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Aspects of Phase Retrieval in X-Ray Crystallography

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Aspects of Phase Retrieval in X-Ray Crystallography Aspects of Phase Retrieval in X-ray Crystallography Romain Arnal A thesis presented for the degree of Doctor of Philosophy in Electrical and Computer Engineering at the University of Canterbury, Christchurch, New Zealand. June 2019 Deputy Vice-Chancellor’s Office Postgraduate Office Co-Authorship Form This form is to accompany the submission of any thesis that contains research reported in co- authored work that has been published, accepted for publication, or submitted for publication. A copy of this form should be included for each co-authored work that is included in the thesis. Completed forms should be included at the front (after the thesis abstract) of each copy of the thesis submitted for examination and library deposit. Please indicate the chapter/section/pages of this thesis that are extracted from co-authored work and provide details of the publication or submission from the extract comes: Chapter 2 describes work by the candidate that is reported in “Millane and Arnal, Uniqueness of the macromolecular crystallographic phase problem, Acta Cryst., A71, 592-598, 2015.” Please detail the nature and extent (%) of contribution by the candidate: This publication describes joint work by the candidate and Millane. The theory was derived jointly by the candidate and Millane. The simulations, the required software development, and presentation of the results were conducted by the candidate. Writing was a joint effort. Certification by Co-authors: If there is more than one co-author then a single co-author can sign on behalf of all The undersigned certifys that: ▪ The above statement correctly reflects the nature and extent of the PhD candidate’s contribution to this co-authored work ▪ In cases where the candidate was the lead author of the co-authored work he or she wrote the text Name: Rick Millane Signature: Date: 31 May 2019 Deputy Vice-Chancellor’s Office Postgraduate Office Co-Authorship Form This form is to accompany the submission of any thesis that contains research reported in co- authored work that has been published, accepted for publication, or submitted for publication. A copy of this form should be included for each co-authored work that is included in the thesis. Completed forms should be included at the front (after the thesis abstract) of each copy of the thesis submitted for examination and library deposit. Please indicate the chapter/section/pages of this thesis that are extracted from co-authored work and provide details of the publication or submission from the extract comes: Chapter 3 is a reproduction of the paper “Arnal and Millane, The phase problem for two-dimensional crystals. I. Theory, Acta Cryst., A73, 438-448, 2017.” Permission to reproduce this paper in the thesis has been obtained from the publisher IUCr. Please detail the nature and extent (%) of contribution by the candidate: The candidate was the lead author of this publication. The candidate developed key aspects of the theory with assistance from Millane. Design of the simulations, algorithm and software development, experimental design, and development and presentation of the results were conducted by the candidate. The candidate wrote most of the paper with assistance from Millane. Certification by Co-authors: If there is more than one co-author then a single co-author can sign on behalf of all The undersigned certifys that: ▪ The above statement correctly reflects the nature and extent of the PhD candidate’s contribution to this co-authored work ▪ In cases where the candidate was the lead author of the co-authored work he or she wrote the text Name: Rick Millane Signature: Date: 31 May 2019 Deputy Vice-Chancellor’s Office Postgraduate Office Co-Authorship Form This form is to accompany the submission of any thesis that contains research reported in co- authored work that has been published, accepted for publication, or submitted for publication. A copy of this form should be included for each co-authored work that is included in the thesis. Completed forms should be included at the front (after the thesis abstract) of each copy of the thesis submitted for examination and library deposit. Please indicate the chapter/section/pages of this thesis that are extracted from co-authored work and provide details of the publication or submission from the extract comes: Chapter 4 is a reproduction of the paper “Arnal, Zhao, Mitra, Spence and Millane, The phase problem for two-dimensional crystals. II. Simulations, Acta Cryst., A74, 537-544, 2018.” Permission to reproduce this paper in the thesis has been obtained from the publisher IUCr. Please detail the nature and extent (%) of contribution by the candidate: The candidate was the lead author of this publication. The candidate developed the algorithms and software described in the paper, and used these to generate, interpret and present the results described. The candidate wrote most of the paper with assistance from Millane. Mitra provided data for the simulations. Spence provided some general guidance for this study. Zhao was Spence’s student and provided little input to this work. Certification by Co-authors: If there is more than one co-author then a single co-author can sign on behalf of all The undersigned certifys that: ▪ The above statement correctly reflects the nature and extent of the PhD candidate’s contribution to this co-authored work ▪ In cases where the candidate was the lead author of the co-authored work he or she wrote the text Name: Rick Millane Signature: Date: 31 May 2019 ACKNOWLEDGEMENTS First and foremost, I must express my sincere gratitude to my supervisor Prof. Rick Millane. For all his support and guidance over the last few years I am especially indebted to him. During the course of my research I had the extraordinary chance to take part in many experiments at the Linac Coherent Light Source (LCLS) at Stanford. I am very grateful to principal investigators Henry Chapman, Carolin Seuring, Richard Bean and Lourdu Paulraj, for giving me the opportunity to participate in and learn from these experiments. I express my thanks to all the people involved in experiments cxils2616, cxilr7416, mfxn1116, amok8916 and amok9916, especially Steve Aplin and David Wojtas. Thanks again to Henry Chapman, for the invitations to visit and work at the Center for Free-electron Laser Science (CFEL) in Hamburg, and to the various people I met during my trips there. I also owe special thanks to my friend Joe Chen, for introducing me to this field and for the many discussions, valuable insights and for being an outstanding teacher. I am also thankful to Rick Kirian, for his invitations to visit and work at Arizona State University, and the help provided by many people there. Finally I would like to thank Andy Tan for always being there for me, Dr. Steve Weddell for inviting me to New Zealand back in 2013, and my family in France for allowing me travel across the world without seeing them for years. Contents 1 INTRODUCTION 1 1.1 Diffraction imaging . .1 1.2 X-ray crystallography . .2 1.2.1 X-ray diffraction crystallography methods . .2 1.2.2 Diffraction by a crystal . .4 1.3 The phase problem . .7 1.3.1 Nature of the phase problem . .7 1.3.2 Phase retrieval . .8 1.3.2.1 Direct methods . .8 1.3.2.2 Isomorphous replacement . .8 1.3.2.3 Anomalous scattering . .9 1.3.2.4 Molecular replacement . .9 1.3.2.5 Ab initio phasing . 10 1.4 Iterative projection algorithms . 11 1.4.1 Constraints and projections . 11 1.4.2 Error reduction algorithm . 12 1.4.3 Hybrid input-output algorithm . 14 1.4.4 Difference map algorithm . 14 1.4.5 Specific constraints, projections and error metrics . 15 1.4.5.1 Support constraint . 15 1.4.5.2 Fourier amplitude constraint . 16 1.5 Uniqueness of the phase problem . 17 1.6 X-ray sources . 18 1.6.1 X-ray tubes and synchrotron light sources . 18 1.6.2 X-ray free electron laser sources . 19 1.7 Serial femtosecond crystallography . 19 1.7.1 Sample delivery . 19 1.7.2 Hit finding . 21 1.7.3 Indexing and integrating . 21 1.8 Open problems . 21 2 ASPECTS OF THE PHASE PROBLEM FOR 3D CRYSTALS 23 2.1 Introduction . 23 2.2 Uniqueness for a crystal . 23 2.3 Real space constraints . 25 2.3.1 Known molecular envelope . 25 2.3.2 Unknown molecular envelope . 26 2.3.2.1 Number of hyperplanes . 27 2.3.2.2 Simulations . 29 2.3.3 Crystallographic symmetry . 31 2.3.4 Noncrystallographic symmetry . 31 2.4 Summary . 33 3 THE PHASE PROBLEM FOR 2D CRYSTALS. I. THEORY 35 4 THE PHASE PROBLEM FOR 2D CRYSTALS. II. SIMULATIONS 47 5 AB INITIO MOLECULAR REPLACEMENT PHASING 57 5.1 Introduction . 57 5.2 Diffraction by multiple crystal forms . 58 5.3 Uniqueness . 59 5.4 Implementations of aiMR.......................... 61 5.4.1 Implementation A . 62 5.4.1.1 Reciprocal space projection . 62 5.4.1.2 Real space projection . 62 5.4.1.3 Two-dimensional simulations of approach A . 63 5.4.2 Implementation B . 66 5.4.2.1 Reciprocal space projection . 66 5.4.2.2 Real space projection . 68 5.5 Simulation methods . 68 5.5.1 Simulated data . 68 5.5.2 Structural homology . 70 5.5.3 Molecular and crystal models . 70 5.5.4 Determination of the envelope . 72 5.5.5 Implementation of the real space merging . 72 5.6 Simulation results . 74 5.7 Discussion . 76 6 CONCLUSIONS AND FURTHER RESEARCH SUGGESTIONS 79 6.1 Further research suggestions . 80 REFERENCES 83 Preface X-ray crystallography is the primary technique for imaging the structures, or the po- sitions of the atoms, of molecules.
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