygens (red) and for side chains. Finally, at
1.6 Å, the map fits the model like a glove, zigzagging precisely in unison with the
backbone of the model and showing well-defined lobes for individual side-chain
atoms.
Look again at the block diagram of Fig. 7.1, p. 148, which gives an overview
of structure determination. Now I can be more specific about the criteria for error
removal or filtering, which is shown in the diagram as horizontal dashed lines in
real and reciprocal space. Real-space filtering of the map entails removing noise or
adding density information, as in solvent flattening or flipping. Reciprocal-space
filtering of phases entails using only the strongest reflections (for which phases are
more accurate) to compute the early maps, and using figures of merit and phase
probabilities to select the most reliable phases at each stage. The molecular model
can be filtered in either real or reciprocal space. Errors are removed in real space by
improving the fit of model to map, and by allowing only realistic bond lengths and
angles when adjusting the model (regularization). Here the criteria are structural
parameters and congruence to the map (real space). Model errors are removed in
reciprocal space (curved arrow in center) by least-squares refinement, which entails
adjusting atom positions in order to bring calculated intensities into agreement with
measured intensities. Here the criteria are comparative structure-factor amplitudes
(reciprocal space). Using the Fourier transform, the crystallographer moves back
and forth between real and reciprocal space to nurse the model into congruence
with the data.
7.6.2 Guides to convergence
Judging convergence and assessing model quality are overlapping tasks. I will
discuss criteria of convergence here. In Chapter 8, I will discuss some of the
criteria further, particularly as they relate to the quality and usefulness of the final
model.
The progress of iterative real- and reciprocal-space refinement is monitored by
comparing the measured structure-factor amplitudes |Fobs| (which are proportional
to (Iobs)1/2) with amplitudes |Fcalc| from the current model. In calculating the new
phases at each stage, we learn what intensities our current model, if correct, would
yield. As we converge to the correct structure, the measured Fs and the calculated
Fs should also converge. The most widely used measure of convergence is the
172 Chapter 7 Obtaining and Judging the Molecular Model
residual index, or R-factor (Sec. 6.5.5, p. 141):
R =
∑ ||Fobs| − |Fcalc||∑ |Fobs| . (7.13)
In this expression, each |Fobs| is derived from a measured reflection intensity
and each |Fcalc| is the amplitude of the corresponding structure factor calculated
from the current model. Values of R range from zero, for perfect agreement of
calculated and observed intensities, to about 0.6, the R-factor obtained when a
set of measured amplitudes is compared with a set of random amplitudes. An
R-factor greater than 0.5 implies that agreement between observed and calculated
intensities is very poor, and many models with R of 0.5 or greater will not respond
to attempts at improvement unless more data are available. An early model with R
near 0.4 is promising and is likely to improve with the various refinement methods
I have presented. A desirable target R-factor for a protein model refined with data
to 2.5 Å is 0.2. Very rarely, small, well-ordered proteins may refine to R-values
as low as 0.1, whereas small organic molecules commonly refine to R values
below 0.05.
Amore demanding and revealing criterion of model quality and of improvements
during refinement is the free R-factor, Rfree. Rfree is computed with a small set of
randomly chosen intensities, the “test set,” which are set aside from the beginning
and not used during refinement. They are used only in cross-validation, a quality
control process that entails assessing the agreement between calculated (from the
model) and observed data. At any stage in refinement, Rfree measures how well
the current atomic model predicts a subset of the measured intensities that were
not included in the refinement, whereas R measures how well the current model
predicts the entire data set that produced the model. You can see a sort of circularity
in R that is avoided in Rfree. Many crystallographers believe that Rfree gives a
better and less-biased measure of overall model. In many test calculations, Rfree
correlates very well with phase accuracy of the atomic model. In general, during
intermediate stages of refinement, Rfree values are higher than R, but in the final
stages, the two often become more similar. Because incompleteness of data can
make structure determination more difficult (and perhaps because the lower values
of R are somewhat seductive in stages where encouragement is welcome), some
crystallographers at first resisted using Rfree. But most now use both Rs to guide
them in refinement, looking for refinement procedures that improve both Rs, and
proceeding with great caution when the two criteria appear to be in conflict. In
Bayesian methods, Rfree is replaced by free log-likelihood gain, Lfree, calculated
over the same test data set as Rfree.
In addition to monitoring R- or L- factors as indicators of convergence, the
crystallographer monitors various structural parameters that indicate whether
the model is chemically, stereochemically, and conformationally reasonable. In
a chemically reasonable model, the bond lengths and bond angles fall near the
expected values for simple organic molecules. The usual criteria applied are the
root-mean-square (rms) deviations of all the model’s bond lengths and angles from
Section 7.7 Sharing the model 173
an accepted set of values. A well-refined model exhibits rms deviations of no more
than 0.02 Å for bond lengths and 4o for bond angles. Bear in mind, however, that
refinement restrains bond lengths and bond angles, making them less informative
indicators of model quality than structural parameters that are allowed to refine
free of restraints.
A stereochemically reasonable model has no inverted centers of chirality (for
instance, no D-amino acids). A conformationally reasonable model meets several criteria. (1) Peptide bonds are nearly planar, and nonproline peptide bonds
are trans-, except where obvious local conformational constraints produce an
occasional cis-peptide bond, which is usually followed by proline. (2) The backbone conformational angles  and  fall in allowed ranges, as judged from
Ramachandran plots of these angles (see Chapter 8). And finally, (3) torsional
angles at single bonds in side chains lie within a few degrees of stable, staggered
conformations. During the progress of refinement, all of these structural parameters
should continually improve. And unlike bond lengths and bond angles, conformational angles are usually not restrained during refinement, so these parameters are
better indicators of model quality, as I will discuss in Chapter 8.
In addition to the guides to convergence described in this section, the model
validation tools described in Section 8.2.5 can also be used to find potential model
errors at the end of refinement. Error correction at this point followed by additional refinement cycles can insure that the published model is as error-free as
possible. There is wide variation in the extent to which these additional validation
tools are applied.
7.7 Sharing the model
If the molecule under study plays an important biological or medical function, an
intensely interested audience awaits the crystallographer’s final molecular model.
The audience includes researchers studying the same molecule by other methods,
such as spectroscopy or kinetics, or studying metabolic pathways or diseases in
which the molecule is involved. The model may serve as a basis for understanding
the properties of the protein and its behavior in biological systems. It may also
serve as a guide to the design of inhibitors or to engineering efforts to modify its
function by methods of molecular biology. But in the case of models produced
by high-throughout crystallography programs, the substance may be of unknown
function and not the subject of current study. In this case, the model may simply be
deposited, without fanfare or further analysis, into a database. Later, researchers
who develop an interest in the substance may find, to their pleasant surprise, that
the structure determination has already been done for them. I advise that they
worry, however, about whether the final model has received the same scrutiny as
one more anxiously awaited.
Most crystallographers would agree that an important part of their work is
to make molecular structures available to the larger community of scientists.
174 Chapter 7 Obtaining and Judging the Molecular Model
This belief is reflected in the policies of many journals and funding organizations that require public availability of the structure as a condition of publication
or financial support. But in the desire to make sure that they capitalize fully on commercially important leads provided by new models, crystallographers sometimes
delay making models available. Research support from industry sometimes carries
the stipulation that the full atomic details of new models be withheld long enough
to allow researchers to explore and perhaps patent ideas of potential commercial
value. Some journals now allow such delays, agreeing to publish announcements
and discussions of new models if public access within a reasonable period is
assured. This assurance can be enforced by requiring, as a condition of publication, that the researcher submit the model to the Protein Data Bank (discussed
later in this section) immediately, but with the stipulation that public access is
denied for a time, usually no more than one year. It is becoming more common
for publication to require immediate public availability of the model.
Crystallographers share the fruits of their work in the form of lists of atomic
coordinates, which can be used to display and study the molecule with molecular
graphics programs (Chapter 11). It is becoming more common, with the improvement of resources to use them, for crystallographers to share also the final structure
factors, from which electron-density maps can be computed. The audience for
structure factors includes other crystallographers developing new techniques of
data handling, refinement, or map interpretation. The audience for electron-density
maps includes anyone who wants to judge the quality of the final molecular image,
and assess the evidence for published conclusions drawn from the model.
Upon request, many authors of published crystallographic structures provide
coordinate lists or structure factors by computer mail directly to interested parties.
But the great majority of models and structure factors are available through the
Protein Data Bank (PDB).1 Crystallographers can satisfy publication and funding
requirements for availability of their structures by depositing coordinates with the
PDB. Depositors are required to run a series of checks for errors and inconsistencies
in coordinates and format, and a validation check that produces a report on various
model properties. These validation checks are not as extensive as can be done with
the online validation tools described in Chapter 8 (p. 189). Once the files are
processed, the PDB makes them available free over the World Wide Web, in a
standard text format. As the size of the PDB has grown, so has the power of its
searching tools, and the number of links from individual entries to other databases,
including the massive protein and nucleic-acid sequence databases that have grown
from worldwide genome projects.
The PDB structure files, which are called atomic coordinate entries, can be
read within editor or word-processor programs. Almost all molecular graphics
programs read PDB files directly or use them to produce their own files in binary
1The Protein Data Bank is described fully in H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland,
T. N. Bhat, H. Weissig, I. N. Shindyalov, P. E. Bourne: The Protein Data Bank. Nucleic Acids Research,
28, pp. 235–242 (2000).
Section 7.7 Sharing the model 175
form for rapid access during display. In addition to the coordinate list, a PDB file
contains a header or opening section with information about published papers on
the protein, details of experimental work that produced the structure, and other
useful information.
Following is a brief description of PDB file contents, based on requirements for
newly deposited files in March of 2005. The line types (also called record types),
given in capital letters, are printed at the left of each line in the file. Some of these
line types do not apply to all models and may be missing. Typical contents of a
coordinate file, in order of appearance, are
 HEADER lines, containing the file name (“ID code”), deposition date, and a
brief title.
 TITLE lines, containing a brief title, usually based on the title of the model
publication.
 COMPND lines, containing the name of the protein, including synonyms.
 SOURCE lines, giving the organism from which the protein was obtained,
and if an engineered protein, the organism in which the protein was expressed.
 KEYWDS lines, giving keywords that would guide a search to this file.
 EXPDTA lines, giving the experimental method (X-ray diffraction or NMR).
 AUTHOR lines, listing the persons who placed this data in the Protein
Data Bank.
 REVDAT lines, listing all revision dates for data on this protein.
 JRNL lines, giving the journal reference to the lead article about this model.
When viewed online, these lines are usually linked to the article in PubMed.
 REMARK lines, containing (1) references to journal articles about the structure of this protein and (2) general information about the contents of this
file, including many specifics about resolution, refinement methods, and final
criteria of model quality (see Chapter 8). The REMARK lines can provide
extensive information about the details of structure determination, including
lists of missing residues. Ranges of REMARK line numbers are reserved for
specific types of information, as detailed in PDB documentation.
 DBREF lines, which cross-reference PDB entries to other databases, particularly sequence databases.
 SEQADV lines, which identify conflicts between the PDB sequence and those
of other databases. Such conflicts might arise because of sequencing revisions
made during the structure determination, but more commonly, the conflicts
reflect engineered sequence changes made to study protein function, or to facilitate structure determination, such as attachment of histidines for purification,
or replacement of methionine with selenomethionine.
 SEQRES lines, giving the amino-acid sequence of the protein, with amino
acids specified by three-letter abbreviation.
176 Chapter 7 Obtaining and Judging the Molecular Model
 HET, HETNAM, and FORMUL lines, listing the names, synonyms, and formulas of cofactors, prosthetic groups, or other nonprotein substances present
in the structure. Online versions of PDB files often contain links to more
information about HET groups, including links to graphics displays of their
structures (see Chapter 11).
 HELIX, SHEET, TURN, CISPEP, LINK, and SITE lines, listing the elements
of secondary structure in the protein, residues involved in cis-peptide bonds
(almost always involving proline as the second residue), cross-links such as
disulfide bonds, and residues in the active site of the protein.
 CRYST lines, giving the unit cell dimensions and space group.
 ORIG and SCALE lines, which give matrix transformations relating the
orthogonal angstrom coordinates in the entry to the submitted coordinates,
which might not have been orthogonal, or might have been in fractions of
unit-cell lengths instead of in angstroms.
This concludes what is loosely called the header of the PDB file. The remaining
lines provide the atomic coordinates and other information needed to display and
analyze the model:
 ATOM lines, containing the atomic coordinates of all protein atoms, plus their
structure factors and occupancies. Atoms are listed in the order given in the
paragraph following this list.
 TER lines, among the ATOM lines, specifying the termini of distinct chains
in the model.
 HETATM lines, which contain the same information as ATOM lines for any
nonprotein molecules (cofactors, prosthetic groups, and solvent molecules,
collectively called heteromers or, loosely, het-groups) included in the structure
and listed in HET, HETNAM, and FORMUL lines above.
 CONECT lines, which list covalent bonds between nonprotein atoms in
the file.
 MASTER and END lines, which provide some data for record keeping, and
mark the end of the file.
After the header comes a list of model atoms in standard order. Atoms in the
PDB file are named and listed according to a standard format in an all-English
version of the Greek-letter conventions used by organic chemists. For each amino
acid, beginning at the N terminus, the backbone atoms are listed in the order
alpha nitrogen N, alpha carbon CA, carbonyl carbon C, and carbonyl oxygen O,
followed by the side chain atoms, beta carbon CB, gamma carbon CG, and so forth.
In branched side chains (or rings), atoms in the two branches are numbered 1 and
2 after the proper Greek letter. For example, the atoms of aspartic acid, in the order
of PDB format, are N, CA, C, O, CB, CG, OE1, and OE2. The terminal atoms of
Section 7.7 Sharing the model 177
the side chain are followed in the file by atom N of the next residue. There are
no markers in the file to tell where one residue begins and another ends; each N
marks the beginning of the next residue.
You will find more information on PDB file contents and other available information, as well as tutorials on using the PDB, at their web site (see CMCC home
page). The coordinate files are only the first level of information about the model,
and related models that you will find at the Protein Data Bank. PDB users may
subscribe to a discussion group, providing an email forum for discussion of all
aspects of PDB use. Questions to the list often reveal attempts to use the PDB in
innovative ways. For example, at the moment, many subscribers are interested in
so-called data mining—finding ways to extract specific information from many
PDB entries automatically.
In this form, as a PDB atomic coordinate entry, a crystallographic structure
becomes a matter of public record. The final model of the molecule can then fall
before the eyes of anyone equipped with a computer and an appropriate molecular
display program. It is natural for the consumer of these files, as well as for anyone
who sees published structures in journals or textbooks, to think of the molecule
as something someone has seen more or less directly. Having read this far, you
know that our crystallographic vision is quite indirect. But you probably still have
little intuition about possible limits to the model’s usefulness. For instance, just
how precise are the relative locations of atoms? How much does molecular motion
alter atomic positions? For that matter, how well does the model fit the original
diffraction data from which it was extracted? These and other questions are the
subject of Chapter 8, in which I will start you off toward becoming a discriminating consumer of the crystallographic product. This entails understanding several
criteria of model quality and being able to extract these criteria from published
accounts of crystallographic structure determination, or from the model and X-ray
data itself, by way of online validation tools. Be aware that the Protein Data Bank
does not check or vouch for the quality of models, but a full PDB entry includes
much of the information that users need in order to verify that a model is good
enough for their purposes. In addition to the coordinate entry, the PDB provides
links to many additional tools for the wise user of models.
 Chapter 8
A User’s Guide to
Crystallographic Models
8.1 Introduction
If you are not a crystallographer, this chapter may well be the most important one
for you. Although structure determination by X-ray crystallography becomes more
and more accessible every year, many structural biologists will never determine a
protein structure by X-ray crystallography. But many will at some time use a crystallographic model in research or teaching. In research, study of molecular models
by computer graphics is an indispensable tool in formulating mechanisms of protein action (for instance, binding or catalysis), searching for modes of interaction
between molecules, choosing sites to modify by chemical methods or site-specific
mutagenesis, and designing inhibitors of proteins involved in disease. Because
protein chemists would like to learn the rules of protein folding, every new model
is a potential test for proposed theories of folding, as well as for schemes for
predicting conformation from amino-acid sequence. Every new model for which
homologous sequences are known is a potential scaffold on which to build homology models (Chapter 11). In education, modern texts in biology and chemistry are
effectively and dramatically illustrated with graphics images, sometimes as stereo
pairs. Projection monitors allow instructors to show “real-time” graphics displays
in the classroom, giving students vivid, animated, three-dimensional views of complex molecules. Any up-to-date structural biology course, such as biochemistry,
molecular biology, or bioinformatics, now includes the study of macromolecules
using molecular graphics programs on personal computers. Free or inexpensive,
yet very powerful, graphics programs for personal computers, combined with easy
179
180 Chapter 8 A User’s Guide to Crystallographic Models
online access to the Protein Data Bank, now make it possible for anyone to study
any available molecular model.
In all of these applications, there is a tendency to treat the model as a physical
entity, as a real object seen or filmed. How much confidence in the crystallographic
model is justified? For instance, how precisely does crystallography establish the
positions of atoms in the molecule? Are all of the atomic positions equally well
established? How does one rule out the possibility that crystallizing the protein
alters it in some significant way? The model is a static image of a dynamic molecule,
a springy system of atoms that breathes with characteristic vibrations, and tumbles
dizzily through solution, as it executes its function. Does crystallography give
us any insight into these motions? Are parts of the molecule more flexible than
others? Are major movements of structural elements essential to the molecule’s
action? How does the user decide whether proposed motions of the molecule are
reasonable?
More seriously, how does the user confirm that the crystallographic data actually justify published conclusions drawn from the model? As I stated earlier, the
Protein Data Bank neither judges nor vouches for the quality of models. The
PDB simply requires that authors provide, in a consistent format, the information needed to make these judgments. Is there really any chance that published
conclusions are not justified by the data? It is rare, but unfortunately, yes. In any
area of science, as in any intellectually or economically competitive endeavor,
there are those who gild the lily, take short cuts, or think wishfully. By far most
PDB models are deposited with diligent attention to detail and accuracy of representation, but within the anonymity of tens of thousands of entries posted in
what is basically an honor system, there are all imaginable shortcomings and all
imaginable reasons for them, so you should thoroughly check any model in the
PDB before relying on it. If you are just planning to make a picture to illustrate
a point in a lecture, then model weaknesses might not affect you. But if you are
going to base research directions and conclusions on a published model, or if you
are going to use a structure determination as a teaching example (as I will do later
in this chapter), you should start by getting to know it and the evidence behind it
very well.
In this chapter, I will discuss the inherent strengths and limitations of molecular
models obtained by X-ray diffraction. My aim is to help you to use crystallographic models wisely and appropriately, and realize just what is known, and what
is unknown, about a molecule that has yielded up some of its secrets to crystallographic analysis. Knowing the limitations that are inherent to the crystallographic
process will also put you in the position to recognize if a specific model really lives
up to its depositor’s claims. To help you learn how to make quality assessments for
yourself, I will review some online model validation tools that are currently available. I will conclude this chapter by discussing a recent structure publication, as it
appeared in a scientific journal. Here my goals are (1) to help you learn to extract
criteria of model quality from published structural reports, and (2) to review some
basic concepts of protein crystallography from previous chapters (repetition is one
of the staples of effective teaching).
Section 8.2 Judging the quality and usefulness of the refined model 181
8.2 Judging the quality and usefulness of
the refined model
8.2.1 Structural parameters
As discussed in Sec. 7.6.2, p. 171, crystallographers monitor the decrease in
some type of R-factor (or the increase in a likelihood factor) as an indicator of
convergence to a final, refined model, with a general target for R of 0.20 or lower
for proteins, and adequate additional cycles of refinement to confirm that R is not
still declining. In addition, various constraints and restraints are relaxed during
refinement, and after these restricted values are allowed to refine freely, they should
remain in, or converge to, reasonable values. Among these are the root-meansquare (rms) deviations of the model’s bond lengths, angles, and conformational
angles from an accepted set of values based upon the geometry of small organic
molecules. A refined model should exhibit rms deviations of no more than 0.02 Å
for bond lengths and 4o for bond angles. These values are routinely calculated
during refinement to be sure that all is going well. Because they are restrained in
earlier stages, they are not as valuable as quality indicators as parameters that are
allowed to vary freely throughout the refinement.
In effect, protein structure determination is a search for the conformation of
a molecule whose chemical composition is known. For this reason, conformational angles about single bonds are not constrained during refinement, and they
should settle into reasonable values. Spectroscopic evidence abundantly implies
that peptide bonds are planar, and some refinements constrain peptide geometry. If unconstrained, peptide bonds should settle down to within a few degrees
of planar.
Aside from peptide bonds, the other backbone conformational angles are ,
along the N − Cα bond and , along the Cα–C bond, as shown in Fig. 8.1.
In this figure,  is the torsional angle of the N − Cα bond, defined by the atoms
C − N − Cα–C (C is the carbonyl carbon) and is the torsional angle of the Cα–C
bond, defined by the atoms N − Cα − C − N. In the figure,  =  = 180o.
Model studies show that, for each amino acid, the pair of angles  and  is
greatly restricted by steric repulsion. The allowed pairs of values are depicted
on a Ramachandran diagram (Fig. 8.2). A point (,) on the diagram represents the conformational angles  and  on either side of the alpha carbon of
one residue. Irregular polygons enclose backbone conformational angles that do
not give steric repulsion (yellow polygons) or give only modest repulsion (blue
polygons). Location of the letters α and β correspond to conformational angles of
residues in right-handed α helix and in β pleated sheet. Although the differences
are relatively small, the shapes of allowed regions are slightly different for each
of the 20 common amino acids, and in addition, Ramachandran diagrams from
different sources will exhibit slight differences in the shapes of allowed regions.
Every year, someone in my biochemistry class, facing the formidable
Ramachandran diagram for the first time, asks me, “Are these diagrams good
for anything?” I am happy to say that they are indeed. During the final stages of
182 Chapter 8 A User’s Guide to Crystallographic Models
Figure 8.1  Backbone conformational angles in proteins (stereo).
Figure 8.2  Ramachandran diagram for nonglycine amino-acid residues in proteins.
Angles  and  are as defined in Fig. 8.1. Diagram produced with DeepView.
Section 8.2 Judging the quality and usefulness of the refined model 183
map fitting and crystallographic refinement, Ramachandran diagrams are a great
aid in spotting conformationally unrealistic regions of the model. Now that the
majority of residues are built automatically and do not receive individual scrutiny,
it is especially important to have easy means of drawing attention to unrealistic
features in the model. Crystallographic software packages and map-fitting programs (and some graphics display programs as well) usually contain a routine for
computing  and  for each residue from the current coordinate list, as well as
for generating the Ramachandran diagram and plotting a symbol or residue number at the position (,). Refinement papers often include the diagram, with an
explanation of any residues that lie in high-energy (“forbidden”) areas. For an
example, see Fig. 8.7, p. 209. Glycines, because they lack a side chain, usually
account for most of the residues that lie outside allowed regions. If nonglycine
residues exhibit forbidden conformational angles, there should be some explanation in terms of structural constraints that overcome the energetic cost of an
unusual backbone conformation.
The conformations of amino-acid side chains are unrestrained during refinement.
In well-refined models, side-chain single bonds end up in staggered conformations
(commonly referred to as rotamers). Distributions of side-chain conformations for
all amino acids are available among the online structure validation tools that I will
discuss later.
8.2.2 Resolution and precision of atomic positions
In microscopy, the phrase “resolution of 2 Å,” implies that we can resolve objects
that are 2 Å apart. If this phrase had the same meaning for a crystallographic model
of a protein, in which bond distances average about 1.5 Å, we would be unable to
distinguish or resolve adjacent atoms in a 2-Å map. Actually, for a protein refined
at 2-Å resolution to an R-factor near 0.2, the situation is much better than the
resolution statement seems to imply.
In X-ray crystallography, “2-Å model” means that analysis included reflections
out to a distance in the reciprocal lattice of 1/(2 Å) from the center of the diffraction
pattern. This means that the model takes into account diffraction from sets of
equivalent, parallel planes spaced as closely as 2 Å in the unit cell. (Presumably,
data farther out than the stated resolution was unobtainable or was too weak to
be reliable.) Although the final 2-Å map, viewed as an empty contour surface,
may indeed not allow us to discern adjacent atoms, prior knowledge in the form of
structural constraints on the model greatly increase the precision of atom positions.
The main constraint is that we know we can fit the map with groups of atoms—
amino-acid residues—having known connectivities, bond lengths, bond angles,
and stereochemistry.
More than the resolution, we would like to know the precision with which atoms
in the model have been located. For years, crystallographers used the Luzzati plot
(Fig. 8.3) to estimate the precision of atom locations in a refined crystallographic
model. At best, this is an estimate of the upper limit of error in atomic coordinates.
The numbers to the right of each smooth curve on the Luzzati plot are theoretical
estimates of the average uncertainty in the positions of atoms in the refined model
184 Chapter 8 A User’s Guide to Crystallographic Models
Figure 8.3  Luzzati diagram.
(more precisely, the rms errors in atom positions). The average uncertainty has been
shown to depend uponR-factors derived from the final model in various resolution
ranges. To prepare data for a Luzzati plot, we separate the intensity data into groups
of reflections in narrow ranges of 1/d (where d is the spacing of real lattice planes).
Then we plot each R-factor (vertical axis) versus the midpoint value of 1/d for
that group of reflections (horizontal axis). For example, we calculate R using only
reflections corresponding to the range 1/d = 0.395 − 0.405 (reflections in the
2.53- to 2.47-Å range) and plot this R-factor versus 1/d = 0.400/Å, the midpoint
value for this group. We repeat this process for the range 1/d = 0.385–0.395, and
so forth. As the theoretical curves indicate, the R-factor typically increases for
lower-resolution data (higher values of 1/d). The resulting curve should roughly
fit one of the theoretical curves on the Luzzati plot. From the theoretical curve
closest to the experimental R-factor curve, we learn the average uncertainty in the
atom positions of the final model. It is now widely accepted that Luzzati plots using
Rfree (Sec. 7.6.2, p. 171) or cross-validated sigma-Avalues give better estimates of
uncertainty in coordinates. Bayesian methods promise more appropriate models of
how phase errors lead to model errors, and hence even more rigorous assessment
of uncertainty in atom positions.
Publications of refined structures may include a Luzzati plot or one of its more
modern equivalents, allowing the reader to assess very roughly the average uncertainty of atom positions in the model. Alternatively, they may simply report the
uncertainty as “determined by the method of Luzzati.” For highly refined models,
rms errors as low as 0.15 Å are sometimes attained. For example, in Fig. 8.6a,
p. 207 (to be discussed in Sec. 8.3.3, p. 198), the jagged curve represents the data
for the refined model of adipocyte lipid binding protein (ALBP). The position of
the curve on the Luzzati plot indicates that rms error for this model is about 0.34 Å,
about one-fifth the length of a carbon-carbon bond.
Section 8.2 Judging the quality and usefulness of the refined model 185
In crystallography, unlike microscopy, the term resolution simply refers to the
amount of data ultimately phased and used in the structure determination. In contrast, the precision of atom positions depends in part upon the resolution limits of
the data, but also depends critically upon the quality of the data, as reflected by
such parameters as R-factors. Good data can yield atom positions that are precise
to within one-fifth to one-tenth of the stated resolution.
8.2.3 Vibration and disorder
Notice, however, that the preceding analysis gives only an upper limit and an
average, or rms value, of position errors, and further, that the errors result from the
limits of accuracy in the data. There are also two important physical (as opposed to
statistical) reasons for uncertainty in atom positions: thermal motion and disorder.
Thermal motion refers to vibration of an atom about its rest position. Disorder
refers to atoms or groups of atoms that do not occupy the same position in every
unit cell, in every asymmetric unit, or in every molecule within an asymmetric unit.
The temperature factor Bj obtained during refinement reflects both the thermal
motion and the disorder of atom j , making it difficult to sort out these two sources
of uncertainty.
Occupancies nj for atoms of the protein (but not necessarily its ligands, which
may be present at lower occupancies) are usually constrained at 1.0 early in refinement, and in many refinements are never released, so that both thermal motion and
disorder show their effects upon the final B values. In some cases, after refinement
converges, a few B values fall far outside the average range for the model. This
is sometimes an indication of disorder. Careful examination of sigma-A weighted
maps or comparison of 2Fo − Fc and Fo − Fc maps may give evidence for more
than one conformation in such a troublesome region. If so, inclusion of multiple conformations followed by refinement of their occupancies may improve the
R-factor and the map, revealing the nature of the disorder more clearly.
If Bj were purely a measure of thermal motion at atom j (and assuming that
occupancies are correct), then in the simplest case of purely harmonic thermal
motion of equal magnitude in all directions (called isotropic vibration), Bj is
related to the magnitude of vibration as follows:
Bj = 8π2{u2j } = 79{u2j }, (8.1)
where {u2j } is the mean-square displacement of the atom from its rest position.
Thus if the measured Bj is 79 Å2, the total mean-square displacement of atom j
due to vibration is 1.0 Å2, and the rms displacement is the square root of {u2j }, or
1.0 Å. The B values of 20 and 5 Å2 correspond to rms displacements of 0.5 and
0.25 Å. But the B values obtained for most proteins are too large to be seen as
reflecting purely thermal motion and must certainly reflect disorder as well.
With small molecules, it is usually possible to obtain anisotropic temperature
factors during refinement, giving a picture of the preferred directions of vibration
for each atom. But a description of anisotropic vibration requires six parameters
186 Chapter 8 A User’s Guide to Crystallographic Models
per atom, vastly increasing the computational task. In many cases, the total number
of parameters sought, including three atomic coordinates, one occupancy, and six
thermal parameters per atom, approaches or exceeds the number of measured
reflections. As mentioned earlier, for refinement to succeed, observations (measured reflections and constraints such as bond lengths) must outnumber the desired
parameters, so that least-squares solutions are adequately overdetermined. For this
reason, anisotropic temperature factors for proteins have not usually been obtained.
The increased resolution possible with synchrotron sources and cryocrystallography will make their determination more common. With this development, it may
become possible to know coordinate uncertainty for individual atoms, rather than
the average uncertainty obtained by methods like that of Luzzati.
Publications of refined structures often include a plot of average isotropic B
values for side-chain and main-chain atoms of each residue, like that shown in
Fig. 8.6b, p. 207, for ALBP. Pictures of the model may be color coded by temperature factor, red (“hot”) for high values of B and blue (“cold”) for low values
of B. Either presentation calls the user’s attention to parts of the molecule that
are vibrationally active and parts that are particularly rigid. Not surprisingly, sidechain temperature factors are larger and more varied (5–60 Å2) than those of
main-chain atoms (5–35 Å2).
As I mentioned earlier, if occupancies are constrained to values of 1.00, variation
in actual occupancies will show up as increased temperature factors. This is true
for other types of constraints as well, and so high temperature factors can mask
other kinds of errors. For this reason, the temperature factor has been uncharitably
referred to as a garbage can for model errors. A less pessimistic assessment is that
B-values are informative only if other kinds of errors have been carefully removed.
Remember that we see in a crystallographic model an average of all the
molecules that diffracted the X-rays. Furthermore, we see a static model representing a stable conformation of a dynamic molecule. It is sobering to realize that the
crystallographic model of ALBP exhibits no obvious path for entry and departure
of its ligands, which are lipid molecules like oleic acid. Similarly, comparison of
the crystallographic models of hemoglobin and deoxyhemoglobin reveals no path
for entry of the tiny O2 molecule. Seemingly simple processes like the binding
of small ligands to proteins often involve conformational changes to states not
revealed by crystallographic analysis.
Nevertheless, the crystallographic model contributes importantly to solving
such problems of molecular dynamics. The refined structure serves as a starting point for simulations of molecular motion. From that starting point, which
undoubtedly represents one common conformation of the protein, and from the
equations of motion of atoms in the force fields of electrostatic and van der Waals
forces, scientists can calculate the normal vibrational motions of the molecules and
can simulate random molecular motion, thus gaining insights into how conformational change gives rise to biomolecular function. Even though the crystallographic
model is static, it is an essential starting point in revealing the dynamic aspects
of structure. As I will show in Chapter 9, time-resolved crystallography offers the
potential of giving us highly detailed views of proteins in motion.
Section 8.2 Judging the quality and usefulness of the refined model 187
8.2.4 Other limitations of crystallographic models
The limitations discussed so far apply to all models and suggest questions that the
user of crystallographic results should ask routinely. Other limitations are special
cases that may or may not apply to a given model. It is important to read the
original publications of a structure, as well as its PDB file header, to see whether
any of the following limitations apply.
Low-resolution models
Not all published models are refined to high resolution. For instance, publication
of a low-resolution structure may be warranted if it displays an interesting and
suggestive arrangement of cofactors or clusters of metal ions, provides possible
insights into conformations of a new family or proteins, or displays the application
of new imaging methods. In some cases, the published structure is only a crude
electron-density model. Or perhaps it contains only the estimated positions of alpha
carbons, so-called alpha-carbon or Cα models. Such models may be of limited use
for comparison with other proteins, but of course, they cannot support detailed
molecular analyses. In alpha-carbon models, there is great deal of uncertainty in
the positions, and even in the number, of alpha carbons. Some graphics programs
(Chapter 11) will open such files but show no model, giving the impression that
the file is empty or damaged. The model appears when the user directs the program
to show the alpha-carbon backbone only. It is not unusual for further refinement
of these models to reveal errors in the chain tracing. Protein Data Bank headers
include important information about model resolution and descriptions of model
contents.
Disordered regions
Occasionally, portions of the known sequence of a protein are never found in
the electron-density maps, presumably because the region is highly disordered
or in motion, and thus invisible on the time scale of crystallography. The usual
procedure is simply to omit these residues from the deposited model. It is not at all
uncommon for residues at termini, especially the N-terminus, to be missing from a
model. Discussions of these structure-specific problems are included in a thorough
refinement paper, and lists of missing residues are provided in PDB header.
Unexplained density
Just as the auto mechanic sometimes has parts left over, electron-density maps
occasionally show clear, empty density after all known contents of the crystal have been located. Apparent density can appear as an artifact of missing
Fourier terms, but this density disappears when a more complete set of data is
obtained. Among the possible explanations for density that is not artifactual are
ions like phosphate and sulfate from the mother liquor; reagents like mercaptoethanol, dithiothreitol, or detergents used in purification or crystallization; or
cofactors, inhibitors, allosteric effectors, or other small molecules that survived
188 Chapter 8 A User’s Guide to Crystallographic Models
the protein purification. Later discovery of previously unknown but important
ligands has sometimes resulted in subsequent interpretation of empty density.
Distortions due to crystal packing
Refinement papers and PDB headers should also mention any evidence that the protein is affected by crystallization. Packing effects may be evident in the model itself.
For example, packing may induce slight differences between what are otherwise
expected to be identical subunits within an asymmetric unit. Examination of the
neighborhood around such differences may reveal that intermolecular contact is a
possible cause. In areas where subunits come into direct contact or close contact
through intervening water, surface temperature factors are usually lower than at
other surface regions.
Functional unit versus asymmetric unit
The symmetry of functional macromolecular complexes in solution is sometimes
important to understanding their functions, as in the binding of regulatory proteins
having twofold rotational symmetry to palindromic DNA sequences. As discussed
in Sec. 4.2.8, p. 65, in the unit cell of a crystal, the largest aggregate of molecules
that possesses no symmetry elements, but can be juxtaposed on other identical
entities by symmetry operations, is called the crystallographic asymmetric unit.
Users of models should be careful to distinguish the asymmetric unit from the functional unit, which the Protein Data Bank currently calls the “biological molecule.”
For example, the functional unit of mammalian hemoglobin is a complex of four
subunits, two each of two slightly different polypeptides, called α and β. We
say that hemoglobin functions as an α2β2 tetramer. In some hemoglobin crystals,
the twofold rotational symmetry axis of the tetramer corresponds to a unit-cell
symmetry axis, and the asymmetric unit is a single αβ dimer. In other cases, the
asymmetric unit may contain more than one biological unit.
For technical reasons having to do with data collection strategies, crystal properties, and other processes essential to crystallography itself, the asymmetric unit
is often mentioned prominently in papers about new crystallographic models. This
discussion is part of a full description of the crystallographic methods for assessment of the work by other crystallographers. It is easy to get the impression that
the asymmetric unit is the functional unit, but frequently it is not. Beyond the
technical methods sections of a paper, in their interpretations and discussions of
the meaning of the model, authors are careful to describe the functional form of
the substance under study (if it is known), and this is the form that holds the most
interest for users.
It is safe to think of functional-unit symmetry as not necessarily having anything
to do with crystallographic symmetry. If the two share some symmetry elements, it
is coincidental and may actually be useful to the crystallographer (see, for example,
Sec. 7.3.3, p. 151 on noncrystallographic symmetry). But in another crystal form
of the same substance, the unit cell and the functional unit may share different
symmetry elements or none. For you as a user of crystallographic models, looking
Section 8.2 Judging the quality and usefulness of the refined model 189
at crystal symmetry and packing is primarily of value in making sure that you
do not make errors in interpreting the model by not allowing for the relatively
rare but possibly disruptive effects of crystallization. Once you are confident that
the portions of the molecule that pique your interest are not affected by crystal
packing, then you can forget about crystal symmetry and the asymmetric unit, and
focus on the functional unit.
As mentioned earlier, the asymmetric unit may be only a part of the functional
unit. This sometimes poses a problem for users of crystallographic models because
the PDB file for such a crystallographic model may contain only the coordinates
of the asymmetric unit. So in the case of hemoglobin, a file may contain only
one αβ dimer, which is only half of what the user would like to see. The Protein Data Bank routinely takes care of this problem by preparing files containing
the coordinates of all atoms in the functional unit (for example, oxy- and deoxyhemoglobin tetramers), and providing online graphics viewers for examining these
models and saving the prepared coordinate files. The additional coordinates are
computed by applying symmetry operations to the coordinates of the asymmetric
unit. In these models, one can study all the important intersubunit interactions of
the full tetramer. Another solution to this problem is for users themselves to compute the coordinates of the additional subunits. Many molecular graphic programs
provide for such calculations (Sec. 11.3.10, p. 287). Users should also beware that
the asymmetric unit, and hence the PDB file, may contain two or several functional
units.
8.2.5 Online validation tools: Do it yourself!
NOTE: To find all of the validation tools discussed in the following section, see
the CMCC home page.
Over the period from 1990 to 2005, the appearance of new crystallographic
models turned from a trickle to a flood. As a result, many papers reporting new
structures no longer provide detailed information about the crystallographic work,
and thus are less helpful in providing measures of model quality. Models produced
by high-throughput crystallography may be deposited quietly in the PDB, without
detailed structural analysis, and without announcement in journals. A researcher
whose interest turns to a previously unstudied gene, and needs to know the structure
of its protein product, may find that a model is already present in the PDB, deposited
more or less without comment. The researcher may then be the first to explore
the new model, and is faced with assessing its quality with perhaps very limited
information. In all of these cases, online validation tools can help.
Here is the first step in do-it-yourself model assessment: READ THE HEADER!
As mentioned repeatedly in the preceding sections, the inherent limitations of a
model should be (usually are) described in the PDB file header. As a subscriber to
the PDB Discussion Forum, I know firsthand that many a model user, puzzled by
breaks in the graphics display of the model or by chains that begin with residue
13 (both signs of residues not visible in density), or by side chains listed twice
(alternative conformations), or by failure of graphics to display a model (C-alpha
190 Chapter 8 A User’s Guide to Crystallographic Models
model), or by the presence of many superimposed models [not a crystallographic
structure at all, but an ensemble of NMR models (Chapter 10)], has sought help
from the Forum when they could have resolved their puzzlement by simply reading
the header. A conscientiously completed PDB header is a good users’ manual for
the model. Once you know what the authors are trying to tell you about the model,
you are ready to further your assessment on your own. READ THE HEADER!
In my opinion, the current best introduction to concepts and tools for assessing
model quality is the tutorial “Model Validation” at Uppsala University, Sweden.
Maintained by Research Scientist Gerard Kleywegt, an outspoken proponent of
vigiliance in using models, this tutorial introduces all of the classical and modern
quality indicators, rates their power in helping you assess models, and guides you
in their proper use. The tutorial includes a rogue’s gallery of models that fall short
in ways both small and large, from bent indole rings in tryptophan and flat beta
carbons in valine to whole ligands that fail to show up in the electron-density map.
Anyone who graduates from this little validation course will be armed with the
know-how and skepticism needed to use models wisely. If this book is your classroom introduction to judging model quality, Kleywegt’s tutorial is, at the moment,
the best lab practical. For global quality indicators that might help you decide if
a model is good enough for comparative modeling, Kleywegt recommends Rfree,
Ramachandran plots, and packing scores, which tell how “comfortable” each
residue is in its protein environment. For assessing local quality, such as whether an
active site is good enough for modeling ligand binding, he recommends real-space
R- (RSR) factors, which measure how well the electron-density map conforms to
a map calculated from the ideal electron density of the model itself, sort of like
comparing the map with a very accurate space-filling rendition of the model. He
also suggests looking at graphs of main-chain and side-chain torsion-angle combinations in comparison to charts of these combinations in high-quality models.
He values these measures above widely used ones such as conventional R values,
RMS deviation of bond lengths and angles from ideal values, and temperature
factors, all of which are more subject to bias from refinement restraints and from
the model.
In my opinion, the most powerful way to assess model quality, bar none, is to
examine the electron-density map and model. This shows you whether the map
supports specific placement of important residues and ligands. When I am curious
about a model, my first stop is the Electron Density Server (EDS), also at Uppsala
University. If the depositor of the model also deposited structure factors with
the PDB, you can use the EDS server to get electron-density maps, which are
automatically prepared for all sets of structure factors in the PDB. In addition, the
EDS allows you to produce interactive charts such as Ramachandran diagrams
and graphs of residue number versus B values, packing scores, RSR factors, or
other measures. With a click on a residue number in a graph, an online graphics
display (a Java applet called Astex Viewer) opens in your browser to allow you to
peruse model and map. In short, the EDS site provides almost every imaginable
tool for assessing model quality. If you have the proper software (for example,
DeepView—see Chapter 11), you can download maps and examine map quality
Section 8.2 Judging the quality and usefulness of the refined model 191
and map/model conformity in critical areas of the model. If you have the map-fitting
program O (that’s the name of it—O), you can download a package including
model, maps, and most of the graphs mentioned above, all for interactive display
within O.
Another useful and eye-opening validation site is the MolProbity Web Service
at Duke University. This service uses all-atom contact analysis to help you find and
correct bumps (bad contacts) and unrealistic geometry in both side chains and main
chains. It provides graphics using KiNG, a modern Java version of the pioneering
molecular graphics program Mage, to allow structure analysis within your web
browser. Users interested in validating or optimizing a model begin their work by
adding hydrogen atoms. Why? Because around 75% of atomic contacts in proteins
involve hydrogen. In addition, even though macromolecular crystallography does
not usually resolve hydrogens, most hydrogen positions (actually, all but -OH) are
well defined by the conformations of the atoms on which hydrogens reside, and
thus hydrogens can be added with confidence. With hydrogens present, contact
analysis can reveal unrealistic side-chain rotamers, and can also determine the
correct rotamers of side chains such as asparagine, glutamine, and histidine, whose
rotamer alternatives are often difficult to determine from electron density (do you
know why?). Common misplacements of main-chain alpha carbons can also be
detected and corrected. These errors can be detected by deviations of beta carbons
from their optimum positions, and can be corrected by small rotations about the
axis between two alpha carbons, the most useful being correction at residue i
by rotation about the axis between Cαi−1 and Cαi+1. If structure factors for the
model are available, then all of the model corrections at the MolProbity site can
be examined within the electron-density map. Of course, this map my be slightly
biased by the uncorrected model. A map based on phases from the corrected model
would be the ideal reference. The MolProbity site includes a detailed users’manual
and an excellent tutorial.
One of the easiest model-validation tools to use is the Biotech Validation Suite
at the European Bioinformatics Institute. Users simply submit a coordinate file in
PDB format, and receive in return an extensive report on the model, the results
of analysis by three programs, PROCHECK, PROVE, and WHAT IF. The report
gives results of checks on more than twenty aspects of the model, including verification of bond angles and bond lengths and checks for buried hydrogen-bond
donors; bumps (bad contacts); flipped peptides; handedness of chiral atoms; conformational alternatives for HIS, GLN, and ASN; proline puckering; plausibility
of water assignments; and atomic occupancy, to name a few. The PROCHECK
suite of checks is made automatically for all PDB entries, and is linked to the
Structure Explorer page for the entry, under the Analyze menu. At the time of this
writing, these validation tools do not provide for online or automated correction
of errors, but they probably provide the most extensive and rigorous analysis performed against the most demanding standards. As a result of these high standards,
error lists are quite long and may include errors that are of little consequence to
most users. Nevertheless, this package of checks leaves no stones unturned, not
even the tiny ones.
192 Chapter 8 A User’s Guide to Crystallographic Models
The prudent crystallographer will use these validation tools before deciding that
the model is fully refined, complete, and ready for publication. In other words, it
makes sense to use validation tools to find and correct errors during the refinement
process, so that subsequent users of the model will find no problems when applying the same validation tools. At the moment, there is no standard way of knowing
what validation tools were applied to the model, beyond the standard checks (like
PROCHECK) required by the PDB. PDB file headers might provide this information in some cases. Because use of these “accessory” validation tools varies
widely among research groups, users of models should assume that validation is
necessary.
As mentioned earlier, data mining, the automated extraction of specific information from many PDB entries, is a growing activity. Data miners should also
be asking questions about model validation, because in many cases, they collect
information from many files on the assumption that all models are error-free. This
suggests the need for greater automation in model validation, so that data spanning
many PDB files are not skewed by errors in unvalidated models.
And another reminder: For quick access to validation web sites, see the CMCC
home page.
8.2.6 Summary
Sensible use of a crystallographic model, as with any complex tool, requires an
understanding of its limitations. Some limitations, like the precision of atom positions and the static nature of the model, are general constraints on use. Others,
like disordered regions, undetected portions of sequence, unexplained density,
and packing effects, are model-specific. If you use a protein model from the PDB
without reading the header information, or without reading the original publications, you may be missing something vital to the appropriate use of the model.
The result may be no more than a crash of your graphics software because of
unexpected input like a file containing only alpha carbons. Or more seriously, you
may devise and publish a detailed molecular explanation based upon a structural
feature that is quite uncertain. In most cases, the PDB model isn’t enough. If
specific structural details of the model are crucial to a proposed mechanism or
explanation, it is prudent to apply the most rigorous validation tools, and to look
at the electron-density map in the important region, in order to be sure that the
map is well defined there and that the model fits it well.
8.3 Reading a crystallography paper
8.3.1 Introduction
Original structure publications, especially older ones or ones in the more technical
crystallographic journals, often provide enough experimental information to help
with assessment of model quality. To help you learn to extract this information
Section 8.3 Reading a crystallography paper 193
from published papers, as well as to review concepts from the preceding chapters,
I will walk you through a detailed structure publication. Following are annotated
portions of two papers announcing the structure of adipocyte lipid binding protein (ALBP, PDB 1alb)), a member of a family of hydrophobic-ligand-binding
proteins. The first paper1 appeared in August 1991, announcing the purification
and crystallization of the protein, and presenting preliminary results of crystallographic analysis. The second paper,2 which appeared in April 1992, presented the
completed structure with experimental details. In examining these papers and the
model that sprang from it, I will focus primarily on the experimental and results
sections of the papers and specifically upon (1) methods and concepts treated earlier in this book and (2) criteria of refinement convergence and quality of the model.
Although I have reproduced parts of the published experimental procedures here
(with the permission of the authors and publisher), you may wish to obtain the full
publications and read them before proceeding with this example (see footnotes 1
and 2).
Readers of earlier editions may wonder why I have not replaced these papers
with more recent examples. In looking for replacements, I found that most of
today’s structure publications in mainstream journals provide much less detail
about methods, in part because structure determination is far more routine now
than in the early 1990s, and in part because many educationally interesting decisions are now made by software. The following papers thus provide insights into
crystallographic decision making that are often missing from more recent papers.
If you can read papers like the following with insight and understanding, you will
be able to extract more than from the much more abbreviated experimental details
provided in more recent papers.
In the following material, sections taken from the original papers are presented
in indented type. Annotations are in the usual type. For convenience, figures
and tables are renumbered in sequence with those of this chapter. For access to
references cited in excerpts, see the complete papers. Stereo illustrations of maps
and models (not part of the papers) are derived from files kindly provided by
Zhaohui Xu. I am indebted to Xu and to Leonard J. Banaszak for allowing me
to use their work as an example and for supplying me with an almost complete
reconstruction of this structure determination project.
8.3.2 Annotated excerpts of the preliminary (8/91) paper
All reprinted parts of this paper (cited in Footnote 1) appear with the permission
of Professor Leonard J. Banaszak and the American Society for Biochemistry and
Molecular Biology, Inc., publisher of Journal of Biological Chemistry.
1Z. Xu, M. K. Buelt, L. J. Banaszak, and G.A. Bernlohr, Expression, purification, and crystallization
of the adipocyte lipid binding protein, J. Biol. Chem. 266, 14367–14370, 1991.
2Z. Xu, D. A. Bernlohr, and L. J. Banaszak, Crystal structure of recombinant murine adipocyte
lipid-binding protein, Biochemistry 31, 3484–3492, 1992.
194 Chapter 8 A User’s Guide to Crystallographic Models
In the August 5, 1991, issue of Journal of Biological Chemistry, Xu, Buelt,
Banaszak, and Berniohr reported the cloning, expression, purification, and crystallization of adipocyte lipid binding protein (ALBP, or rALBP for the recombinant
form), along with preliminary results of crystallographic analysis. Even into the
mid-1990s, this type of preliminary paper sometimes appeared as soon as a research
team had carried a structure project far enough to know that it promised to produce a good model. An important aim of announcing that work was in progress
on a molecule was to avoid duplication of effort in other laboratories. Although
one might cynically judge that such papers constitute a defense of territory, and
a grab for priority in the work at hand, something much more important was at
stake. Crystallographic structure determination is a massive and expensive undertanding. The worldwide resources, both equipment and qualified scientists, for
structure determination were, and in some respects still are, inadequate for the
many molecules we would like to understand. Duplication of effort on the same
molecule squanders limited resources in this important field. So generally, as soon
as a team had good evidence that they could produce a structure, they alerted the
crystallographic community to prevent parallel work from beginning in other labs.
Because structure determination has become so much more rapid, it is now common for all elements of a structure project—discovery, expression, purification,
crystallization, data collection, structure determination, and structure analysis—to
be reported in a single paper.
The following paragraph is an excerpt from the preliminary (8/91) paper,
“Experimental Procedures” section:
Crystallization—Small crystals (0.05×0.1×0.1 mm) were obtained
using the hanging drop/vapor equilibrium method (18). 10-µl drops
of 2.5 mg/ml ALBP in 0.05 M Tris, 60% ammonium sulfate, 1 mM
EDTA, 1 mM dithiothreitol, 0.05% sodium azide buffer with a pH of
7.0 (crystallization buffer) were suspended over wells containing the
same buffer with varying concentrations of ammonium sulfate, from
75 to 85% saturation. Small, well shaped crystals were formed within
a month at an 80% saturation and 19oC. These crystals were isolated, washed with mother liquid, and used as seeds by transferring
them into a 10-µl drop of 4 mg/ml fresh ALBP in the 80% saturation crystallization buffer over a well containing the same buffer. Large
crystals, 0.3×0.4×0.4 mm, grew in 2 days at a constant temperature
of 19oC.
The precipitant used here is ammonium sulfate, which precipitates proteins by
salting out. Notice that Xu and coworkers tried a range of precipitant concentrations, probably after preliminary trials over a wider range. Crystals produced by
the hanging drop method (Sec. 3.3.2, p. 38) were too small for X-ray analysis but
were judged to be of good quality. The small crystals were used as seeds on which
to grow larger crystals under the same conditions that produced the best small
crystals. This method, called repeated seeding, was also discussed in Chapter 3.
The initial unseeded crystallization probably fails to produce large crystals because
Section 8.3 Reading a crystallography paper 195
many crystals form at about the same rate, and soluble protein is depleted before
any crystals become large. The seeded crystallization is probably effective because
it decreases the number of sites of crystal growth, causing more protein to go into
fewer crystals. Notice also how much faster crystals grow in the seeded drops
(2 days) than in the unseeded (1 month). The preformed crystals provide nucleation sites for immediate further growth, whereas the first crystals form by random
nucleation events, which are usually rate-limiting in unseeded crystallizations.
Data Collection and Processing—Crystals were analyzed with the
area detector diffractometer from Siemens/Nicolet. A 0.8-mm collimator was used, and the crystal to detector distance was set at 12 cm with
the detector midpoint at 2θ = 15o. One φ scan totaling 90o and three
scans of 68o with χ at 45o were collected with the Rigaku Ru200
operating at 50 kV and 180 mA. Each frame consisted of a 0.25o rotation taken for 120 s. The diffractometer data were analyzed with the
Xengen package of programs (19). Raw data within 50 frames were
searched to find about 100 strong reflections which were then indexed,
and the cell dimensions were refined by least squares methods. Data
from different scans were integrated separately and then merged
together.
The angles φ, χ, ω, and 26 refer to the diffractometer angles shown in Fig. 4.26,
p. 82. The Rigaku Ru200 is the X-ray source, a rotating-anode tube. Each frame
of data collection is, in essence, one electronic film on which are recorded all
reflections that pass through the sphere of reflection during a 0.25o rotation of
the crystal. This rotation size is chosen to collect as many reflections as possible
without overlap. As mentioned in Chapter 4, diffractometer measurements are
almost fully automated. In this instance, cell dimensions were worked out by
a computer program that finds 100 strong reflections and indexes them. Then
the program employs a least-squares routine (Sec. 7.6.1, p. 153) to refine the
unit-cell dimensions, by finding the cell lengths and angles that minimize the
difference between the actual positions of the 100 test reflections and the positions
of the same reflections as calculated from the current trial set of cell dimensions.
(Least-squares procedures are used in many areas of crystallography in addition
to structure refinement.) Using accurate cell dimensions, the program indexed all
reflections, and then integrated the X-ray counts received at each location to obtain
reflection intensities.
The following excerpt is from the “Results and Discussion” section of the 8/91
paper:
Crystallization experiments using rALBP were immediately successful. With seeding, octahedral crystals of the apo-protein grew to a
length of 0.4 mm and a height of 0.3 mm. These crystals give diffraction data to 2.4 Å. An entire data set was collected to 2.7-Å resolution
using the area detector system. Statistical details of the combined
X-ray data set are presented in Table 8.1.
196 Chapter 8 A User’s Guide to Crystallographic Models
TABLE 8.1  X-Ray Data Collection Statistics for Crystalline ALBP
Merging R-factor based on I 0.0426
Resolution limits 2.2 Å
Number of observations 20,478
Number of unique X-ray reflections collected 5,473
Average number of observation for each reflection 4.0
% of possible reflections collected to 2.7 Å 98
% of possible reflections collected to 2.4 Å 36
From Z. Xu et al. (1991) J. Biol. Chem. 266, pp. 14367–14370, with permission.
Xu and colleagues had exceptionally good fortune in obtaining crystals of the
recombinant form of ALBP. Efforts to crystallize a desirable protein can give
success in a few days, or never, or anything in between. Modern commercial
crystallization screens have made quick success more common. The extent of
diffraction in preliminary tests (2.4 Å) is a key indicator that the crystals might
yield a high-quality structure.
Table 8.1 provides you with a glimpse into the quality of the native data set.
The 0.25o frames of data from the area detector are merged into one data set by
multiplying all intensities in each frame by a scale factor. A least-squares procedure determines scale factors that minimize the differences between intensities
of identical reflections observed on different frames. The merging R-factor [see
Eq. (7.13)] gives the level of agreement among the different frames of data after
scaling. In this type of R-factor, |Fobs|s are derived from averaged, scaled intensities for all observations of one reflection, and corresponding |Fcalc|s are derived
from scaled intensities for individual observations of the same reflection. The better the agreement between these two quantities throughout the data set, the lower
the merging R-factor. In this case, individual scaled intensities agree with their
scaled averages to within about 4%.
You can see from Table 8.1 that 98% of the reflections available out to 2.7 Å
[those lying within a sphere of radius 1/(2.7 Å) centered at the origin of the reciprocal lattice] were measured, and on the average, each reflection was measured
four times. Additional reflections were measured out to 2.4 Å. The number of
available reflections increases with the third power of the radius of the sampled
region in the reciprocal lattice (because the volume of a sphere of radius r is proportional to r3), so a seemingly small increase in resolution from 2.7 to 2.4 Å
requires 40% more data. [Compare (1/2.4)3 with (1/2.7)3.] For a rough calculation
of the number of available reflections at specified resolution, see annotations of the
4/92 paper.
The lattice type was orthorhombic with unit cell dimension of
a =34.4 Å, b =54.8 Å, c =76.3 Å. The X-ray diffraction data were
examined for systematic absences to determine the space group.
Such absences were observed along the a*, b*, and c* axes.
Only reflections with h, k, or l = 2nwere observed along the reciprocal
Section 8.3 Reading a crystallography paper 197
axes. This indicated that the space group is P 212121 (25). A unit cell
with the dimensions described above has a volume of 1.44×105 Å3.
Assuming that half of the crystal volume is water, the volume of protein is approximately 7.2×104 Å3. Considering the space group here,
the volume of protein in 1 asymmetric unit would be 1.8×104 Å3. By
averaging the specific volume of constituent amino acids, the specific volume of ALBP is 0.715 mL/g. This led to the conclusion that
the molecular mass in one asymmetric unit is 15,155 daltons. Since
the molecular mass of ALBP is approximately 15 kDa, there is only
1 molecule of ALBP in an asymmetric unit.
Recall from Sec. 5.4.3, p. 105, that for a twofold screw axis along the b edge,
all odd-numbered 0k0 reflections are absent. In the space group P 212121, the unit
cell possesses twofold screw axes on all three edges, so odd-numbered reflections
on all three principal axes of the reciprocal lattice (h00, 0k0, and 00l) are missing.
The presence of only even-numbered reflections on the reciprocal-lattice axes
announces that the ALBP unit cell has P 212121 symmetry.
The number of molecules per asymmetric unit can be determined from unit-cell
dimensions and a rough estimate of the protein/water ratio. Since this number
is an integer, even a rough calculation can give a reliable answer. The assumption that ALBP crystals are 50% water is no more than a guess taken from near
the middle of the range for protein crystals (30 to 78%). The unit-cell volume is
(34.4 Å)× (54.8 Å)× (76.3 Å)= 1.44× 105 Å3, and if half that volume is protein, the protein volume is 7.2× 104 Å3. In space group P 212121, there are four
equivalent positions (Sec. 4.2.8, p. 65), so there are four asymmetric units per unit
cell. Each one must occupy one-fourth of the protein volume, so the volume of the
asymmetric unit is one-fourth of 7.2× 104, or 1.8× 104 Å3. The stated specific
volume (volume per gram) of the protein is the weighted average of the specific
volumes of the amino-acid residues (which can be looked up), weighted according
to the amino-acid composition of ALBP. The molecular mass of one asymmetric
unit is obtained by converting the density of ALBP in grams per milliliter (which
is roughly the inverse of the specific volume) to daltons per cubic angstrom, and
then multiplying by the volume of the asymmetric unit, as follows:
1 g
0.715 mL
· 1 ml
cm3
· cm
3
(108)3 Å3
· 6.02 × 10
23
g daltons · 1.8 × 104 Å3
= 1.5 × 104 daltons.
This result is very close to the known molecular mass of ALBP, so there is
one ALBP molecule per asymmetric unit. This knowledge is an aid to early map
interpretation.
As indicated, ALBP belongs to a family of low molecular weight fatty
acid binding proteins. The sequences of the proteins in the family
198 Chapter 8 A User’s Guide to Crystallographic Models
have been shown to be very similar and in particular in the aminoterminal domain where Y193 resides. Among them, the structure of
myelin P2 and IFABP has been solved. Since the amino acid identity between ALBP and myelin P2 is about 69%, P2 should be a
good starting structure to obtain phase information for ALBP using
the method of molecular replacement. Preliminary solutions to the
rotation and translation functions have been obtained. Seeding techniques will allow us to obtain large crystals for further study of the
holo- and phosphorylated protein. By comparing the crystal structures
of these different forms, it should be possible to structurally determine
the effects of protein phosphorylation on ligand binding and ligand
binding on phosphorylation.
Because ALBP is related to several proteins of known structure, molecular
replacement is an attractive option for phasing. The choice of a phasing model is
simple here: just pick the one with the amino-acid sequence most similar to ALBP,
which is myelin P2 protein. Solution of rotation and translation functions refers to
the search for orientation and position of the phasing model (P2) in the unit cell
of ALBP (Section 6.5.4, p. 139). The subsequent paper provides more details.
8.3.3 Annotated excerpts from the full structure-determination
(4/92) paper
All reprinted parts of this paper (cited in Footnote 2) appear with the permission
of Professor Leonard J. Banaszak and the American Chemical Society, publisher
of Biochemistry.
In April 1992, the structure determination paper appeared in Biochemistry. This
paper contains a full description of the experimental work, and a complete analysis
of the structure. The following is from the 4/92 paper, “Abstract” section:
Adipocyte lipid-binding protein (ALBP) is the adipocyte member of an
intracellular hydrophobic ligand-binding protein family. ALBP is phosphorylated by the insulin receptor kinase upon insulin stimulation.The
crystal structure of recombinant murine ALBP has been determined
and refined to 2.5 Å.The final R-factor for the model is 0.18 with good
canonical properties.
A 2.5-Å model refined to an R-factor of 0.18 should be a detailed model. “Good
canonical properties” means good agreement with accepted values of bond lengths,
bond angles, and planarity of peptide bonds.
The following excerpts are from the “Materials and Methods” section of the
4/92 paper:
Crystals and X-ray Data Collection. Detailed information concerning
protein purification, crystallization, and X-ray data collection can be
3Y19 is tyrosine 19, a residue considered important to the function of ALBP.
Section 8.3 Reading a crystallography paper 199
found in a previous report (Xu et al., 1991) and will be mentioned here
in summary form. Recombinant murine apo-ALBP crystallizes in the
orthorhombic space group P 212121 with the following unit cell dimensions a =34.4 Å, b =54.8 Å, and c =76.3 Å. The asymmetric unit
contains one molecule with a molecular weight of 14,500. The entire
diffraction data set was collected on one crystal. In the resolution range
∞–2.5 Å, 5115 of the 5227 theoretically possible reflections were
measured. Unless otherwise noted the diffraction data with intensities
greater than 2σ were used for structure determination and refinement. As can be seen in Table 8.2, this included about 96% of the
measured data.
This section reviews briefly the results of the preliminary paper. Full data
collection from a single crystal was relatively rare at the time of this paper, but with
cryocrystallography and more powerful X-ray sources, it is almost the rule today.
In the early stages of the work, reflections weaker than two times the standard
deviation for all reflections (2σ ) were omitted from Fourier syntheses, because of
greater uncertainty in the measurements of weak reflections. Table 8.2 is discussed
later. The diffractometer software computes the number of reflections available at
2.5-Å resolution by counting the number of reciprocal-lattice points that lie within
a sphere of radius [1/(2.5 Å)], centered at the origin of the reciprocal lattice. This
number is roughly equal to the number of reciprocal unit cells within the 1/(2.5 Å)
sphere, which is, again roughly, the volume of the sphere (Vrs) divided by the
volume of the reciprocal unit cell (Vrc). The volume of the reciprocal unit cell
is the inverse of the real unit-cell volume V . So the number of reflections available at 2.5-Å resolution is approximately (Vrs) · (V ). Because of the symmetry
of the reciprocal lattice and of the P 212121 space group, only one-eighth of the
reflections are unique (Sec. 4.3.7, p. 88). So the number of unique reflections is
approximately (Vrs) · (V )/8, or
4
3
π
(
1
2.5 Å
)3 (
1.44 × 105 Å3
)
8
= 4825 reflections.
The 8% difference between this result and the stated 5227 reflections is due to
the approximations made here and to the sensitivity of the calculation to small
round-off in unit-cell dimensions.
Molecular Replacement. The tertiary structure of crystalline ALBP
was solved by using the molecular replacement method incorporated into the XPLOR computer program (Brunger et al., 1987). The
refined crystal structure of myelin P2 protein without solvent and
fatty acid was used as the probe structure throughout the molecular replacement studies. We are indebted to Dr. A. Jones and his
colleagues for permission to use their refined P2 coordinates before
publication.
200 Chapter 8 A User’s Guide to Crystallographic Models
Myelin P2 coordinates were not yet available from the Protein Data Bank and
were obtained directly from the laboratory in which the P2 structure was determined. In this project, the search for the best orientation and position of P2 in the
ALBP unit cell was divided into three parts: a rotation search to find promising
orientations, refinement of the most promising orientations to find the best orientation, and a translation search to find the best position. Here are the details of the
search:
(1) Rotation Search. The rotation search was carried out using the
Patterson search procedures in XPLOR. The probe Patterson maps
were computed from structure factors calculated by placing the P2
coordinates into an orthorhombic cell with 100-Å edges. One thousand
highest Patterson vectors in the range of 5–15 Å were selected and
rotated using the pseudoorthogonal Eulerian angles (θ+, θ2, θ−) as
defined by Lattman (1985). The angular search interval for θ2 was
set to 2.5o; intervals for θ+ and θ− are functions of θ2. The rotation
search was restricted to the asymmetric unit θ− = 0 − 180o, θ2 =
0−90o, θ+ = 0−720o for the P 212121 space group (Rao, et al., 1980).
XPLOR produces a sorted list of the correlation results simplifying final
interpretation (Brunger 1990).
XPLOR is a package of refinement programs that includes powerful procedures
for energy refinement by simulated annealing, in addition to more traditional tools
like least-squares methods and molecular-replacement searches. The package is
available for use on many different computer systems.
The P2 phasing model is referred to here as the probe. For the rotation search,
the probe was placed in a unit cell of arbitrary size and Fcalcs were obtained from
this molecular model, using Eq. (5.15). Then a Patterson map was computed
from these Fcalcs using Eq. (6.10), p. 125. Recall that Patterson maps reflect the
molecule’s orientation but not its position. All peaks in the Patterson map except the
strongest 1000 were eliminated. Then the resulting simplified map was compared
to a Patterson map calculated from ALBP reflection intensities. The probe Patterson was rotated in a three-dimensional coordinate system to find the orientation that
best fit the ALBP Patterson, as illustrated in Fig. 6.18, p. 141 (the system of angles
employed here and that shown in the figure are just two of at least nine different
ways of specifying angles in a rotation search). The search was monitored by a rotation function like Eq. (6.17), p. 142, producing a plot of the angles versus a criterion
of coincidence between peaks in the two Patterson maps (see page 139). Peaks
in the rotation function occur at sets of angles where many coincidences occur.
The coincidences are not perfect because there is a finite interval between angles
tested, and the exact desired orientation is likely to lie between test angles. The
interval is made small enough to avoid missing promising orientations altogether.
(2) Patterson Correlation Refinement. To select which of the orientations determined from the rotation search is the correct solution a
Patterson correlation refinement of the peak list of the rotation function
was performed. This was carried out by minimization against a target
Section 8.3 Reading a crystallography paper 201
function defined by Brunger (1990) and as implemented in XPLOR.
The search model, P2, was optimized for each of the selected peaks
of the rotation function.
As discussed later in the “Results” section, the rotation function contains many
peaks. The strongest 100 peaks were selected, and each orientation was refined by
least squares to produce the best fit to the ALBP Patterson map. For each refined
orientation, a correlation coefficient was computed. The orientation giving the
highest correlation coefficient was chosen as the best orientation for the phasing
model.
(3) Translation Search. A translation search was done by using the P2
probe molecule oriented by the rotation function studies and refined
by the Patterson correlation method.The translation search employed
the standard linear correlation coefficient between the normalized
observed structure factors and the normalized calculated structure
factors (Funinaga & Read, 1987; Brunger, 1990). X-ray diffraction
data from 10–3 Å resolution were used. Search was made in the
range x=0-0.5, y=0-0.5, and z=0-0.5, with the sampling interval
0.0125 of the unit cell length.
The last step in molecular replacement was to find the best position for the probe
molecule in the ALBP unit cell. The P2 orientation obtained from the rotation
search and refinement was tried in all unique locations at intervals of one-eightieth
(0.0125) of the unit-cell axis lengths. The symmetry of the P 212121 unit cell
allowed this search to be confined to the region bound by one-half of each cell
axis. The total number of positions tested was thus (40)(40)(40) or 64,000. For
each position, Fcalcs were computed [Eq. (5.15)] from the P2 model and their
amplitudes are compared with the |Fobs|s from the ALBP native data set. An
unspecified correlation coefficient, probably similar to an R-factor, was computed
for each P2 position, and the position that gave P2 |Fcalc|s in best agreement
with ALBP |Fobs|s was chosen as the best position for P2 as a phasing model.
The starting phase estimates for the refinement were thus the phases of Fcalcs
computed [Eq. (5.15)] from P2 in the final orientation and position determined by
the three-stage molecular-replacement search.
Structure Refinement. The refinement of the structure was based
on an energy function approach (Brunger et al., 1987): arbitrary
combinations of empirical and effective energy terms describing crystallographic data as implemented in XPLOR.Molecular model building
was done on an IRIS Workstation (Silicon Graphics) with the software
TOM, a version of FRODO (Jones, 1978).
The initial model of ALBP was built by simply putting the amino acid
sequence of ALBP into the molecular structure of myelin P2 protein.
After a 20-step rigid-body refinement of the positions and orientations
of the molecule, crystallographic refinement with simulated annealing
was carried out using a slow-cooling protocol (Brunger et al., 1989,
202 Chapter 8 A User’s Guide to Crystallographic Models
Figure 8.4  ALPB electron-density map calculated with molecular-replacement
phases before any refinement, shown with the final model. Compare with Fig. 2.3, page 11,
which shows the final, much improved, electron-density map in the same region.
1990).Temperature factor refinement of grouped atoms, one for backbone and one for side-chain atoms for each residue, was initiated after
the R-factor dropped to 0.249.
The first electron-density map was computed [Eq. (7.03), p. 150] with |Fobs|s
from the ALBP data set and αcalcs from the oriented P2 molecule. Fig. 8.4 shows
a small section of this map superimposed upon the final model.
An early map like Fig. 8.4, computed from initial phase estimates, harbors many
errors, where the map does not agree with the model ultimately derived from
refinement. In this section, you can see both false breaks and false connections in
the density. For example, there are breaks in density at C-β of the phenylalanine
residue (side chain ending with six-membered ring) on the right, and along the
protein backbone at the upper left. The lobe of density corresponding to the valine
side chain (center front) is disconnected and out of place. There is a false connection
between density of the carbonyl oxygen (red) at lower left and side chain density
above. Subsequent refinement was aimed at improving this map.
Next, the side chains of P2 were replaced with the side chains of ALBP at corresponding positions in the amino-acid sequence to produce the first ALBP model.
The position and orientation of this model were refined by least squares, treating
the model as a rigid body. Subsequent refinement was by simulated annealing.
At first, all temperature factors were constrained at 15.0 Å2. After the first round
of simulated annealing, temperature factors were allowed to refine for atoms in
groups, one value of B for all backbone atoms within a residue and another for
side-chain atoms in the residue.
The new coordinates were checked and adjusted against a (2 |Fo| −
|Fc|) and a (|Fo| − |Fc|) electron density map, where |Fo| and |Fc|
are the observed and calculated structure factor amplitudes. Phases
are calculated from the crystal coordinates. The Fourier maps were
Section 8.3 Reading a crystallography paper 203
calculated on a grid corresponding to one-third of the high-resolution
limit of the input diffraction data. All residues were inspected on the
graphics system at several stages of refinement. The adjustments
were made on the basis of the following criteria: (a) that an atom
was located in low electron density in the (2 |Fo| − |Fc|) map or
negative electron density in the (|Fo| − |Fc|) map; (b) that the parameters for the ,  angles placed the residue outside the acceptable
regions in the Ramachandran diagram. Iterative refinement and model
adjustment against a new electron density map was carried out until
the R-factor appeared unaffected. Isotropic temperature factors for
individual atoms were then included in the refinement.
In between rounds of computerized refinement, maps were computed using
|Fobs|s from the ALBP data set and αcalcs from the current model [taken from
|Fcalc|s computed by Eq. (5.15)]. The model was corrected where the fit to maps
was poor, or where the Ramachandran angles  and  were forbidden. Notice
that the use of 2Fo −Fc and Fo −Fc maps [Eq. (7.5) and Eq. (7.4)] is as described
in Sec. 7.4.2, p. 154. When alternating rounds of refinement and map fitting
produced no further improvement in R-factor, temperature factors for each atom
were allowed to refine individually, leading to further decrease in R.
The next stage of the crystallographic study included the location of
solvent molecules. They were identified as well-defined peaks in the
electron-density maps within hydrogen-bonding distance of appropriate protein atoms or other solvent atoms. Solvent atoms were
assigned as water molecules and refined as oxygen atoms. Those
that refined to positions too close to other atoms, ended up located
in low electron density, or had associated temperature factors greater
than 50 Å2 were removed from the coordinate list in the subsequent
stage. The occupancy for all atoms, including solvent molecules,
was kept at 1.0 throughout the refinement. Detailed progress of the
crystallographic refinement is given in Table 8.2, p. 204.
Finally, ordered water molecules were added to the model where unexplained
electron-density was present in chemically feasible locations for water molecules.
Temperature factors for these molecules (treated as oxygen atoms) were allowed to
refine individually. If refinement moved these molecules into unrealistic positions
or increased their temperature factors excessively, the molecules were deleted
from the model. Occupancies were constrained to 1.0 throughout the refinement.
This means that B values reflect both thermal motion and disorder (Sec. 8.2.3,
p. 185). Because all B values fall into a reasonable range, the variation in B
can be attributed to thermal motion. Table 8.2 (p. 204) shows the progress of the
refinement. Note that R drops precipitously in the first stages of refinement after
ALBP side chains replace those of P2. Note also that R and the deviations from
ideal bond lengths, bond angles, and planarity of peptide bonds decline smoothly
throughout the later stages of refinement. The small increase in R at the end is due
to inclusion of weaker reflections in the final round of simulated annealing.
204 Chapter 8 A User’s Guide to Crystallographic Models
TABLE 8.2  Progress of Refinement
RMS deviations
Bond Bond
Number of Solvent length angle Planarity
Stagea reflections R-factor B(Å2) included (Å) (deg) (deg)
1 2976 0.458 15.0 0.065 4.12 9.015
2 2976 0.456 15.0 0.065 4.12 9.012
3 4579 0.235 Group 0.019 3.17 1.506
4
5 4579 0.220 Indiv. 0.018 3.77 1.408
6
7 4579 0.197 Indiv. 31 0.018 3.73 1.366
8
9 4579 0.172 Indiv. 88 0.016 3.47 1.139
10
11 4773 0.183 Indiv. 69 0.017 3.46 1.070
Reprinted with permission from Z. Xu et al. (1992). Biochemistry 31, 3484–3492. Copyright 1992 American
Chemical Society.
aKey to stages of refinement:
Stage Action
1 Starting model
2 Rigid-body refinement
3 Simulated annealing
4 Model rebuilt using (2Fo − Fc) and (Fo − Fc) electron-density maps
5 Simulated annealing
6 Model rebuilt using (2Fo − Fc) and (Fo − Fc) electron-density maps, H2O included
7 Simulated annealing
8 Model rebuilt using (2Fo − Fc) and (Fo − Fc) electron-density maps, H2O included
9 Simulated annealing
10 Model rebuilt using (2Fo − Fc) and (Fo − Fc) electron-density maps, H2O included
11 Simulated annealing
The following excerpts are from the “Results” section of the 4/92 paper:
Molecular Replacement. From the initial rotation search, the 101
highest peaks were chosen for further study. These are shown in
Fig. 8.4, p. 206. The highest peak of the rotation function had a value
4.8 times the standard deviation above the mean and 1.8 times the
standard deviation above the next highest peak. The orientation was
consistently the highest peak for diffraction data within the resolution
ranges 10–5, 7–5, and 7–3 Å. Apart from peak number 1, six strong
peaks emerged after PC4 refinement, as can be seen in Fig. 8.4b,
p. 206. These peaks all corresponded to approximately the same orientation as peak number 1.Three of them were initially away from that
orientation and converged to it during the PC refinement.
4Patterson correlation.
Section 8.3 Reading a crystallography paper 205
A translation search as implemented in XPLOR was used to find
the molecular position of the now oriented P2 probe in the ALBP
unit cell. Only a single position emerged at x = 0.250, y=0.425,
z = 0.138 with a correlation coefficient of 0.419. The initial R-factor for
the P2 coordinates in the determined molecular orientation and position was 0.470 including X-ray data in the resolution range of 10–3 Å.
A rigid-body refinement of orientation and position reduced the starting R-factor to only 0.456, probably attesting to the efficacy of the
Patterson refinement in XPLOR.
In Fig. 8.5a, the value of the rotation function, which indicates how well the
probe and ALBP Patterson maps agree with each other, is plotted vertically against
numbers assigned to the 101 orientations that produced best agreement. Then each
of the 101 orientations were individually refined further, by finding the nearby
orientation having maximum value of the rotation function. In some cases, different
peaks refined to the same final orientation. Each refined orientation of the probe
received a correlation coefficient that shows how well it fits the Patterson map of
ALBP. The orientation receiving the highest correlation coefficient was taken as
the best orientation of the probe, and then used to refine the position of the probe
in the ALBP unit cell. The orientation and position of the model obtained from
the molecular replacement search was so good that refinement of the model as a
rigid body produced only slight improvement inR. The authors attribute this to the
effectiveness of the Patterson correlation refinement of model orientation, stage
two of the search.
Refined Structure of apo-ALBP. The refined ALBP structure has
a R-factor 0.183 when all observed X-ray data (4773 reflections)
between 8.0 and 2.5 Å are included. The rms deviation of bond
lengths, bond angles, and planarity from ideality is 0.017 Å, 3.46o, and
1.07o, respectively. An estimate of the upper limit of error in atomic
coordinates is obtained by the method of Luzzati (1952). Figure 8.6
summarizes the overall refined model.
The plot presented in Fig. 8.6a suggests that the upper limit for
the mean error of the refined ALBP coordinates is around 0.34 Å.The
mean temperature factors for main-chain and side-chain atoms are
plotted in Fig. 8.6b.
The final R-factor and structural parameters exceed the standards described in
Sec. 8.1, p. 179 and attest to the high quality of this model. Atom locations are
precise to an average of 0.34 Å, about one-fifth of a carbon-carbon covalent bond
length. The plot of temperature factors shows greater variability and range for
side-chain atoms, as expected, and shows no outlying values. The model defines
the positions of all amino-acid residues in the protein.
Careful examination of (2 |Fo| − |Fc|) and (|Fo| − |Fc|) maps at
each refinement step led to the conclusion that no bound ligand was present. There was no continuous positive electron density present near the ligand-binding site as identified in both P2
206 Chapter 8 A User’s Guide to Crystallographic Models
Figure 8.5  Rotation function results: P2 into crystalline ALBP. (a) Plot of the 101
best solutions to the rotation function, each peak numbered in the horizontal direction
(abscissa). The correlation between the Patterson maps of the probe molecule and the
measured ALBP X-ray results are shown in the vertical direction (ordinate) and are given in
arbitrary units. (b) Description of the rotation studies after Patterson correlation refinement.
The peak numbers plotted in both panels (a) and (b) are the same. Reprinted with permission
from Z. Xu et al. (1992) Biochemistry 31, 3484–3492. Copyright 1992 American Chemical
Society.
Section 8.3 Reading a crystallography paper 207
Figure 8.6  ALBP refinement results. (a) Theoretical estimates of the rms positional
errors in atomic coordinates according to Luzzati are shown superimposed on the curve for
the ALBP diffraction data. The coordinate error estimated from this plot is 0.25 Å with an
upper limit of about 0.35 Å.(b) Mean values of the main-chain and side-chain temperature
factors are plotted versus the residue number. The temperature factors are those obtained
from the final refinement cycles. Reprinted with permission from Z. Xu et al. (1992)
Biochemistry 31, 3484–3492. Copyright 1992 American Chemical Society.
208 Chapter 8 A User’s Guide to Crystallographic Models
(Jones, et al., 1988) and IFABP (Sacchettini et al., 1989a). The
absence of bound fatty acid in crystalline ALBP is consistent with
the chemical modification experiment which indicates ALBP purified
from E. coli is devoid of fatty acid (Xu et al., 1991). The final refined
coordinate list includes 1017 protein atoms and 69 water molecules.
The final maps exhibit no unexplained electron density. This is of special concern because ALBP is a ligand-binding protein (its ligand is a fatty acid), and
ligands sometimes survive purification and crystallization, and are found in the
final electron-density map. It is implied by references to apo-protein and holoprotein that attempts to determine the structure of an ALBP-ligand complex were
under way at the time of publication. If it is desired to detect conformational
changes upon ligand binding, then it is crucial to know that no ligand is bound to
this apo-protein, so that conformational differences between apo- and holo-forms,
if found, can reliably be attributed to ligand binding.
To compare apo- and holo-forms of proteins after both structures have been
determined independently, crystallographers often compute difference Fourier
syntheses (Sec. 7.4.2, p. 154), in which each Fourier term contains the structurefactor difference Fholo − Fapo. A contour map of this Fourier series is called a
difference map, and it shows only the differences between the holo- and apoforms. Like the Fo − Fc map, the Fholo − Fapo map contains both positive and
negative density. Positive density occurs where the electron density of the holoform is greater than that of the apo-form, so the ligand shows up clearly in positive
density. In addition, conformational differences between holo and apo-forms result
in positive density where holo-protein atoms occupy regions that are unoccupied
in the apo-form, and negative density where apo-protein atoms occupy regions
that are unoccupied in the holo-form. The standard interpretation of such a map is
that negative density indicates positions of protein atoms before ligand binding,
and positive density locates the same atoms after ligand binding. In regions where
the two forms are identical, Fholo − Fapo = 0, and the map is blank.
Structural Properties of Crystalline ALBP. A Ramachandran plot of
the main chain dihedral angle  and  is shown in Fig. 8.7. In
the refined model, 13 residues have positive  angles, 9 of which
belong to glycine residues. There are 11 glycine residues in ALBP, all
associated with good quality electron density.
Most of the residues having forbidden values of  and  are glycines, represented by small squares in Fig. 8.7, whereas all other amino acids are represented
by + symbols. Succeeding discussion reveals that these unusual conformations
are also found in P2 and other members of this protein family, strengthening
the argument that these conformations are not errors in the model, and suggesting that they might be important to structure and/or function in this family of
proteins.
Figure 2.3, p. 11 shows, at the end of refinement, the same section of map as
in Fig. 8.4. By comparing Fig. 2.3 with Fig. 8.4, you can see that the map errors
Section 8.4 Summary 209
Figure 8.7  Ramachandran plot of the crystallographic model of ALBP, generated
with DeepView (see Chapter 11). The main-chain torsional angle  (N-Cα bond) is plotted versus  (C-Cα bond). The following symbols are used: (+) nonglycine residues; ()
glycine residues. The enclosed areas of the plot show sterically allowed angles for nonglycine residues. The symbols are colored according to their inclusion in secondary
structural elements: red, alpha helix; yellow, beta sheet; gray, coil.
described earlier were eliminated, and that the map is a snug fit to a chemically,
stereochemically, and conformationally realistic model.
8.4 Summary
All crystallographic models are not equal. The noncrystallographer can assess
model quality by carefully reading original publications of a macromolecular
structure and by using the latest online validation tools. The kind of reading and
210 Chapter 8 A User’s Guide to Crystallographic Models
interpretation implied by my annotations is essential to wise use of models. Don’t
get me wrong. There is no attempt on the part of crystallographers to hide the
limitations of models. On the contrary, refinement papers often represent almost
heroic efforts to make plain what the final model says and leaves unsaid. These
efforts are in vain if the reader does not understand them, or worse, never reads
them. These efforts are often undercut by the simple power of the visual model.
The brightly colored stereo views of a protein model, which are in fact more akin
to cartoons than to molecules, endow the model with a concreteness that exceeds
the intentions of the thoughtful crystallographer. It is impossible for the crystallographer, with vivid recall of the massive labor that produced the model, to forget
its shortcomings. It is all too easy for users of the model to be unaware of them. It is
also all too easy for the user to be unaware that, through temperature factors, occupancies, undetected parts of the protein, and unexplained density, crystallography
reveals more than a single molecular model shows.
Even the highest-quality model does not explain itself. If I showed you a perfect
model of a protein of unknown function, it is unlikely that you could tell me what
it does, or even pinpoint the chemical groups critical to its action. Using a model
to explain the properties and action of a protein means bringing the model to bear
upon all the other available evidence. This involves gaining intimate knowledge
of the model, a task roughly as complex as learning your way around a small city.
In Chapter 11, I will discuss the exploration of macromolecular models by computer graphics. But first, I must carry out two other tasks. In the next chapter, I will
build on your understanding of X-ray diffraction to introduce you to other means
of structure determination using diffraction, including X-ray diffraction of fibers
and powders, and diffraction by neutrons and electrons. Then, in Chapter 10, I will
briefly introduce other, noncrystallographic, methods of structure determination,
in particular NMR and homology modeling. With each method, I will discuss how
to assess the quality of the resulting models, by analogy with the criteria described
in this chapter.
 Chapter 9
Other Diffraction Methods
9.1 Introduction
The same principles that underlie single-crystal X-ray crystallography make other
kinds of diffraction experiments understandable. In this chapter, I provide brief,
qualitative descriptions of other diffraction methods. First is X-ray diffraction
by fibers rather than crystals, which reduces reciprocal space from three dimensions to two. Next we go down to one dimension, with diffraction by amorphous
materials like powders and solutions. Then I will look at diffraction using other
forms of radiation, specifically neutrons and electrons. I will show how each type
of diffraction experiment provides structural information that can complement or
supplement information from single-crystal X-ray diffraction. Finally, returning to
X-rays and crystals, I will discuss Laue diffraction, which allows the collection of
a full diffraction data set from a single brief exposure of a crystal to polychromatic
X-radiation. Laue diffraction opens the door to time-resolved crystallography,
yielding crystallographic models of intermediate states in chemical reactions. For
each of the crystallographic methods, I provide references to research articles that
exemplify the method.
9.2 Fiber diffraction
Many important biological substances do not form crystals. Among these are most
membrane proteins and fibrous materials like collagen, DNA, filamentous viruses,
and muscle fibers. Some membrane proteins can be crystallized in matrices of lipid
and studied by X-ray diffraction (Sec. 3.3.4, p. 41), or they can be incorporated
into lipid films (which are in essence two-dimensional crystals) and studied by
211
212 Chapter 9 Other Diffraction Methods
electron diffraction. I will discuss electron diffraction later in this chapter. Here
I will examine diffraction by fibers.
Like crystals, fibers are composed of molecules in an ordered form. When irradiated by an X-ray beam perpendicular to the fiber axis, fibers produce distinctive
diffraction patterns that reveal their dimensions at the molecular level. Because
many fibrous materials are polymeric and of known chemical composition and
sequence, their molecular dimensions are sometimes all that is needed to build a
feasible model of their structure.
Some materials (for example, certain muscle proteins) form fibers spontaneously
or are naturally found in fibrous form. Many other polymeric substances, like
DNA, can be induced into fibers by pulling them from an amorphous gel with
tweezers or a glass rod. For data collection, the fiber is simply suspended between
a well-collimated X-ray source and a detector (Fig. 9.1).
The order in a fiber is one-dimensional (along the fiber) rather than threedimensional, as in a crystal. You can think of molecules in a fiber as being stretched
Figure 9.1  Fiber diffraction. Molecules in the fiber are oriented parallel to the beam
axis but aligned at random. X-ray beams emerging from the fiber strike the detector in layer
lines perpendicular to the fiber axis.
Section 9.2 Fiber diffraction 213
out parallel to the fiber axis but having their termini occurring at random along
the fiber, as shown in the expanded detail of the fiber in Fig. 9.1. Because the
X-ray beam simultaneously sees all molecules in all possible rotational orientations about the fiber axis, Bragg reflections from a fiber are cylindrically averaged,
and irradiation of the fiber by a beam perpendicular to the fiber axis gives a complete, but complex, two-dimensional diffraction pattern from a single orientation
of the fiber.
Fibers can be crystalline or noncrystalline. Crystalline fibers are actually composed of long, thin microcrystals oriented with their long axis parallel to the fiber
axis. When a crystalline fiber is irradiated with X-rays perpendicular to the fiber
axis, the result is the same as if a single crystal were rotated about its axis in the
X-ray beam during data collection, sort of like a rotation photograph taken over
360o instead of the usual very small rotation angle (Sec. 4.3.4, p. 80). All Bragg
reflections are registered at once, on layer lines perpendicular to the fiber axis. All
fiber diffraction patterns have two mirror planes, parallel (the meridian) and perpendicular (the equator) to the fiber axis. Many of the reflections overlap, making
analysis of the diffraction pattern very difficult.
In noncrystalline fibers, all the molecules (as opposed to oriented groups of them
in microcrystals) are parallel to the fiber axis but aligned along the axis at random.
This arrangement gives a somewhat simpler diffraction pattern, also consisting of
layer lines, but with smoothly varying intensity rather than distinct reflections. In
the diffraction patterns from both crystalline and noncrystalline materials, spacings
of layer lines are related to the periodicity of the individual molecules in the fiber,
as I will show.
A simple and frequently occurring structural element in fibrous materials is the
helix. I will use the relationship between the dimensions of simple helices and
that of their diffraction patterns to illustrate how diffraction can reveal structural
information. As a further simplification, I will assume that the helix axis is parallel
to the fiber axis. As in all diffraction methods, the diffraction pattern is a Fourier
transform of the object in the X-ray beam, averaged over all the orientations present
in the sample. In the case of fibers, this means that the transform is averaged
cylindrically, around the molecular axis parallel to the fiber axis.
Figure 9.2 shows some simple helices and their transforms. The transform of
the helix in Fig. 9.2a exhibits an X pattern that is always present in transforms of
helices. I will explain the mathematical basis of the X pattern later. Although each
layer line looks like a row of reflections, it is actually continuous intensity. This
would be apparent if the pattern were plotted at higher overall intensity. The layer
lines are numbered with integers from the equator (l = 0). Because of symmetry,
the first lines above and below the equator are labeled l = 1, and so forth.
Compare helix (a) with (b), in which the helix has the same radius, but a longer
pitch P (peak-to-peak distance). Note that the layer lines for (b) are more closely
spaced. The layer-line spacing is inversely proportional to the helix pitch. The
relationship is identical to that in crystals between reciprocal-lattice spacing and
unit-cell dimensions [(Eq. (4.10), p. 87]. As a result, precise measurement of
layer-line spacing allows determination of helix pitch.
214 Chapter 9 Other Diffraction Methods
Figure 9.2  Helices and their Fourier transforms. (a) Simple, continuous helix. The
first intensity peaks from the centers of each row form a distinctive X pattern. (b) Helix
with longer pitch than (a) gives smaller spacing between layer lines. (c) Helix with larger
radius than (a) gives narrower X pattern. (d) Helix of same dimensions as (a) but composed
of discrete objects gives overlapping X patterns repeated along the meridian.
Section 9.2 Fiber diffraction 215
Helix (c) has the same pitch as (a) but a larger radius r . Notice that the X pattern
in the transform is narrower than that of helices (a) and (b), which have the same
radius. The angle formed by the branches of the X with the meridian, shown as the
angle δ, is determined by the helix radius. But it appears at first that the relationship
between δ and helix radius must be more complex, because angles δ for helices (a)
and (b), which have the same radius, appear to be different. However, we define
δ as the angle whose tangent is the distance w from the meridian to the center of
the first intensity peak divided by the layer-line number l. You can see that the
distance w at the tenth layer line is the same in (a) and (b). Defined in this way,
and measured at relatively large layer-line numbers (beyond which the tangent of
an angle is simply proportional to the length of the side opposite), δ is inversely
proportional to the helix radius. Because the layer-line spacing is the same in (a)
and (c), it is clear that an increase in radius decreases δ, and that we can determine
the helix radius from δ.
Helix (d) has the same pitch and radius as helix (a), but is a helix of discrete
objects or “repeats,” like a polymeric chain of repeating subunits. The transform
appears at first to be far more complex, but it is actually only slightly more so. It
is merely a series of overlapping X patterns distributed along the meridian of the
transform. To picture how multiple X patterns arise from a helix of discrete objects,
imagine that the helix beginning with arbitrarily chosen object number 1 produces
the X at the center of the transform. Then imagine that the same helix beginning
with object number 2 produces an X of its own. The distance along the meridian
between centers of the two Xs is inversely proportional [Eq. (4.10) again] to the
distance between successive discrete objects in the helix. So careful measurement
of the distance between successive meridional “reflections” (remember that the
intensities on a layer line are actually continuous) allows determination of the
distance between successive subunits of the helix. In a polymeric helix like that of
a protein or nucleic acid, this parameter is called the rise-per-residue orp. Dividing
the pitch P by the rise-per-residue p gives the number of residues per helical turn
(rise-per-turn divided by rise-per-residue gives residues-per-turn).
In addition, for the simplest type of discrete helix, in which there is an integral
number of residues per turn of the helix, this integer P/p is the same as number
l of the layer line on which the first meridional intensity peak occurs. Note that
helix d contains exactly six residues per turn, and that the first meridional intensity
peak above or below the center of the pattern occurs on layer line l = 6. To review,
the layer-line spacing Z is proportional to 1/P , and the distance from the origin
to the first meridional reflection is proportional to 1/p.
If the number of residues per turn is not integral, then the diffraction pattern is
much more complex. For example, a protein alpha helix has 3.6 residues per turn,
which means 18 residues in five turns. The diffraction pattern for a discrete helix
of simple objects (say, points) with these dimensions has layer lines at all spacings
Z = (18m + 5α)/5P , where m and α are integers, and the diffraction pattern
will repeat every 18 layer lines. But of course, protein alpha helix does not contain 3.6 simple points per turn, but instead 3.6 complex groups of atoms per turn.
Combined with the rapid drop-off of diffraction intensity at higher diffraction
216 Chapter 9 Other Diffraction Methods
angles, this makes for diffraction patterns that are too complex for detailed
analysis.
Now let’s look briefly at just enough of the mathematics of fiber diffraction
to explain the origin of the X patterns. Whereas each reflection in the diffraction
pattern of a crystal is described by a Fourier sum of sine and cosine waves, each
layer line in the diffraction pattern of a noncrystalline fiber is described by one or
more Bessel functions, graphs that look like sine or cosine waves that damp out
as they travel away from the origin (Fig. 9.3). Bessel functions appear when you
apply the Fourier transform to helical objects. A Bessel function is of the form
Jα(x) =
∞∑
n=0
(−1)n
n!(n+ α)! (x/2)
(2n+α). (9.1)
Figure 9.3  (a) Bessel functions J (x) of order α = 0 (red), 1 (green), and 2 (blue),
showing positive values of x only. Note that, as the order increases, the distance to the
first peak increases. (b) Enlargement of a few layer lines from Fig. 9.2a, showing the
correspondence between diffraction intensities and the squares of Bessel functions having
the same order as the layer-line number.
Section 9.2 Fiber diffraction 217
The variable α is called the order of the function, and the values of n are integers.
To plot the Bessel function of order zero, you plug in the values α = 0 and n = 0
and then plot J as a function of x over some range −x to +x. Next you plug in
α = 0 and n = 1, plot again, and add the resulting curve to the one for which
n = 0, just as curves were added together to give the Fourier sum in Fig. 2.16, p.
25. Continuing in this way, you find that eventually, for large values of n, the new
curves are very flat and do not change the previous sum. Bessel functions of orders
zero, one, and two, for positive values of x, are shown in Fig. 9.3a. Notice that
as the order increases, the position of the first peak of the function occurs farther
from the origin.
Francis Crick showed in his doctoral dissertation that in the transform of a
continuous helix, the intensity along a layer line is described by the square of the
Bessel function whose order α equals the number l of the layer line, as shown in
Fig. 9.3b, which is an enlargement of three layer lines from the diffraction pattern
of Fig. 9.2a. Thus, the intensity of the central line, layer line zero, varies according
to [(J0(x)]2, which is the square of Eq. (9.1) with α = 0 (red). The intensity of
the first line above (or below) center varies according to [(J1(x)]2 (green), and so
forth. This means that, for a helix, the first and largest peak of intensity lies farther
out from the meridian on each successive layer line. The first peaks in a series of
layer lines thus form the X pattern described earlier. The distance to the first peak
in each layer line decreases as the helix radius increases, so thinner helices give
wider X patterns.
For helices with a nonintegral number of residues per turn, the intensity functions, like the layer-line spacing, are also more complex, with two or more Bessel
functions contributing to the intensities on each layer line. For the α helix, with 18
residues in five turns, the Bessel functions that contribute to layer line l are those
for which α can be combined with some integral value of m (positive, negative or
zero) to make l = 18α + 5m an integer. For example, for layer line l = 0, one
solution to this equation is α = m = 0, so J0(x) contributes to layer line 0. So also
does J5(x), because α = 5, m = −18 also gives l = 0. You can use l = 18α+5m
also to show that J2(x) (m = −7) and J7(x) (m = −25) contribute to layer line
l = 1, but that J1(x) does not.
Probably the most famous fiber diffraction patterns are those of A-DNA and
B-DNA obtained by Rosalind Franklin and shown in Fig. 9.4. Franklin’s sample
of A-DNA was microcrystalline, so its diffraction pattern (a) contains discrete
reflections, many of them overlapping at the higher diffraction angles. Her B-DNA
was noncrystalline, so the intensities in its diffraction pattern (b) vary smoothly
across each layer line. Considering the B-helix, the narrow spacing between layer
lines is inversely proportional to its 34-Å pitch. The distance from the center to the
strong meridional intensity near the edge of the pattern is inversely proportional to
the 3.4-Å rise per subunit (a nucleotide pair, we now know). Dividing pitch by rise
gives 10 subunits per helical turn, as implied by the strong meridional intensity at
the tenth layer line. Finally, the angle of the X pattern implies a helix radius of 20 Å.
Francis Crick recognized that Franklin’s data implied a helical structure
for B-DNA, a wonderful example of Pasteur’s dictum: “Chance favors the
218 Chapter 9 Other Diffraction Methods
Figure 9.4  Fiber diffraction patterns from A-DNA (left half of figure) and B-DNA
(right). A-DNA was microcrystalline and thus gave discrete, but overlapping, Bragg reflections. B-DNA was noncrystalline and thus gave continuous variation in intensity along each
layer line. Image kindly provided by Professor Kenneth Holmes.
prepared mind.” Using helical parameters deduced from the pattern, knowing the
chemical composition of DNAand the structures of DNA’s molecular components,
and finally, using the tantalizing observation that certain pairs of bases occur in
equimolar amounts, James Watson was able to build a feasible model of B-DNA.
Why could Watson and Crick not simply back-transform the diffraction data,
get an electron-density map, and fit a model to it? As in any diffraction experiment, we obtain intensities, but not phases. In addition, if we could somehow learn
the phases, the back-transform would be the electron density averaged around the
molecular axis parallel to the fiber axis, which would show only how electron
density varies with distance from the center of the helix. Instead of map interpretation, structure determination by fiber diffraction usually entails inferring the
dimensions of chains from the diffraction pattern and then building models using
these dimensions plus prior knowledge: the known composition and the constraints
on what is stereochemically allowable. As Watson’s and Crick’s success showed,
for a helical substance, being able to deduce from diffraction the pitch, radius, and
number of residues per helical turn puts some very strong constraints on a model.
The Fourier transform does, however, provide a powerful means of testing proposed models. With a feasible model in hand, researchers compute its transform
and compare it to the pattern obtained by diffraction. If the fit is not perfect,
they adjust the model, and again compare its transform with the X-ray pattern.
They repeat this process until a model reproduces in detail the experimental
Section 9.3 Diffraction by amorphous materials (scattering) 219
diffraction pattern. Thus you can see why a very successful practitioner in this
field, who may prefer to remain nameless, said, “Fiber diffraction is not what
you’d do if you had a choice.” Sometimes it is simply the only way to get structural
information from diffraction.
For an example of structure analysis by fiber diffraction, see H. Wang and G.
Stubbs, Structure determination of cucumber green mottle mosaic virus by X-ray
fiber diffraction. Significance for the evolution of tobamoviruses, J. Mol. Biol.
239, 371–384, 1994.
9.3 Diffraction by amorphous materials
(scattering)
With fibers, the diffraction pattern, and hence any structural information contained
therein, is averaged cylindrically about the molecular axis parallel to the fiber axis.
This means that the transform of the diffraction pattern (computation of which
would require phases) is an electron-density function showing only how electron
density varies with distance from the molecular axis. For a helix of points aligned
with the fiber axis, there would be a single peak of density at the radius of the
helix. For a polyalanine helix (Fig. 9.5), there would be a density peak for the
backbone, because atoms C, O, and CA are all roughly the same distance from
the center of the helix (inner circle), and a second peak for the beta carbons, which
are farther from the helix axis (outer circle). This averaging greatly reduces the
structural information that can be inferred from the diffraction pattern, but it does
imply that distance information is present, despite rotational averaging.
Imagine now a sample, such as a powder or solution, in which all the
molecules are randomly oriented. Diffraction by such amorphous samples is
Figure 9.5  Polyalanine α helix, viewed down the helix axis (stereo). An electrondensity map averaged around this axis would merely show two circular peaks of electron
density, one at the distance of the backbone atoms (inner circle) and another at the distance
of the β carbons (outer circle).
220 Chapter 9 Other Diffraction Methods
usually called scattering. The diffraction pattern is averaged in all directions—
spherically—because the X-ray beam encounters all possible orientations of the
molecules in the sample. But the diffraction pattern still contains information about
how electron density varies with distance from the center of the molecules that
make up the sample. Obviously, for complex molecules, this information would
be singularly uninformative. But if the molecule under study contains only a few
atoms, or only a few that dominate diffraction (like metal atoms in a protein), then
it may be possible to extract useful distance information from the way that scattered
X-ray intensity varies with the angle of scattering from the incident beam of radiation. As usual, when we try to extract information from intensity measurements,
we work without knowledge of phases.
I have shown that, in simple systems, Patterson functions can give us valuable
clues about distances, even when we know nothing about phases (see Sec. 6.3.3,
p. 124). Diffraction from the randomly oriented molecules in a solution or powder
would give a spherically averaged diffraction pattern, from which we can compute a spherically averaged Patterson map. Is this map interpretable? As in the
case of heavy-atom derivatives, we can interpret a Patterson map if there are just
a few atoms or a few very strong diffractors. Consider a linear molecule containing only three atoms (Fig. 9.6).We can see what to expect in a Patterson function
computed from diffraction data on this molecule by constructing a Patterson function for the known structure. First, construct a Patterson function for the structure
shown in Fig. 9.6a, using the procedure described in Fig. 6.12, p. 126. The result
is (b). Spherical averaging of (b) gives a set of spherical shells of intensity, with
Figure 9.6  Radial Patterson function. (a) Linear triatomic molecule. (b) Patterson
function constructed from (a). For construction procedure, see Sec. 6.3.3, p. 124 and
Fig. 6.12, p. 126. (c) Cross section of Patterson function (b) averaged by rotation to produce
spherical shells. (d) Typical presentation of radial Patterson function as calculated from
scattering measurements on an amorphous sample.
Section 9.3 Diffraction by amorphous materials (scattering) 221
cross-section shown in (c). This cross section contains information about distances
between the atoms in (a). The radius of the first circle is the bond length r , and the
radius of the second circle is the length of the molecule (2r). A plot of the magnitude of the Patterson function as a function of distance from the origin (d), called a
radial Patterson function, contains peaks that correspond to vectors between atoms.
Furthermore, the intensity of each peak will depend on how many vectors of that
length are present. In the molecule of Fig. 9.6a, there are four vectors of length r
and two vectors of length 2r , so the Patterson peak at r is stronger than the one at
2r . You can see that, for a small molecule, the radial Patterson function computed
from scattering intensities may contain enough information to determine distances
between atoms. For larger numbers of atoms, the radial Patterson function would
contain peaks corresponding to all the interatomic distances.
The radial Patterson function of an amorphous (powder or dilute solution) sample of a protein contains an enormous number of peaks. But imagine a protein
containing a cluster of one or two metal ions surrounded by sulfur atoms. These
atoms may dominate the powder diffraction data, and the strongest peaks in the
radial Patterson function may reveal the distances among the metal ions and sulfur
atoms. Remember that we obtain distance information but no geometry because
diffraction is spherically averaged and all directional information is lost. Sometimes powder or solution diffraction can be used to extract distance information
relating to a cluster of the heavier atoms in a protein. These distances put constraints on models of the cluster. Researchers can compare spherically averaged
back-transforms of plausible models with the experimental diffraction data to guide
improvements in the model, as in fiber diffraction. In some cases, model building
and comparison of model back-transforms with data allows identification of ligand
atoms and estimation of bond distances.
At very small diffraction angles, additional information can be obtained about
the size and shape of a molecule. A detailed treatment of this method, called lowangle scattering, is beyond the scope of this book, but it can be shown that, for very
small angles, the variation in scattering intensity is related to the radius of gyration,
RG, of the molecule under study. The radius of gyration is defined as the root-meansquare average of the distance of all scattering elements from the center of mass
of the molecule. For two proteins having the same molecular mass, the one with
the larger radius of gyration is the more extended or less spherical one. Combined
knowledge of the molecular mass and the radius of gyration of a molecule allows
an estimate of its shape. The precise relationship between scattering intensity and
radius of gyration is
〈I (θ)〉 = n2e(1 − 16π2R2G sin2 θ/3λ2). (9.2)
〈I (θ)〉 is the intensity of scattering at angle θ , ne is the number of electrons in the
molecule, and λ is the X-ray wavelength. Equation (9.2) implies that a graph of
radiation intensity versus sin2 θ has a slope that is directly proportional to the square
of RG. Note also that such a graph can be extrapolated to θ = 0, where the second
term in parentheses disappears, and 〈I (θ)〉 is equal to the square of ne, or roughly,
222 Chapter 9 Other Diffraction Methods
to the square of the molecular mass. So for molecules about which very little is
known, measurement of scattering intensity at low angles provides estimates of
both molecular mass and shape.
This kind of information about mass, shape, and distance can be obtained on
amorphous samples not only from X-ray scattering but also from scattering by
other forms of radiation, including light, and as I will discuss in Sec. 9.4, p. 222,
neutrons. The choice of radiation depends on the size of the objects under study.
The variable-wavelength X-rays available at synchrotron sources give researchers an additional, powerful way to obtain precise distance information from
amorphous samples. At wavelengths near the absorption edge of a metal atom
(Sec. 4.3.2, p. 73), there is rapid oscillation of X-ray absorption as a function of
wavelength. This oscillation results from interference between diffraction from the
absorbing atom and that of its neighbors. The Fourier transform of this oscillation,
in comparison with transforms calculated from plausible models, can reveal information on the number, types, and distances of the neighboring atoms. Distance
information in favorable cases can be much more precise than atomic distances
determined by X-ray crystallography. Thus this information can anticipate or add
useful detail to crystallographic models. The measurement of absorbance as a
function of wavelength is, of course, a form of spectroscopy. In the instance
described here, it is called X-ray absorption spectroscopy, or XAS. Fourier analysis
of near-edge X-ray absorption is called extended X-ray absorption fine structure,
or EXAFS.
For an example of structure analysis by X-ray scattering, see D. I. Svergun,
S. Richard, M. H. Koch, Z. Sayers, S. Kuprin, and G. Zaccai, Protein hydration in
solution: Experimental observation by X-ray and neutron scattering, Proc. Natl.
Acad. Sci. USA 95, 2267–2272, 1998.
9.4 Neutron diffraction
The description of diffraction or scattering as a Fourier transform applies to all
forms of energy that have wave character, including not only electromagnetic
energy like X-rays and light but also subatomic particles, including neutrons and
electrons, which have wavelengths as a result of their motion. The de Broglie
equation (9.3) gives the wavelength λ of a particle of massmmoving at velocity v:
λ = h/mv, (9.3)
where h is Planck’s constant. We can use the de Broglie wavelength to describe
diffraction of particles by matter. In this section, I will describe single-crystal
neutron diffraction and neutron scattering by macromolecules, emphasizing the
type of information obtainable. In the next section, I will apply these ideas to
electron crystallography.
Section 9.4 Neutron diffraction 223
Recall that X-rays are diffracted by the electrons that surround atoms, and that
images obtained from X-ray diffraction show the surface of the electron clouds
that surround molecules. Recall also that the X-ray diffracting power of elements
in a sample increases with increasing atomic number. Neutrons are diffracted by
nuclei, not by electrons. Thus a density map computed from neutron diffraction
data is not an electron-density map, but instead a map of nuclear mass distribution,
a “nucleon-density map” of the molecule (nucleons are the protons and neutrons
in atomic nuclei).
The neutron crystallography experiment is much like X-ray crystallography
(see Figs. 2.6, p. 13, and 2.12, p. 19). A crystal is held in a collimated beam
of neutrons. Diffracted beams of neutrons are detected in a diffraction pattern
that is a reciprocal-lattice sampling of intensities, but as usual not phases, of the
Fourier transform of the average object in the crystal. Structure determination is, in
principle, similar to that in X-ray crystallography, involving estimating the phases,
back-transforming a set of structure factors composed of experimental intensities
and estimated phases, improving the phases, and refining the structure.
There are two common ways to produce a beam of neutrons. One is steady-state
nuclear fission in a reactor, which produces a continuous output of neutrons, some
of which sustain fission, while the excess are recovered as a usable neutron beam.
The second type is a pulsed source in which a cluster of protons or other charged
particles from a linear accelerator are injected into a synchrotron, condensed into
a tighter cluster or pulse, and allowed to strike a target of metal, such as tungsten.
The high-energy particles drive neutrons from the target nuclei in a process called
spallation. Neutrons from both fission and spallation carry too much energy for
use in diffraction, so they are slowed or cooled (“thermalized”) by passage through
heavy water (D2O) at 300◦K, producing neutrons with De Broglie wavelengths
ranging from 1 to 2 Å. This wavelength is in the same range as X-rays used in
crystallography.
Thermal neutrons then enter a collimator followed by a monochromator, which
selects a narrow range of wavelengths to emerge and strike the sample. Monochromators are single crystals of graphite, zinc, or copper. They act like diffraction
gratings to direct neutrons of different energies (and hence, wavelengths) in different directions, as a prism does with light. The collimated, monochromatic neutron
beam is then delivered to the sample mounted on a goniometer, and diffraction is
detected by an area detector (Sec. 4.3.3, p. 77). One common type is an imageplate that employs gadolinium oxide, which absorbs a neutron and emits a gamma
ray, which in turn exposes the image plate.
The great advantage of neutron diffraction is that small nuclei like hydrogen are
readily observed. By comparison with carbon and larger elements, hydrogen is a
very weak X-ray diffractor and is typically not observable in electron-density maps
of proteins. But hydrogen and its isotope deuterium (2H or D) diffract neutrons
very efficiently in comparison with larger elements.
The concept of scattering length is used to compare diffracting power of elements. The scattering length b (do not confuse it with temperature factor B or
the atomic scattering factor f ) is the amplitude of scattering at an angle of zero
224 Chapter 9 Other Diffraction Methods
degrees to the incident beam and is the absolute measure of scattering power of an
atom. Measured diffraction intensities are proportional to b2, which is why X-ray
diffraction from a single heavy atom can be easily detected above the diffraction of all the small atoms in a massive macromolecule. Table 9.1 gives neutron
and X-ray scattering lengths for various elements and allows us to compare the
scattering power of elements in neutron and X-ray diffraction.
Note that, even though H and D are weak X-ray diffractors compared to other
elements in biomolecules (they have so few electrons around them), they are
comparable to other elements when it comes to neutron diffraction. So hydrogen
will be a prominent feature in density maps from neutron diffraction. Note also
that the sign of b is negative for H. This means that H diffracts with a phase that
is opposite to that of other elements. As mentioned before, measured diffraction
intensities are related to b2, so the negative sign of b has no observable effect on
diffraction patterns. But in density maps, H gives negative density, which makes
it stand out. In addition, the large magnitude of the difference between scattering
lengths for H and D allows some powerful ways to use D as a label in diffraction
or scattering experiments.
First, let us consider experiments analogous to single-crystal X-ray crystallography. In most electron-density maps of macromolecules, we cannot observe
hydrogens. Thus we cannot distinguish the amide nitrogen and oxygen in glutamine and asparagine side chains [although online validation tools (p. 189) like
MolProbity can assign them on the theoretical basis of hydrogen-bonding possibilities]. Nor can we determine the locations of hydrogens on histidine side chains,
whose pKa values allow both protonated and unprotonated forms at physiological pH. And we cannot detect critical hydrogens involved in possible hydrogen
bonding with ligands like cofactors and transition-state analogs. Some hydrogens,
including those on hydroxyl OH and amide N, can exchange with hydrogens of
the solvent if they are exposed to solvent, and if they are not involved in tight
TABLE 9.1  X-Ray and Neutron Scattering Lengths of Various Elements
Element X-rays b × 1013(cm) Neutrons b × 1013(cm)
H 2.8∗ −3.74
D 2.8 6.67
C 16.9 6.65
N 19.7 9.40
O 22.5 5.80
P 42.3 5.10
S 45 2.85
Mn 70 −3.60
Fe 73 9.51
Pt 220 9.5
Data from C. R. Cantor and P. R. Schimmel (1980). Bio-physical Chemistry, Part II: Techniques for the study of
Biological Structure and Function. W. H. Freeman and Company, San Francisco, p. 830.
*Value corrected from original.
Section 9.4 Neutron diffraction 225
hydrogen bonds. Distinguishing H from D would mean being able to determine
which hydrogens are exchangeable. Neutron diffraction, it would appear, gives us
a way to answer these questions.
We can collect diffraction intensities from macromolecular crystals, but can we
phase them and thus obtain maps that include clear images of hydrogen atoms?
How about heavy atoms for phasing? According to Table 9.1, there is no such thing
as a heavy atom. In other words, no nucleus diffracts so strongly that we can detect
it above all others in a Patterson map. That rules out MIR, SIR with anomalous
dispersion, and MAD as possible phasing methods. But if we are trying to find
hydrogens in a structure known, or even partially known, from X-ray work, we
have a source of starting phases in hand.
A crystal has the same reciprocal lattice for all types of diffraction, because the
construction of the reciprocal lattice (Sec. 4.2.4, p. 57) does not depend in any
way upon the type of radiation involved. So if we know the positions of all or most
of the nonhydrogen atoms from X-ray structure determination, we can compute
their contributions to the neutron-diffraction phases. These contributions depend
only on atomic positions in the unit cell, and like reciprocal lattice positions, they
are independent of the type and wavelength of radiation. We start phasing by
assigning the final phases computed from the X-ray model of nonhydrogen atoms
to the reflections obtained by neutron diffraction. From a Fourier sum combining
neutron-diffraction intensities and X-ray phases, we compute the first map. Then
we proceed as usual, alternating cycles of examining the map, building a model (in
this case, adding hydrogens whose images we see in the map), back-transforming
the model to get better phases, and so forth. In essence, this is isomorphous
molecular replacement (Sec. 6.5, p. 136), using a hydrogen-free model that is
identical to the fully hydrogenated model we seek.
Neutron diffraction has been used to detect critical hydrogen bonds in proteinligand complexes. For example, the presence of a hydrogen bond between
dioxygen and the distal histidine of myoglobin is known because of neutron
diffraction studies. In highly refined neutron-diffraction models, hydrogen positions are determined as precisely as positions of other atoms. In the best cases,
not only can hydrogens in hydrogen bonds be seen, but it is even possible to tell
whether hydrogen-containing groups like methyl or hydroxyl can rotate. If methyl
groups are conformationally locked, density maps will show three distinct peaks
of density for the three hydrogens of a methyl group, whereas if dynamic rotation
is possible, or if alternative static conformations occur on different molecules, then
we observe a smooth donut of density around the methyl carbon. For hydroxyls,
it is possible to determine the conformation angle H-O-C-H in hydrogen-bonding
and nonhydrogen-bonding situations. In the former, neutron diffraction sometimes reveals unexpected eclipsed conformations of the hydroxyl. Finally, the
structure of networks of water molecules on the surface of macromolecules have
been revealed by neutron diffraction. Being able to image the hydrogens means
learning the orientation of each water molecule and the exact pattern of hydrogen
bonding. In electron-density maps from X-ray work, we usually learn only the
location of the oxygen atom of water molecules.
226 Chapter 9 Other Diffraction Methods
Because of the difference in signs of b for H and D, it is easy to distinguish
them in density maps. This makes it possible to detect exchangeable hydrogens
in proteins by comparing the density maps from crystals in H2O and in D2O.
Amide hydrogens exchange with solvent hydrogens only if the amide group is
exposed to solvent and is not involved in tight hydrogen bonding to other atoms.
Because X-ray data are taken over a long period of time compared to rapid proton
exchange, proton exchange rates can usually only be assigned to three categories:
no exchange, slow exchange (10–60% during the time of data collection), and fast
exchange (more than 60%).
Turning to scattering by amorphous samples, and to studies at lower resolution
than crystallography, the negative scattering length for H makes possible interesting and useful scattering experiments on macromolecular complexes in solution.
For example, neutron scattering in solution can be used to measure distances
between the various protein components of very large macromolecular complexes
like ribosomes and viruses. Understanding these methods requires bringing up a
point that I have been able to avoid until now. Specifically, all forms of scattering
depend on contrast between the scatterer and its surroundings. Even in singlecrystal X-ray crystallography, we get our first leg up on phases by finding the
molecular boundary, in essence distinguishing protein from solvent. This is possible because of the contrast in density between ordered protein and disordered
water. If the protein and water had exactly the same scattering power, we could
not find this crucial boundary.
The importance of contrast to scattering is analogous to the importance of refractive index to refraction, such as the curving of light beams when they enter water.
Light travels in a straight line through a pure liquid, but changes direction abruptly
when it crosses a boundary into a new medium having a different refractive index.
If two liquids have identical refractive indices, the boundary between them does
not bend a light ray. Analogously, if X-rays or neutrons pass from one medium
(say, solvent) into another (say, protein), scattering occurs only if the average
scattering lengths of the media differ.
Because H and D have scattering lengths of different sign, it is possible to make
mixtures of H2O and D2O with average scattering lengths over a wide range.
In addition, by preparing proteins containing varying amounts of D substituted
for H (by growing protein-producing cells in H2O/D2O mixtures), researchers
can prepare proteins of variable average scattering length. A protein that scatters
identically with the solvent is invisible to scattering. Imagine then reconstituting a
bacterial ribosome, which is a complex of 3 RNA molecules and about 50 proteins,
from partially D-labeled proteins and RNAs whose scattering power matches the
H2O/D2O mixture in which they are dissolved. This mixture will not scatter neutrons. But if two of the components are unlabeled, only those two will scatter. At
low resolution, it is as if the two proteins constitute one molecule made of two
large atoms. The variation in intensity of neutron scattering with scattering angle
will reveal the distance between the two proteins, just as radial Patterson functions
reveal interatomic distances (see Fig. 9.6, p. 220). Repeating the experiment with
different unlabeled pairs of proteins gives enough interprotein distances to direct
the building of a three-dimensional map of protein locations.
Section 9.5 Electron diffraction and cryo-electron microscopy 227
Finally, low-angle neutron scattering can provide information about the shape
of these molecules within the ribosome, which may not be the same as their shape
when free in solution.
Neutron diffraction experiments are generally more difficult than those involving X-rays. There are fewer neutron sources worldwide. Available beam intensities,
along with the low neutron-scattering lengths of all elements, translate into long
exposure times. But for most macromolecules, neutron diffraction is the only
source of detailed information about hydrogen locations and the exact orientation
of hydrogen-carrying atoms.
For an example of neutron diffraction applied to a crystallographic problem, see D. Pignol, J. Hermoso, B. Kerfelec, I. Crenon, C. Chapus, and J. C.
Fontecilla-Camps, The lipase/colipase complex is activated by a micelle: Neutron
crystallographic evidence, Chem. Phys. Lipids 93, 123–129, 1998.
9.5 Electron diffraction and cryo-electron
microscopy
Electrons, like X-rays and neutrons, are scattered strongly by matter and thus
are potentially useful in structure determination. Electron microscopy (EM) is the
most widely known means of using electrons as structural probes. Scanning electron microscopes give an image of the sample surface, which is usually coated
with a thin layer of metal. Sample preparation techniques for scanning EM are not
compatible with obtaining images of molecules at atomic resolution. Transmission electron microscopes produce a projection of a very thin sample or section.
In the most familiar electron micrographs of cells and organelles, the sample is
stained with metals to outline the surfaces of membranes and large multimolecular assemblies like ribosomes. Unfortunately, staining results in distortion of the
sample that is unacceptable at high resolution. The most successful methods of
applying electron microscopes to molecular structure determination involve transmission microscopes used to study unstained samples by either electron diffraction
or cryo-electron microscopy (cryo-EM).
The electrons produced by transmission electron microscopes, whose design is
analogous to light microscopes, have de Broglie wavelengths of less than 0.1 Å,
so they are potentially quite precise probes of molecular structure. Unlike X-rays
and neutrons, electrons can be focused (by electric fields rather than glass lenses)
to produce an image, although direct images of objects in transmission EM do
not approach molecular resolution. However, electron microscopes can be used
to collect electron-diffraction patterns from two-dimensional arrays of molecules,
such as closely packed arrays of membrane proteins in a lipid layer. Analysis of
diffraction by such two-dimensional “crystals” is called electron crystallography.
Among the main difficulties with electron crystallography are (1) sample damage
from the electron beam (a 0.1-Å wave carries a lot of energy), (2) low contrast
between the solvent and the object under study, and (3) weak diffraction from
228 Chapter 9 Other Diffraction Methods
the necessarily very thin arrays that can be studied by this method. Despite these
obstacles, cryoscopic methods (Sec. 3.5, p. 46) and image processing techniques
have made electron crystallography a powerful probe of macromolecular structure,
especially for membrane proteins, many of which resist crystallization.
Transmission electron microscopy is analogous to light microscopy, with visible
light replaced by a beam of electrons produced by a heated metal filament, and
glass lenses replaced by electromagnetic coils to focus the beam. An image of
the sample is projected onto a fluorescent screen or, for a permanent record, onto
film or a CCD detector (Sec. 4.3.3, p. 77). Alternatively, an image of the sample’s
diffraction pattern can be projected onto the detectors.
To see how we can observe either an image or a diffraction pattern, look again
at Fig. 2.1, p. 8, which illustrates the action of a simple lens, such as the objective
(lower) lens of a microscope. Recall that the lens produces an image at I of an
object O placed outside the front focal point F of the lens. In a light microscope,
an eyepiece (upper) lens is positioned so as to magnify the image at I for viewing.
If we move the eyepiece to get an image of what lies at the back focal plane F ′,
we see instead the diffraction pattern of the sample. Analogously, in EM, we can
adjust the focusing power of the lenses to project an image of the back focal plane
onto the detector, and thus see the sample’s diffraction pattern. If the image is
nonperiodic (say, a section of a cell) the diffraction pattern is continuous, as in
Fig. 6.1, p. 110. If the sample is a periodic two-dimensional array, the diffraction
pattern is sampled at reciprocal-lattice points, just as in X-ray diffraction by crystals
(Fig. 2.19e, p. 29). The Fourier transform of this pattern is an image of the average
object in the periodic array. As with all diffraction, producing such an image
requires knowing both diffraction intensities and phases.
The possibility of viewing both the image and the diffraction pattern is unique to
electron crystallography and, in favorable cases, can allow phases to be determined
directly. For an ordered array (a 2-D crystal) of proteins in a lipid membrane, the
direct image is, even at the highest magnification, a featureless gray field. But
phase estimates can be obtained from this singularly uninteresting image in the
following manner. The image is digitized to pixels at high resolution, producing a
two-dimensional table of image-intensity values (not diffraction intensities). This
table is then Fourier transformed. Computing the FT of a table of values is the
same process as used to produce the images in Fig. 2.19, p. 29, Fig. 6.1, p. 110,
and Fig. 6.17, p. 138, in which the “samples” are in fact square arrays of pixels
with different numerical values. If the gray EM image is actually a periodic array,
the result of the FT on the table of pixel values is the diffraction pattern of the
average object in the array, sampled at reciprocal-lattice points. But because this
transform is computed from “observed” objects, it includes both intensities and
phases. The intensities are not as accurate as those that we can measure directly in
the diffraction plane of the EM, but the phases are often accurate enough to serve
as first estimates in a refinement process.
Single-crystal X-ray crystallography requires measuring diffraction intensities
at many closely spaced crystal orientations, so as to measure most of the unique
reflections in the reciprocal lattice. The Fourier transform (with correct phases) of
Section 9.5 Electron diffraction and cryo-electron microscopy 229
reflections from a single orientation gives only a projection of the unit cell in a
plane perpendicular to the beam. The transform of the full data set gives a threedimensional image. Electron microscopes allow samples to be tilted, which for
diffraction is analogous to rotating the crystal. Tilting gives diffraction patterns
whose transforms are projections of the sample contents from a different angle.
(In the same way, the fall of leaves during a slow, steady wind produces a pattern
of fallen leaves that is like a projection of the tree from an angle off the vertical.)
With diffraction patterns taken from a wide enough range of angles, you can
obtain a three-dimensional image of the sample contents. The EM sample can be
tilted about 75o at most, which may or may not allow a sufficiently large portion of
the reciprocal lattice to be sampled. It is not unusual for EM data sets to be missing
data in parts of reciprocal space that cannot be brought into contact with the sphere
of reflection (see Fig. 4.12, p. 62, and Fig. 4.13, p. 63) by tilting. Most common
is for a cone of reflections to be missing, with the result that maps computed from
these data do not have uniform resolution in all directions.
Armed with sets of data measured at different sample tilt angles, each set including (1) intensities measured at the diffraction plane and (2) phases from the Fourier
transform of direct images, the crystallographer can compute a map of the unit cell.
Recall that X-rays are scattered by electrons around atoms, producing a map of
electron density, and neutrons are scattered by nuclei, producing a map of mass or
nucleon density. Electrons, in turn, are scattered by electrostatic interactions, producing maps of electrostatic potential, sometimes called electron-potential maps,
or just potential maps. Maps from electron diffraction, like all other types, are
interpreted by building molecular models to fit them and refined by using partial
models as phasing models.
One of the special features of electron crystallography is the possibility of detecting charge on specific atoms or functional groups. The probing electrons interact
very strongly with negative charge, with the result that negatively charged atoms
have negative scattering factors at low resolution. (Recall that the negative scattering length for H in neutron diffraction can make H particularly easy to detect
in neutron-density maps.) At high resolution, the negative sign of the scattering
factor weakens the signal of negatively charged groups, but not usually enough to
be obvious. However, a comparison between maps computed with and without the
low-angle data can reveal charged groups. If the two maps are practically identical
around a possibly charged group like a carboxyl, then the group is probably neutral.
If, however, the map computed without low-angle data shows a stronger density
for the functional group than does the map computed from all data, then the functional group probably carries a negative charge. For example, functional groups
in proton pumps like rhodopsin are apparently involved in transferring protons
across a membrane by way of a channel through the protein. Determining the ionization state of functional groups in the channel is essential to proposing pumping
mechanisms.
Cryo-EM combines cryoscopic electron microscopy with another group of powerful structural methods collectively called image enhancement. These methods
do not involve diffraction directly, but they take advantage of the averaging power
230 Chapter 9 Other Diffraction Methods
of Fourier transforms to produce direct images at low resolution (10–25 Å). Image
enhancement can greatly improve the resolution of supramolecular complexes like
ribosomes, viruses, and multienzyme complexes that can be seen individually, but
at low resolution, as direct EM images. Samples for cryo-EM are unstained. They
are flash frozen in small droplets of cryoprotected aqueous buffers, so that the
water freezes as a glass, rather than crystalline ice, just as in cryocrystallography.
Imagine a direct EM image of, say, virus particles or ribosomes, all strewn across
the viewing field. Particles lay before you in all orientations, and the images show
very little detail. In image enhancement, we digitize the individual images and
sort them into images that appear to share the same orientation. Then we can
compute the Fourier transforms of the images. These transforms should be similar
in appearance if indeed the images in a set share the same orientation. Next, we
align the transforms, add them together, and back-transform. The result is an
averaged image, in which details common to the component images are enhanced,
and random differences, a form of “noise,” are reduced or eliminated. This process
is then repeated for sets of images at different orientations, the transforms are
all combined into a three-dimensional set, and the back-transform gives a threedimensional image.
Recall that the Fourier transform of a crystal diffraction pattern gives an image
of the average molecule in the crystal. Because any averaging process tends to
eliminate random variations and enhance features common to all components, the
image is much more highly resolved than if it were derived from a single molecule.
In like manner, the final back-transform of Fouriers from many EM images is an
image of the average particle in the direct EM field, and the resolution can be
greatly enhanced in comparison to any single particle image. In favorable cases,
enhanced EM images can be used as phasing models in crystallographic structure
determination by molecular replacement, where the particles or even components
of the particles can be crystallized.
Another powerful combination of cryo-EM and crystallography is making possible structure determination of supramolecular complexes that are not subject to
crystallization, such as bacteriophages and virus-receptor complexes. Components
of such complexes are purified and crystallized, and their individual structures are
determined at atomic resolution by single-crystal X-ray crystallography. Structures
of the intact complexes are determined at resolutions of 10–20 Å by cryo-EM.
Then the crystallographic component structures are fitted into the low-resolution
complex structure, and their positions refined by real-space fitting of component
electron density into the cryo-EM density. If a majority of components can be
fitted this way, then subtraction of their density from the cryo-EM density reveals
the shape and orientation of remaining components. The resulting low-resolution
models of the remaining components might be useful as molecular replacement
models in their crystallographic structure determination. Such studies are yielding
detailed structures of supramolecular complexes and insights into their action.
The European Bioinformatics Institute (EBI) is host for the Electron Microscopy
Database (EMD), a repository of molecular structures determined by electron
microscopy. A typical EMD entry contains sample identity, details of the EM
Section 9.6 Laue diffraction and time-resolved crystallography 231
experiment and data processing, density maps, and images of the model. To find
the EMD, go to the CMCC home page.
For an example of electron diffraction in structure determination of a membrane
protein, see Y. Kimura, D. G. Vassylyev, A. Miyazawa, A. Kidera, M. Matsushima,
K. Mitsuoka, K. Murata, T. Hirai, and Y. Fujiyoshi, Surface of bacteriorhodopsin
revealed by high-resolution electron crystallography, Nature 389, 206–211, 1997.
For an example of crystallography combined with cryo-EM and applied to a
supramolecular complex, see M. Rossmann, V. Mesyanzhinov, F. Arisaka, and
P. Leiman, The bacteriophage T4 DNA injection machine, Current Opinion in
Structural Biology, 14, 171–180, 2004 and references cited therein. To see EMD
entries related to this publication, search the EMD for “Rossmann.”
9.6 Laue diffraction and time-resolved
crystallography
Now I return to X-ray diffraction to describe probably the oldest type of diffraction
experiment, but one whose stock has soared with the advent of synchrotron radiation and powerful computer techniques for the analysis of complex diffraction
data. The method, Laue diffraction, is already realizing its promise as a means
to determine the structures of short-lived reaction intermediates. This method
is sometimes called time-resolved crystallography, implying an attempt to take
snapshots of a chemical reaction or physical change in progress.
Laue crystallography entails irradiating a crystal with a powerful polychromatic
beam of X rays, whose wavelengths range over a two- to threefold range, for
example, 0.5–1.5 Å. The resulting diffraction patterns are more complex than those
obtained from monochromatic X-rays, but they sample a much larger portion of
the reciprocal lattice from a single crystal orientation. To see why this is so, look
again at Fig. 4.12, p. 62, which demonstrates in reciprocal space the conditions
that satisfy Bragg’s law, and shows the resulting directions of diffracted rays
when the incident radiation is monochromatic. Recall that Bragg’s law is satisfied
when rotation of the crystal (and with it, the reciprocal lattice) brings point P
(in Fig. 4.12a) or P ′ [in (b)] onto the surface of the sphere of reflection. The
results are diffracted rays R and R′. Also recall that the sphere of reflection, whose
radius is the reciprocal of the X-ray wavelength λ, passes through the origin of the
reciprocal lattice, and its diameter is coincident with the X-ray beam.
Figure 9.7 extends the geometric construction of Fig. 4.12, p. 62 to show the
result of diffraction when the X-ray beam provides a continuum of wavelengths.
Instead of one sphere of reflection, there are an infinite number, covering the gray
region of the figure, with radii ranging from 1/λmax to 1/λmin, where λmax and
λmin are the maximum and minimum wavelengths of radiation in the beam. In the
figure, spheres of reflection corresponding to λmax and λmin are shown, along with
two others that lie in between. The four points lying on the incident beam X are the
232 Chapter 9 Other Diffraction Methods
Figure 9.7  Geometric construction for Laue diffraction in reciprocal space. Each
diffracted ray R1 through R4 is shown in the same color as that of the sphere of reflection
and the lattice point that produces it.
centers of the four spheres of reflection. Each of the four spheres is shown passing
through one reciprocal-lattice point (some pass through others also), producing a
diffracted ray (R1 through R4).
Note, however, that because there are an infinite number of spheres of reflection,
every reciprocal-lattice point within the gray region lies on the surface of some
sphere of reflection, and thus for every such point Bragg’s law is satisfied, giving
rise to a diffracted ray. Of course, any crystal has its diffraction limit, which
depends on its quality. The heavy arc labeled d∗max represents the resolution limit
Section 9.6 Laue diffraction and time-resolved crystallography 233
of the crystal in this illustration. Diffraction corresponding to reciprocal-lattice
points outside this arc (for example, R3 and R4), even though they satisfy Bragg’s
law, will not be detected simply because this crystal does not diffract out to those
high angles. Because the detector is at the geometric equivalent of an infinite
distance from the crystal, we can picture all rays as emerging from a single point,
such as the origin. To show the relative directions of all reflections in the Laue
pattern from this diagram, we move all the diffracted rays to the origin, as shown
for rays R1 through R4 in Fig. 9.7b.
Note also that because of the tapering shape of the gray area near the origin,
the amount of available data drops off at low angles. One limitation of Laue
diffraction is the scarcity of reflections at small angles. An added complication
of Laue diffraction is that some rays pass through more than one point. When
this occurs, the measured intensity of the ray is the sum of the intensities of both
reflections. Finally, at higher resolution, many reflections overlap.
You can see from Fig. 9.8 that a Laue diffraction pattern is much more complex
than a diffraction pattern from monochromatic X-rays. But modern software can
index Laue patterns and thus allow accurate measurement of many diffraction
intensities from a single brief pulse of X-rays through a still crystal. If the crystal
Figure 9.8  A typical Laue image. From I. J. Clifton, E. M. H. Duke, S. Wakatsuki,
and Z. Ren, Evaluation of Laue Diffraction Patterns, in Methods in Enzymology 277B,
C. W. Carter and R. M. Sweet, eds., Academic Press, New York, 1997, p. 453. Reprinted
with permission.
234 Chapter 9 Other Diffraction Methods
has high symmetry and is oriented properly, a full data set can in theory be collected
in a single brief X-ray exposure. In practice, this approach usually does not provide
sufficiently accurate intensities because the data lack the redundancy necessary for
high accuracy. Multiple exposures at multiple orientations are the rule.
The unique advantage of the Laue method is that data can be collected rapidly
enough to give a freeze-frame picture of the crystal’s contents. Typical X-ray
data are averaged over the time of data collection, which can vary from seconds
to days, and over the sometimes large number of crystals required to obtain a
complete data set. Laue data has been collected with X-ray pulses shorter than 200
picoseconds. Such short time periods for data collection are comparable to halftimes for chemical reactions, especially those involving macromolecules, such as
enzymatic catalysis. This raises the possibility of determining the structures of
reaction intermediates.
Recall that a crystallographic structure is the average of the structures of all
diffracting molecules, so structures of intermediates can be determined when all
molecules in the crystal react in unison or exist in an intermediate state simultaneously. Thus the crystallographer must devise some way to trigger the reaction
simultaneously throughout the crystal. Good candidate reactions include those triggered by light. Such processes can be initiated throughout a crystal by a laser pulse.
Some reactions of this type are reversible, so the reaction can be run repeatedly
with the same crystal.
A strategy for studying enzymatic catalysis entails introducing into enzyme
crystals a “caged” substrate—a derivative of the substrate that is prevented from
reacting by the presence of a light-labile protective group. If the caged substrate
binds at the active site, then the stage is set to trigger the enzymatic reaction by a
light pulse that frees the substrate from the protective group, allowing the enzyme
to act. Running this type of reaction repeatedly may mean replacing the crystal, or
maybe just introducing a fresh supply of caged substrate to diffuse into the crystal
for the next run.
Other possibilities are reactions that can be triggered by sudden changes in temperature or pressure. Reactions that are slow in comparison to diffusion rates in
the crystal may be triggered by simply adding substrate to the mother liquor surrounding the crystal and allowing the substrate to diffuse in. If there is a long-lived
enzyme-intermediate state, then during some interval after substrate introduction,
a large fraction of molecules in the crystal will exist in this state, and a pulse of
radiation can reveal its structure.
In practice, multiple Laue images are usually necessary to give full coverage
of reciprocal space with the required redundancy. Here is a hypothetical strategy for time-resolved crystallography of a reversible, light-triggered process.
From knowledge of the kinetics of the reaction under study, determine convenient reaction conditions (for example, length and wavelength of triggering pulse
and temperature) and times after initiation of reaction that would reveal intermediate states. These times must be long compared to the shortest X-ray pulses that
will do the job of data collection. From knowledge of crystal symmetry, determine the number of orientations and exposures that will produce adequate data
Section 9.7 Summary 235
for structure determination. Then set the crystal to its first orientation, trigger the
reaction, wait until the first data-collection time, pulse with X-rays, and collect
Laue data. Allow the crystal to equilibrate, which means both letting the reaction
come back to equilibrium and allowing the crystal to cool, since the triggering
pulse and the X-ray pulse heat the crystal. Then again trigger the reaction, wait
until the second data-collection time, pulse with X-rays, and collect data. Repeat
until you have data from this crystal orientation at all the desired data-collection
times. Move the crystal to its next orientation, and repeat the process. The result is
full data sets collected at each of several time intervals during which the reaction
was occurring.
For an example of Laue diffraction applied to time-resolved crystallography,
see V. Srajer, T. Teng, T. Ursby, C. Pradervand, Z. Ren, S. Adachi, W. Schildkamp, D. Bourgeois, M. Wulff, and K. Moffat, Photolysis of the carbon monoxide
complex of myoglobin: Nanosecond time-resolved crystallography, Science 274,
1726–1729, 1996.
9.7 Summary
The same geometric and mathematical principles lie at the root of all types of
diffraction experiments, whether the samples are powders, solutions, fibers, or
crystals, and whether the experiments involve electromagnetic radiation (X-rays,
visible light) or subatomic particles (electrons, neutrons). My aim in this chapter was to show the common ground shared by all of these probes of molecular
structure. Note in particular how the methods complement each other and can
be combined to produce more inclusive models of macromolecules. For example,
phases from X-ray work can serve as starting phase estimates for neutron work, and
the resulting accurate coordinates of hydrogen positions can then be added to the
X-ray model. As another example, direct images obtained from EM work (sometimes after image enhancement) can sometimes be used as molecular-replacement
models for X-ray crystallography, or to determine the organization of components
in supramolecular complexes.
As a user of macromolecular models, you are faced with judging whether each
model really supports the insights it appears to offer. The principles presented in
Chapter 7, on how to judge the quality of models, apply to models obtained from
all types of diffraction experiments. But today’s structural databases also contain
a growing number of models obtained by methods other than diffraction. In the
next chapter, I will describe the origin of the major types of nondiffraction models
and provide some guidance on how to use them wisely.
 Chapter 10
Other Kinds of
Macromolecular Models
10.1 Introduction
When you go looking for models of macromolecules that interest you, crystallographic models are not the only type you will find. As of mid-2005, about 15%
of the models in the Protein Data Bank were derived from NMR spectroscopy
of macromolecules in solution. Of over 4000 NMR models, fewer than 2% were
proteins of more than 200 residues, although the number of larger models is sure
to increase. Also proliferating rapidly are homology models, which are built by
computer algorithms that work on the assumption that proteins of homologous
sequence have similar three-dimensional structures. Massive databases of homology models are constantly growing, with the goal of providing homology models
for all known protein sequences that are homologous to structures determined by
crystallography or NMR. Although this effort is just beginning, the number of
homology models available in one database, the SWISS-MODEL Repository (see
the CMCC home page), contains more models than does the Protein Data Bank.
Furthermore, it is quite easy for you to determine protein structures by homology
modeling on your personal computer, as I will describe in Chapter 11. Finally,
there are various means of producing theoretical models, for example, based on
attempts to simulate folding of proteins. As with crystallographic models, other
types of models vary widely in quality and reliability. The user of these models
is faced with deciding whether the quality of the model allows confidence in the
apparent implications of the structure.
In this chapter, I will provide brief descriptions of how protein structures are
determined by NMR and by homology modeling. In addition, I will provide
some guidance on judging the quality of noncrystallographic models, primarily
237
238 Chapter 10 Other Kinds of Macromolecular Models
by drawing analogies to criteria of model quality in crystallography. Recall that
in all protein-structure determination, the goal is to determine the conformation
of a molecule whose chemical composition (amino-acid content and sequence)
is known.
10.2 NMR models
10.2.1 Introduction
Models of proteins in solution can be derived from NMR spectroscopy. In brief,
the process entails collecting highly detailed spectra; assigning spectral peaks
(resonances) to all residues by chemical shift and by decoupling experiments;
deriving distance restraints from couplings, which (a) reveal local conformations
and (b) pinpoint pairs of atoms that are distant from each other in sequence, but
near each other because of the way the protein is folded; and computing a chemically, stereochemically, and energetically feasible model that complies with all
distance restraints. That’s it. There is no image, no “seeing” the model in a map.
NMR structure determination amounts to building a model to fit the conformational
and distance restraints learned from NMR, and of course, to fit prior knowledge
about protein structure. The power of NMR spectroscopy has grown immensely
with the development of pulse-Fourier transform instruments, which allow rapid
data collection; more powerful magnets, which increase spectral resolution; and
sophisticated, computer-controlled pulse sequences, which separate data into subsets (somewhat misleadingly called “dimensions”) to reveal through-bond and
through-space couplings.
Typical sample volumes for NMR spectroscopy are around 0.5 mL of protein
in H2O/D2O (95/5 to 90/10). Typical protein concentrations are 0.5 to 2.0 mM.
For a protein of 200 amino-acid residues, this means that each run consumes 5 to
20 mg of protein. The protein must be stable at room temperature for days to
weeks, and must not be prone to aggregation at these relatively high concentrations. Depending on the thermal stability of the protein, a sample may be used
only once or many times. In addition, proteins must be labeled with NMR-active
isotopes such as 13C and 15N. Like crystallography, NMR spectroscopy reaps
the benefits of modern molecular biology’s powerful systems for expression and
purification of desired proteins in quantity, labeled with appropriate isotopes from
their growth media.
In this section, I provide a simplified physical picture of pulse NMR spectroscopy, including a simple conceptual model to help you understand multidimensional NMR. Then I briefly discuss the problems of assigning resonances and
determining distance restraints for molecules as large and complex as proteins,
and the methods for deriving a structure from this information. Finally, I discuss
the contents of coordinate files from NMR structure determination and provide
some hints on judging the quality of models.
Section 10.2 NMR models 239
10.2.2 Principles
I assume that you are conversant with basic principles of 1H or proton NMR
spectroscopy as applied to small molecules. In particular, I assume that you
understand the concepts of chemical shift (δ) and spin-spin coupling, classical
continuous-wave methods of obtaining NMR spectra, and decoupling experiments
to determine pairs of coupled nuclei. If these ideas are unfamiliar to you, you may
wish to review NMR spectroscopy in an introductory organic chemistry textbook
before reading further.
Chemical shift and coupling
Many atomic nuclei, notably 1H,13C, 15N, 19F, and 31P, have net nuclear spins as
a result of the magnetic moments of their component protons and neutrons. These
spins cause the nuclei to behave like tiny magnets, and as a result, to adopt preferred
orientations in a magnetic field. A nucleus having a spin quantum number of 1/2
(for example, 1H) can adopt one of two orientations in a magnetic field, aligned
either with or against the field. Nuclei aligned with the field are slightly lower in
energy, so at equilibrium, there are slightly more nuclei (about 1 in 10,000) in the
lower energy state. The orientations of nuclear spins can be altered by pulses of
electromagnetic radiation in the radio-frequency (RF) range.
Nuclei in different chemical environments absorb different frequencies of
energy. This allows specific nuclei to be detected by their characteristic absorption energy. This energy can be expressed as an RF frequency (in hertz), but because
the energy depends on the strength of the magnetic field, it is expressed as a frequency difference between that of the nucleus in question and a standard nucleus
(like hydrogen in tetramethylsilane, a common standard for 1H NMR) divided by
the strength of the field. The result, called the chemical shift δ and expressed in parts
per million (ppm), is independent of field strength, but varies informatively with
the type of nucleus and its immediate molecular environment. Figure 10.1 shows
the 1H-NMR spectrum of human thioredoxin,1 a small protein of 105 amino-acid
residues. (In humans, thioredoxin plays a role in activating certain transcriptional
and translational regulators by a dithiol/disulfide redox mechanism. More familiar
to students of biochemistry is the role of thioredoxin in green plants: mediating
light activation of enzymes in the Calvin cycle of photosynthesis.) The spectrum
is labeled with ranges of δ for various types of hydrogen atoms in proteins.
In addition to exhibiting characteristic chemical shifts, nuclear spins interact
magnetically, exchanging energy with each other by a process called spin-spin
coupling. Coupling distributes or splits the absorption signals of nuclei about their
characteristic absorption frequency, usually in a distinctive pattern that depends
on the number of equivalent nuclei that are coupled, giving the familiar multiplets
in NMR spectra of simple molecules. The spacing between signals produced by
1J. D. Porman-Kay, G. M. Clore, P. T. Wingfield, and A. M. Gronenbom, High-resolution threedimensional structure of reduced recombinant human thioredoxin in solution, Biochemistry 30, 2685,
1991 (PDB files 2trx and 3trx).
240 Chapter 10 Other Kinds of Macromolecular Models
Figure 10.1  1H-NMR spectrum of thioredoxin, reduced form. Signal intensity
(vertical axis) is plotted against frequency (horizontal axis). Labels show chemical-shift
values typical of various hydrogen types in protein chains having random coil conformation. Some signals lie outside these ranges because of specific interactions not present
in random coils. Atom labels are as found in PDB coordinate files (p. 176). Spectrum
generously provided by Professor John M. Louis.
splitting is called the coupling constant J , which is expressed in hertz because its
magnitude does not depend on field strength. Because nuclei must be within a few
bonds of each other to couple, this effect can be used to determine which nuclei
are neighbors in the molecule under study.
Absorption and relaxation
You can view the nuclear spins in a sample as precessing about an axis, designated z, aligned with the magnetic field H , as shown in Fig. 10.2, which depicts
the spins for a large number of equivalent nuclei in (a), and only the slight equilibrium excess of spins at the lower energy in (b). The slight excess of spins
aligned with the field means that there is a net magnetization vector pointing in
the positive direction along z at equilibrium. Spins precess at their characteristic RF absorption frequency, but the phases of precession are random. In other
words, equivalent, independent spins precess at the same rate, but their positions
are randomly distributed about the z-axis.
In Fig. 10.3, a highly simplified sketch of an NMR instrument shows the orientation of the x-, y-, and z-axes with respect to the magnets that apply the field H ,
the sample, and wire coils that transmit (green) and detect (red) radio-frequency
Section 10.2 NMR models 241
Figure 10.2  (a) Nuclear spins precess around the z-axis, which is parallel to the
applied field H , indicated by the arrow in the center of the figure. Each spin precesses at a
rate that depends on its RF absorption frequency. (b) Excess spins in the lower energy state.
At equilibrium, there are slightly more spins aligned withH than againstH . (c) Immediately
after a 90o pulse, equal numbers of spins are aligned with and against H , and initially all
spins lie in the yz-plane, giving a net magnetic vector on the y-axis. This vector precesses
in the xy plane, inducing a signal in the receiving coils. As time passes after the pulse, spins
precess about z and spread out, due to their different precession frequencies, ultimately
returning to the state shown in a. The result is a decay in signal intensity. (d) Immediately
after a 180o pulse, or two successive 90o pulses, excess spins shown in (b) are inverted, and
lie in the yz-plane as shown. The net magnetization vector in the xy-plane has a magnitude
of zero, and thus induces no signal in the receiver coils.
signals. The receiver coils, which encircle the x-axis, detect RF radiation in the
xy-plane only. A single nuclear spin precessing alone around the z-axis has a
rotating component in the xy-plane, and transmits RF energy at its characteristic
absorption frequency to the receiver coils, in theory revealing its chemical shift.
But when many equivalent spins in a real sample are precessing about the z-axis
with random phase, the xy components of their magnetic vectors cancel each other
out, the net magnetic vector in the xy plane has a magnitude of zero, and no RF
signals are detected. Detection of RF radiation from the sample requires that a
net magnetization vector be moved into the xy-plane. This is accomplished by a
wide-band (multifrequency) pulse of RF energy having just the right intensity to
242 Chapter 10 Other Kinds of Macromolecular Models
Figure 10.3  Diagram of an NMR experiment. The sample lies between the poles
of a powerful magnet, and is spun rapidly around its long (x) axis in order to compensate
for any unevenness (inhomogeneity) of the magnetic field. Radio frequency receiver coils
(red) form a helix around the x-axis, and transmitter coils (green) spiral around the y-axis.
tip the net magnetization vector into the xy-plane. This pulse is applied along x,
and it equalizes the number of spins in the higher and lower energy states, while
aligning them onto the yz-plane (Fig. 10.2c). A pulse of this intensity is call a 90o
pulse. The result is that a net magnetization vector appears in the xy-plane and
begins to rotate, generating a detectable RF signal. Because nuclei of different
chemical shifts precess at different frequencies, their net magnetization vectors in
the xy-plane rotate at different frequencies, and the resulting RF signal contains
the characteristic absorption frequencies of all nuclei in the sample, from which
chemical shifts can be derived.
As shown in Fig. 10.2c, the 90o pulse equalizes the number of nuclei aligned
with and against the applied field along z, giving a net magnetization vector along
z of zero. This means that the system of spins is no longer at equilibrium. It will
return to equilibrium as a result of spins losing energy to their surroundings, a
process called spin-lattice relaxation, in which the term lattice simply refers to
the surroundings of the nuclei. This relaxation is a first-order process whose rate
constant I will call RL (L for lattice). The inverse of RL is a time constant I will
call TL, the spin-lattice relaxation time constant. TL is also called the longitudinal
relaxation time constant, because relaxation occurs along the z-axis, parallel to
the magnetic field. (TL is traditionally called T1, but I adopt a more descriptive
symbol to reduce confusion with other symbols in the following discussion.)
After a 90o pulse, another relaxation process also occurs. Although all spins
have the same phase just after the pulse (aligned along y), the phases of identical
nuclei spread out due to exchanges of energy that result from coupling with other
spins. The result is that the net magnetization vector in the xy-plane for each
distinct set of nuclei diminishes in magnitude, as individual spin magnetizations
move into orientations that cancel each other. Furthermore, the RF frequencies
Section 10.2 NMR models 243
of chemically distinct nuclei disappear from the overall signal at different rates,
depending on their coupling constants, which reflect the strength of coupling,
or in other words, how effectively the nuclei exchange energy. Phases of pairs
of nuclei coupled to each other spread out at the same rate because their spin
energies are simply being exchanged. The rate constant for this process, which is
called spin-spin relaxation, is traditionally called R2, but I will call it RS (S for
spin). Its inverse, the time constant TS, is the spin-spin relaxation time constant.
It is also called the transverse relaxation time constant because the relaxation
process is perpendicular or transverse to the applied magnetic field H . Each set
of chemically identical nuclei has a characteristic TS. (Often, and confusingly, the
rate constants R are loosely called rates, and time constants T are simply called
relaxation times.)
So after a 90o RF pulse applied along x tips the net magnetization vector onto
the y-axis, an RF signal appears at the detector because there is net magnetization
rotating in the xy-plane. This signal is a composite of the frequencies of all the
precessing nuclei, each precessing at its characteristic RF absorption frequency.
In addition, this signal is strong at first but decays because of spin-spin relaxation.
Signals for different nuclei diminish at different rates that depend on their individual spin-spin relaxation rates. This complex signal is called a free-induction decay
or FID: free because it is free of influence from the applied RF field, which is
turned off after the pulse; induction because the magnetic spins induce the signal
in the receiver coil; and decay because the signal decays to an equilibrium value
of zero. A typical hydrogen NMR FID signal decays in about 300 ms, as shown in
Fig. 10.4.
One-dimensional NMR
How do we extract the chemical shifts of all nuclei in the sample from the freeinduction decay signal? The answer is our old friend the Fourier transform. The FID
is called a time-domain signal because it is a plot of the oscillating and decaying
RF intensity versus time, as shown in Fig. 10.4 (the time axis is conventionally
labeled t2, for reasons you will see shortly). Fourier transforming the FID produces
a frequency-domain spectrum, a plot of RF intensity versus the frequencies present
in the FID signal, with the frequency axis labeled v2 for frequency or F2 for
chemical shift, as shown in Fig. 10.1. So the Fourier transform decomposes the
FID into its component frequencies, revealing the chemical shifts of the nuclei in
the sample.
NMR spectroscopists collect this type of classical or one-dimensional (1-D)
NMR spectra on modern FT-NMR instruments by applying a 90o pulse and collecting the FID signal that is induced in the receiving coil. To make a stronger
signal, they collect FIDs repeatedly and add them together. Real signals appear
at the same place in all the FIDs and add up to a large sum. On the other hand,
random variation or “noise” appears in different places in different FIDs, so their
signals cancel each other. The summed FIDs thus have a high signal-to-noise ratio,
and FT produces a clean spectrum with well-resolved chemical shifts. From here
on, I may not always mention that pulse/FID collection sequences are repeated to
244 Chapter 10 Other Kinds of Macromolecular Models
Figure 10.4  Free induction decay signal, which appears in the receiver coils after
a 90o pulse. The FID is a time-domain spectrum, showing RF intensity as a function of
time t2. It is a composite of the RF absorption frequencies of all nuclei in the sample. The
Fourier transform decomposes an FID into its component frequencies, giving a spectrum
like that shown in Fig. 10.1. Figure generously provided by Professor John M. Louis.
improve the signal, but you can assume that all sequences I describe are carried
out repeatedly, commonly 64 times, for this purpose.
Two-dimensional NMR
Couplings between nuclei influence the rate at which their characteristic frequencies diminish in the FID. If we could measure, in addition to the frequencies
themselves, their rates of disappearance, we could determine which pairs of nuclei
are coupled, because the signals of coupled pairs fade from the FID at the same
rate. This is the basis of two-dimensional (2-D) NMR, which detects not only
the chemical shifts of nuclei but also their couplings. The 2-D NMR employs
computer-controlled and -timed pulses that allow experimenters to monitor the
progress of relaxation for different sets of nuclei.
Figure 10.5 illustrates, with a system of four nuclei, the principles of 2-D NMR
in its simplest form, when we want to assign pairs of 1H nuclei that are spin-spin
coupled through a small number of bonds, such as hydrogens on adjacent carbon
atoms. You may find this figure daunting at first, but careful study as you read the
following description will reward you with a clearer picture of just what you are
seeing when you look at a 2-D NMR spectrum.
Section 10.2 NMR models 245
The 2-D experiment is much more complex than obtaining a conventional “onedimensional” spectrum. The experimenter programs a sequence of pulses, delays,
and data (FID) collections. In each sequence, the program calls for a 90o pulse (the
“preparation” pulse), followed by a delay or “evolution” time t1, and then a second
90o “mixing” pulse, followed by “detection” or data collection (all repeated 64
times to enhance the signal). In successive preparation/evolution/mixing/detection
(PEMD) sequences, the evolution time t1 is increased in equal steps. In a typical
experiment of this type, t1 might be incremented 512 times in 150-µs steps from
0 µs up to a maximum delay of about 77 ms, giving a full data set of 512 signalenhanced FIDs, five of which are shown with their FTs in Fig. 10.5a. Each FID is
a plot of RF signal intensity versus time t2, and its FT shows intensity as a function
of frequency F2. In each of the FTs (except the first one, in which the signals are
too weak—see Fig. 10.1d to see why), you can see the RF signals of the four nuclei
in the sample.
What is the purpose of the second or mixing pulse? Recall that the first pulse
tips the precessing magnetic vectors toward the y-axis and aligns their phases,
thus putting rotating net magnetic vectors in the xy-plane and an FID signal into
the detector. But in 2-D NMR, we do not record this FID. Instead, we wait for a
specified evolution time t1 (between 0 and 77 ms in the example described here),
and then pulse again. During this interval—the evolution time—magnetic vectors
spread out in the xy-plane, and the RF signal diminishes in intensity. The second
pulse tips any magnetization that currently has a component in the yz-plane (as
a result of vector spreading) back onto the xy-plane, and aligns the spins on the
y-axis. Then relaxation occurs again, and this time we record the FID. In the
first sequence, with t1 = 0, essentially no relaxation occurs between pulses, no
magnetization enters the yz-plane, and after the second pulse, the FID contains
weak or no signals. As t1 is increased, more relaxation occurs between the first
and second pulses, magnetization representative of the precession state of each
nucleus at time t1 is present in the yz-plane, and it is detected after the second
pulse.
The FIDs are Fourier transformed to produce 512 frequency spectra, each from
a different time t1, as shown on the right in Fig.10.5a. Just as in 1-D NMR, each
spectrum gives the intensities of RF signals as a function of frequency F2 but, in
this case, recorded from all nuclei after relaxation for a time t1. To assign couplings,
we are interested in finding out which pairs of RF frequencies F2 vary in intensity
at the same rate. This can be accomplished by plotting the signal at each frequency
F2 versus time t1. The data for such a plot for nucleus 1 is shown within the narrow
vertical rectangle (red) in Fig. 10.5a, and the plot itself is shown by the FID symbol
at the top left in Fig. 10.5b. As implied by the symbol for this plot, at a frequency
F2 at which an RF signal occurs, a plot of its intensity versus t1 is a time-domain
spectrum, actually a pattern of interference among all frequencies present. This
pattern is mathematically like an FID, and for convenience, it is usually called
an FID. Recall that FIDs are composed of one or more component frequencies.
If a nucleus is coupled to more than one other nucleus, its FID taken over time
246 Chapter 10 Other Kinds of Macromolecular Models
Figure 10.5  Two-dimensional NMR for a hypothetical system of four nuclei.
(a) FIDs are collected after each of a series of sequences [90o pulse/t1 delay/90
o pulse],
with t1 varied. Each FID decays over time t2. The Fourier transform of an FID gives the RF
intensities of each frequency F2 in the FID at time t1. Each signal occurs at the chemical
shift F2 of a nucleus in the sample. Chemical shifts of the nuclei are labeled 1 through 4.
Section 10.2 NMR models 247
Figure 10.5  (Continued) Intensity at a single frequency F2 evolves over time t1,
as shown in the red vertical rectangle for nucleus 1. (b) A plot of this variation is like an FID
that decays over time t1 [top of (b)]. Fourier transforms of t1-FIDs reveal the frequencies
F1 present in the signal, such as the red frequencies that make up the signal from nucleus 1.
Coupled nuclei have frequencies F1 in common, as shown for nuclei 1, 2, and 4 in the green
horizontal rectangle. They share decay frequencies because they are coupled, which means
they are exchanging energy with each other. (c) Contour plot presenting the information
of FTs in (b). Peaks on the diagonal correspond to the 1-D NMR spectrum, with F1 and
F2 both corresponding to chemical shifts. Rows of peaks off the diagonal indicate sets of
nuclei that have frequencies F1 in common due to coupling. The third horizontal row of
signals in (c) corresponds to the information in the green rectangle in (b). In this example,
the couplings are 1 to 2, 2 to 1 and 4, 3 to 4, and 4 to 2 and 3. From Principles of Biophysical
Chemistry by Van Holde et al., copyright 1998; adapted by permission of Prentice-Hall,
Inc., Upper Saddle River, NJ.
t1 will contain frequencies (corresponding to the spreading of vectors in the xyplane) corresponding to all of its couplings. To determine these frequencies, we
once again use the Fourier transform. The FT of this FID will reveal the different
frequencies for each coupling. This FT is a frequency-domain spectrum, a plot
of RF signal intensity versus phase-spreading frequencies or spin-spin-relaxation
frequencies F1, shown for the four F2 signals in the lower part of Fig. 10.5b. The
frequencies F1 in each spectrum are relaxation frequencies for a given nucleus.
The intensities in these spectra give the strength of coupling, or to put it another
way, they give the amount of correlation between nuclei whose F1 FIDs contain
the same frequency.
A 2-D NMR spectrum is a plot of F1 versus F2, that is, it is our set of 512 F1
spectra laid side by side at a spacing corresponding to the frequency F2 at which
each F1 FID was taken, as in Fig. 10.5b. Looking along horizontals across the F1
spectra in (b), we see the frequencies F2 of sets of nuclei that relaxed at the same
rate. For example, the narrow horizontal rectangle (green) in Fig. 10.5b shows
that spins 1, 2, and 4 share a common relaxation frequency. Thus we conclude
that these nuclei are coupled to each other and are on neighboring atoms. The 2-D
spectrum is usually presented as shown in Fig. 10.5c. The position and intensity of
a spot in this spectrum corresponds to the position and intensity of a signal in (b).
The off-diagonal or “cross” peaks have finer details, not shown here, that correspond to the simple and familiar, but information-rich, J -splitting patterns,
such as doublets, triplets, and quartets, seen in conventional 1H-NMR spectra
of small molecules. These patterns tell us how many chemically equivalent nuclei
are involved in these couplings. Of course, each signal correlates most strongly
with itself over time t1, so the strongest signals lie on the diagonal. The spectrum
on the diagonal corresponds to the one-dimensional NMR spectrum.
Interpretation of the 2-D spectrum in Fig. 10.5c is simple. The spins of the
four nuclei give strong signals on the diagonal (numbered 1 through 4). All offdiagonal signals indicate couplings. Each such signal is aligned horizontally along
248 Chapter 10 Other Kinds of Macromolecular Models
F2 and vertically along F1 with signals on the diagonal that correspond to the two
nuclei, or sets of nuclei, that are coupled. The signals labeled 2–4 denote coupling between nuclei 2 and 4. This type of 2-D NMR spectroscopy, which reveals
spin-spin or J couplings by exhibiting correlations between spin-spin relaxation
times of nuclei, is called correlation spectroscopy or COSY. In a sense, it spreads
the information of the one-dimensional spectrum into two dimensions, keeping
coupled signals together by virtue of their correlated or shared relaxation times.
By far the most widely useful nucleus for NMR COSY study of proteins is 1H.
Obtaining 1H spectra (rather than 13C or 15N spectra, for example) simply means
carrying out the NMR experiments described here in the relatively narrow range
of RF frequencies over which all 1H atoms absorb energy in a magnetic field.
Nuclear Overhauser effect
The COSY spectra reveal through-bond couplings because these couplings are
involved in the relaxation process revealed by this PEMD sequence of operations:
[90◦ pulse/t1 delay/90◦ pulse/data collection]. In a sense, this pulse sequence
stamps the t1 coupling information about relaxation rates onto the t2 RF absorption
signals. The second set of Fourier transforms extracts this information. Other pulse
sequences can stamp other kinds of information onto the t2 signal.
In particular, there is a second form of coupling between nuclear dipoles that
occurs through space, rather than through bonds. This interatomic interaction is
called the nuclear Overhauser effect (NOE), and the interaction can either weaken
or strengthen RF absorption signals. Nuclei coupled to each other by this effect will
show the effect to the same extent, just as J -coupled nuclei share common spinspin relaxation rates. Appropriate pulse sequences can impress NOE information
onto the t2 absorption signals, giving 2-D NMR spectra in which the off-diagonal
peaks represent NOE couplings, rather than J couplings, and thus reveal pairs
of nuclei that are near each other in space, regardless of whether they are near
each other through bonds. This form of 2-D NMR is called NOESY. Though the
off-diagonal signals represent different kinds of coupling, NOESY spectra look
just like COSY spectra.
A partial NOESY spectrum of human thioredoxin is shown in Fig. 10.6. This
2-D spectrum shows the region from δ = 6.5 to δ = 10. First, compare this
spectrum with the 1-D spectrum in Fig. 10.1, p. 240. Recall that the signals on
the diagonal of a 2-D spectrum are identical to the 1-D spectrum, but they are in
the form of a contour map, sort of like looking at the 1-D spectrum from above,
or down its intensity axis. For example, notice the two small peaks near δ = 10
on the 1-D spectrum, and near F1 = F2 = 10 on the 2-D spectrum. Notice
that there are off-diagonal peaks aligned horizontally and vertically with these
peaks. These off-diagonal signals align with other diagonal peaks corresponding
to nuclei NOE-coupled to these nuclei. Several coupling assignments that were
used to define distance restraints are indicated by pairs of lines—one horizontal,
one vertical—on the spectrum. The pairs converge on an off-diagonal peak and
diverge to the signals of coupled nuclei on the diagonal.
Section 10.2 NMR models 249
Figure 10.6  NOESY spectrum of thioredoxin in the region δ = 6.5 to δ = 10.
Pairs of lines—one horizontal, one vertical—converge at off-diagonal peaks indicating
NOE couplings and diverge to the signals on the diagonal for the coupled nuclei. Assigning
resonances on the diagonal to specific hydrogens in the protein requires greater resolution
and simplification of the spectrum than shown here. The off-diagonal peak labeled “F89N,δ”
indicates a specific NOESY interaction that is described and shown in Fig. 10.10, p. 256.
Spectrum generously provided by Professor John M. Louis.
NMR in higher dimensions
For a large protein, even spectra spread into two dimensions by J or NOE coupling
are dauntingly complex. Further simplification of spectra can be accomplished
by three- and higher-dimensional NMR, in which the information of a complex
2-D spectrum is separated onto sets of spectra that can be pictured as stacked
planes of 2-D spectra, as in Fig. 10.7. One technique separates the 2-D spectrum
into separate planes, each corresponding to a specific chemical shift of a different
NMR-active nucleus, such as 13C or 15N, to which the hydrogens are bonded. This
method, called 13C- or 15N-editing, requires that the protein be heavily labeled
with these isotopes, which can be accomplished by obtaining the protein from
cells grown with only 13C- and 15N-containing nutrients. In 13C-edited COSY or
NOESY spectra, each plane contains crosspeaks only for those hydrogens attached
250 Chapter 10 Other Kinds of Macromolecular Models
Figure 10.7  3-D and 4-D NMR. (a) 2-D NMR of H-Ca/H-Cb/H-Na/H-Nb. Chemical shifts of H-Ca and H-Cb are identical, as are those of H-Na and H-Nb. so only two peaks
appear on the diagonal. One cross-peak could arise from four possible couplings, listed on
the figure. (b) Spectrum of (a) separated onto planes, each at a different 13C chemical shift
δ. The δ-13C axis represents the third NMR dimension. Gray lines represent the position of
the diagonal on each plane. The cross-peak occurs only on the plane at δ-13Cb, revealing
that coupling involves H-Cb and eliminating two possible couplings. (c) Spectrum as in (b),
separated onto planes, each at a different 15N chemical shift δ. The δ-15N axis represents
the fourth NMR dimension. The cross-peak occurs only on the plane at δ-15Na, revealing
that coupling involves H-Na. Thus the coupling indicated by the-cross peak in (a) is H-Cb
to H-Na.
to 13C atoms having the chemical shift corresponding to that plane. Because you
can picture these planes as stacked along a 13C chemical-shift axis perpendicular
to F1 and F2 (as illustrated in Fig. 10.7b), this type of spectroscopy is referred to
as three-dimensional NMR. Further separation of information on these planes onto
additional planes, say of 15N chemical shifts (Fig. 10.7c), is the basis of so-called
4-D and higher dimensional NMR. The dimensions referred to are not really spatial
dimensions, but are simply subsets of the data observed in 2-D spectra.
Figure 10.7 shows how 3-D and higher dimensional NMR can simplify spectra and reveal the details of couplings. Consider a hypothetical example of four
hydrogens—two bonded to nitrogen, Na-H and Nb-H, and two bonded to carbon,
Ca-H and Cb-H. The two hydrogens of each pair have the same chemical shift, so
the 1-D NMR spectrum shows two peaks, shown on the diagonal in Fig. 10.7a.
In addition, there is one cross-peak, indicating that one H-C is coupled to one
H-N. There are four possibilities for this coupling, listed in (a). If we separate
Section 10.2 NMR models 251
these spectra, as shown in (b), into planes that show only cross-peaks for protons
attached to 13C atoms having a narrow range of chemical shifts, the cross-peak
occurs only on the plane corresponding to Cb. This means that Cb carries one
of the coupled hydrogens and reduces the possible couplings to those listed as 3
and 4 in (a). If we again separate the spectra, as in (c), into planes based on 15N
shifts, we find that the cross-peak occurs only on the Na plane, eliminating all but
possibility 3, coupling of Cb-H to Na-H. Separating the information onto additional planes in this manner does not really produce any new information. Instead
it divides the information into subsets that may be easier to interpret.
10.2.3 Assigning resonances
The previous section describes methods that can provide an enormous amount of
chemical-shift and coupling information about a protein. Recall that our goal is to
determine the protein’s conformation. We hope that we can use couplings to decide
which pairs of hydrogens are neighbors, and that this information will restrict our
model’s conformations to one or a few similar possibilities. But before we can use
the couplings, we must assign all the resonances in the 1-D spectrum to specific
protons on specific residues in the sequence. This is usually the most laborious
task in NMR structure determination, and I will provide only a brief sketch of
it here.
The 1-D NMR spectrum of thioredoxin in Fig. 10.1, p. 240 shows the range
of chemical shifts δ for various proton types found in a protein. Because there
are only about 20 different types of residues present, and because residues have
many proton types in common (most have one proton on amide N, one on Cα ,
two on Cβ , and so forth), the spectrum is composed of several very crowded
regions. Obviously, we could never assign specific signals to specific residues
without greatly expanding or simplifying the spectrum. A variety of through-bond
correlation and NOE experiments allow us to resolve all these signals. Given the
necessary resolution, we are faced first with identifying individual residue types,
say, distinguishing alanines from valines.
The cross-peaks in through-bond spectra guide us in this task, because each
residue type has a characteristic set of splittings that makes it identifiable. This
allows us to determine which of the many Cα , Cβ , and Cγ signals, each in its
own characteristic region of the spectrum, belong to the same residue. Thus spin
systems that identify specific residue types can be identified and grouped together.
This is usually the first stage of analysis: identification of spin systems and thus of
groups of signals that together signify individual residues. There is conformational
information in these couplings also, because the precise values of J -coupling
constants depend on the conformational angles of bonds between atoms carrying
coupled atoms. So the splitting patterns reveal residue identities, and actual values
of splitting constants put some restrictions on local conformations.
Next, of all those spin systems identified as, say, valine, how do we determine which valine is valine-35, and which is valine-128? The next part of the
analysis is making this determination, for which we must examine connectivities
between residues. You can imagine that we might step from atom to atom through
252 Chapter 10 Other Kinds of Macromolecular Models
the protein chain by finding successively coupled proton pairs. But because proton spin-spin couplings are usually observed through at most three bonds (for
example, H-C-N-H), and because the carbonyl carbon of each residue carries no
hydrogens, proton spin-spin couplings do not extend continuously through the
chain. In between two main-chain carbonyl carbons, the spin-spin coupling systems are isolated from each other. But there are interresidue spin-spin couplings
because the N-H of residue n can couple to the Cα-H of residue n − 1.
Covalent connectivities between residues can also be determined using proteins
labeled fully with 13C and 15N, by way of various three-dimensional double and
triple resonance experiments, each of which reveals atoms at opposite ends of a
series of one-bond couplings. In essence, a sequence of pulses transfers magnetization sequentially from atom to atom. Coupling sequences such as H-N-Cα,
H-N-(CO)-Cα , H-N-(CO), H-Cα-(CO), and H-Cα-(CO)-N can be used to “walk”
through the couplings in a protein chain, establishing neighboring residues. Of
course, as in crystallography, much of this work can be automated, but the operator
must intervene when software cannot make the decisions unequivocally.
Taking NOE connectivities into account brings more potential correlations to
our aid. There is a high probability that at least one proton among the N-H, Cα-H,
or Cβ -H of residue n will be within NOE distance of the N-H of residue n + 1.
This means that NOE correlations can also help us determine which residues are
adjacent. In the end, this kind of analysis yields a complete set of resonance
assignments. In addition, it reveals much about backbone and sidechain conformations because knowing which protons interact by NOE greatly restricts the
number of possible conformations. These restrictions are a key to determining the
conformation of the protein.
As you might suspect, assigning closely spaced chemical shifts to many similar atoms and assigning myriad correlation peaks to many similar atom pairs is
fruitful turf for Bayesian methods (Sec. 7.5.6, p. 164). Indeed, these methods
are being applied successfully in sorting out the enormous numbers of chemicalshift and correlation signals that must be assigned in order to proceed to structure
determination. Bayesian methods are also useful in assessing the likelihoods of
models that fit all or most of these assignments. As you will see in the next section,
constructing models that conform to all of the distance restrictions obtained from
analysis of signals is the essence of NMR structure determination.
10.2.4 Determining conformation
Recall that our goal (you might well have forgotten by now) is to determine, in
detail, the conformation of the protein. From the analysis described in the previous
section, we know which signals in the spectra correspond to which residues in the
sequence. In addition, the magnitudes of J couplings put some restraints on local
conformational angles, both main-chain and side-chain, for each residue. What
is more, the specific NOE couplings between adjacent residues also restrain local
conformations to those that bring the coupled atoms to within NOE distance of
each other (the range of distances detected by NOESY can be determined by the
details of pulse timing during data collection).
Section 10.2 NMR models 253
Nuclear Overhauser effect correlations must be interpreted with care, because
the NOE couples atoms through space, not through bonds. So NOE coupling
between protons on two residues may not mean that they are adjacent in sequence.
It may instead mean that the protein fold brings the residues near to each
other. Distinguishing NOE cross-peaks between sequential neighbors from those
between residues distant in the sequence sets the stage for determining the conformation of the protein. Once we have assigned all neighboring-residue NOESY
cross-peaks properly, then the remaining cross-peaks tell us which hydrogens of
sequentially distant residues are interacting with each other through space. This
adds many of the most powerful and informative entries to the list of distance
restraints with which our final model must agree.
The end result of analyzing NMR spectra is a list of distance restraints. The
list tells us which pairs of hydrogens are within specified distances of each other
in space. This is really about all that we learn about the protein from NMR.
But as in X-ray crystallography, we already have a lot of prior knowledge about
the protein. We know its chemical structure in detail because we know the full
sequence of amino-acid residues, as well as the full structures of any cofactors
present. We know with considerable precision all of the covalent bond lengths and
bond angles. We know that most single-bond dihedral angles will lie within a few
degrees of the staggered conformations we call rotamers. We know that amide
single-bond dihedral angles will be close to 180o. We know that most main-chain
conformational angles  and  will lie in the allowed zones of the Ramachandran
diagram. We know all these things about any protein having a known sequence. The
question is, does all this add up to knowing the three-dimensional conformation
of the protein?
Next we would like to build a model of the protein that fits all we know, including
the distance restraints we learn from NMR. This is no trivial task. Much research
has gone into developing computer algorithms for this kind of model building.
One general procedure entails starting from a model of the protein having the
known sequence of residues, and having standard bond lengths and angles but random conformational angles. This starting structure will, of course, be inconsistent
with most of the distance and conformational restraints derived from NMR. The
amount of inconsistency can be expressed as a numerical parameter that should
decline in value as the model improves, in somewhat the same fashion as the
R-factor decreases as a crystallographic model’s agreement with diffraction data
improves during refinement.
Starting from a random, high-temperature conformation, simulated annealing
or some form of molecular dynamics is used to fold our model under the influence of simulated forces that maintain correct bond lengths and angles, provide
weak versions of van der Waals repulsions, and draw the model toward allowed
conformations, as well as toward satisfying the restraints derived from NMR.
Electrostatic interactions and hydrogen-bonding are usually not simulated, in order
to give larger weight to restraints based on experimental data; after all, we want
to discover these interactions in the end, not build them into the model before the
data have had their say. The simulated folding process entails satisfying restraints
254 Chapter 10 Other Kinds of Macromolecular Models
locally at first, and then gradually satisfying them over greater distances. At some
points in the procedure, the model is subjected to higher-temperature dynamics,
to shake it out of local minima of the consistency parameter that might prevent
it from reaching a global best fit to the data. Near the end, the strength of the
simulated van der Waals force is raised to a more realistic value, and the model’s
simulated temperature is slowly brought down to about 25oC.
Next the model is examined for serious van der Waals collisions, and for large
deviations from even one distance or conformational restraint. Models that suffer
from one or more such problems are judged not to have converged to a satisfactory final conformation. They are discarded. The entire simulated folding process
is carried out repeatedly, each time from a different random starting conformation,
until a number of models are found that are chemically realistic and consistent
with all NMR-derived restraints. If some of these models differ markedly from
others, the researchers may try to seek more distance restraints in the NMR spectra that will address specific differences, and repeat the process with additional
restraints. It may be possible at various stages in this process to use the current
models to resolve previous ambiguities in NOE assignments and to include them
in further model building. When the group of models appears to contain the full
range of structures that satisfy all restraints, this phase of structure determination
is complete.
The result of the NMR structure determination is thus not a single model, but
a group or ensemble of from half a dozen to over 30 models, all of which agree
with the NMR data. The models can be superimposed on each other by leastsquares fitting (Sec. 7.5.1, p. 159, and Sec. 11.4.1, p. 288) and displayed as shown
in Fig. 10.8 for thioredoxin (PDB 4trx). In this display of 10 of the 33 models
Figure 10.8  Ten of the 33 NMR models of thioredoxin (stereo, PDB 4trx). Image:
DeepView/POV-Ray.
Section 10.2 NMR models 255
obtained, it is readily apparent that all models agree with each other strongly in
some regions, often in the interior, whereas there is more variation in other regions,
usually on the surface.
Researchers interested in understanding this protein will immediately want to
use the results of the NMR work as an aid to understanding the function of this
protein. Which of these perhaps two or three dozen models should we use? For
further studies, as well as for purposes of illustration, it is appealing to desire a
single model, but one that will not let us forget that certain regions may be more
constrained by data than others, and that certain regions may be different in different models within the ensemble. One way to satisfy the common, but perhaps
uninformed (see Sec. 10.2.6, p. 257), desire for a single model is to compute the
average position for each atom in the model and to build a model of all atoms in
their average position. This model may be unrealistic in many respects. For example, bond lengths and angles involving atoms in their averaged positions may not
be the same as standard values. This averaged model is then subjected to restrained
energy minimization, which in essence brings bond lengths and angles to standard
values, minimizes van der Waals repulsions, and maximizes noncovalent interactions, with minimal movement away from the averaged atomic coordinates. The
result is a single model, the restrained minimized average structure. The rms deviation in position from the average of the ensemble is computed for each atom in
this model, and the results are recorded in the B-factor column of the atomiccoordinate file. Deviations are low in regions where models in the ensemble agree
well with each other, and deviations are high where the models diverge. Figure 10.9
Figure 10.9  Energy-minimized averaged model (stereo, PDB 3trx), colored by rms
deviation of atom positions in the ensemble from the average position. For each residue,
main-chain colors reflect the average rms deviation for C, O, N, and CA, and all side-chain
atoms are colored to show the average rms deviation for atoms in the whole side chain.
Image: DeepView/POV-Ray.
256 Chapter 10 Other Kinds of Macromolecular Models
shows the final model of thioredoxin (PDB 3trx) derived from the full ensemble
partially represented in Fig. 10.8. Colors of the model represent rms deviations
of atom positions from the ensemble average, with blue assigned to the smallest deviations, red to the largest, and colors across the visible spectrum for
intermediate values.
How much structural information must we obtain from NMR in order to derive
models like those shown in Fig. 10.8? As summarized in the PDB 3trx file
header for thioredoxin (remember, READ THE HEADER!), the models shown in
Fig. 10.8 were determined from 1983 interproton distance restraints derived from
NOE couplings. In addition, the researchers used 52 hydrogen-bonding distance
restraints defining 26 hydrogen bonds. Detection of these hydrogen bonds was
based on H/D exchange experiments, in which spectra in H2O and D2O were
compared, and NOEs involving exchangeable H disappeared or were diminished
in D2O (deuterium does not give an NMR signal). All hydrogen bonds detected in
this manner involved exchangeable amide hydrogens. Once these hydrogen bonds
were detected in early model construction, they were included in the restraints used
in further calculations. Finally, there were 98 and 71 backbone dihedral-angle
restraints, and 72 χ1(Cβ -Cγ ) side-chain dihedral-angle restraints, derived from
NOE and J -coupling. Thus the conformation of each of the 33 final thioredoxin
models is defined by a total of 2276 restraints. Recall that thioredoxin is a relatively
small protein of 105 residues, so these models are based on about 20 restraints per
residue. Finally, for an example of the effect of a single distance restraint on the
final model, take another look at Fig. 10.6, p. 249, and Fig. 10.10. In Fig. 10.6,
p. 249, one of the NOESY off-diagonal signals is labeled “F89N,δ.” This notation
Figure 10.10  Detail (stereo) of the averaged model at phenylalanine-89, showing the averaged distance between the two hydrogens inolved in the distrance restraint
indicated as “F89N,δ” in the NOESY spectrum, Fig. 10.6. Image: DeepView/POV-Ray.
Section 10.2 NMR models 257
means that the correlated nuclei are in phenylalanine-89 (F89), and specifically,
that the hydrogen on the amide nitrogen is correlated with the hydrogen on one of
the Cβ carbons, which are on the aromatic ring, ortho to the side-chain connection.
If you compare this figure with Fig. 10.1, p. 240, you can identify the diagonal
signals: the signal at approximately δ = 9.0 is the amide hydrogen, and the signal
at δ = 7.3 is the aromatic hydrogen. Figure 10.10 shows the distance between
these two hydrogens in the final averaged model. You can see how this distance
restraint confines the conformation of the phenylalanine side chain in this model.
10.2.5 PDB files for NMR models
Like crystallographers, NMR spectroscopists share their models with the scientific
community by depositing them in the Protein Data Bank. For NMR models, two
PDB files sometimes appear—one containing all models in the ensemble, and one
containing the coordinates of the restrained minimized average (although the trend
is away from depositing averaged structures). For human thioredoxin, these two
files are 4trx (ensemble of 33 models) and 3trx (averaged model). In parallel with
deposition of structure factors for crystallographic models, the most responsible
research groups also deposit an accessory file listing all distance restraints used in
arriving at the final models (PDB id code 3trx.mr for this model).
At first glance, the averaged model would appear to serve most researchers
who are looking for a molecular model to help them explain the function of
the molecule and rationalize other chemical, spectroscopic, thermodynamic, and
kinetic data. On the other hand, you might think that the ensemble and distancerestraint files are of most use to those working to improve structure determination
techniques. There are good reasons however, for all researchers to look carefully
at the ensemble, as discussed in the next section.
As with PDB files for crystallographic models, NMR coordinate files also
include headers containing citations to journal articles about the structure determination work, as well as brief descriptions of specific techniques used in producing
the model presented in the file, including the numbers and types of restraints. When
you view PDB files in web browsers, the literature citations contain convenient
live links to PubMed abstracts of the listed articles.
10.2.6 Judging model quality
The most obvious criterion of quality of an NMR model is the level of agreement of
models in the ensemble. You can make this assessment qualitatively by viewing the
superimposed models in a molecular graphics program like the one I will describe
the Chapter 11. You should be particularly interested in agreement of the models in
regions of functional importance, including catalytically active or ligand-binding
sites. The PDB file for the averaged model usually contains information to help
you make this assessment. In particular, the data column that contains B-factors
for crystallographic models contains, for NMR models, the rms deviations from
average ensemble coordinate positions. As with B-factors, rms deviations are
smaller in the main chain than in side chains. The best quality models exhibit
main-chain deviations no greater than 0.4 Å, with side-chain values below 1.0 Å.
258 Chapter 10 Other Kinds of Macromolecular Models
Many graphics programs will color the averaged model according to these deviations, as illustrated in Fig. 10.9. This coloring is a useful warning to the user, but
it is not as informative as the deviations themselves because the graphics program,
depending on its settings, may give colors from blue to red relative to the range
of values in the PDB file, regardless of how narrow or wide the range. Thus two
models colored by rms deviations may appear similar in quality, but one will have
all rms deviations smaller than 3 Å, while the other will have much higher values.
You can best compare the quality of two different models by coloring rms deviations on an absolute scale, rather than the relative scale of the contents of a single
PDB file.
It is also useful to think about why rms deviations might vary from region to
region in the averaged model. In crystallographic models, higher B-factors in
sections of a well-refined model can mean that these sections are dynamically
disordered in the crystal, and thus moving about faster than the time scale of the
data collection. The averaged image obtained by crystallography, just like a photo
of a moving object, is blurred. On the other hand, high B-factors may mean static
disorder, in which specific side chains or loops take slightly different conformations
in different unit cells. Recall that the electron-density map sometimes reveals
alternative conformations of surface side chains. Are there analogous reasons for
high rms deviations in sections of an NMR model?
The proximate reason for high rms values is the lack of sufficient distance
restraints to confine models in the ensemble to a particular conformation. Thus
various models converge to different conformations that obey the restraints, but
several or many conformations may fill the bill. At the molecular level, there
are several reasons why NMR spectroscopy may not provide enough distance
restraints in certain regions of the model. The physically trivial reason is spectral
resolution. Either resolution may simply not be good enough for all couplings to
be resolved and assigned, or some spectral peaks may overlap simply because the
structures and environments they represent are practically identical. This problem
is worst in the most crowded regions of the spectrum and might persist despite
multidimensional simplification. Because a protein NMR spectrum is composed
of many similar spin systems, like many valines, some may simply have such similar chemical shifts and couplings that portions of them cannot be distinguished.
Alternatively, the width of spectral peaks may be too wide to allow closely spaced
peaks to be resolved, and if peak widths exceed coupling constants, correlations
cannot be detected. Peak width is inversely related to the rate at which the protein
tumbles in solution, and larger proteins tumble more slowly, giving broader peaks.
This is unfortunate, of course, because for larger proteins, you need better resolution. But three- and four-dimensional spectra simplify these complex spectra,
and the use of one-bond couplings circumvents the correlation problem because
one-bond coupling constants are generally much larger than three-bond proton
coupling constants. By the late 1990s, top NMR researchers were claiming that
it is now potentially feasible to determine protein structures in the 15- to 35-kDa
range (135 to 320 residues) at an accuracy comparable to crystallographic models
at 2.5-Å resolution. They see the upper limit of applicability of methods described
Section 10.3 Homology models 259
here as probably 60–70 kDa. In early 2005, when I searched the PDB for the NMR
models of the largest molecules, I found 10 models of more than 300 residues and
28 of more than 250 residues, out of the approximately 4500 NMR models in the
database.
Beyond limitations in resolution are more interesting reasons for high-rms
variations within an ensemble of NMR models. Lack of observed distance
restraints may in fact mean a lack of structural restraints in the molecule itself.
Large variations among the models may be pointing us to the more flexible and
mobile parts of the structure in solution. In fact, if we are satisfied that our models
are not limited by resolution or peak overlap, we can take seriously the possibility
that variations in models indicate real dynamic processes. (Relaxation experiments
can be used to determine true flexibility.) Sometimes, mobility may take the form
of two or a few different conformations of high-rms regions of the real molecule,
each having a long enough lifetime to produce NOE signals. The distance inferred
from NOE intensities would be an average figure for the alternative conformations.
The ensemble of final models would then reveal conformations that are compatible
with averaged distance restraints, and thus point to the longer-lived conformations
in solution.
If some or all of the ensemble conformations reveal actual alternative conformations in solution, then these models contain useful information that may be lost in
producing the averaged model. If the most important conformations for molecular
function are represented in subsets of models within the final ensemble, then an
averaged model may mislead us about function. Just like crystallographic models,
NMR models do not simply tell us what we would like to know about the inner
workings of molecules. Evidence from other areas of research on the molecule are
necessary in interpreting what NMR models have to say.
As for other indicators of model quality, we expect that NMR models, just like
crystallographic models, should agree with prior knowledge about protein structure. So Ramachandran diagrams and distributions of side-chain conformations
should meet the same standards of quality as those for any other type of model.
For an example of a recent NMR structure determination, see NMR structure of Mistic, a membrane-integrating protein for membrane protein expression,
T. Roosild, et al., Science 307, 1317 (2005).
10.3 Homology models
10.3.1 Introduction
I have repeatedly reminded you that protein structure determination is a search
for the conformation of a molecule whose chemical composition is known. Much
experience supports the conclusion that proteins with similar amino-acid sequences
have similar conformations. These observations suggest that we might be able to
use proteins of known structure as a basis for building models of proteins for
260 Chapter 10 Other Kinds of Macromolecular Models
which we know only the amino-acid sequence. This type of structure determination has been called knowledge-based modeling but is now commonly known as
comparative protein modeling or homology modeling. We refer to the protein we
are modeling as the target, and to the proteins used as frameworks as templates.
If, in their core regions, two proteins share 50% sequence identity, the alpha
carbons of the core regions can almost always be superimposed with an rms deviation of 1.0 Å or less. This means that the core region of a protein of known
structure provides an excellent template for building a model of the core region of
a target protein having 50% or higher sequence homology. The largest structural
differences between homologous proteins, and thus the regions in which homology models are likely to be in error, lie in surface loops. So comparative modeling
is easiest and most reliable in the core regions, and it is the most difficult and unreliable in loops. The structures resulting from homology modeling are, in a sense,
low-resolution structures, but they can be of great use, for example, in guiding
researchers to residues that might be involved in protein function. Hypotheses
about the function of these residues can then be tested by looking at the effects
of site-directed mutagenesis. Homology models may also be useful in explaining
experimental results, such as spectral properties; in predicting the effects of mutations, such as site-directed mutations aimed at altering the properties or function
of a protein; and in designing drugs aimed at disrupting protein function.
Because much of the homology modeling process can be automated, it is possible to develop databases of homology models automatically as genome projects
produce new protein sequences. Users of such databases should be aware that automatically generated homology models can often be improved by user intervention
in the modeling process (see Sec. 10.3.4, p. 263).
10.3.2 Principles
Comparative protein modeling entails the following steps: (1) constructing an
appropriate template for the core regions of the target, (2) aligning the target
sequence with the core template and producing a target core model, (3) building surface loops, (4) adding side chains in mutually compatible conformations,
and (5) refining the model. In this section, I will give a brief outline of a typical modeling strategy. Although a variety of such strategies have been devised,
my discussion is based on the program ProMod and on procedures employed by
SWISS-MODEL, a public service homology modeling tool on the World Wide
Web. SWISS-MODEL is one of many tools available on the ExPASy Molecular
Biology Server operated by the Swiss Institute of Bioinformatics. You will find a
link to ExPASy on the CMCC home page.
Templates for modeling
The first step in making a comparative protein model is the selection of appropriate
templates from among proteins of known structure (that is, experimental models
derived from crystallographic or NMR data) that exhibit sufficient homology with
the target protein. Even though it might seem that the best template would be the
Section 10.3 Homology models 261
protein having highest sequence homology with the target, usually two or several
proteins of high percent homology are chosen. Multiple templates avoid biasing
the model toward one protein. In addition, they guide the modeler in deciding
where to build loops and which loops to choose. Multiple templates can also aid
in the choice of side-chain conformations for the model.
Users find templates by submitting the target sequence for comparison with
sequences in databases of known structures. Programs like BLAST or FastA carry
out searches for sequences similar to the target. A BLAST score lower than 10−5
(meaning a probability lower than one in 100,000 that the sequence similarity is
a coincidence), or a FastA score 10 standard deviations above the mean score for
random sequences, indicates a potentially suitable template. A safe threshold for
automated modeling is 35% homology between target and templates. Below this
threshold, alignment of template and target sequences may be unreliable, even
though it may turn out that their three-dimensional structures are quite similar.
Structural homology can, and often does, lurk at homology percentages too low
for homology to be detectable.
Next, programs like SIM align the templates. Because many proteins contain
more than one chain, and many chains are composed of more than one domain,
modelers use databases, extracted from the Protein Data Bank, of single chains.
There are also means to find and extract single domains homologous to the target
from within larger protein chains. If only one homolog of known structure can be
found, it is used as the template. If several are available, the template will be an
averaged structure based on them. The chain most similar to the target is used as the
reference, and the others are superimposed on it by means of least-squares fitting,
with alignment criteria emphasizing those regions that are most similar (referred
to as the “conserved” regions because sequence similarities presumably represent
evolutionary conservation of sequence within a protein family). To obtain the best
alignment, a program like ProMod starts by aligning alpha carbons from only those
regions that share the highest homology with the reference. Alpha carbons from
the other (nonreference) templates are added to the aligned, combined template if
they lie within a specified distance, say 3.0 Å, of their homologous atoms in the
reference. The result is called a structurally corrected multiple-sequence alignment.
Modeling the target core
To build the core regions of the target protein, its sequence is first aligned with that
of the template or, if several templates are used, with the structurally corrected
multiple sequence alignment. The procedure aligns the target with all regions,
including core and loops, that give high similarity scores, with the result that the
core of the target aligns with the core regions of all models, whereas loop regions
of the target align only with individual models having very similar loop sequences.
This means that subsequent modeling processes will take advantage not only of the
general agreement between target and all templates in core regions but also of the
specific agreement between target and perhaps a single template that shares a very
similar loop sequence.
262 Chapter 10 Other Kinds of Macromolecular Models
With this sequence alignment, a backbone model of the target sequence is
threaded—folded onto the aligned template atoms—producing a model of the
target in the core regions and in any loops for which a highly homologous template loop was found. When multiple templates are used, the target atom positions
are the best fit to the template atom positions for a target model that keeps correct
bond lengths and angles, perhaps weighted more heavily toward the templates of
highest local sequence homology with the target. As for residue side chains in the
averaged template, in SWISS-MODEL, side-chain atom positions are averaged
among different templates and added so that space is occupied more or less as
in the templates. After averaging, the program selects, for each side chain, the
allowed side-chain conformation or rotamer that best matches the averaged atom
positions.
Modeling loops
At this stage, we have a model of the core backbone in which the atom positions
are the average of the atom positions of the templates, and in which some core
side chains are included. Perhaps some loops are also modeled, if one or more
templates were highly homologous to the target. But most of the surface loops
are not yet modeled, and in most cases, the templates give us no evidence about
their structures. This is because the most common variations among homologous
proteins lie in their surface loops. These variations include differences in both
sequence and loop size. The next task is to build reasonable loops containing the
residues specified by the target sequence.
One approach is to search among high-resolution (≤2.5-Å) structures in the Protein Data Bank for backbone loops of the appropriate size (number of residues) and
end-point geometry. In particular, we are looking for loops whose conformation
allows their own neighboring residues to superimpose nicely on the target loop’s
neighboring residues in the already modeled target core region. To hasten this
search, modelers extract and keep loop databases from the PDB, containing the
alpha-carbon coordinates of all loops plus those of the four neighboring residues,
called “stem” residues, on both ends of each loop.
To build a loop, a modeling program selects loops of desired size and scores them
according to how well their stem residues can be superimposed on the stem residues
of the target core, aligning each prospective loop as a rigid body. The program
might search for loops whose rms stem deviations are less than 0.2 Å and, finding
none, might search again with a criterion of 0.4 Å, continuing until a small number,
say five, candidates are found. The target loop is then modeled, its coordinates
based on the average of five database loops with lowest rms-deviation scores. This
process builds only an alpha-carbon model, so the amide groups must be added.
This might involve another search through even higher-resolution structures in the
PDB, looking for short peptides (not necessarily loops) whose alpha carbons align
well with short stretches of the current loop atoms. Again, if several good fits are
found, the added atoms are modeled on their average coordinates, thus completing
the backbone of our model.
Section 10.3 Homology models 263
Modeling side chains
Our model now consists of a complete backbone, with side-chains only present
on those residues where templates and target are identical. In regions of high
sequence homology, target side chains nonidentical to templates might be modeled
on the template side chains out to the gamma atoms. The remaining atoms and the
remaining full side chains called for by the target sequence are then added, using
rotamers of the side chains that do not clash with those already modeled, or with
each other.
Refining the model
Our model now contains all the atoms of the known amino-acid sequence. Because
many atom positions are averages of template positions, it is inevitable that the
model harbors clashing atoms and less-than-optimal conformations. The end of all
homology modeling is some type of structure refinement, including energy minimization. SWISS-MODEL uses the program GROMOS to idealize bond lengths
and angles, remove unfavorable atom contacts, and allow the model to settle into
lower-energy conformations lying near the final modeled geometry. The number
of cycles of energy minimization is limited so that the model does not drift too
far away from the modeled geometry. In particular, loops tend to flatten against
the molecular surface upon extended energy minimization, probably because the
model’s energy is being minimized in the absence of simulated interactions with
surrounding water.
A modeling process like the one I have described is applied automatically—but
with the possibility of some user modifications—to target sequences submitted to
the SWISS-MODEL server at ExPASy. This tool also allows for intervention at
various stages, during which the user can apply special knowledge about the target
protein or can examine and adjust sequence or structural alignments. DeepView
(formerly called Swiss-PdbViewer)—an excellent free program for viewing, analyzing, and comparing models—is specifically designed to carry out or facilitate
these interventions for SWISS-MODEL users, thus allowing a wider choice of
tools for template searching, sequence alignment, loop building, threading, and
refinement. You will learn more about DeepView in Chapter 11.
10.3.3 Databases of homology models
As a result of the many genome-sequencing projects now under way, an enormous
number of new structural genes (genes that code for proteins) are being discovered.
With the sequences of structural genes comes the sequences of their product proteins. Thus new proteins are being discovered far faster than crystallographers and
NMR spectroscopists can determine their structures. Although homology models
are almost certain to be less accurate than those derived from experimental data,
they can be obtained rapidly and automatically. Though they cannot guide detailed
analysis of protein function, these models can guide further experimental work on
a protein, such as site-directed mutagenesis to pinpoint residues essential to function. The desirability of at least “low-resolution” structures of new proteins has led
264 Chapter 10 Other Kinds of Macromolecular Models
to initiatives to automatically model all new proteins as they appear in specified
databases. Thus the number of homology models has already exceeded the number
of experimentally determined structures in the Protein Data Bank.
One of the first automated modeling efforts was carried out in May of 1998.
Called 3D-Crunch, the project entailed submission of all sequences in two major
sequence databases, SWISS-PROT and trEMBL, to the SWISSMODEL server.
The result was about 64,000 homology models (compared to about 9000 models in
the Protein Data Bank at the time), which are now available to the public through
the SWISS-MODEL Repository (see the CMCC home page). Aversatile searching
tool allows users to find and download final models. Alternatively, users can download entire modeling projects, containing the final coordinates of the homology
model along with aligned coordinates of the templates. These project files enable
the user to carry the modeling project beyond the automated process and to use
other tools or special knowledge about the protein to further improve the model.
Project files are most conveniently opened in DeepView, which provides both builtin modeling tools and interfaces to additional programs for further improvement of
models. Figure 10.11 shows a modeling project returned from SWISS-MODEL.
For more discussion of this figure, see the next section and the figure caption.
Figure 10.11  Homology modeling project returned from SWISS-MODEL (stereo).
The target protein is a fragment of FasL, a ligand for the widely expressed mammalian
protein Fas. Interaction of Fas with FasL leads to rapid cell death by apoptosis. The template
proteins are (1) tumor necrosis factor receptor P55, extracellular domain (PDB 1tnr, black)
and (2) tumor necrosis factor-alpha (PDB 2tun, gray). The modeled FasL fragment is
shown as ribbon and colored by model B-factors. Only the alpha carbons of the templates
are shown. Image: DeepView/POV-Ray.
Section 10.3 Homology models 265
In addition to general homology-modeling tools like SWISS-MODEL and
DeepView, there are specialized modeling tools and servers devoted to specific
proteins of wide interest, including antibodies and membrane proteins such as
G-protein-coupled receptors. See the CMCC home page for links to such sites.
10.3.4 Judging model quality
Note that the entire comparative protein-modeling process is based on structures of
proteins sharing sequence homology with the target, and that no experimental data
about the target is included. This means that we have no criteria, such as R-factors,
that allow us to evaluate how well our model explains experimental observations
about the molecule of interest. The model is based on no experimental observations.
At the worst, in inept hands, a homology modeling program is capable of producing
a model of any target from even the most inappropriate template. Even the most
respected procedures may produce dubious models. By what criteria do we judge
the quality of homology models?
We would like to ask whether the model is correct. We could say it is correct if
it agrees with the actual molecular structure. But we do not have this kind of assurance about any model, even one derived from experiment. It is more reasonable for
us to define correct as agreeing to within experimental error with an experimental
(X-ray or NMR) structure. Most of the time, we accept a homology model because
we do not have an experimental structure, but not always. Researchers working
to improve modeling methods often try to model known structures starting from
related known structures. Then they compare the model with the known structure
to see how the modeling turned out. In this situation, a correct model is one in
which the atom positions deviate from those of the experimental model by less
than the uncertainty in the experimental coordinates, as assessed by, for example,
a Luzzati plot (Sec. 8.2.2, p. 183). Areas in which the model is incorrect are arenas
for improving modeling tools.
Typically, however, we want to use a homology model because it is all we have.
In some cases, even without an experimental model for comparison, we can recognize incorrectness. A model is incorrect if it is in any sense impossible. What
are signs of an incorrect model? One is the presence of hydrophobic side chains
on the surface of the model, or buried polar or ionic groups that do not have their
hydrogen-bonding or ionic-bonding capabilities satisfied by neighboring groups.
Another is poor agreement with expected values of bond lengths and angles.
Another is the presence of unfavorable noncovalent contacts or “clashes.” Still
another is unreasonable conformational angles, as exhibited in a Ramachandran
plot (Sec. 8.2.1, p. 181). We know that high-quality models from crystallography
and NMR do not harbor these deficiencies, and we should not accept them in
a homology model. Many molecular graphics programs can compute deviations
from expected bond geometry; highlight clashes with colors, dotted lines, or overlapping spheres; and display Ramachandran diagrams, thus giving us immediate
visual evidence of problems with models. We can also say that the model is incorrect if the sequence alignment is incorrect or not optimal. The details of sequence
comparison are beyond the scope of this book, but we can test the alignment of
266 Chapter 10 Other Kinds of Macromolecular Models
target with templates by using different alignment procedures, or by altering the
alignment parameters to see if the current alignment is highly sensitive to slight
changes in method. If so, it should shake our confidence in the model.
Beyond criteria for correctness, we can also ask how well the model fits its
templates. The rms deviation of model from template should be very small in
the core region. If not, we say that the model is inaccurate. An inaccurate model
implies that the modeling process did not go well. Perhaps the modeling program
simply could not come up with a model that aligns well with the coordinates
of the template or templates. Perhaps during energy minimization, coordinates
of the model drifted away from the template coordinates. Another possibility is
poor choice of templates. For instance, occasionally a crystallographic model is
distorted by crystal contacts, or an NMR template model is distorted by the binding
of a salt ion. If we unwittingly use such models as templates, energy refinement in
the absence of the distorting effect would introduce inaccuracy, as defined here,
while perhaps actually improving the model. A good rule of thumb is that if the
templates share 30–50% homology with the target, rms differences between final
positions of alpha carbons in the model and those of corresponding atoms in the
templates should be less than 1.5 Å. But it is also essential to look at the template
structures and make sure that they are really appropriate. An NMR structure of an
enzyme-cofactor complex is likely to be a poor model for a homologous enzyme
in the absence of the cofactor.
The rms deviations apply only to corresponding atoms, which means mainly the
core regions. Loop regions often cannot be included in such assessment because
there is nothing to compare them to. Again, we should demand correctness, that
is, the lack of unfavorable contacts or conformations. But beyond this kind of
correctness, our criteria are limited. If surface loops contain residues known to
be important to function, we must proceed with great caution in using homology
models to explain function.
If the model appears to be correct (not harboring impossible regions like clashes)
and accurate (fitting its templates well), we can also ask if it is reasonable, or
in keeping with expectations for similar proteins. Researchers have developed
several assessments of reasonableness that can sometimes signal problems with a
model or specific regions of a model. One is to sum up the probabilities that each
residue should occur in the environment in which it is found in the model. For all
Protein Data Bank models, each of the 20 amino acids has a certain probability
of belonging to one of the following classes: solvent-accessible surface, buried
polar, exposed nonpolar, helix, sheet, or turn. Regions of a model that do not fit
expectations based on these probabilities are suspect.
Another criterion of reasonableness is to look at how often pairs of residues interact with each other in the model in comparison to the same pairwise interactions
in templates or proteins in general. The sum of pairwise potentials for the model,
usually expressed as an “energy” (smaller is better) should be similar to that for the
templates. One form of this criterion is called threading energy. Threading energy
indicates whether the environment of each residue matches what is found for the
same residue type in a representative set of protein folds. Such criteria ask, in a
Section 10.4 Other theoretical models 267
sense, whether a particular stretch of residues is “happy” in its three-dimensional
setting. If a fragment is “unhappy” by these criteria, then that part of the model
may be in error.
To be meaningful, all of these assessments of reasonableness of a model must
be compared with the same properties of the templates. After all, the templates
themselves, even if they are high-quality experimental structures, may be unusual
in comparison to the average protein.
Homology model coordinate files returned from SWISS-MODELcontain, in the
B-factor column, a confidence factor, which is based on the amount of structural
information that supports each part of the model. Actually, it would be better to call
this figure an uncertainty factor, or a model B-factor, because a high value implies
high uncertainty, or low confidence, about a specific part of the model. (Recall
that higher values of the crystallographic B-factor imply greater uncertainty in
atom positions.) The model B-factor for a residue is higher if fewer template
structures were used for that residue. It is also higher for a residue whose alphacarbon position deviates by more than a specified distance from the alpha carbon
of the corresponding template residue. This distance is called the distance trap.
In SWISS-MODEL (or more accurately, in ProMod II, the program that carries
out the threading at SWISS-MODEL), the default distance trap is 2.5 Å, but the
user can increase or decrease it. However, if the user increases the distance trap,
all of the model B-factors increase, so they still reflect uncertainty in the model,
even if the user is willing to accept greater uncertainty. Finally, all atoms that are
built without a template, including loops for which none of the template models
had a similar loop size or sequence, are assigned large model B-factors, reflecting
the lack of template support for those parts of the model. Computer displays of
homology models can be colored by these model B-factors to give a direct display
of the relative amount of information from X-ray or NMR structures that was
used in building the model (Sec. 11.3.5, p. 281). Figure 10.11, p. 264 shows
a homology model and its templates. The target model is colored by the model
B-factors assigned by SWISS-MODEL. The templates are black and gray. With
this color scheme, it is easy to distinguish the parts of the model in which we can
have the most confidence. Blue regions were built on more templates and fit the
templates better. Red regions were built completely from loop databases, without
template contributions. Colors of the visible spectrum between blue and red may
align well with none or only a subset of the templates. (For more information about
these proteins, see the legend to Fig. 10.11, p. 264.)
10.4 Other theoretical models
Structural biologists produce other types of theoretical models as they pursue
research in various fields. Work that produces new models includes developing schemes to predict protein conformation from sequence; attempts to simulate
268 Chapter 10 Other Kinds of Macromolecular Models
folding or other dynamic processes; and attempts to understand ligand binding by
building ligands into binding sites and then minimizing the energy of the resulting combined model. The Protein Data Bank once contained some homology and
theoretical models, but they were removed in 2002. According to online documentation, “The Protein Data Bank (PDB) is an archive of experimentally determined
three-dimensional structures . . ..” The presence of theoretical models in the Protein Data Bank was only a temporary measure due to the lack of a data bank for
homology and other theoretical models. As of this writing, the only database for
theoretical models is the SWISS-MODEL Repository mentioned in Sec. 10.3.4,
p. 263.
At least two distributed-computing projects promise to produce large numbers
of theoretical models. The Folding@Home and Human Proteome Folding projects
are both attempts to simulate protein folding and discover folding principles that
might eventually make it possible to predict protein conformation from aminoacid sequence. Both projects will produce many theoretical structures, but it is not
known at the moment whether results will be available in searchable databases like
the PDB or the SWISS-MODEL Repository. Look for links to model databases
on the CMCC home page.
My goals in this chapter are to make you aware of the variety of model types
available to the structural biologist, to give you a start toward understanding other
methods of structure determination, and to guide you in judging the quality of noncrystallographic models, primarily by drawing your attention to analogies between
criteria of quality in crystallography.
Models are not molecules observed. No matter how they are obtained, before
we ask what they tell us, we must ask how well macromolecular models fit with
other things we already know. A model is like any scientific theory: it is useful
only to the extent that it supports predictions that we can test by experiment. Our
initial confidence in it is justified only to the extent that it fits what we already
know. Our confidence can grow only if its predictions are verified.
 Chapter 11
Tools for Studying
Macromolecules
11.1 Introduction
There is an old line about a dog who is finally cured of chasing cars—when it
catches one. What now? In this chapter, I discuss what to do when you catch
a protein. My main goal is to inform you about some of the tools available for
studying protein models and to suggest strategies for learning your way around
the unfamiliar terrain of a new protein. I will begin with a very brief glimpse
of the computations that underlie molecular graphics displays. Then I will take
you on a tour of molecular modeling by detailing the features present on most
modeling programs. Finally, I will briefly introduce other computational tools for
studying and comparing proteins. My emphasis is on tools that you can use on
your personal computer, although today’s personal computers do not limit your
possibilities very much.
11.2 Computer models of molecules
11.2.1 Two-dimensional images from coordinates
Computer programs for molecular modeling provide an interactive, visual environment for displaying and exploring models. The fundamental operation of computer
programs for studying molecules is producing vivid and understandable displays—
convincing images of models. Although the details of programming for graphics
269
270 Chapter 11 Tools for Studying Macromolecules
displays vary from one program (or programming language, or computer operating system) to another, they all produce an image according to the same geometric
principles.
A display program uses a file of atomic coordinates to produce a drawing on
the screen. Recall that a PDB coordinate file contains a list of all atoms located
by crystallographic, NMR, or theoretical analysis, with coordinates x, y, and z
for each atom. When the model is first displayed, the coordinate system is usually
shifted by the modeling program so that the origin is the center of the model. This
origin lies at the center of the screen, becoming the origin of a new coordinate
system, the screen coordinates, which I will designate xs, ys, and zs. The xs-axis
is displayed horizontally, ys is vertical, and zs is perpendicular to the computer
screen. (In a right-handed coordinate system, positive xs is to the right, positive ys
is toward the top of the screen, and positive zs is toward the viewer.) As the model
is moved and rotated, the screen coordinates are continually updated.
The simplest molecular displays are stick models with lines connecting atoms,
and atoms simply represented by vertices where lines meet (look ahead to Fig. 11.2,
p. 274 for examples of various model displays or renderings). It is easy to imagine
a program that simply plots a point at each position (xs, ys, zs) and connects the
points with lines according to a set of instructions about connectivity of atoms in
amino acids.
But the computer screen is two-dimensional. How does the computer plot in
three dimensions? It doesn’t; it plots a projection of the three-dimensional stick
model. Mathematically, projecting the object into two dimensions involves some
simple trigonometry, but graphically, projecting is even simpler. The program plots
points on the screen at positions (xs, ys, 0); in other words, the program does not
employ the zs coordinate in producing the display. This produces a projection of the
molecule on the xsys plane of the screen coordinate system, which is the computer
screen itself (see Fig. 11.1). You can imagine this projection process as analogous
to casting a shadow of the molecule on the screen by holding it behind the screen
and lighting it from behind. Another analogy is the image of a tree projected
onto the ground by its leaves when they fall during a cold windless period. The
program may use the z-coordinate to produce shading (by scaling color intensity in
proportion to zs, thus making foreground objects brighter than background objects)
or perspective (by scaling the xs and ys coordinates in proportion to zs, thus making
foreground objects larger than background objects), or to allow foreground objects
to cover background objects (allowing objects with larger zs , to overwrite those
with smaller zs), and thus to make the display look three-dimensional.
11.2.2 Into three dimensions: Basic modeling operations
The complexity of a protein model makes it essential to display it as a threedimensional object and move it around (or move our viewpoint around within it).
The first step in seeing the model in three dimensions is rotating it, which gives
many three-dimensional cues and greatly improves our perception of it. Rotating
the model to a new orientation entails calculating new coordinates for all the
Section 11.2 Computer models of molecules 271
Figure 11.1  Geometry of projection. (a) Model viewed from off to the side of
the screen coordinate system. Each atom is located by screen coordinates xs, ys, and zs.
(b) Model projected onto graphics screen. Each atom is displayed at position (xs, ys, 0),
producing a projection of the model onto the xsys-plane, which is the plane of the graphics
screen.
atoms and redisplaying by plotting on the screen according to the new (xs, ys)
coordinates.
The arithmetic of rotation is fairly simple. Consider rotating the model by
θ degrees around the xs (horizontal) axis. It can be shown that this rotation
transforms the coordinates of point p, [xs(p), ys(p), zs(p)], to new coordinates
[x′s(p), y′s(p), z′s(p)] according to these equations:
x′s(p) = xs(p)
y′s(p) = [ys(p) · cos θ ] − [zs(p) · sin θ ] (11.1)
z′s(p) = [ys(p) · sin θ ] + [zs(p) · cos θ ].
272 Chapter 11 Tools for Studying Macromolecules
Notice that rotating the model about the xs-axis does not alter the xs coordinates,
[x′s(p) = xs(p)], but does change ys and zs. A similar set of equations provides
for rotation about the ys- or zs-axis. If we instruct the computer to rotate the
model around the xs-axis by θ degrees, it responds by converting all coordinates
(xs, ys, zs) to new coordinates (x′s, y′s, z′s) and plotting all points on the screen at
the new positions (x′s, y′s, 0). Graphics programs allow so-called real-time rotation
in which the model appears to rotate continuously. This requires fast computation,
because to produce what looks like smooth rotation, the computer must produce a
new image about every 0.05 second. So in literally the blink of an eye, the computer
increments θ by a small amount, recalculates coordinates of all displayed atoms,
using equations such as 11.1, and redraws the screen image. Fast repetition of this
process gives the appearance of continuous rotation.
11.2.3 Three-dimensional display and perception
Complicated models become even more comprehensible if seen as threedimensional (3-D) objects even when not in motion. So graphics display programs
provide some kind of full-time 3-D display. This entails producing two images like
the stereo pairs used in this book, and presenting one image to the left eye and
the other to the right eye, which is the function of a stereo viewer. The right-hand
view is just like the left-hand view, except that it is rotated about 5o about its ys
axis (clockwise, as viewed from above). Molecular modeling programs display
the two images of a stereo pair side by side on the screen, for viewing in the same
manner as printed pairs. (To learn how to see the three-dimensional image using
side-by-side pairs, see Appendix, p. 293, or the CMCC home page.)
With proper hardware, modeling programs can produce full-screen 3-D objects.
The technique entails flashing full-size left and right views alternately at high
speed. The viewer wears special glasses with liquid-crystal lenses that alternate
rapidly between opaque and transparent. When the left-eye view is on the screen,
the left lens is transparent and the right lens is opaque. When the screen switches to
the right-eye view, the right lens becomes transparent and the left becomes opaque.
In this manner, the left eye sees only the left view, and the right eye sees only the
right view. The alternation is fast, so switching is undetectable, and a full-screen
3-D model appears to hang suspended before the viewer. Several viewers, each
wearing the glasses, can see 3-D at the same time, without having to align their
eyes with the screen, as is required for stereo pairs.
Both types of stereo presentation mimic the appearance of objects to our two
eyes, which produce images on the retina of objects seen from two slightly different viewpoints. The two images are rotated about a vertical axis located at the
current focal point of the eyes. From the difference between the two images, called
binocular disparity, we obtain information about the relative depth of objects in
our field of view. For viewers with normal vision, two pictures with a 5o binocular
disparity, each presented to the proper eye, gives a convincing three-dimensional
image.
Unfortunately, a small but significant percentage of people cannot obtain depth
information from binocular disparity alone. In any group of 20 students, there is
Section 11.2 Computer models of molecules 273
a good chance that one or more will not be able to see a three-dimensional image
in printed stereo pairs, even with a viewer. In the world of real objects, we decode
depth not only by binocular disparity but also by relative motion, size differences
produced by perspective, overlap of foreground objects over background objects,
effects of lighting, and other means. Molecular modeling programs provide depth
information to a wider range of viewers by providing depth cues in the form of
shading, perspective, and movement, as described in the next section.
11.2.4 Types of graphical models
Modeling and graphics programs provide many ways to render or represent a
molecular model. Figure 11.2 shows several different representations of three
strands of beta sheet from human thioredoxin (PDB 1ert). The simplest representation is the wireframe model (a). Although other renderings have greater visual
impact, the wireframe model is without question the most useful when you are
exploring a model in detail, for several reasons. First, you can see the model, yet
see through it at the same time. Foreground and background parts of the model
do not block your view of the part of the model on which you center your attention. Second, in wireframe models, you can see atom positions, bond angles, and
torsional angles clearly. Third, wireframe models are the simplest and hence the
fastest for your computer to draw. This means that, as you rotate or zoom a model,
the atom positions are recalculated and the images redrawn faster, so that the model
moves more rapidly and smoothly on the screen. Rapid and smooth motion give
the model a tangible quality that improves your ability to grasp and understand
it. Combined with colorings that represent element types, or other properties like
B-factors or solvent accessibility (Sec. 11.3.8, p. 282), wireframe models can be
very informative. There is one disadvantage: wireframe models do not contain
obvious depth cues; therefore, they are best viewed in stereo.
Other renderings, some of which are also illustrated in Fig. 11.2, have their
strengths and weaknesses. All of them make vivid illustrations in textbooks and
on web pages, when conveying fine structural details is not necessary. It takes
programs longer to draw these more complex models, so they might not move as
smoothly on the screen. Ball-and-stick models are similar to wireframe models, but
they provide better depth cues, because they are shaded and appear as solid objects
that obscure objects behind them. Aside from providing depth cues, the balls
primarily produce clutter, but they look nice. Figure 11.2b shows a ball-and-stick
model of the beta strands shown in (a).
Various cartoon renderings, like the ribbon diagram in Fig. 11.2c, show main
chains as parallel strands or solid ribbons, perhaps with arrows on strands of
pleated sheet to show their direction, and barrel-like alpha helices. These are some
of the most vivid renderings and are excellent for giving an overview of protein
secondary and tertiary structure. Combined images, for example, with most of the
molecule in cartoon form and an important binding site in ball and stick, make nice
printed illustrations that guide the reader from the forest of the whole molecule to
the trees of functionally important structural details. For example, this particular
image also shows more clearly than either (a) or (b) that successive side chains lie
274 Chapter 11 Tools for Studying Macromolecules
Figure 11.2  Common types of computer graphics models (stereo), all showing
the same three strands of pleated-sheet structure from human thioredoxin (PDB 1ert).
(a) Wireframe; (b) ball and stick; (c) ribbon backbone with ball-and-stick side chains;
(d) space filling. Image: DeepView/POV-Ray.
Section 11.3 Touring a molecular modeling program 275
on opposite sides of a pleated sheet. A disadvantage of more complex images is
that rotation on a personal computer can be choppy and slow.
The same atoms are shown again in Fig. 11.2d as a space-filling model, which
gives a more realistic impression of the density of a model, and of the extent of
its surface. The surface of a space-filling model can be either the van der Waals
(shown) or the solvent-accessible surface, and can be colored according to various
properties, like surface charge. Space-filling models are excellent for examining
the details of atomic contacts within and between models. Note, for example, in
this figure, how side chains nestle together on the top of this sheet. Space-filling
models of an enzyme and its inhibitor appear to fit each other like hand and glove.
A disadvantage: you cannot make out exact atom positions and bond angles in
space-filling models.
As the renderings of Fig. 11.2 show, there is no single model that shows everything you might like to see. Graphics programs allow you to mix model types
and switch quickly among them, to highlight whatever you are trying to see or
illustrate. Browsing a model with computer graphics is far superior to viewing
even a large number of static images.
11.3 Touring a molecular modeling
program
As you might infer from a brief description of the computing that underlies molecular graphics, the computer must be fast. But today’s personal computers are up to
the task of dealing with quite large and complex graphics. The current generation
of personal computers can allow you to conduct highly satisfying and informative
expeditions into the hearts of the most complex macromolecular models.
The basic operations of projecting and rotating a screen image of the molecular model make the foundation of all molecular graphics programs. Upon these
operations are built many tools for manipulating the display. These tools give
viewers the feeling of actively exploring a concrete model. Now I will discuss
tools commonly found in modeling programs.
Although this is a general discussion of modeling tools, I will use as my example
an excellent molecular viewing and modeling program that you can obtain free of
charge in Linux, Macintosh, Silicon Graphics, or Windows versions. My example
is DeepView, originally and still often called Swiss-PdbViewer, an easy-to-learn
yet very powerful molecular viewing and modeling tool. Figure 11.3 provides a
picture of a personal computer screen during use of DeepView, and the caption
gives a description of its main sets of functions. If you currently own no modeling
program and want to learn to use a program that will allow you start easily, yet
grow painlessly into very sophisticated modeling and structural analysis, DeepView is an excellent choice. In fact, it compares favorably with costly commercial
276 Chapter 11 Tools for Studying Macromolecules
programs. As of this writing, the number of free macromolecular graphics programs is growing constantly. But I still know of no single program that does as
many things, and as many of them very well, as DeepView. To learn how to download this program from the World Wide Web and to find a complete, self-guiding
tutorial for modeling beginners, see the CMCC home page. Whatever the program
you decide you use, search its documentation or the World Wide Web for tutorials
on how to get started with it. There is no faster way to learn molecular exploration
and analysis than by a hands-on tutorial. The CMCC home page also contains
links to help you find other molecular graphics programs and tutorials for learning
how to use them.
11.3.1 Importing and exporting coordinate files
As indicated earlier, image display and rotation requires rapid computing. Reading
coordinates from a text file like a PDB file is slow because every letter or number
in the file must be translated into binary or machine language for the computer’s
internal processing. For this reason, graphics programs work with a machinelanguage version of the coordinate file, which they can read and recompute faster
than a text file. So the first step in exploring a model is usually converting the PDB
file to binary form. With almost all modeling programs, this operation is invisible
to the user.
Often users produce revealing views of the model and wish to use the coordinates
in other programs, such as energy calculations or printing for publication. For this
purpose, molecular modeling programs include routines for writing coordinate
files in standard formats like PDB, using the current binary model coordinates as
input.
Figure 11.3  Screen shot of DeepView in use on a Macintosh OS X computer. Controls for manipulating the model are at the top of the main graphics window, which can
be expanded to fill the screen. The Control Panel lists residues in the model and allows
selection of residues for display, coloring, labeling, and surface displays. The Align window shows residue sequence also, including alignment of multiple models if present.
The Ramachandran Plot window shows main-chain torsional angles for residues currently
selected (red in the Control Panel, purple in the Align window). Dragging dots on the Rama
plot changes torsion angles interactively. The Layer Infos window (not shown) allows control of display features for multiple models (layers) in any combination. In the graphics
window is a stereo display of the heme region of cytochrome b5 (PDB 1cyo). Selected
hydrogen bonds are shown in green, and measured distances in yellow. A TORSION operation is in progress (darkened button, top of graphics window). The side-chain conformation
of phenylalanine-58 is being changed. Clashes between the side chain and other atoms are
shown in pink. The user can display and read the PDB file of the currently active model by
clicking the document icon at the upper right of the graphics windows. Clicking an ATOM
line in the PDB file display centers the graphics model on that residue and reduces the
display to the residue and its nearest neighbors.
Section 11.3 Touring a molecular modeling program 277
278 Chapter 11 Tools for Studying Macromolecules
11.3.2 Loading and saving models
The coordinate files produced by graphics programs can be loaded and saved
just like any other files. As the model is manipulated or altered, coordinates are
updated. At any point, the user can save the current model, or replace it with one
saved earlier, just as you can do with a word-processor document. To facilitate
studying intermolecular interactions, comparing families of models, or scanning
through NMR ensembles, modeling programs can handle several or many models
at once, each as separate layers that you can view separately or superimpose.
The models currently in memory can be viewed and manipulated individually or
together. You can save them with their current relative orientations so that you can
resume a complicated project later.
Although almost all macromolecular model coordinates are available in PDB
format, modeling programs provide for loading and saving files in a variety of
formats. These might include Cambridge Structure Database format, used by the
largest database for small molecules, or others from among the dizzying list of
small-molecule formats used by organic and inorganic chemists. Other special
file formats include those used by energy minimization, dynamics simulation,
and refinement programs, many of which are now also available for personal
computers. Full-featured modeling programs like DeepView usually provide for
exporting coordinates for these powerful programs, shipping them over networks
for completion of complex computing tasks, retrieving the resulting model files,
and converting them back to PDB format for viewing. As personal computers
get faster, these transfers will become unnecessary. Useful adjuncts to graphics
programs are online tools for interconverting various types of files. For example,
the Dundee PRODRG2 Server allows users to paste or upload coordinates in almost
any known format, and then to convert the file to any other format. See the CMCC
home page for links to this and other file-conversion web sites.
11.3.3 Viewing models
Standard viewing commands allow users to rotate the model around screen xs, ys,
and zs axes and move (translate) the model for centering on areas of interest. Viewers magnify or zoom the model by moving it along the zs-axis toward the viewer,
or by using a command that magnifies the image without changing coordinates
by simply narrowing the viewing angle. “Clip” or “slab” commands simplify the
display by eliminating foreground and background, producing a thin slab of displayed atoms. With most programs, including DeepView, all of these operations
are driven by a mouse or other point-and-drag device (trackball, for example), after
selecting the desired operation by buttons (top of graphics window in Fig. 11.3)
or key commands.
Figure 11.4 shows two views of the small protein cytochrome b5 (PDB 1cyo), an
electron-transport protein containing an iron-heme prosthetic group (shown edgeon with gray iron at center). In (a), you are looking into the heme binding pocket,
with much of the protein in the background. Even in stereo, the background clutter
makes it difficult to see the heme environment clearly. In (b), the background is
Section 11.3 Touring a molecular modeling program 279
Figure 11.4  Heme region of cytochrome b5 (stereo, PDB 1cyo). (a) View without
clipping; (b) same view after “slab” command to eliminate all except contents of a 12-Å
slab in the zs direction. Image: DeepView.
removed using a “slab” command to give a clearer view of the heme and protein
groups above and below it. The program clips by displaying only those atoms
whose current zs-coordinates lie within a specified range (12.5 Å, in this case),
which is chosen visually by sliding front and rear clipping planes together until
unwanted background and foreground atoms disappear.
Centering commands allow the user to select an atom to be made the center
of the display. Upon selection by pointing to the atom and clicking a mouse, or
by naming the atom, the program moves the model so as to center the desired
atom on the screen and within the viewing slab, and also to make it the center
of subsequent rotations. For example, DeepView provides a very handy centering
feature. Simply pressing the “=” key centers the part of the model currently on
display, and resizes it to fit the screen. It is not unusual for novice viewers to
accidentally move the model completely out of view and be unable to find it.
Nothing is more disconcerting to a beginner than completely losing sight of the
model. When the model disappears, it may be off to the side of the display, above
or below the display, or still centered, but outside the slab defined by clipping
planes. In any event, automatic recentering can often help viewers find the model
and regain their bearings. As a last resort, there is usually a reset command,
which brings model and clipping planes back to starting positions. The viewer
280 Chapter 11 Tools for Studying Macromolecules
pays a price for resetting, losing the sometimes considerable work of finding a
particularly clear orientation for the model, centering on an area of interest, and
clipping away obscuring parts of the structure.
Viewing commands usually also include selection of stereo or mono viewing
and offer various forms of depth cueing to improve depth perception, either by
mimicking the effects of perspective (front of model larger than rear), shading
(front of model brighter than rear), or rocking the model back and forth by a few
degrees of rotation about the ys-axis.
11.3.4 Editing and labeling the display
A display of every atom in a protein is often forbidding and incomprehensible.
Viewers are interested in some particular aspect of the structure, such as the active
site or the path of the backbone chain, and may want to delete irrelevant parts of
the model from the display. Display commands allow viewers to turn atoms on and
off. Atoms not on display continue to be affected by rotation and translation, so
they are in their proper places when redisplayed. Viewers might eliminate specific
atoms by pointing to them and clicking a mouse, or they might eliminate whole
blocks of sequence by entering residue numbers. They may display only alpha
carbons to show the folding of the protein backbone (see Fig. 3.4, p. 36), or only
the backbone and certain side chains to pinpoint specific types of interactions.
Editing requires knowledge of how atoms are named in the coordinate file,
which is often, but not always, the same as PDB atom labels (Sec. 7.7, p. 173).
Thus viewers can produce an alpha-carbon-only model by limiting the display to
atoms labeled CA. Each program has its own language for naming atoms, residues
(by number or residue name), distinct chains in the model (like the α and β chains
of hemoglobin), and distinct models. Viewers must master this language in order
to edit displays efficiently. In DeepView, editing the view is greatly simplified by
a Control Panel (Fig. 11.3, right side) that provides a scrollable list of all residues
in the PDB file. A menu at the top allows you to switch to other PDB files if you
are currently viewing more than one model. You can select, display, label, and
color residues, and add surface displays as well, all with mouse operations.
Most programs provide powerful selection tools to allow you to pick specific
parts of the model for displaying, labeling, or coloring. For example, DeepView
provides, in its Control Panel, the means to select main chain, side chains, or
both; individual residues; multiple residues (contiguous or not); individual elements of secondary structure; and individual chains. In addition, DeepView menu
commands allow selecting residues by type (for example, select all histidines, or
select water, or select heteromeric group), property (select acidic residues), or secondary structure (select helices, beta strands, or coils); surface or buried residues;
residues with cis-peptide bonds; problem residues in a model, like those with 
and  values outside the range of allowed values (Sec. 8.2.1, p. 181), or residues
making clashes; neighbors of the current selection (select within 4.5 Å of current
selection); and residues in contact with another chain, to name just a few.
Even with an edited model, it is still easy for viewers to lose their bearings. Label
commands attach labels to specified atoms, signifying element, residue number,
Section 11.3 Touring a molecular modeling program 281
or name. Labels like the one for PHE-58 in Fig. 11.3 float with the atom during
subsequent viewing, making it easy to find landmarks in the model.
11.3.5 Coloring
Although you may think that color is merely an attractive luxury, adding color
to model displays makes them dramatically more understandable. Most programs
allow atoms to be colored manually, by selecting a part of the model and then choosing a color from a color wheel or palette. Additional color commands allow the use
of color to identify elements or specific residues, emphasize structural elements,
or display properties. For example, DeepView provides, among others, commands
for coloring the currently selected residues by CPK color (as in Fig. 11.3), residue
type, B-factor (sometimes called coloring by temperature), secondary structure
(different colors for helix, sheet, and turns), secondary structure in sequence (blue
for first helix or beta strand, red for last one, and colors of the visible spectrum for
each secondary structural element in between), chain (a different color for each
monomer in an oligomeric protein), layer (different color for each model currently
on display), solvent accessibility, threading or force-field energies (Sec. 10.3.4,
p. 263), and various kinds of model problems.
WARNING: Coloring by B-factor uses whatever information is in the B-factor
column of the PDB file. To interpret the colors that result, it is crucial for you to
know what kind of model you are viewing, because this information tells you different things about different kinds of models. Coloring a good crystallographic model
by B-factor reveals relative uncertainty in atom positions due to static or dynamic
disorder (Sec. 8.2.3, p. 185). Coloring an averaged NMR model by “B-factor”
actually reveals relative rms deviation of atom positions from the average positions of the corresponding atoms in the ensemble of NMR models (Sec. 10.2.5,
p. 257). And coloring a homology model by “B-factor” distinguishes parts of the
model according to how much support exists for the model in the form of X-ray
or NMR structural information (Sec. 10.3.4, p. 263).
Combined with selecting tools, color commands can be powerful tools simply
for finding features of interest. If you are interested in cysteines in a protein, you
can select all cysteines, choose a vivid color for them, and immediately be able to
find them in the forest of a large protein.
11.3.6 Measuring
Measurements are necessary in identifying interactions within and between
molecules. In fact, noncovalent interactions like hydrogen bonds are defined by
the presence of certain atoms at specified distances and bond angles from each
other. In Fig. 11.3, p. 276, yellow dotted lines connect two carbon atoms, a valine
methyl and a heme methyl. The distance between the atoms is displayed and is
approximately the distance expected for carbon atoms involved in a hydrophobic
contact.
Modeling programs allow display of distances, bond angles, and dihedral angles
between bonded and nonbonded atoms. These measurements float on dotted lines
282 Chapter 11 Tools for Studying Macromolecules
within the model (just like labels) and are often active; that is, they are continually
updated as the model is changed, as described in the next section.
11.3.7 Exploring structural change
Modeling programs allow the viewer to explore the effects of various changes in
the model, including conformational rotation, change in bond length or angle, and
movement of fragments or separate chains. Used along with active measurements,
these tools allow viewers to see whether side chains can move to new positions
without colliding with other atoms, or to examine the range of possible movements of a side chain. In Fig. 11.3, a change of torsion angles is in progress for
phenylalanine-58. In its current conformation, the side chain is in collision with
the heme, as shown by a pink “clash” line.
Like rotation and translation, changes of model conformation, bond angles,
or bond lengths are reflected by changes in the coordinate file. The changes are
tentative at first, while users explore various alterations of the model. After making changes, users have a choice of saving the changes, removing the changes,
or resetting in order to explore again from the original starting point. In Fig.
11.3, notice that the torsions button is darkened, showing that the operation is in
progress, and that the user will have an opportunity to keep or discard the changes
being made.
11.3.8 Exploring the molecular surface
Stick models of the type shown in Fig. 11.2a, p. 274 are the simplest and fastest
type of model to compute and display because they represent the molecule with the
smallest possible number of lines drawn on the screen. Stick models are relatively
open, so the viewer can see through the outer regions of a complex molecule
into the interior or into the interface between models of interacting molecules.
But when the viewer wants to explore atomic contacts, a model of the molecular
surface is indispensable.
Published structure papers often contain impressive space-filling computer
images of molecules, with simulated lighting and realistic shadows and reflections.
These images require the computer to draw hundreds of thousands of multicolored lines, and so most computers cannot redraw such images fast enough for
continuous movements. Some of these views require from seconds to hours to
draw just once. Although such views show contacts between atomic surfaces, they
are not practical for exploring the model interactively. They are used primarily as
snapshots of particularly revealing views.
How then can you study the surface interactively? The most common compromise is called a dotted surface (Fig. 11.5), in which the program displays dots
evenly spaced over the surface of the molecule. This image reveals the surface
without obscuring the atoms within and can be redrawn rapidly as the viewer
manipulates the model. Several types of surfaces can be computed, each with its
own potential uses. One type is the van der Waals surface (Fig. 11.5a), in which all
dots lie at the van der Waals radius from the nearest atom, the same as the surface
Section 11.3 Touring a molecular modeling program 283
Figure 11.5  Dotted surface displays of heme in cytochrome b5 (stereo, PDB 1cyo).
(a) Van der Waals surface enclosing the entire heme. (b) Solvent-accessible surface of heme,
showing only the portion of heme surface that is exposed to surrounding solvent. Most of
heme is buried within the protein. Image: DeepView.
of space-filling models. This represents the surface of contact between nonbonded
atoms. Any model manipulations in which van der Waals surfaces penetrate each
other are sterically forbidden. Van der Waals surfaces show packing of structural
elements with each other, but the display is complicated because all internal and
external atomic surfaces are shown. Elimination of the internal surfaces produces
the molecular surface, which is simpler, but still shows some surfaces that are not
accessible to even the smallest molecules, like water.
A very useful surface display is the solvent-accessible surface, which shows all
parts of the molecule that can be reached by solvent (usually water) molecules.
284 Chapter 11 Tools for Studying Macromolecules
This display omits all internal atomic surfaces, including crevices that are open
to the outside of the model, but too small for solvent to enter. Some modeling
programs, like DeepView, contain routines for calculating this surface, whereas
others can take as input the results of surface calculations from widely available
programs. Calculating the solvent-accessible surface entails simulating the movement of a sphere, called a probe, having the diameter of a solvent molecule over
the entire model surface, and computing positions of evenly spaced dots wherever
model and solvent come into contact. Figure 11.5b, p. 283, shows the solventaccessible surface for the heme group in cytochrome b5. It is clear from this view
that most of the heme is buried within the protein and not accessible to the solvent.
Carrying out the same simulation that produces solvent-accessible surface displays, but locating the dots at the center of the probe molecule, produces the
extended surface of the model. This display is useful for studying intermolecular
contacts. If the user brings two models together—one with extended surface displayed, the other as a simple stick model—the points of intermolecular contact are
where the extended surface of one model touches the atom centers of the second
model.
The default color of the displayed surface is usually the same as the color selected
for the underlying atoms. In Fig. 11.5b, for example, two oxygens of one of the
heme carboxyl groups produces the large red bulge of accessible surface (the other
carboxyl is hydrogen bonded to serine-64 and is much less accessible to solvent).
Alternatively, color can reflect surface charge (commonly, blue for positive, red
for negative, with lighter colors for partial charges) or surface polarity (contrasting
colors for hydrophobic and hydrophilic regions). These displays facilitate finding
regions of the model to which ligands of specified chemical properties might bind.
The surface of proteins carry many charged and polar functional groups that
confer electrostatic potentials between the protein and its surroundings. Graphics
programs can display such potentials in two ways, with isopotential surfaces or
with potentials mapped onto the molecular surface. Figure 11.6 presents one subunit of the dimeric enzyme acetylcholine esterase (PDB 2ack) with isopotential
surfaces (calculated by DeepView) showing its electrostatic properties. The red
surface is at a constant (that’s what iso means) negative potential and the blue is the
same level of positive potential. The height of the isopotential surface above the
molecular surface is greatest where the surface potential is highest. As you can see,
the surface directly above the substrate binding site (that is, toward the top of the
figure), which is marked by a white competitive inhibitor, has a very high negative
potential. The substrate, acetylcholine, is positively charged, and thus is strongly
attracted by the negatively charged residues that surround the active site region.
Attraction of the active site region for the substrate gives this enzyme a very high
on-rate for substrate binding. The potential surface gives an almost tactile feeling
for the attraction the enzyme would have for a passing substrate molecule; a positively charged substrate would be repelled by the back side of the enzyme, but even
a grazing encounter with the face of the enzyme should result in substrate binding.
In Fig. 11.6b, the model of (a) has been rotated 90◦ around xs toward the viewer,
and the electrostatic potential has been mapped onto the molecular surface of the
Section 11.3 Touring a molecular modeling program 285
Figure 11.6  Electrostatic potentials and surfaces (stereo). (a) Electrostatic isopotential surface of one subunit of acetylcholine esterase (PDB 2ack). (b) Electrostatic potential
mapped onto molecular surface, looking down on the top of the view in (a). A competitive
inhibitor bound at the active site is visible in both views. Image: DeepView/POV-Ray.
enzyme. You can see the competitive inhibitor (space-filling model) through an
opening in the enzyme surface. The shade of the surface reflects the surface potential, with darkest red for highest negative potential, corresponding to locations
above which an isopotential surface would be farthest away. The view shows that
the substrate is deeply buried inside the enzyme, but the electrostatic potential
explains how it quickly finds its way to this secluded site.
Combining models with surfaces can make dramatic illustrations of molecular
properties. In Fig. 11.7, the full dimeric structure is shown as an alphacarbon model, with the isopotential surface for the dimer rendered as a smooth,
286 Chapter 11 Tools for Studying Macromolecules
Figure 11.7  Acetylcholine esterase dimer (stereo). Alpha-carbon stick model of
protein, space filling model of inhibitor, and isopotential surface. Subrates enter at lower
left and upper right. Image: OpenGL rendering in DeepView.
transparent surface. This view helps to explain how acetylcholine esterase can bind
substrate on almost every encounter—its on-rate for binding is very close to the
diffusion rate, the rate at which two species at 1.0 M concentrations are expected
to collide with each other in solution.
11.3.9 Exploring intermolecular interactions: Multiple models
Formulating proposed mechanisms of protein action requires investigating how
proteins interact with ligands of all kinds, including other proteins. Molecular
modeling programs allow the user to display and manipulate several models, either
individually or together. With DeepView and many other modeling programs, the
number of models is limited only by computer memory and speed. Tools for this
purpose usually allow all of the same operations as the viewing tools but permit
selection of models affected by the operations. In docking experiments (a term
taken from satellite docking in the space program), one model can be held still while
another is moved into possible positions for intermolecular interaction. Labeling,
measurement, and surface tools are used simultaneously during docking to ensure
that the proposed interactions are chemically realistic. Some programs include
computational docking, in which the computer searches for optimal interaction,
usually from a user-specified starting point. Such calculations are quite slow, and
usually done by stand-alone programs that produce output coordinate files for
Section 11.3 Touring a molecular modeling program 287
viewing on programs like DeepView. Many graphics programs can prepare files
for external docking programs. See the CMCC home page for links to programs
and servers for protein-protein and protein-ligand docking.
11.3.10 Displaying crystal packing
Many molecular modeling programs include the capacity to display models of
the entire unit cell. All the program needs as input is a set of coordinates for one
molecule, the unit-cell dimensions, and a list of equivalent positions for the crystal
space group. The user can display one cell or clusters (say, 2× 2× 2) of cells. The
resulting images, particularly when teamed with surface displays, reveal crystalpacking interactions, allowing the user to see which parts of the crystallographic
model might be altered by packing, and might thus be different from the solution
structure. For examples of crystal-packing displays, see Fig. 4.18, p. 70, and
Fig. 4.19, p. 71. Unit-cell tools usually allow the user to turn equivalent positions
on and off individually, making them useful for teaching the topics of equivalent
positions and symmetry. For example, DeepView allows the user to create new
models by specifying symmetry operations or selecting them from a list or from
symmetry lines in PDB files.
11.3.11 Building models from scratch
In addition to taking coordinate files as inputs, modeling programs allow the user to
build peptides to specification and to change amino-acid residues within a model.
To build new models, users select amino acids from a palate or menu and direct the
program to link the residues into chains. Users can specify conformation for the
backbone by entering backbone angles and, by selecting a common secondary
structure or by using the tools described earlier for exploring structural change.
Model-builder tools are excellent for making illustrations of common structural
elements like helices, sheet, and turns. DeepView allows you to start building a
model by simply providing a text file of the desired sequence in one-letter abbreviations; it imports the sequence and models it as an alpha helix for compactness. Then
you can select parts of the model and either select their secondary structural type
or set their Ramachandran angles. Alternatively, you can display a Ramachandran
diagram for the model and change torsional angles by dragging residue symbols to
new locations on the diagram, watching the model change as you go—a great tool
for teaching the meaning of the  and  angles. Similar tools are used to replace
one or more side chains in a model with side chains of different amino acids, and
thus explore the local structural effects of mutation.
11.3.12 Scripts and macros: Automating routine structure analysis
In many areas of structure analysis, you find yourself repeating sequences of operations. For example, in comparing a number of related proteins, you might routinely
open a model file, align it with a previously loaded model (your reference model),
and color main-chain atoms by rms deviation from the reference. You might even
want to capture the results by constructing a table of sequence alignments and
288 Chapter 11 Tools for Studying Macromolecules
rms deviations by residue. Such analysis of dozens of models quickly becomes
drudgery. Sounds like a job for a programmer, and some modeling programs
make such programming easy by providing for writing and executing command
scripts, or for recording and playing back sequences of operations (sometimes
called macros). For example, DeepView incorporates a scripting language that is
similar to Perl or C++ . You can write scripts with any text editor, and execute
them with menu commands within the program. Sample scripts are included with
program documentation. Scripting and macros allow you to turn your modeling
program loose on large repetitive tasks, and then retrieve the results in easily
usable form.
11.4 Other tools for studying structure
It is beyond the scope of this little book to cover all the tools available for studying
protein structure. I will conclude by listing and briefly describing additional tools,
especially ones used in conjunction with modeling on graphics systems.
11.4.1 Tools for structure analysis and validation
In addition to molecular graphics, a complete package of tools for studying protein structure includes many accessory programs for routine structure analysis
and judging model quality. The chores executed by such programs include the
following:
 Calculating  and  angles and using the results to elements of secondary
structure as well as to display a Ramachandran diagram, which is useful in
finding model errors during structure refinement (crystallography or NMR)
or homology modeling. As I mentioned earlier, DeepView has a unique interactive Ramachandran plot window that allows the user to change main-chain
conformational angles in the model (Fig. 11.3, p. 276).
 Using distance and angle criteria to search for hydrogen bonds, salt links,
and hydrophobic contacts, and producing a list of such interactions. DeepView calculates and displays hydrogen bonds according to user specifications
of distance and angle. A very powerful pair of menu commands in DeepView allows you to show hydrogen bonds only from selected residues or
hetero groups and then to show only residues with visible hydrogen bonds. In
these two operations, you can eliminate everything from the view except, for
example, a prosthetic group and its hydrogen-bonding neighbors.
 Comparing homologous structures by least-squares superposition of one protein backbone on another. The result is a new coordinate set for one model that
best superimposes it on the other model. I used such a tool in DeepView to
compare the X-ray and NMR structures of thioredoxin in Fig. 3.4, p. 36 (alignment was instantaneous). DeepView provides superposition tools combined
Section 11.4 Other tools for studying structure 289
Figure 11.8  X-ray and NMR models of human thioredoxin (stereo, PDB 1ert and
3trx), aligned by least-squares superposition of corresponding alpha carbons. X-ray model
is gray. NMR model is colored by deviation from X-ray model. Greatest deviations are red,
and smallest deviations are blue.
with sequence comparison to improve the structural alignment between proteins that are only moderately homologous. In addition, you can color a protein
by the rms deviation of its atoms from those of the reference protein on which
it was superimposed, giving a vivid picture of areas where the structures
are alike and different. For example, in Fig. 11.8, full backbone models of
the X-ray and NMR structures of human thioredoxin are superimposed. The
X-ray model is gray, and the NMR model is colored by rms deviations of corresponding residues from the X-ray model. Residues for which the two models
deviate least are colored blue. Those exhibiting the greatest deviations are red.
Residues with intermediate deviations are assigned spectral colors between
blue and red. Even if the X-ray model were not shown, it would be easy to see
that the most serious disagreement between the two models lies in the surface
loop at the bottom, and that the two models agree best in the interior residues.
 Building additional subunits using symmetry operations, either to complete a
functional unit (Sec. 8.2.4, p. 187) or to examine crystal packing. DeepView
allows you to build additional subunits by selecting symmetry operations from
a palette, clicking on symmetry operations listed in a PDB file, or typing in
the components of a transformation matrix.
 Carrying out homology modeling. As mentioned in Chapter 10, DeepView is
a full homology modeling program. Using DeepView, a web browser, and
electronic mail, you can obtain template files from SWISS-PROT, align,
average, and thread your sequence onto the template, build loops or select
them from a database, find and fix clashes, submit modeling projects to
SWISS-MODEL, and retrieve them to examine the results or apply other
290 Chapter 11 Tools for Studying Macromolecules
Figure 11.9  Model and portion of electron-density map of bovine Rieske ironsulfur protein (stereo, PDB 1rie). The map is contoured around selected residues only.
Image: DeepView/POV-Ray.
modeling tools. GROMOS energy minimization is included in DeepView.
Figure 10.11, p. 264, shows a homology model started in DeepView and completed at SWISS-MODEL. The homology model is shown as ribbon colored
according to model B-factors (see Sec. 10.3.4, p. 265; also see the warning
about B-factor coloring on p. 281). Two templates used in the modeling are
shown as black and gray alpha-carbon displays. (For more information about
these proteins, see the legend to Fig. 10.11.)
 Displaying an electron-density map and adjusting the models to improve its
fit to the map (see Fig. 11.9). DeepView can display maps of several types
(CCP4, X-PLOR, DN6). Outside of full crystallographic computation packages, I am aware of no programs currently available for computation of maps
from structure factors on personal computers, but I am sure this will change.
Structure factors are available for some of the models in the PDB, and can
usually be obtained from the depositors of a model. As described in Sec. 8.2.5,
p. 189, the Electron Density Server at Uppsala University provides electrondensity maps for most models for which structure factors are deposited in the
Protein Data Bank.
11.4.2 Tools for modeling protein action
The crystallographic model is used as a starting point for further improvement
of the model by energy minimization and for simulations of molecular motion.
Additional insight into molecular function can be obtained by calculating charge
densities and bond properties by molecular orbital theory. For small molecules,
some of these calculations can be done “on the fly” as part of modeling. For the
more complex computations, and for larger molecules, such calculations are done
outside the graphics program, often as separate tasks on computers whose forte
is number crunching rather than graphics. DeepView can write files for export
Section 11.5 Final note 291
to several widely used programs for energy minimization and molecular dynamics. But as personal computers get faster, these operations will no longer require
transfer to specialized machines.
11.5 Final note
Making computer images and printed pictures of molecular models endows them
with the concreteness of everyday objects. While exploring models, viewers can
easily forget the difficult and indirect manner by which they are obtained. I wrote
this book in hopes of providing an intellectually satisfying understanding of the
origin of molecular models, especially those obtained from single-crystal X-ray
crystallography. I also hope to encourage readers to explore the many models
now available, but to approach them with full awareness of what is known and
what is unknown about the molecules under study. Just as good literature depicts
characters and situations in a manner that is “true to life,” a sound model depicts
a molecule in a manner that is true to the data from which it was derived. But
just as real life is more multifarious than the events, settings, and characters of
literature, not all aspects of molecular truth (or even of crystallographic, NMR,
or modeling truth) are reflected in the colorful model floating before us on the
computer screen. Users of models must probe more deeply into the esoterica of
structure determination to know just where the graphics depiction is not faithful
to the data. The user must probe further still—by using validation tools and by
reading wider literature on the molecule—to know whether the model is faithful
to other evidence about structure and action. The conversation between structural
models and evidence on all sides continually improves models as depictions of
molecules.
Perhaps I have stimulated your interest in crystallography itself, and have made
you wonder if you might jump in and determine the structure of that interesting
protein you are studying. I am happy that I can encourage you by reiterating that
crystallography, though still one of structural biology’s more challenging callings,
is faster and easier than ever before. Screening for crystal growth conditions does
not require expensive equipment or chemicals. Most research universities have at
least enough crystallographic instrumentation to allow you to assess the diffracting
power of your crystals, or perhaps even to collect preliminary data. You can take
promising crystals to synchrotron sources for data collection. Finally, you can
do the computation on reasonably modern personal computers. Look for a local
crystallography research group to help you get started.
Of course, if you pursue crystallography, there are many more details to learn.
As your next step toward a truly rigorous understanding of the method, I suggest Introduction to Macromolecular Crystallography by Alexander McPherson
(Wiley-Liss, 2002); Crystal Structure Analysis for Chemists and Biologists: Methods in Stereochemical Analysis by Jenny Glusker, Mitchell Lewis, and Miriam
Rossi (John Wiley & Sons, 1994); X-ray Structure Determination: A Practical
Guide, 2nd edition, by George H. Stout and Lyle H. Jensen (John Wiley and
292 Chapter 11 Tools for Studying Macromolecules
Sons, Inc., 1989); and Practical Protein Crystallography, by Duncan E. McRee
(Academic Press, Inc., 1993). On the World Wide Web, I recommend Crystallography 101 by Bernhard Rupp. Finally, if you are really serious about being a
crystallographer, and don’t mind getting very little sleep for about 16 days, apply
for admission to the course X-Ray Methods in Structural Biology, taught each fall at
Cold Spring Harbor Laboratory. You will find links to this and other crystallography
resources at the CMCC home page, www.usm.maine.edu/∼rhodes/CMCC.
 Appendix
Viewing Stereo Images
To see three-dimensional images using divergent stereo pairs in this book, use a
stereo viewer such as item #D8-GEO8570, Carolina Biological Supply Company
(1-800-334-5551). Or better, you can view stereo pairs without a viewer by training
yourself to look at the left image with your left eye and the right image with
your right eye (called divergent viewing). This is a very useful skill for structural
biologists, and is neither as difficult nor as strange as it sounds. (According to my
ophthamologist, it is not harmful to your eyes, and may in fact be good exercise
for eye muscles.)
Try it with this stereo image:
Figure A.1  Divergent stereo pair of interior, Dom St. Stephan, Passau, Deutschland.
293
294 Appendix Viewing Stereo Images
Try putting your nose on the page between the two views. With both eyes open,
you will see the two images superimposed, but out of focus, because they are too
close to your eyes. Slowly move the paper away from your face, trying to keep the
images superimposed until you can focus on them. (Keep the line between image
centers parallel to the line between your eyes.) When you can focus, you will
see three images. The middle one should exhibit convincing depth. Try to ignore
the flat images on either side. This process becomes easier and more comfortable
with practice. If you have difficulty, try it with a very simple image, such as
Fig. 4.18, p. 70.
For additional help with viewing stereo images in books or on computers,
click on Stereo Viewing at the CMCC home page, http://www.usm.maine.edu/
∼rhodes/CMCC.

Index
Numerics
3D-Crunch project, 264
A
A-DNA diffraction patterns, 217
absences in diffraction patterns,
73, 105–107
absorption, nuclear spin and, 240–241
absorption edge, 128
absorption of X rays, 74. See also
detectors, X-ray
anomalous scattering, 128–136
direct phasing methods, 135–136
extracting phase, 130–132
hand problem, 135
measurable effects, 128–130
multiwavelength (MAD), 133–134
amorphous materials, diffraction by,
219–222
amplitude of electronic-density map, 150
amplitude, wave, 21
angles within cells, 49–50
anomalous scattering (anomalous
dispersion), 128–136
direct phasing methods, 135–136
extracting phase, 130–132
hand problem, 135
measurable effects, 128–130
multiwavelength (MAD), 133–134
applying prior knowledge to models, 152,
154, 253
area detectors, 78
arrays, diffraction by, 16, 18
assessing model quality. See quality of
models
assigning NMR model resonances,
251–252. See also NMR models
asymmetric units, 68
asymmetric vs. functional units, 188
ATOM lines (PDB file), 176
atom occupancy, 161, 185
atomic coordinates
refinement of, 147–148
resolution and precision, 183–185
atomic coordinates entries, 174–176
atomic coordinates, modeling from,
269–270
atomic plane indices, 50–55
atomic structure factors, 98
AUTHOR lines (PDB file), 175
automating routine structure analysis,
287–288
average spacing of reciprocal-lattice
points, 86
B
B-DNA diffraction patterns, 217
B-factors (homology models), 267, 281
B-factors (NMR models), 257–258
back-transforms, 96
295
296 Index
ball-and-stick models, 273
Bayesian refinement models, 164–168
beam stops, 77
Bessel functions, 216
bias, minimizing, 154–156
Bayesian inference, 164
bichromatic X-ray sources, 74
binocular disparity, 272, 293–294
Biotech Validation Suite, 191
BLAST program, 261
body-centered (internal) lattices, 66
bootstrapping
density modification, 151–153
error removal, 147, 171
overview of, 146–149
refinement of atomic coordinates,
147–148
Bragg’s law, 55–57, 111
polychromatic rays, 231
in reciprocal space, 60–64
Bravais lattices, 67
brightness of reflections, 16–17, 63
electron density as function of,
115–117
scaling and postrefinement, 85–86
broken crystals, recognizing, 45
building models. See map fitting
C
13C-editing, 250
caged substrates, 234
calculated intensities, 105
cameras, X-ray, 80–85
capillary mounting, 45
cartoon renderings, 273
CCDs (charge-coupled devices), 78–80
cells. See unit cells
center of symmetry, 88
characteristic lines, 73
charge-coupled devices, 78–80
charge detection, 229
chemical shift, 239
chemically reasonable models, 172–173
CISPEP lines (PDB file), 176
CMCC web page, 4–5
cocrystallization, 40
cofactors, to grow protein crystals, 45
collimators, 77
coloring models (DeepView program),
281
comparative protein modeling, 260–263
complex numbers, 92–93
in two dimensions, 112
complex objects, diffraction by, 17
complex vectors, structure factors as,
112–115
COMPND lines (PDB file), 175
computer models of molecules, 269–275
conditions for crystal growth, 41–46
CONECT lines (PDB file), 176
confidence factor (homology models), 267
conformation, determining for NMR
models, 252–257
conformationally reasonable models,
173, 183
constraints, defined, 163
contour levels, 94
contour maps. See maps of electron
density
convergence to final model, 168–173
coordinate systems, 19–20, 50
depositing with PDB, 173–177
importing and exporting, 276–277
refinement of atomic coordinates,
147–148
two-dimensional computer images
from, 269–270
correlation spectroscopy (COSY), 248
COSY (correlation spectroscopy), 248
coupling, nuclear, 239–240
Crick, Francis, 217
cross-validation, 172
cryo-electronic microscopy, 227–231
cryocrystallography, 45–46
CRYST lines (PDB file), 176
crystal packing, 188, 287
crystallation techniques. See growing
crystals
crystalline fibers, 213
crystallographic asymmetric units, 188
crystallographic refinement. See structure
refinement
Crystallography Made Crystal Clear
web page, 4–5
crystallography models. See models
crystallography papers, reading, 192–208
Index 297
crystals, in general, 10–13. See also
protein crystals
growing, 11–13, 37–46
optimal conditions for, 41–46
mounting for data collection, 46–47
quality of, judging, 46
cubic cells, 50
D
data collection, 13–15, 73–89
cameras, 80–85
detectors, 77–80
intensity scaling and postrefinement,
85–86
mounting crystals for, 46–47
symmetry and strategy, 88–89
unit-cell dimension determination,
86–88
X-ray sources, 73–77
data mining, 177
database of proteins. See PDB
databases of homology models, 263–265
DBREF lines (PDB file), 175
de Broglie equation, 222
dead time, 78
DeepView program, 275–288
density modification, 151–153
density, unexplained, 187
depositing coordinates with PDB,
174–177
depth cuing, 280
derivative crystals, 37
growing, 40–41
selenomet derivatives, 41
detectors, X-ray, 77–80
difference Patterson functions, 125
diffracted X-ray reflections, 13, 49
as Fourier terms, 101–104
intensities of, 16–17
measurable, number of, 64–65
measured, electronic density from,
28–30
phases of. See phase
sphere of reflection, 63–64
symmetry elements and, 71–73
wave descriptions of, 24–26
diffraction, 15–19, 211–235
by amorphous materials (scattering),
219–222
calculating electronic density, 91–107
data collection, 13–15, 73–89
cameras, 80–85
detectors, 77–80
intensity scaling and
postrefinement, 85–86
mounting crystals for, 46–47
symmetry and strategy, 88–89
unit-cell dimension determination,
86–88
X-ray sources, 73–77
electron diffraction, 227–231
fiber diffraction, 211–219
geometric principles of, 49–73
judging crystal quality, 46
Laue diffraction, 231–235
neutron diffraction, 222–227
systematic absences in patterns, 73,
105–107
diffraction patterns, 30
diffractometry, 82
digestion of proteins, 45
dimensions of unit cells, 65, 86–88
direct phasing methods, 135–136
disorder, atomic, 185–186
regions of disorder, 187
disordered water, 34–35
dispersive differences, 134
displaying electron-density maps,
150–151
distance trap, 267
distortion from crystal packing, 188
divergent viewing, 272, 293–294
DNA diffraction patterns, 217
dynamic range, 78
E
edges, cell, 49–50. See also unit cells
editing display (DeepView program), 280
EDS (Electron Density Server), 190
electron cloud imaging, 11, 26
electron crystallography, 227
electron density
calculating from diffraction data,
91–107
determining from measured
reflections, 28–30
298 Index
electron density (continued )
determining from structure factors,
27–28
as Fourier sum, 99–100
as function of intensities and phases,
115–117
maps of, 26–27
amplitude of, 150
developing model from, 153–159
final models, 168–173
first maps, 149–153
improving. See bootstrapping
structure refinement, 159–168
unexplained, 187
Electron Density Server (EDS), 190
electron diffraction, 227–231
electron-potential maps, 229
EM (electron microscopy), 227–231
EMD (Electron Microscope Database),
230–231
enantiomeric arrangements, 127–128
END lines (PDB file), 176
energy-minimized averaged models,
255
energy refinement, 163
equivalent positions (symmetry), 69
error removal (filtering), 147, 171
etched crystals, 40
evaluating model quality. See quality of
models
Ewald sphere, 63–64
examining electron-density maps,
150–151
ExPASy tool, 263
EXPDTA lines (PDB file), 175
experimental models, 7
exporting coordinate files, 276–277
expression vectors, 41
extended surfaces, 284
F
face-centered lattices, 67
FastA program, 261
fiber diffraction, 211–219
FID (free-induction decay), 243.
See also NMR models
figure of merit, 156
figure of merit (phase), 152
film, X-ray-sensitive, 78
filtering, 147, 171
final model, converging to, 168–173
first maps, 149–153
Fo – Fc maps, 155–156, 169
disorder, indications of, 185
focusing mirrors, 77
FORMUL lines (PDB file), 176
Fourier analysis, defined, 96, 97
Fourier series, 23–24, 97
Fourier sums (summations), 26, 92–97
electron density as, 99–100
reflections as terms in, 101–104
structure factors as, 98–99
Fourier synthesis, 24, 97
Fourier terms, defined, 92
Fourier transforms, 28, 96–97
of FID (free-induction decay),
243, 247
of helices, 214
testing models with, 217–218
frames of data, 83
free-induction decay (FID), 243.
See also NMR models
free log-likelihood gain, 172
free R-factor, 172
freezing crystals, 45–46
frequency, wave, 22
Friedel pairs, 114
anomalous scattering, 131–133
Friedel’s law, 88, 114
FTs. See Fourier transforms
functional vs. asymmetric units, 188
functions of proteins, structure and, 35
G
generalized unit cells, 49–50
geometric principles of diffraction,
49–73
Bragg’s law, 55–57, 111
polychromatic rays, 231
in reciprocal space, 60–64
global minimum (structure refinement),
162
goniometer heads, 80
goniostats, 80
graphical models, types of, 273–275
graphing three-dimensional functions, 94
GROMOS program, 263, 290
Index 299
growing crystals, 11–13, 37–46
optimal conditions for, 41–46
gyration, radius of, 221
H
hand problem
anomalous scattering, 135
isomorphous replacement, 128
handing-drop method, 38–40
Harker sections (Harker planes), 127
harvest buffers, 46
header, PDB files, 175–176, 189
heavy-atom derivatives, 37. See also
isomorphous replacement
hand problem, solving, 135
preparing, 117–119
heavy-atom method. See isomorphous
replacement
heavy atoms, neutron scattering with,
225
helices, in fibrous materials, 213–214
HELIX lines (PDB file), 176
HET and HETNAM lines
(PDB file), 176
high-angle reflections, 101
higher-dimensional NMR spectra,
249–251
homology models, 237, 259–267
basic principles, 260–263
databases, 263–265
judging quality of, 265–267
hydrogen bonds, 31
I
image enhancement with cryo-EM,
229–230
image plate detectors, 78
imaginary numbers. See entries at
complex
imaging microscopic objects, 8
importing coordinate files, 276–277
improving maps and models.
See bootstrapping
impurity of samples, 41
indices (coordinates), 19, 50–55
atomic planes, 52–55
lattice indices, 50–52
integrity, structural, 31–33
intensity of reflections, 16–17, 63
electron density as function of,
115–117
scaling and postrefinement, 85–86
intermolecular interactions, exploring
(DeepView program), 286
internal lattices, 66
internal symmetry of unit cells, 65–73
data collection strategies, 88–89
functional-unit, 188
noncrystallographic (NCS), 153
Patterson map searches, 127
International Tables for X-Ray
Crystallography, 69
interplanar spacing. See Bragg’s law
inverse Fourier transforms, 96
isomorphism, 37, 118
isomorphous phasing models, 137–139
isomorphous replacement, 117–128
isotropic vibration, 185
iterative improvement of maps and
models. See bootstrapping
J
JRNL lines (PDB file), 175
judging crystal quality, 46
judging model quality. See quality
of models
justifiable restrictions on model,
152, 154, 253
K
KEYWS lines (PDB file), 175
knowledge-based modeling, 260
knowledge (prior), applying,
152, 154, 253
known properties, improving model with,
152, 154, 253
L
labeling display (DeepView program),
280
lack of closure (isomorphous
replacement), 124
lattice indices, 50–52
lattices, 10
real, 16
Bragg’s law, 55–57
300 Index
lattices (continued )
reciprocal, 16
average point spacing, 86
how to construct, 57–60
limiting sphere, 64
types of, 50, 66–67
Laue diffraction, 231–235
Laue groups, 89
layer lines, 213–216
least-squares refinement methods,
159–161
lenses, 8
ligands, 37
cocrystallization, 40
to grow protein crystals, 45
limited digestion of proteins, 45
limiting sphere, 64
LINK lines (PDB file), 176
liquid nitrogen. See cryocrystallography
loading models (DeepView program), 278
local minima (structure refinement),
162–163
location of phasing model, 139–141
log-likelihood gain, 168, 172
longitudinal relaxation, 242
loops, homology models, 262
low-angle reflections, 101
low-angle scattering, 221, 227
low-resolution models, 187
Luzzati plots, 183–184
M
macros with DeepView program, 287–288
MAD (multiwavelength anomalous
dispersion) phasing, 133–134
manual model building, 168–169
map fitting, 154, 156–159
convergence to final model, 168–173
with DeepView program, 287
manual, 168–169
maps of electron density, 26–27
amplitude of, 150
developing model from, 153–159
final models, 168–173
first maps, 149–153
improving. See bootstrapping
structure refinement, 159–168
unexplained, 187
maps of electron potential, 229
maps of nucleon density, 223
MASTER lines (PDB file), 176
mathematics of crystallography,
20–30
maximum-density lines, 154
measurements, taking (DeepView
program), 281
membrane-associated proteins, 45, 211
microscopic imaging, 8
Miller indices, 60
minima, local (structure refinement),
162–163
minimaps, 151
minimizing bias from models, 154–156
MIR (multiple isomorphous replacement),
123
mirror plane of symmetry, 67
model validation tools, 189–192,
288–290
Model Validation tutorial, 190
modeling after known proteins. See
molecular replacement
modeling tools, 269–292
computer modeling, 269–275
DeepView program (example),
275–288
model validation tools, 189–192,
288–290
models
computer models, 269–275
improving electronic-density maps
density modification, 151–153
error removal, 147, 171
overview of, 146–149
refinement of atomic coordinates,
147–148
judging quality of. See quality of
models
modifying to obtain phase information,
147
non-crystallographic, 237–268
homology models, 237, 259–267
NMR models, 238–259
other theoretical models, 267–268
obtaining and improving, 30, 145,
153–159
convergence to final model,
168–173
Index 301
judging convergence, 171–173
minimizing bias, 154–156
model building. See map fitting
regularization, 157, 171
structure refinement, 159–168
structure refinement parameters,
161–162, 189–192
phasing models. See molecular
replacement
reading crystallography papers,
192–208
sharing, 173–177
types of, 7
understanding, in general, 1–3
molecular dynamics, 163, 186
molecular envelopes, 151
molecular models. See models
molecular models, types of, 7
molecular replacement, 136–143
imposing prior knowledge,
152, 154, 253
isomorphous phasing models,
137–139
location and orientation searches,
139–143
nonisomorphous phasing models,
139
molecular surface exploration (DeepView
program), 282–286
molecules, obtaining images of, 9
MolProbity Web Service, 191
monochromatic X-ray sources, 74
monoclinic cells, 50, 59–60
monodispersity, 45
mosaicity (mosaic spread), 33
mother liquor, 33
cryoprotected, 46
with ligand, 40–41
motion, molecular, 163
mounting crystals for data collection,
46–47
multidimensional NMR models, 249–251
multiple forms for crystals, 33
multiple isomorphous replacement
(MIR), 123
multiwavelength anomalous diffraction
(MAD) phasing, 133–134
multiwavelength X-ray beams, 231–235
multiwire area detectors, 78
N
15N-editing, 250
native crystals, 37
NCS (noncrystallographic symmetry), 153
negative indices, 54
neutron diffraction, 222–227
NMR models, 238–259
basic principles, 239–251
conformation determination, 252–257
judging quality of, 257–259
PDB files for, 257
resonances, assigning, 251–252
NOE (nuclear Overhauser effect),
248, 252
conformation determination, 252–253
NOESY (NMR model), 248
noncrystalline fibers, 213
noncrystallographic symmetry (NCS), 153
nonisomorphous phasing models,
137–139
NSLS (National Synchrotron Light
Source), 75–77
nuclear Overhauser effect, 248
nuclear spin, 239–240
nucleation, 38
nucleon-density maps, 223
O
objective lens, 8
obtaining and improving models, 30, 145,
153–159
convergence to final model, 168–173
judging convergence, 171–173
map fitting (model building), 154,
156–159
convergence to final model,
168–173
with DeepView program, 287
manual, 168–169
minimizing bias, 154–156
regularization, 157, 171
structure refinement, 159–168
parameters for, 161–162,
189–192
obtaining phases, 104–105, 109–143
anomalous scattering, 128–136
isomorphous replacement, 117–128
302 Index
obtaining phases (continued )
molecular replacement, 136–143
two-dimensional representation of
structure factors, 112–117
occupancy, atom, 161, 185
omit maps, 156
one-dimensional NMR models, 243, 251
one-dimensional waves, 92–93
online databases, 44–45
online model validation tools, 189–192
ordered water, 34–35, 169
orientation of phasing model, 139–141
ORIG lines (PDB file), 176
orthorhombic cells, 50
lattice indices of (example), 51–52
oscillation cameras, 82–84
overall map amplitude, 150
Overhauser effect (nuclear), 248
P
packing effects, 36–37
packing scores, defined, 190
papers (crystallography), reading,
192–208
parameters for structure refinement,
161–162, 189–192
partial molecular models, building, 154
particle storage rings, 75–76
Patterson functions and maps, 124–128
phasing model location and
orientation, 139–143
spherically averaged, 220–221
PDB (Protein Data Bank), 7, 174–177
file headers, 175–176, 189
NMR models, 257
quality of publications in, 180
PEG (polyethylene glycol), 11–12
periodic functions, 21–24
phage display, 166–167
phase, 98, 101
allusions to “Phase” poem, 23,
229, 270
electron density as function of,
115–117
filtering, 171
improving. See bootstrapping
mathematics of, 22–23
obtaining, 104–105, 109–143
anomalous scattering, 128–136
isomorphous replacement, 117–128
molecular replacement, 136–143
two-dimensional representation of
structure factors, 112–117
quality of (figure of merit), 152
phase angle, 113
phase extension, 152
phase maps, obtaining, 154–156
“Phase” (poem), xxiv
allusions to, 23, 229, 270
phase probabilities, 124
phase problem, defined, 116
phasing models. See molecular
replacement
phasing power, 118
pitch, helix, 213
plane indices, 50–55
plane-polarized light, observing, 45
pleated sheets, 159
point of inversion, 88
polychromatic X-ray beams, 231–235
polydispersity, 45
polyethylene glycol (PEG), 11–12
position of wave, 23. See also phase
positive vs. negative indices, 54
posterior possibilities (Bayesian
inference), 164, 166
postrefinement of X-ray intensities,
85–86
precession photography, 84
precipitating proteins, 11–12, 38. See also
growing crystals
precision of atomic positions, judging,
183–185
primitive lattices, 66
prior knowledge restrictions on model,
152, 154
NMR models, 253
prior possibilities (Bayesian inference),
164, 165
probability distributions (Bayesian
inference), 165
PROCHECK program, 191
protein action modeling tools, 290
protein conformation, determining
(NMR models), 252–257
protein crystallography, sketch of,
9–10
Index 303
protein crystals, 31–47
growing. See growing crystals
mounting crystals for data collection,
46–47
properties of, 31–34
quality of, judging, 46
related, as phasing models. See
molecular replacement
solution vs. crystal structures, 34–37
Protein Data Bank. See PDB
protein function, structure and, 35
protein precipitation, 11–12, 38. See also
growing crystals
protein structure. See entries at structure
publishing models, 173–177
purity of samples, 41
Q
quality of crystals, judging, 46
quality of models. See also models,
obtaining and improving
assessing convergence, 171–173
homology models, 265–267
how to judge, 180–192
atomic positions, 183–185
miscellaneous limitations of,
187–189
model validation tools, 189–192
structural parameters, 161–162,
181–183, 189–192
vibration and disorder, 185–187
NMR models, 257–259
reading crystallography papers,
192–208
quantity necessary for growing
crystals, 41
R
R-factor, 142
R-factor, 172, 181
Luzzati plots and, 184
radial Patterson functions, 220–221
radius of convergence, 162–163
radius of gyration, 221
Ramachandran diagrams, interpreting,
181–183
reading crystallography papers,
192–208
real lattices, 16
Bragg’s law, 55–57
real-space refinement algorithms,
147–148
manual model building, 168–169
phase filtering, 171
reasonableness of models, 172–173, 183
reciprocal angstroms, 20
reciprocal lattices, 16
average point spacing, 86
how to construct, 57–60
limiting sphere, 64
reciprocal space, 19–20, 57–64
Bragg’s law, 60–64
reciprocal-space refinement algorithms,
147–148, 159
phase filtering, 171
refinement of atomic coordinates,
147–148
refinement of map structure, 159–168
parameters for, 161–162, 189–192
refinement target, defined, 166
reflections (X-ray), 13, 49
as Fourier terms, 101–104
intensities of, 16–17
measurable, number of, 64–65
measured, electronic density from,
28–30
phases of. See phase
sphere of reflection, 63–64
symmetry elements and, 71–73
wave descriptions of, 24–26
regularization, 157, 171
related proteins. See molecular
replacement
relaxation, nuclear, 242–243
REMARK lines (PDB file), 175
removing error from models, 147, 171
rendering computer models. See computer
models
residual index (R-factor), 172, 181
Luzzati plots and, 184
resolution limit, 65
resolution of atomic positions, judging,
183–185, 187
resonances, assigning in NMR models,
251–252. See also NMR models
response-surface procedure, 41–42
304 Index
restrained minimized average structures,
255
restraints, defined, 163
REVDAT lines (PDB file), 175
reversibility of Fourier transforms, 96
ridge lines, drawing, 154
rise-per-residue (helix), 215
rms deviations (NMR models), 257–258
rotamers, 183
rotating anode tubes, 75
rotation, 68
rotation functions, 142
self-rotation, 153
rotation of computer models, 271–272
rotation/oscillation method, 82–84
S
safety, X-ray, 77
sampled diffraction patterns, 17
saving models (DeepView program), 278
SCALE lines (PDB file), 176
scaling X-ray data, 85–86
scattering, diffraction by, 219–222
scattering factor, 98
scattering length, 223–224
scintillation counters, 77–78
screen coordinates (computer models),
270
screw axis, 68
scripts with DeepView program, 287–288
seed crystals, 40
selenomet derivatives, 41
self-rotation functions, 153. See also
rotation functions
SEQADV lines (PDB file), 175
SEQRES lines (PDB file), 175
Shake-and-Bake direct phasing, 136
sharing models, 173–177
SHEET lines (PDB file), 176
side chains, homology models, 263
side chains, identifying, 158
sigma-A weighted maps, 156, 169
disorder, indications of, 185
simple objects, diffraction by, 15–17
simulated annealing, 163
single-crystal C-ray crystallography, 7
single isomorphous replacement (SIR),
118, 131
SIR (single isomorphous replacement),
118
with anomalous scattering (SIRAS),
131
SITE lines (PDB file), 176
size, crystal, 31–33
skeletonizing maps, 154
slow precipitation, 11–12, 38. See also
growing crystals
solution structure vs. crystalline structure,
35–37
solvent-accessible surfaces, 283–284
solvent leveling/flattening/flipping,
151–153. See also density
modification
SOURCE lines (PDB file), 175
sources, X-ray, 73–77
space-filling models, 275
space groups, 66–67
spallation, 223
spectroscopy. See NMR models
sphere of reflection, 63–64
spheres, diffraction by, 15
spherically averaged diffraction patterns,
220–221
spin-lattice relaxation, 242–243
spin, nuclear, 239–240
spin-spin relaxation, 243
square wave, 24
step function, 24
stereo images, viewing, 272, 293–294
stereochemically reasonable models,
173
strategies for data collection, 88–89
structural change exploration (DeepView
program), 282
structural integrity, 31–33
structural parameters, 161–162, 189–192
structure, solution vs. crystalline, 35–37
structure analysis tools, 288–290
structure-factor patterns, 30
structure factors, 24–26
computing from model, 104–105
electron density from, 27–28
as Fourier sums, 98–99, 102–103
Friedel pairs, 88, 114
two-dimensional representation,
112–117
structure files, PDB, 174–176
Index 305
structure refinement, 159–168
parameters for, 161–162, 189–192
summations, Fourier, 26, 92–97
electron density as, 99–100
reflections as terms in, 101–104
structure factors as, 98–99
SWISS-MODEL Repository, 237, 260,
264, 268
Swiss-PdbViewer program. See
DeepView program
symmetry averaging, 153
symmetry elements, 67
symmetry of unit cells, 65–73
data collection strategies, 88–89
functional-unit, 188
noncrystallographic (NCS), 153
Patterson map searches, 127
symmetry-related atoms, 105–106
synchrotrons, 75–77, 222
systematic absences, 73, 105–107
T
target core (homology models), 261–262
temperature factors, 161, 169
vibration measurements, 185–186
temperature of mounted crystals, 45–46
templates for homology models, 260–261
TER lines (PDB file), 176
tetragonal cells, 50
theoretical models, 7. See also models
thermal motion, 185–186
three-dimensional arrays, diffraction by,
18
three-dimensional computer models,
270–273
three-dimensional NMR models, 250
three-dimensional waves, 94–95
time-domain signals, 243
time-resolved crystallography, 231–235
TITLE lines (PDB file), 175
tools for studying macromolecules,
269–292
computer modeling, 269–275
DeepView program (example),
275–288
model validation tools, 189–192,
288–290
touring molecular modeling programs,
275–288
translation, 68
translation searches, 139–141
transmission electron microscopes, 227
transverse relaxation time, 243
triclinic cells, 50
triplet relationship (direct phasing),
135–136
truncated Fourier series, 102
tubes, X-ray, 75–76
TURN lines (PDB file), 176
twinned crystals, 31, 45
two-dimensional arrays, diffraction
by, 18
two-dimensional computer models,
269–270
two-dimensional NMR models, 244–248
twofold rotation axis, defined, 67
U
uncertainty factor (homology models),
267
unexplained density, 187
unit cells, 10, 49–50
asymmetric vs. functional, 188
dimensions of, 65, 86–88
heavy atoms in, locating, 124–128
reciprocal, 59
symmetry of, 65–73
data collection strategies, 88–89
functional-unit, 188
noncrystallographic (NCS), 153
Patterson map searches, 127
unit translation, 68
unreasonableness of models, 172–173,
183
upper-level planes, 59
user’s guide to crystallographic models,
179–210
judging model quality, 180–192
atomic positions, 183–185
miscellaneous limitations of,
187–189
model validation tools, 189–192
structural parameters, 161–162,
181–183, 189–192
vibration and disorder, 185–187
reading crystallography papers,
192–208
306 Index
V
validation tools, 189–192, 288–290
validation tools, online, 189–192
van der Walls forces, 253–254
vapor diffusion, 38
variable-wavelength X rays.
See synchrotrons
vectors, complex, 112–115
vibration, atomic, 185–186
viewing electronic-density maps, 151
viewing models (DeepView program),
278–280
W
water content of crystals, 34–35
wave equations, 21–23
one-dimensional waves, 92–93
three-dimensional waves, 94–95
wavelength, 22
wavelengths of X-ray emissions, 73
weighted maps, 156
wigglers, 75
wireframe models, 273
X
X-ray absorption, 74. See also detectors,
X-ray
anomalous scattering, 128–136
direct phasing methods, 135–136
extracting phase, 130–132
hand problem, 135
measurable effects, 128–130
multiwavelength (MAD), 133–134
X-ray analysis, 9
X-ray data collection, 13–15, 73–89
cameras, 80–85
detectors, 77–80
intensity scaling and postrefinement,
85–86
mounting crystals for, 46–47
symmetry and strategy, 88–89
unit-cell dimension determination,
86–88
X-ray sources, 73–77
X-ray reflections, 13, 49
as Fourier terms, 101–104
intensities of, 16–17
measurable, number of, 64–65
measured, electronic density from,
28–30
phases of. See phase
sphere of reflection, 63–64
symmetry elements and, 71–73
wave descriptions of, 24–26
X-ray scattering lengths, 224
X-ray tubes, 75–76
Z
zero-level planes, 59
zinc fingers, 1

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