Crystals, X-rays and Proteins

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1 Crystals, X-rays and Proteins Comprehensive Protein Crystallography Dennis Sherwood MA (Hons), MPhil, PhD Jon Cooper BA (Hons), PhD OXFORD UNIVERSITY PRESS

2 Contents List of symbols xiv PART I FUNDAMENTALS 1 The crystalline state and its study States of matter Anisotropy The significance of order Crystals Solids which are not crystals Crystal defects Analysing the structure of crystals and molecules Why do we use X-rays? Why do we use diffraction? Protein crystals 20 Summary 21 Bibliography 21 2 Vector analysis and complex algebra 23 VECTORS What is a vector? Vector addition Multiplication by a scalar Unit vectors Components Vector subtraction Multiplication by a vector to give a scalar Multiplication by a vector to give a vector The scalar triple product and the vector triple product 39 COMPLEX ALGEBRA What is a complex number? The Argand diagram The addition of complex numbers Multiplication of complex numbers The complex conjugate The complex exponential representation Complex exponentials and trigonometric functions 51 Summary 52 Appendix: Determinants 55 Bibliography 57

3 viii Contents 3 Crystal systematics What is a crystal? Symmetry The description of the lattice Crystal directions Lattice planes Symmetry operations and symmetry elements Point groups and Laue groups Space groups 87 Summary 91 Bibliography 92 4 Waves and electromagnetic radiation Mathematical functions What is a wave? The mathematical description of a wave The wave equation The solution of the wave equation The principle of superposition Phase Waves and complex exponentials Intensity Waves which are not plane Electromagnetic waves The form of electromagnetic waves The interaction of electromagnetic radiation with matter 126 Summary 131 Bibliography Fourier transforms and convolutions Integrals Curve sketching Fourier transforms Mathematical conventions and physical reality The inverse transform Real space and Fourier space Delta functions Fourier transforms and delta functions Symmetrical and antisymmetrical functions Convolutions The Fourier transform of a convolution The Patterson function 173 Summary 176 Appendix I: Proof of Fourier's theorem 179 Appendix II: Proof of convolution theorem 180 Bibliography 183

4 Contents ix 6 Diffraction The interaction of waves with obstacles The diffraction of water waves Diffraction and information The diffraction of light X-ray diffraction The mathematics of diffraction Diffraction and Fourier transforms The significance of the Fourier transform Fourier transforms and phase Fourier transforms and the wave equation Fourier transforms and information The inverse transform The significance of the inverse transform Experimental limitations 213 Summary 214 Bibliography 215 Review I 216 PART II DIFFRACTION THEORY 7 Diffraction by one-dimensional obstacles The geometrical arrangement One narrow slit One wide slit Two narrow slits Young's experiment Two wide slits Three narrow slits Three wide slits N narrow slits N wide slits An infinite number of narrow slits An infinite number of wide slits The significance of the diffraction pattern Another way of looking at N wide slits 246 Summary 250 Bibliography Diffraction by a three-dimensional lattice The diffraction pattern of a crystal Non-normally incident waves The diffraction pattern of a finite three-dimensional lattice The diffraction pattern of an infinite lattice The Laue equations 265

5 x Contents 8.6 The solution of the Laue equations The reciprocal lattice Reciprocal-lattice vectors and real-lattice planes Bragg's law The Ewald circle The reciprocal lattice and diffraction Why X-ray diffraction works The Ewald sphere The Ewald sphere and diffraction Bragg's law and crystal planes The effect of finite crystal size 293 Summary The contents of the unit cell The scattering of X-rays by a single electron The scattering of X-rays by a distribution of electrons The diffraction pattern of the motif The calculation of the electron density function Fourier synthesis The calculation of structure factors Atomic scattering factors Anomalous scattering Crystal symmetry and X-ray diffraction Diffraction pattern symmetry EriedePslaw The breakdown of Friedel's law Friedel's law and electron density calculations Systematic absences The determination of crystal symmetry 335 Summary 336 Review II 339 PART III STRUCTURE SOLUTION 10 Experimental techniques: sample preparation Protein expression Protein purification Crystallisation Crystal mounting 356 Summary 360 References Experimental techniques: data collection and analysis The origin of X-rays Laboratory X-ray sources Synchrotron sources 368

6 Contents xi 11.4 Optimising the X-ray beam The rotation method Electronic detectors Other aspects of data collection Data processing The basis of intensity data corrections The polarisation factor The Lorentz factor Absorption The temperature factor Scaling and merging intensity measurements Conversion of intensities to structure factor amplitudes Normalised structure factors Completeness of the data Estimating the solvent content Misindexing and twinning 422 Summary 426 References The phase problem and the Patterson function The nature of the problem Why is phase not detectable? The Fourier transform of the intensities The Patterson function and the crystal structure The form of the Patterson function The meaning of the Patterson function Patterson maps Patterson map symmetry The use of Patterson maps 445 Summary 447 Bibliography Molecular replacement Solving the phase problem when the structure of a related protein is known The rotation function Choice of variables in the rotation function Testing the rotation function Refining the rotation function solution Symmetry of the rotation function The translation function Patterson-based translation methods Reciprocal-space translation searches Asymmetric unit of the translation function Non-crystallographic symmetry 468

7 xii Contents The packing function Verifying the results Wider applications of molecular replacement 472 Summary 473 References Solving the phase problem experimentally The techniques of solution Isomorphism and the preparation of heavy-atom derivatives Scaling and analysing derivative data The difference Patterson function The methods of Patterson solution Direct methods for locating sites Refinement of heavy-atom sites Cross-phasing The isomorphous replacement method Exploiting anomalous scattering effects in phasing Density modification 530 Summary 538 References Refinement 15.1 The necessity for refinement Obtaining the trial structure Assessing the trial structure Least-squares refinement Theory of the least-squares method The use of stereochemical restraints The benefits of non-crystallographic symmetry Modelling rigid-group displacement Simulated annealing Cross-validation Use of Fourier maps in refinement The difference Fourier synthesis The maximum-likelihood method in refinement Validation and deposition 587 Summary 589 References Complementary diffraction methods Finding hydrogen atoms in X-ray structures Neutron protein crystallography Nevitron data collection Neutron applications Advantages of perdeuteration X-ray Laue diffraction 607

8 16.7 Laue data processing Summary References Review III General bibliography Index

DIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY. Regents' Professor enzeritus Arizona State University

DIFFRACTION PHYSICS THIRD REVISED EDITION JOHN M. COWLEY Regents' Professor enzeritus Arizona State University 1995 ELSEVIER Amsterdam Lausanne New York Oxford Shannon Tokyo CONTENTS Preface to the first