Structure Determination by X-ray Crystallography Analysis by X-rays and Neutrons

Mục Lục

 

Periodic Table …………………………………………………. ii
Physical Constants and Other Numerical Data ………………… xxxi
Notation …………………………………………………………xxxiii
1 Crystal Morphology and Crystal Symmetry ……………….. 1
1.1 Brief Historical Introduction ……………………………. 1
1.2 The Crystalline State …………………………………… 7
1.2.1 Crystallographic Reference Axes . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Equation of a Plane …………………………….. 7
1.2.3 Indices of Planes and the Law
of Rational Intercepts …………………………… 8
1.2.4 Axial Ratios …………………………………….. 12
1.2.5 Zones …………………………………………… 12
1.3 Stereographic Projection: Brief Survey . . . .. . . . .. . . . . .. . . . .. . 15
1.4 External Symmetry of Crystals …………………………. 17
1.4.1 Two-Dimensional Point Groups …………………. 19
1.4.2 Three-Dimensional Point Groups . . . . . . . . . . . . . . . . . . . . 22
1.4.3 Quasicrystals, Buckyballs, and Icosahedral
Symmetry ………………………………………. 32
1.5 Problems ………………………………………………. 39
References and Bibliography …………………………………. 49
2 Lattices and Space-Group Theory ………………………….. 51
2.1 Introduction ……………………………………………. 51
2.2 Lattices ………………………………………………… 51
2.2.1 Two-Dimensional Lattices ………………………. 52
2.2.2 Choice of Unit Cell ….. ….. ….. ….. ….. ….. ….. 53
2.2.3 Three-Dimensional Lattices .. .. .. … .. .. .. .. .. … .. .. 54
2.3 Families of Planes and Interplanar Spacings …………….. 62
2.4 Reciprocal Lattice: Geometrical Treatment ……………… 63
2.5 Unit-Cell Transformations ……………………………… 65
2.5.1 Bravais Unit-Cell Vectors . .. … .. .. .. … .. .. .. … .. .. 65
2.5.2 Directions (Zone Symbols) ……………………… 66
2.5.3 Coordinates of Sites in the Unit Cell …………….. 67
2.5.4 Miller Indices …………………………………… 67
2.5.5 Reciprocal Unit-Cell Vectors ……………………. 68
2.6 Rotational Symmetries of Lattices . . . .. . . . . . .. . . . . . .. . . . . . .. . 71
xv

2.7 Space Groups …………………………………………. 72
2.7.1 Two-Dimensional Space Groups
(Plane Groups) ……………………………….. 73
2.7.2 Plane Groups Related to 2mm . . . . . . . . . . . . . . . . . . . . . 79
2.7.3 Three-Dimensional Space Groups ……. ………. 81
2.7.4 Screw Axes …………………………………… 84
2.7.5 Glide Planes ………………………………….. 87
2.7.6 Analysis of the Space-Group Symbol . . . . . . . . . . . . . 90
2.7.7 Orthorhombic Space Groups ………………….. 91
2.7.8 Relative Orientations of Symmetry
Elements in Space Groups . .. … .. … .. .. … .. … .. 93
2.7.9 Tetragonal and Hexagonal Space Groups . . . . . . . . . 95
2.8 Matrix Representation of Symmetry Operations . . . . . . . . . . . 98
2.8.1 Matrices in Point-Group Symmetry . . . . . . . . . . . . . . . 98
2.8.2 Matrices in Space-Group Symmetry …………… 100
2.9 Diffraction Symbols …………………………………… 101
2.10 Some Other Types of Symmetry ………………………. 103
2.10.1 Black-White Symmetry ………………………. 103
2.10.2 Color Symmetry ……………………………… 104
2.11 Problems ……………………………………………… 106
References …………………………………………………… 109
3 X-Rays and X-Ray Diffraction ……………………………… 111
3.1 Generation and Properties of X-Rays ………………….. 111
3.1.1 X-Rays and White Radiation ………………….. 111
3.1.2 Characteristic Radiation ………………………. 113
3.1.3 Absorption of X-Rays ………………………… 114
3.1.4 Monochromatic Radiation …………………….. 116
3.1.5 Collimation …………………………………… 116
3.1.6 Synchrotron Sources ………………………….. 118
3.2 X-Ray Scattering ……………………………………… 121
3.2.1 Scattering by a Single Electron ……………….. 122
3.2.2 Scattering by Two or More Electrons . . . . . . . . . . . . . 122
3.2.3 Waves and Wave Sums .. .. .. … .. … .. .. … .. … .. 123
3.2.4 Coherent and Incoherent Scattering . . . . . . . . . . . . . . . 127
3.2.5 Scattering by an Atom ………………………… 128
3.3 Scattering by Regular Arrays of Atoms . . . . . . . . . . . . . . . . . . . . 130
3.3.1 Laue Equations …… ……….. ……….. ……… 130
3.3.2 Bragg Equation ………………………………. 132
3.3.3 Equivalence of the Laue and Bragg
Equations …………………………………….. 134
3.3.4 Further Analysis of the Path Difference ……….. 135
3.4 Reciprocal Lattice: Analytical Treatment . . . . . . . . . . . . . . . . . . 135
3.4.1 Reciprocal Lattice Properties . . .. . .. . .. . .. . .. . .. . .. 137
3.4.2 Reciprocal Lattice and Reflection
Condition: Ewald Sphere ……………………… 138
3.5 Scattering by a Crystal Structure ………………………. 139
3.5.1 Structure Factor Equation .. .. … .. … .. … .. … .. … 140
3.6 Using the Structure Factor Equation …………………… 140
3.6.1 Friedel’s Law ………………………………….. 140
3.6.2 Structure Factor for a Centrosymmetric Crystal . . . 141
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3.7 Limiting Conditions and Systematic Absences .. . .. .. .. . .. .. 142
3.7.1 Body-Centered Unit Cell . .. . .. .. .. .. .. .. .. . .. .. .. .. .. 142
3.7.2 Screw Axes and Glide Planes . .. . . . . . .. . . . . . .. . . . . . .. 143
3.8 Practical Determination of Space Groups
from Diffraction Data ………………………………….. 152
3.8.1 Monoclinic Space Groups . .. .. . .. .. .. .. .. .. .. .. .. .. .. 153
3.8.2 Orthorhombic Space Groups . . . . . . . . . . . .. . . . . . . . . . . . . 154
3.8.3 Tetragonal Space Groups ……………………….. 155
3.8.4 Hexagonal Space Groups ……………………….. 155
3.9 Problems ………………………………………………. 155
References …………………………………………………… 159
4 Intensities and Intensity Statistics ………………………….. 161
4.1 Intensity Expressions and Factors Affecting
Intensities ……………………………………………… 161
4.1.1 Polarization and Lorentz Factors ………………… 162
4.1.2 Extinction ………………………………………. 164
4.1.3 Absorption Measurement and Correction ………… 165
4.1.4 Scaling …………………………………………. 167
4.1.5 Merging Equivalent Reflections …………………. 167
4.1.6 Practical Intensity Expression
and its Standard Deviation ………………………. 168
4.1.7 Scale Factor for Fo ……………………………… 169
4.1.8 Thermal Vibrations and the Temperature
Factor ………………………………………….. 169
4.2 Intensity Statistics ……………………………………… 172
4.2.1 Determining Scale and Temperature
Factors …………………………………………. 172
4.2.2 Other Aspects of the Wilson Plot . . . . . . . . . . . . . . . . . . . . 175
4.2.3 Statistics of Reciprocal Space …. ….. ….. ….. ….. 175
4.2.4 Acentric and Centric Distributions
of Structure Factors …………………………….. 177
4.2.5 Normalized Structure Factors . . . . . . . . . . . . . . . . . . . . . . . . 182
4.3 Problems ………………………………………………. 185
References …………………………………………………… 186
5 Examination of Single Crystals: Optical and X-Ray
Diffraction Practice ………………………………………… 187
5.1 Introduction ……………………………………………. 187
5.2 Crystal Growing ……………………………………….. 187
5.2.1 Growing Crystals for X-Ray Diffraction . . . . . . . . . . . . 187
5.2.2 Crystallization from Solution ……………………. 188
5.2.3 Crystallization by Diffusion …. …… ….. …… ….. 188
5.2.4 Crystallization by Sublimation .. .. .. .. .. .. .. … .. .. .. 188
5.2.5 Other Issues …………………………………….. 188
5.3 Optical Techniques …………………………………….. 189
5.3.1 Polarized Light …………………………………. 189
5.3.2 Optical Classification of Crystals . . . . . . . . . . . . . . . . . . . . 190
5.3.3 Uniaxial Crystals ……………………………….. 190
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5.3.4 Birefringence ………………………………… 192
5.3.5 Biaxial Crystals ………………………………. 194
5.4 Single-Crystal X-Ray Diffraction Techniques:
Intensity Data Collection ……………………………… 197
5.4.1 Laue Method …………………………………. 197
5.4.2 Symmetry in Laue Photographs . … … … … … … 200
5.4.3 Laue Method and Synchrotron Radiation ……… 200
5.4.4 Oscillation Method …………………………… 205
5.5 Measurement of the Intensities of Diffraction Data …….. 208
5.5.1 Single Counter or Serial Diffractometers . . . . . . . . . 209
5.6 Single-Crystal X-Ray Diffractometry ………………….. 209
5.6.1 Instrument Geometry …………………………. 209
5.6.2 Rotation of the Crystal into a
Diffracting Position …………………………… 210
5.6.3 Transformation from Miller Indices
to Diffractometer Angles ……………………… 211
5.6.4 Data Collection ………………………………. 211
5.6.5 Scanning Over a Peak: o/y
Versus o Scans ………………………………. 212
5.7 Area Detectors (Position-Sensitive Detectors) . . . . . . . . . . . . . 213
5.7.1 Multiwire Proportional Counter … … …. … … … 213
5.7.2 FAST Area Detector (Enraf–Nonius
FAST) ……………………………………….. 215
5.7.3 Image Plate ………………………………….. 215
5.7.4 Charge-Coupled Device Area Detectors . . . . . . . . . . 217
5.7.5 The Tiled CCD ………………………………. 219
5.7.6 Charge-Coupled Device Including Tiled
CCD Versus Image Plate . .. .. .. .. .. . .. .. .. .. .. .. .. 219
5.7.7 Data Collection Strategies …. ….. …… ….. ….. 219
5.7.8 The CMOS Detector, Pilatus 1M Detector
System, and Continuous Rotation . …. …. …. …. 221
5.7.9 Data Processing Software …………………….. 222
5.7.10 Detectors and Diffractometers ………………… 222
5.7.11 Other Diffractometer Systems ………………… 223
5.8 Monochromators ……………………………………… 223
5.8.1 Single-Type Crystal Monochromators . . . . . . . . . . . . 224
5.8.2 Double-Type Crystal Monochromators . . . . . . . . . . . 224
5.8.3 Monochromators for Synchrotron
Radiation …………………………………….. 225
5.9 Focusing Mirrors ……………………………………… 225
5.10 Twinning ……………………………………………… 226
5.10.1 Morphology of Twinning .. … … … … … … … … 226
5.10.2 Twinning and X-Ray Diffraction ……………… 228
5.11 Problems ……………………………………………… 229
References …………………………………………………… 232
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6 Fourier Series and Fourier Transforms …………………… 235
6.1 Image Formation and Focusing ………………………. 235
6.2 Fourier Series ………………………………………… 236
6.2.1 Analysis of the Square Wave … ………. ……… 238
6.2.2 Exponential Forms of Fourier Series .. …. …. …. 240
6.3 Fourier Series in X-Ray Crystallography ……………… 241
6.3.1 One-Dimensional Function .. . . . . . .. . . . . . . .. . . . . . .. 241
6.3.2 Two- and Three-Dimensional Functions .. . . . . . . .. 243
6.3.3 Units of Electron Density …………………….. 245
6.4 Holes and Atoms …………………………………….. 245
6.5 Generalized Fourier Transform . … … … … … … … … … 246
6.5.1 Fourier Transform of a Molecule ……………… 248
6.5.2 Fourier Transform of a Unit Cell ……………… 248
6.6 Practice with Transforms …………………………….. 249
6.6.1 Optical Diffractometer …… …… …… …… ….. 249
6.6.2 Single Hole …………………………………… 249
6.6.3 Two or More Holes …………………………… 250
6.6.4 Change of Origin …………………………….. 252
6.6.5 Systematic Absences …………………………. 252
6.6.6 Reconstruction of the Image ………………….. 252
6.6.7 Transforms and Inverse Transforms …………… 255
6.6.8 Delta Function ……………………………….. 258
6.6.9 Weighted Reciprocal Lattice ………………….. 259
6.7 Some General Properties of Transforms ………………. 261
6.8 Convolution …………………………………………. 261
6.8.1 Convolution and Diffraction ………………….. 261
6.8.2 Convolution Integral . .. .. .. … .. .. .. .. .. .. .. … .. .. 262
6.8.3 Convolution and Crystal Structure . . . . . . . . . . . . . . . . 264
6.9 Structure Solution in Brief .. ….. …. ….. …. …. ….. …. 266
6.9.1 Use of Heavy Atoms …………………………. 266
6.9.2 General Phase-Free Transform:
Patterson Function ……………………………. 267
6.9.3 Sign Relationships ……………………………. 268
6.10 Problems …………………………………………….. 270
References ………………………………………………….. 272
7 Fourier Techniques in X-Ray Structure
Determination ……………………………………………… 273
7.1 Introduction ………………………………………….. 273
7.2 Analysis of the Unit-Cell Contents .. . . . . . .. . . . . . .. . . . . . .. . 273
7.2.1 Papaverine Hydrochloride,
C20H21NO4·HCl … …… ….. …… …… ….. ….. 274
7.2.2 Naphthalene, C10H8 …….. …………. ……….. 275
7.2.3 Molecular Symmetry …………………………. 275
7.2.4 Special Positions ……………………………… 276
7.2.5 Nickel Tungstate, NiWO4 …………………….. 276
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7.3 Interpretation of Electron Density Distributions …………. 278
7.3.1 Peak Heights and Weights ………………………. 279
7.3.2 Computation and Display of Electron
Density Distributions ……………………………. 279
7.3.3 Projections ……………………………………… 279
7.4 Methods of Solving the Phase Problem ……… …………. 281
7.4.1 Number of Reflections in the Data Set …………… 281
7.4.2 The Patterson Function .. .. .. .. … .. .. .. … .. .. .. … .. 282
7.4.3 Positions and Weights of Peaks
in the Patterson Function . .. .. .. .. . .. .. .. .. .. . .. .. .. .. 285
7.4.4 Sharpened Patterson Function …………………… 287
7.4.5 Symmetry of the Patterson Function
for a Crystal of Space Group Pm ………………… 288
7.4.6 Vector Interactions in Other Space
Groups …………………………………………. 289
7.4.7 Examples of the Use of the Patterson Function
in Solving the Phase Problem .. . . .. . . . .. . . .. . . . .. . . .. 289
7.4.8 Determination of the Chlorine Atom Positions
in Papaverine Hydrochloride … .. .. .. … .. .. .. .. … .. 296
7.4.9 Determination of the Mercury Atom
Positions in KHg2 ………………………………. 296
7.5 Heavy-Atom Method and Partial Fourier Synthesis … … … 301
7.5.1 Reliability Factor ……………………………….. 303
7.5.2 Pseudosymmetry in Electron Density Maps . . . . . . . . . 308
7.5.3 Successive Fourier Refinement ………………….. 309
7.5.4 Difference-Fourier Synthesis … …….. ……. ……. 309
7.5.5 Limitations of the Heavy-Atom Method …………. 310
7.5.6 Patterson Selection ……………………………… 310
7.5.7 Isomorphous Replacement ………………………. 312
7.5.8 Further Details of the Isomorphous
Replacement Phasing Procedure . . . . . . . . . . . . . . . . . . . . . 319
7.6 Anomalous Scattering ………………………………….. 325
7.6.1 The Flack x Parameter ………………………….. 326
7.6.2 Effect of Anomalous Scattering
on the Symmetry of Diffraction Patterns . . . . . . . . . . . . 330
7.6.3 Form of the Structure Factor for a Structure
Composed of Heavy-Atom Anomalous
Scattering Species ………………………………. 332
7.6.4 Phasing by Use of Anomalous Scattering . . . . . . . . . . . 334
7.6.5 Resolution of the Phase Problem for Proteins
Using Anomalous Scattering Measurements
(SIRAS Method) ……… ………. ………. ……… 335
7.6.6 Protein Phasing Using the Multiple-Wavelength
Anomalous Dispersion Technique (MAD) with
Synchrotron Radiation (SR) … … …. …. …. …. …. 337

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7.7 Charge flipping ………………………………………… 338
7.8 Location of Hydrogen Atoms …………………………… 339
7.9 Problems ………………………………………………. 340
References …………………………………………………… 347
8 Direct Methods and Refinement ……………………………. 351
8.1 Introduction ……………………………………………. 351
8.2 Direct Methods of Phase Determination .. . . .. . . .. . .. . . .. . . .. 351
8.2.1 Normalized Structure Factors … ……. ……. …… 351
8.2.2 Structure Invariants and Origin-Fixing
Reflections …………………………………….. 352
8.2.3 Sign Determination: Centrosymmetric
Crystals ……………………………………….. 355
8.2.4 Amplitude Symmetry and Phase Symmetry . . . . . . . 358
8.2.5 P
2-Listing ……………………………………. 358
8.2.6 Symbolic-Addition Procedure: Example . . . . . . . . . . . 359
8.2.7 Calculation of E Maps …………………………. 360
8.2.8 Phase Determination: Non-centrosymmetric
Crystals ……………………………………….. 361
8.2.9 Enantiomorph Selection ……………………….. 367
8.2.10 Phase Determination in Space Group P21 ………. 368
8.2.11 Advantages and Disadvantages
of Symbolic Addition ………………………….. 371
8.2.12 Signs of Trouble, and Past Remedies
When the Structure Failed to Solve …………….. 372
8.2.13 Triplets, Quartets, and the SHELX
Program Strategy ………………………………. 372
8.2.14 The SHELX Computer Program System . . . . . . . . . . . 374
8.2.15 The WinGX Program System ………………….. 375
8.2.16 Direct Methods in the Program SHELX-97
for Small Molecules …………………………… 375
8.2.17 Example of a SHELX-97 Structure Solution:
Crystal Code Name BW202W92(R) ……………. 377
8.3 Patterson Search Methods ………………………………. 380
8.3.1 General Comments for Small Molecules
and Macromolecules …………………………… 381
8.3.2 Intramolecular Interatomic Vectors
and Molecular Orientation . . . . . . . . . . . .. . . . . . . . . . . .. . 382
8.3.3 Intermolecular Interatomic Vectors:
Translation Stage of MR .. .. .. .. .. … .. .. .. .. .. … .. 384
8.3.4 Crystal Packing and Refinement
of the Structure ………………………………… 385
8.3.5 Patterson Search Methods for Small
Molecules ……………………………………… 386
8.3.6 The Program PATSEE …………………………. 387
8.3.7 Examples of Structure Solution
Using PATSEE ………………………………… 388
8.3.8 Shake and Bake ……………………………….. 399
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8.4 Least-Squares Refinement …………………………….. 400
8.4.1 Unit-Cell Dimensions ………………………… 401
8.4.2 Least-Squares Parameters …………………….. 401
8.4.3 Theory of Least-Squares Refinement
and Strategies to Use …………………………. 405
8.4.4 Least-Squares Refinement Against Fo

2 . . . . . . . . . . . . 407
8.4.5 Constraints and Restraints . … … … … … … … … 408
8.5 Molecular Geometry ………………………………….. 408
8.5.1 Bond Lengths and Angles . .. … .. .. .. .. .. .. … .. .. 408
8.5.2 Torsion Angles ………………………………. 411
8.5.3 Conformational Analysis .. . .. .. .. .. .. .. .. . .. .. .. .. 412
8.5.4 Mean Planes …………………………………. 414
8.6 Precision ……………………………………………… 415
8.7 Correctness of a Structure Analysis ……………………. 416
8.7.1 Databases ……………………………………. 417
8.8 Limitations of X-Ray Structure Analysis ………………. 419
8.9 Disorder in Single Crystals ……………………………. 419
8.10 Computer Prediction of Crystal Structures . . . . . . . . . . . . . . . . . 422
8.10.1 Crystal Structure of 5-Azauracil ………………. 422
8.10.2 Developments in Computer Crystal Structure
Prediction ……………………………………. 425

8.11 Blind Structure Prediction of the Flexible
Molecule 1-Benzyl-1H-Tetrazole .. ….. ….. ….. ….. ….. 426
8.12 Problems ……………………………………………… 433
References …………………………………………………… 435
9 Examples of Crystal Structure Determination ……………… 439
9.1 Introduction …………………………………………… 439
9.2 Crystal Structure of 2-Bromobenzo[b] Indeno[1,2-e] Pyran …………………………………… 439
9.2.1 Preliminary Physical and X-Ray
Measurements ………………………………… 439
9.2.2 Intensity Measurement and Correction ………… 444
9.2.3 Structure Analysis in the xz Projection ………… 446
9.2.4 Three-Dimensional Structure
Determination ………………………………… 447
9.2.5 Refinement …………………………………… 449
9.2.6 Molecular Geometry . .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. 451
9.3 Crystal Structure of Potassium
2-Hydroxy-3,4-Dioxocyclobut-1-ene-1-Olate
Monohydrate (KHSQ) ………………………………… 455
9.3.1 Preliminary X-Ray and Physical
Measurements ………………………………… 455
9.3.2 Intensity Measurement and Correction ………… 456
9.3.3 P
2-Listing …………………………………… 456
9.3.4 Specifying the Origin …………………………. 457
9.3.5 Sign Determination …………………………… 458
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9.3.6 The E Map …………………………………… 459
9.3.7 Completion and Refinement
of the Structure ………………………………. 462

9.4 Crystal and Molecular Structure and Absolute
Configuration of 3b-Acetoxy-6,7-Epidithio-19-
Norlanosta-5,7,9,11-Tetraene ………………………….. 465
9.4.1 Preparation and Preliminary Optical
and X-Ray Examinations . .. . . . . . . . . . .. . . . . . . . . . .. . 466
9.4.2 X-Ray Measurement of the Unit-Cell
Dimensions and Intensities . . . . . . . . . . . . . . . . . . . . . . . . 466
9.4.3 Structure Determination and Refinement . . . . . . . . . 468
9.4.4 Absolute Configuration ……………………….. 468
9.5 Discussion of the Structure ……………………………. 468
9.6 Some Remarks on X-Ray Structure
Determination ………………………………………… 470
9.7 Biomolecular Modeling: Bioinformatics . . . . . . . . . . . . . . . . . . . 471
9.8 Docking Oligomycin into ATP Synthase:
Ligand and Receptor ………………………………….. 471
9.8.1 Why Modeling Studies? .. .. . .. .. . .. .. . .. .. . .. . .. .. 471
9.9 X-Ray Structures and Absolute Configurations
of the Antibiotics Oligomycins A,B, and C:
Inhibitors of ATP Synthase ……………………………. 472
9.9.1 Summary …………………………………….. 473
9.9.2 Background ………………………………….. 474
9.9.3 Experimental …………………………………. 474
9.9.4 Structure Determination and Refinement . . . . . . . . . 475
9.9.5 Results ………………………………………. 475
9.9.6 Discussion …………………………………… 475
9.9.7 Conformational Variations
in the Macrocyclic Structures . .. . . .. . .. . . .. . .. . . .. 478
9.10 Structure of ATP Synthase (ATPase): The Receptor . . . . . . 480
9.11 Docking Oligomycin into ATPase . ….. ….. ….. ….. ….. 481
9.11.1 ATP Synthase FO Model . .. . . . .. . . .. . . . .. . . .. . . . .. 481
9.11.2 Homology Modeling ….. ….. ….. …… ….. ….. 482
9.11.3 Refining the Model: Energy Minimization . . . . . . . 482
9.11.4 Creation of a Pocket for Docking
Oligomycin into the ATP Synthase FO . . . . . . . . . . . 483
9.12 Problems ……………………………………………… 484
References …………………………………………………… 487
10 Proteins and Macromolecular X-Ray Analysis . . . . . . . . . . . . . . . . . 489
10.1 Introduction …………………………………………… 489
10.1.1 What Is a Protein? ……………………………. 489
10.2 Crystallization of Proteins and Complexes
for X-Ray Analysis ……………………………………. 491
10.2.1 Introduction ………………………………….. 491
10.2.2 Crystallization Conditions
for Macromolecules ………………………….. 492
Table of Contents xxiii

10.2.3 Properties of Protein Crystals . . . . . . . . . . . . . . . . . . . . 492
10.2.4 Crystallization of Proteins …………………… 492
10.2.5 Molecular Purity .. …… ……. …… ……. …… 493
10.2.6 Practical Considerations … …… ….. …… …… 493
10.2.7 Batch Crystallization ………………………… 493
10.2.8 Microbatch Screening ……………………….. 493
10.2.9 Vapor Diffusion Techniques . . . . . . . . . . . . . . . . . . . . . 494
10.2.10 Co-crystallization .. … …. … …. … …. … … …. 496
10.2.11 How to Improve the Crystals ………………… 496
10.2.12 Heavy-Atom Derivatives for MIR . . . . . . . . . . . . . . . 497
10.2.13 Protein Complex Crystals with Small Molecules 498
10.3 Crystal Mounting for X-Ray Data Collection . . . . . . . . . . . . . . 499
10.3.1 Mounting at Room Temperature . .. . .. . .. . .. .. . .. 499
10.3.2 Cryo-Crystallography ……………………….. 499
10.4 Macromolecular Crystallography . .. . .. .. . .. . .. .. . .. . .. .. . .. 501
10.4.1 Space Groups ……………………………….. 501
10.4.2 X-Ray Diffraction from Macromolecular
Crystals …………………………………….. 501
10.4.3 Recording X-Ray Diffraction
from Macromolecular Crystals ………………. 503
10.4.4 Measurement of X-Ray Diffraction
from Macromolecular Crystals ………………. 505
10.4.5 Problems with Data Collection
and Suggested Cures ………………………… 508
10.4.6 Preliminary Structure Determination:
Unit Cell and Symmetry …………………….. 509
10.4.7 Ricin Agglutinin …………………………….. 509
10.5 Types of Fourier Synthesis for Protein Analysis . . . . . . . . . . . 512
10.5.1 Reconstruction of the Molecular Structure . . . . . . 512
10.5.2 Difference Electron Density …………………. 513
10.5.3 The 2FoðhklÞjFcðhklÞj Map ………………… 514
10.6 Determination of the Phases for Protein Crystals . . . . . . . . . . 514
10.6.1 Introduction …………………………………. 514
10.6.2 Isomorphous Replacement (MIR) ……………. 514
10.6.3 Preparation and Screening
of Heavy-Atom Derivatives .. . .. .. .. .. .. . .. .. .. .. 515
10.6.4 Molecular Replacement (MR) ……………….. 516
10.6.5 Example of a Self-Rotation Function:
Ricin Agglutinin …………………………….. 520
10.6.6 Molecular Replacement in Practice .. .. … .. .. … 521
10.6.7 Application of the AmoRe Algorithms
to Ricin Agglutinin ………………………….. 525
10.7 SIRAS and MAD Phasing …………………………….. 526
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10.8 Use of Phase Information and Density
Modification …………………………………………. 528
10.8.1 Properties of r(xyz) for Proteins . . . . . . . . . . . .. . . . 528
10.8.2 Programs for Density Modification … …. … … 528
10.8.3 Preparing to Refine the Structure . . . . . . . . . . . . . . . 529
10.9 Macromolecular Structure Refinement
and Solvent and Ligand Fitting ………………………. 531
10.9.1 Refinement Techniques …………………….. 531
10.9.2 Simulated Annealing .. .. .. .. .. . .. .. .. .. . .. .. .. .. 533
10.9.3 Least-Squares Refinement: Constrained,
Restrained and Other Protocols . ….. …… ….. 534
10.10 Structure Validation: Final Checks …. ….. ….. ….. ….. 537
10.10.1 R-Factors ………………………………….. 537
10.10.2 Evaluation of Errors ……………………….. 539
10.11 Geometry Validation: Final Checks . .. .. .. .. .. .. .. .. .. .. .. 539
10.11.1 Bond Lengths, Bond Angles,
Planarity, and Chirality …………………….. 539
10.11.2 Conformation …. ……….. ……….. ………. 540
10.12 Humidity Control and the Use of Cryoprotectants
in Protein Crystallography . … … … …. … … … … …. … 545
10.13 Problems …………………………………………….. 545
References …………………………………………………… 546
11 Neutron Diffraction from Single Crystals … ……….. ……… 549
11.1 Introduction …………………………………………. 549
11.1.1 Refinement of Hydrogen Atom Positions . . . . . . 550
11.2 Neutrons, Neutron Sources, and Data Collection . . . . . . . . . 551
11.2.1 Neutrons …………………………………… 551
11.2.2 Neutron Sources . … .. … … .. … … .. … … .. … 551
11.2.3 Neutron Data Collection ……………………. 551
11.2.4 Thermal Neutrons ………………………….. 553
11.3 Neutron Scattering …………………………………… 553
11.3.1 Neutron Scattering Lengths . . . . . . . . . . . . . . . . . . . . . 554
11.4 Experimental Neutron Diffraction Data Collection ……. 554
11.4.1 LADI-III and VIVALDI at ILL, Grenoble . . . . . 555
11.4.2 Oak Ridge National Laboratory
(ORNL) ……………………………………. 556
11.4.3 Other Neutron Sources . . . . .. . . . . .. . . . . . .. . . . . . .. 559
11.5 Deuteration and Perdeuteration ………………………. 559
11.6 Examples of Structure Determination
by Neutron Crystallography ………………………….. 560
11.7 X-Ray and Neutron Structure of 1,8-(3,6,9-
Trioxaundecane-1,11-diyldioxy)-9,10-
dihydro-10-10 dimethylanthracene-9-ol ………………. 560
11.7.1 Experimental ………………………………. 561
11.7.2 Structure Analysis and Refinement . .. .. .. .. .. .. 564
11.7.3 Discussion of the Structure …………………. 565
11.7.4 Hydrogen Bonding …………………………. 566
11.8 The Pointless Program in CCP4 . .. … .. … … .. … … .. … 567
Table of Contents xxv

11.9 Determination of the Positions of the Deuterium
Atoms of the Bound Water Molecules in the Lectin
Protein Concanavalin A by Neutron Laue
Crystallography ……………………………………… 567
11.9.1 Introduction ……………………………….. 567
11.9.2 Deuteration of the Concanavalin
A Crystals …………………………………. 568
11.9.3 Data Collection and Analysis .. . .. . .. .. . .. . .. . .. 568
11.9.4 X-Ray Model Refinement ………………….. 569
11.9.5 Neutron Structure Refinement . . . . . . . . . . . . . . . . . . 569
11.9.6 The Bound Water Structure ………………… 570
11.9.7 The Metal Sites ……………………………. 570
11.9.8 The Saccharide Binding Site ……………….. 571
11.9.9 Conclusion ………………………………… 572
11.10 The Neutron Structure of the Formyl Peptide
Receptor Antagonist Cyclosporin H (CsH)
Unambiguously Determines the Solvent
and Hydrogen Bonding Structure for Crystal
Form II ……………………………………………… 574
11.10.1 Introduction ……………………………….. 574
11.10.2 Experimental ………………………………. 576
11.10.3 Structure Refinement …….. ……….. ……… 576
11.10.4 Description of the Neutron Structure
and Comparison with the X-Ray Structure ….. 579
11.10.5 Conclusion ………………………………… 580
11.11 Problems …………………………………………….. 582
References …………………………………………………… 583
12 Powder Diffraction …………………………………………. 585
12.1 Introduction …………………………………………. 585
12.1.1 Identification ………………………………. 585
12.1.2 Crystallinity: Size and Strain Broadening . . . . . . 585
12.1.3 Unit-Cell Parameters … … … .. … … … … .. … 586
12.1.4 Expansion Properties ……. ……….. ………. 586
12.1.5 Phase Transitions and Alloy Systems . . . . . . . . . . 586
12.2 Crystal Structure Analysis with Powders ……………… 586
12.2.1 Crystal Structure Determination Scheme ……. 586
12.3 Basis of the Powder Method …………………………. 588
12.4 Data Collection ……………………………………… 590
12.4.1 Guinier-Type Cameras . .. .. .. .. .. .. . .. .. .. .. .. .. 590
12.4.2 Image Plate Camera .. …………… ………… 592
12.4.3 Powder Diffractometers …….. …………….. 593
12.4.4 Diffractometry at a Neutron Source . . . . . . . . . . . . 594
12.5 Indexing Powder Patterns ……………………………. 598
12.5.1 General Indexing …………………………… 599
12.5.2 Reduced and Conventional Unit Cells . . . . . . . . . 601
12.5.3 Computer Indexing of the
Diffraction Pattern …………………………. 602

xxvi Table of Contents

12.6 Extracting Integrated Intensities from
a Powder Pattern …………………………………….. 605
12.7 The Rietveld Procedure ………………………………. 605
12.7.1 The Le Bail Method ………………………. 607
12.7.2 The Pawley Method ………………………. 608
12.8 Examples of Solved Structures ……………………….. 608
12.8.1 Traditional Methods ………………………. 609
12.8.2 SIR Program System ……. ……………….. 611
12.8.3 EXPO Program System . .. .. .. .. .. .. .. … .. .. .. 612
12.9 Direct-Space Methods ……………………………….. 613
12.9.1 Zeolites and the FOCUS Algorithm ……….. 614
12.9.2 Zinc–Silicate Complex VIP-9 . . . . . . . . . . . . . . . . . 614
12.10 Monte Carlo Method ….. ……….. ………… ……….. 617
12.10.1 Simulated Annealing … …. …. …. …. …. …. 621
12.11 ESPOIR Program System ……………………………. 621
12.12 Powder Diffraction with Proteins …………………….. 623
12.12.1 T3R3 Zinc–Insulin Complex ………………. 623
12.13 Maximum Entropy in Crystal Structure Analysis ……… 624
12.13.1 Most Probable Distribution . . . . . . . . . . . . . . . . . . . . 624
12.13.2 Electron Density Map . ……. …… …… …… 625
12.14 Log-Likelihood Gain in the Phase Problem …………… 626
12.14.1 Basis Set and Expansion of Reflections . . . . . . . 626
12.14.2 Log-Likelihood Gain ……………………… 627
12.14.3 Centroid Maps ……………………………. 627
12.15 Genetic Algorithms ………………………………….. 628
12.16 Energy Minimization Techniques …………………….. 628
12.17 Concluding Remarks . …………. ………….. ……….. 629
12.18 Problems …………………………………………….. 630
References …………………………………………………… 632
13 Computer-Aided Crystallography ………………………….. 635
13.1 Introduction …………………………………………. 635
13.1.1 Collaborative Computational Projects ……… 635
13.1.2 Structure of the Web Program Packages . . . . . . 636
13.2 Derivation of Point Groups (EULR) ………………….. 636
13.3 Point-Group Recognition (SYMM) …………………… 637
13.4 Structure Determination Simulation (XRAY) …………. 640
13.4.1 Patterson Function ………………………… 641
13.4.2 Superposition Function ……………………. 642
13.4.3 Structure Factor Calculation … … … …. … … 642
13.4.4 Least-Squares Refinement . . . . . . . . . . . . . . . . . . . . . 642
13.4.5 Electron Density Maps ……………………. 643
13.4.6 Direct Methods: Calculation of jEj Values . . . 643
13.4.7 Calculation of E Maps . .. . . . . .. . . . .. . . . .. . . . . .. 644
13.4.8 Bond Lengths and Bond Angles …………… 645
13.4.9 Scale and Temperature Factors
by Wilson’s Method ………………………. 645
13.4.10 jEj Values Calculated from
the Structure ………………………………. 645
Table of Contents xxvii

13.5 Crystal Structure Analysis Problems …………………… 646
13.5.1 Ni o-Phenanthroline Complex (NIOP) ……….. 647
13.5.2 2-Amino-4,6-dichloropyrimidine (CL1P) . . . . . . . 648
13.5.3 2-Amino-4-methyl-6-chloropyrimidine
(CL2P) ……………………………………… 648
13.5.4 m-Tolidine Dihydrochloride (MTOL) . . . . . . . . . . . 649
13.5.5 Nitroguanidine (NO2G) ……………………… 649
13.5.6 Bis(6-sulfanyloxy-1,3,5-triazin-2(1H)-one)
(BSTO) …………………………………….. 650
13.5.7 2-S-methylthiouracil (SMTX and SMTY) . . . . . . . 650
13.6 General Crystal Structure and Other Programs …………. 650
13.6.1 One-Dimensional Fourier Summation
(FOUR1D) ………………………………….. 650
13.6.2 Two-Dimensional Fourier Summation
(FOUR2D) ………………………………….. 650
13.6.3 One-Dimensional Fourier Transform
(TRANS1) ………………………………….. 651
13.6.4 Reciprocal Unit Cell (RECIP) ……………….. 651
13.6.5 Molecular Geometry (MOLGOM) …………… 651
13.6.6 Internal and Cartesian Coordinates
(INTXYZ) ………………………………….. 652
13.6.7 Linear Least Squares (LSLI) . . . .. . . . .. . . . .. . . . . .. 653
13.6.8 Matrix Operations (MATOPS) …. …. … …. …. 653
13.6.9 Q Values (QVALS) …………………………. 653
13.6.10 Le Page Unit-Cell Reduction
(LEPAGE) ………………………………….. 654
13.6.11 Zone symbols/Miller indices (ZONE) . . . . . . . . . . . 654
13.7 Automatic Powder Indexing: ITO12 …………………… 654
13.8 Automatic Powder Structure Solving: ESPOIR . . . . . . . . . . . . 655
13.8.1 Aragonite …………………………………… 655
13.8.2 a-Alumina (Corundum) ……………………… 656
13.9 Problems ……………………………………………… 658
References …………………………………………………… 658
Appendix A: Stereoviews and Crystal Models .. …. … …. … … …. 659
A.1 Stereoviews ………………………………………….. 659
A.2 Model of a Tetragonal Crystal ………………………… 659
Appendix B: Scho ̈nflies’ Symmetry Notation ……………………. 663
B.1 Alternating Axis of Symmetry ………………………… 663
B.2 Symmetry Notations ………………………………….. 663
Appendix C: Cartesian Coordinates …………………………….. 665
C.1 Cartesian to Crystallographic Transformation
and Its Inverse ……………………………………….. 665
Appendix D: Crystallographic Software ………………………… 669
D.1 Single Crystal Suites …………………………………. 669
D.2 Single Crystal Structure Solution Programs . . . . . . . . . . . . . . . 670
xxviii Table of Contents

D.3 Single Crystal Twinning Software …………………….. 670
D.4 Freestanding Structure Visualization Software . . . . . . . . . . . . 670
D.5 Powder Diffraction Data: Powder Indexing
Suites (Dedicated and Other) ……… ………… ………. 671
D.6 Powder Pattern Decomposition ……………………….. 671
D.7 Structure Solution from Powder
Diffraction Data ………………………………………. 671
D.8 Software for Macromolecular
Crystallography ………………………………………. 672
D.8.1 Data Processing … …… ….. …… ….. ….. ….. 672
D.8.2 Fourier and Structure
Factor Calculations ………………………….. 672
D.8.3 Molecular Replacement ……………………… 672
D.8.4 Schematic Structure Plots ……………………. 673

 

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25 days ago
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