Crystals, X-rays and Proteins Comprehensive Protein Crystallography Dennis Sherwood MA (Hons), MPhil, PhD Jon Cooper BA (Hons), PhD OXFORD UNIVERSITY PRESS
Contents List of symbols xiv PART I FUNDAMENTALS 1 The crystalline state and its study 3 1.1 States of matter 3 1.2 Anisotropy 4 1.3 The significance of order 6 1.4 Crystals 9 1.5 Solids which are not crystals 11 1.6 Crystal defects 13 1.7 Analysing the structure of crystals and molecules 13 1.8 Why do we use X-rays? 17 1.9 Why do we use diffraction? 20 1.10 Protein crystals 20 Summary 21 Bibliography 21 2 Vector analysis and complex algebra 23 VECTORS 23 2.1 What is a vector? 23 2.2 Vector addition 25 2.3 Multiplication by a scalar 27 2.4 Unit vectors 27 2.5 Components 30 2.6 Vector subtraction 32 2.7 Multiplication by a vector to give a scalar 33 2.8 Multiplication by a vector to give a vector 36 2.9 The scalar triple product and the vector triple product 39 COMPLEX ALGEBRA 42 2.10 What is a complex number? 43 2.11 The Argand diagram 44 2.12 The addition of complex numbers 46 2.13 Multiplication of complex numbers 47 2.14 The complex conjugate 48 2.15 The complex exponential representation 49 2.16 Complex exponentials and trigonometric functions 51 Summary 52 Appendix: Determinants 55 Bibliography 57
viii Contents 3 Crystal systematics 58 3.1 What is a crystal? 58 3.2 Symmetry 61 3.3 The description of the lattice 62 3.4 Crystal directions 72 3.5 Lattice planes 72 3.6 Symmetry operations and symmetry elements 76 3.7 Point groups and Laue groups 84 3.8 Space groups 87 Summary 91 Bibliography 92 4 Waves and electromagnetic radiation 93 4.1 Mathematical functions 93 4.2 What is a wave? 94 4.3 The mathematical description of a wave 97 4.4 The wave equation 104 4.5 The solution of the wave equation 106 4.6 The principle of superposition 111 4.7 Phase 114 4.8 Waves and complex exponentials 118 4.9 Intensity 120 4.10 Waves which are not plane 121 4.11 Electromagnetic waves 122 4.12 The form of electromagnetic waves 124 4.13 The interaction of electromagnetic radiation with matter 126 Summary 131 Bibliography 132 5 Fourier transforms and convolutions 133 5.1 Integrals 133 5.2 Curve sketching 135 5.3 Fourier transforms 141 5.4 Mathematical conventions and physical reality 146 5.5 The inverse transform 148 5.6 Real space and Fourier space 151 5.7 Delta functions 152 5.8 Fourier transforms and delta functions 154 5.9 Symmetrical and antisymmetrical functions 164 5.10 Convolutions 166 5.11 The Fourier transform of a convolution 172 5.12 The Patterson function 173 Summary 176 Appendix I: Proof of Fourier's theorem 179 Appendix II: Proof of convolution theorem 180 Bibliography 183
Contents ix 6 Diffraction 184 6.1 The interaction of waves with obstacles 184 6.2 The diffraction of water waves 185 6.3 Diffraction and information 189 6.4 The diffraction of light 190 6.5 X-ray diffraction 192 6.6 The mathematics of diffraction 192 6.7 Diffraction and Fourier transforms 198 6.8 The significance of the Fourier transform 200 6.9 Fourier transforms and phase 203 6.10 Fourier transforms and the wave equation 205 6.11 Fourier transforms and information 206 6.12 The inverse transform 208 6.13 The significance of the inverse transform 210 6.14 Experimental limitations 213 Summary 214 Bibliography 215 Review I 216 PART II DIFFRACTION THEORY 7 Diffraction by one-dimensional obstacles 221 7.1 The geometrical arrangement 221 7.2 One narrow slit 224 7.3 One wide slit 226 7.4 Two narrow slits 227 7.5 Young's experiment 228 7.6 Two wide slits 232 7.7 Three narrow slits 235 7.8 Three wide slits 236 7.9 N narrow slits 237 7.10 N wide slits 238 7.11 An infinite number of narrow slits 239 7.12 An infinite number of wide slits 240 7.13 The significance of the diffraction pattern 241 7.14 Another way of looking at N wide slits 246 Summary 250 Bibliography 253 8 Diffraction by a three-dimensional lattice 254 8.1 The diffraction pattern of a crystal 254 8.2 Non-normally incident waves 255 8.3 The diffraction pattern of a finite three-dimensional lattice 258 8.4 The diffraction pattern of an infinite lattice 260 8.5 The Laue equations 265
x Contents 8.6 The solution of the Laue equations 266 8.7 The reciprocal lattice 269 8.8 Reciprocal-lattice vectors and real-lattice planes 274 8.9 Bragg's law 277 8.10 The Ewald circle 278 8.11 The reciprocal lattice and diffraction 281 8.12 Why X-ray diffraction works 285 8.13 The Ewald sphere 286 8.14 The Ewald sphere and diffraction 288 8.15 Bragg's law and crystal planes 291 8.16 The effect of finite crystal size 293 Summary 294 9 The contents of the unit cell 297 9.1 The scattering of X-rays by a single electron 297 9.2 The scattering of X-rays by a distribution of electrons 300 9.3 The diffraction pattern of the motif 304 9.4 The calculation of the electron density function 307 9.5 Fourier synthesis 308 9.6 The calculation of structure factors 311 9.7 Atomic scattering factors 317 9.8 Anomalous scattering 322 9.9 Crystal symmetry and X-ray diffraction 323 9.10 Diffraction pattern symmetry 324 9.11 EriedePslaw 326 9.12 The breakdown of Friedel's law 328 9.13 Friedel's law and electron density calculations 331 9.14 Systematic absences 332 9.15 The determination of crystal symmetry 335 Summary 336 Review II 339 PART III STRUCTURE SOLUTION 10 Experimental techniques: sample preparation 343 10.1 Protein expression 343 10.2 Protein purification 347 10.3 Crystallisation 352 10.4 Crystal mounting 356 Summary 360 References 360 11 Experimental techniques: data collection and analysis 362 11.1 The origin of X-rays 362 11.2 Laboratory X-ray sources 363 11.3 Synchrotron sources 368
Contents xi 11.4 Optimising the X-ray beam 370 11.5 The rotation method 375 11.6 Electronic detectors 378 11.7 Other aspects of data collection 381 11.8 Data processing 383 11.9 The basis of intensity data corrections 392 11.10 The polarisation factor 397 11.11 The Lorentz factor 398 11.12 Absorption 401 11.13 The temperature factor 405 11.14 Scaling and merging intensity measurements 413 11.15 Conversion of intensities to structure factor amplitudes 416 11.16 Normalised structure factors 418 11.17 Completeness of the data 419 11.18 Estimating the solvent content 420 11.19 Misindexing and twinning 422 Summary 426 References 428 12 The phase problem and the Patterson function 431 12.1 The nature of the problem 431 12.2 Why is phase not detectable? 432 12.3 The Fourier transform of the intensities 433 12.4 The Patterson function and the crystal structure 434 12.5 The form of the Patterson function 437 12.6 The meaning of the Patterson function 438 12.7 Patterson maps 439 12.8 Patterson map symmetry 443 12.9 The use of Patterson maps 445 Summary 447 Bibliography 448 13 Molecular replacement 449 13.1 Solving the phase problem when the structure of a related protein is known 449 13.2 The rotation function 451 13.3 Choice of variables in the rotation function 455 13.4 Testing the rotation function 460 13.5 Refining the rotation function solution 460 13.6 Symmetry of the rotation function 461 13.7 The translation function 461 13.8 Patterson-based translation methods 462 13.9 Reciprocal-space translation searches 464 13.10 Asymmetric unit of the translation function 467 13.11 Non-crystallographic symmetry 468
xii Contents 13.12 The packing function 470 13.13 Verifying the results 471 13.14 Wider applications of molecular replacement 472 Summary 473 References 473 14 Solving the phase problem experimentally 475 14.1 The techniques of solution 475 14.2 Isomorphism and the preparation of heavy-atom derivatives 476 14.3 Scaling and analysing derivative data 479 14.4 The difference Patterson function 483 14.5 The methods of Patterson solution 493 14.6 Direct methods for locating sites 496 14.7 Refinement of heavy-atom sites 505 14.8 Cross-phasing 507 14.9 The isomorphous replacement method 509 14.10 Exploiting anomalous scattering effects in phasing 519 14.11 Density modification 530 Summary 538 References 541 15 Refinement 15.1 The necessity for refinement 545 15.2 Obtaining the trial structure 548 15.3 Assessing the trial structure 551 15.4 Least-squares refinement 554 15.5 Theory of the least-squares method 555 15.6 The use of stereochemical restraints 565 15.7 The benefits of non-crystallographic symmetry 571 15.8 Modelling rigid-group displacement 572 15.9 Simulated annealing 572 15.10 Cross-validation 574 15.11 Use of Fourier maps in refinement 576 15.12 The difference Fourier synthesis 578 15.13 The maximum-likelihood method in refinement 586 15.14 Validation and deposition 587 Summary 589 References 591 545 16 Complementary diffraction methods 595 16.1 Finding hydrogen atoms in X-ray structures 595 16.2 Neutron protein crystallography 597 16.3 Nevitron data collection 601 16.4 Neutron applications 604 16.5 Advantages of perdeuteration 606 16.6 X-ray Laue diffraction 607
16.7 Laue data processing Summary References Review III General bibliography Index