* Helps readers to understand where models come from, so they don't use them blindly and
inappropriately
* Provides many visual and geometric models for understanding a largely mathematical method
* Allows readers to judge whether recently published models are of sufficiently high quality and detail to be useful in their own work
* Allows readers to study macromolecular structure independently and in an open-ended fashion on their own computers, without being limited to textbook or journals illustrations
* Provides access to web tools in a format that will not go out of date. Links will be updated and added as existing resources change location or are added
Gale Rhodes earned a B.S. in applied mathematics at North Carolina State University, and then a Ph.D. in Chemistry at the University of North Carolina. He is currently a professor of chemistry at the University of Southern Maine, Portland. His main duty, and first love, is teaching undergraduate biochemistry. He has received awards for outstanding teaching at three different colleges. His best known publication is the first edition of Crystallography Made Crystal Clear, which received very complimentary reviews in several journals. He has also published three book chapters, three book reviews, and about 30 articles on diverse subjects, including research articles in biochemistry, and articles on chemistry, science, and interdisciplinary education.
Crystallography Made Crystal Clear makes crystallography accessible to readers who have no prior knowledge of the field or its mathematical basis. This is the most comprehensive and concise reference for beginning Macromolecular crystallographers, written by a leading expert in the field. Rhodes' uses visual and geometric models to help readers understand the mathematics that form the basis of x-ray crystallography. He has invested a great deal of time and effort on World Wide Web tools for users of models, including beginning-level tutorials in molecular modeling on personal computers. Rhodes' personal CMCC Home Page also provides access to tools and links to resources discussed in the text. Most significantly, the final chapter introduces the reader to macromolecular modeling on personal computers-featuring SwissPdbViewer, a free, powerful modeling program now available for PC, Power Macintosh, and Unix computers. This updated and expanded new edition uses attractive four-color art, web tool access for further study, and concise language to explain the basis of X-ray crystallography, increasingly vital in today's research labs. Helps readers to understand where models come from, so they don't use them blindly andinappropriately Provides many visual and geometric models for understanding a largely mathematical method Allows readers to judge whether recently published models are of sufficiently high quality and detail to be useful in their own work Allows readers to study macromolecular structure independently and in an open-ended fashion on their own computers, without being limited to textbook or journals illustrations Provides access to web tools in a format that will not go out of date. Links will be updated and added as existing resources change location or are added
Front cover 1
Title page 6
Copyright page 7
Table of contents 10
Preface to the Third Edition 16
Preface to the Second Edition 20
Preface to the First Edition 24
1 Model and Molecule 28
2 An Overview of Protein Crystallography 34
Introduction 34
Obtaining an image of a microscopic object 35
Obtaining images of molecules 36
A thumbnail sketch of protein crystallography 36
Crystals 37
The nature of crystals 37
Growing crystals 38
Collecting X-ray data 40
Diffraction 42
Simple objects 42
Arrays of simple objects: Real and reciprocal lattices 43
Intensities of reflections 43
Arrays of complex objects 44
Three-dimensional arrays 45
Coordinate systems in crystallography 46
The mathematics of crystallography: A brief description 47
Wave equations: Periodic functions 48
Complicated periodic functions: Fourier series and sums 50
Structure factors: Wave descriptions of X-ray reflections 51
Electron-density maps 53
Electron density from structure factors 54
Electron density from measured reflections 55
Obtaining a model 57
3 Protein Crystals 58
Properties of protein crystals 58
Introduction 58
Size, structural integrity, and mosaicity 58
Multiple crystalline forms 60
Water content 61
Evidence that solution and crystal structures are similar 62
Proteins retain their function in the crystal 62
X-ray structures are compatible with other structural evidence 63
Other evidence 64
Growing protein crystals 64
Introduction 64
Growing crystals: Basic procedure 65
Growing derivative crystals 67
Finding optimal conditions for crystal growth 68
Judging crystal quality 73
Mounting crystals for data collection 73
4 Collecting Diffraction Data 76
Introduction 76
Geometric principles of diffraction 76
The generalized unit cell 76
Indices of the atomic planes in a crystal 77
Conditions that produce diffraction: Bragg's law 82
The reciprocal lattice 84
Bragg's law in reciprocal space 87
Number of measurable reflections 91
Unit-cell dimensions 92
Unit-cell symmetry 92
Collecting X-ray diffraction data 100
Introduction 100
X-ray sources 100
Detectors 104
Cameras 107
Scaling and postrefinement of intensity data 112
Determining unit-cell dimensions 113
Symmetry and the strategy of collecting data 115
Summary 116
5 From Diffraction Data to Electron Density 118
Introduction 118
Fourier sums and the Fourier transform 119
One-dimensional waves 119
Three-dimensional waves 121
The Fourier transform: General features 123
Fourier this and Fourier that: Review 124
Fourier mathematics and diffraction 125
Structure factor as a Fourier sum 125
Electron density as a Fourier sum 126
Computing electron density from data 127
The phase problem 128
Meaning of the Fourier equations 128
Reflections as terms in a Fourier sum: Eq. (5.18) 128
Computing structure factors from a model: Eq. (5.15) and Eq. (5.16) 131
Systematic absences in the diffraction pattern: Eq. (5.15) 132
Summary: From data to density 134
6 Obtaining Phases 136
Introduction 136
Two-dimensional representation of structure factors 139
Complex numbers in two dimensions 139
Structure factors as complex vectors 139
Electron density as a function of intensities and phases 142
Isomorphous replacement 144
Preparing heavy-atom derivatives 144
Obtaining phases from heavy-atom data 146
Locating heavy atoms in the unit cell 151
Anomalous scattering 155
Introduction 155
Measurable effects of anomalous scattering 155
Extracting phases from anomalous scattering data 157
Summary 159
Multiwavelength anomalous diffraction phasing 160
Anomalous scattering and the hand problem 162
Direct phasing: Application of methods from small-molecule crystallography 162
Molecular replacement: Related proteins as phasing models 163
Introduction 163
Isomorphous phasing models 164
Nonisomorphous phasing models 166
Separate searches for orientation and location 166
Monitoring the search 168
Summary of molecular replacement 170
Iterative improvement of phases (preview of Chapter 7) 170
7 Obtaining and Judging the Molecular Model 172
Introduction 172
Iterative improvement of maps and models - overview 173
First maps 176
Resources for the first map 176
Displaying and examining the map 177
Improving the map 178
The Model becomes molecular 180
New phases from the molecular model 180
Minimizing bias from the model 181
Map fitting 183
Structure refinement 186
Least-squares methods 186
Crystallographic refinement by least squares 187
Additional refinement parameters 188
Local minima and radius of convergence 189
Molecular energy and motion in refinement 190
Bayesian methods: Ensembles of models 191
Convergence to a final model 195
Producing the final map and model 195
Guides to convergence 198
Sharing the model 200
8 A User’s Guide to Crystallographic Models 206
Introduction 206
Judging the quality and usefulness of the refined model 208
Structural parameters 208
Resolution and precision of atomic positions 210
Vibration and disorder 212
Other limitations of crystallographic models 214
Online validation tools: Do it yourself! 216
Summary 219
Reading a crystallography paper 219
Introduction 219
Annotated excerpts of the preliminary (8/91) paper 220
Annotated excerpts from the full structure-determination (4/92) paper 225
Summary 236
9 Other Diffraction Methods 238
Introduction 238
Fiber diffraction 238
Diffraction by amorphous materials (scattering) 246
Neutron diffraction 249
Electron diffraction and cryo-electron microscopy 254
Laue diffraction and time-resolved crystallography 258
Summary 262
10 Other Kinds of Macromolecular Models 264
Introduction 264
NMR models 265
Introduction 265
Principles 266
Assigning resonances 278
Determining conformation 279
PDB files for NMR models 284
Judging model quality 284
Homology models 286
Introduction 286
Principles 287
Databases of homology models 290
Judging model quality 292
Other theoretical models 294
11 Tools for Studying Macromolecules 296
Introduction 296
Computer models of molecules 296
Two-dimensional images from coordinates 296
Into three dimensions: Basic modeling operations 297
Three-dimensional display and perception 299
Types of graphical models 300
Touring a molecular modeling program 302
Importing and exporting coordinate files 303
Loading and saving models 305
Viewing models 305
Editing and labeling the display 307
Coloring 308
Measuring 308
Exploring structural change 309
Exploring the molecular surface 309
Exploring intermolecular interactions: Multiple models 313
Displaying crystal packing 314
Building models from scratch 314
Scripts and macros: Automating routine structure analysis 314
Other tools for studying structure 315
Tools for structure analysis and validation 315
Tools for modeling protein action 317
Final note 318
Appendix: Viewing Stereo Images 320
Index 322
Complementary Science Series 335
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.
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.
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).
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, 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).
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.
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).
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).
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.
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.
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.
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...
Erscheint lt. Verlag | 4.8.2010 |
---|---|
Sprache | englisch |
Themenwelt | Naturwissenschaften ► Biologie |
Naturwissenschaften ► Chemie ► Physikalische Chemie | |
Naturwissenschaften ► Geowissenschaften ► Mineralogie / Paläontologie | |
Technik | |
ISBN-10 | 0-08-045554-9 / 0080455549 |
ISBN-13 | 978-0-08-045554-9 / 9780080455549 |
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