X-ray crystallography is the primary technique for imaging the structures, or the positions of the atoms, of molecules. Knowledge of the geometrical atomic structures of molecules is key information in physics, chemistry, biology, geology and many other areas of science and technology. Structures are determinants of the properties of molecular systems. In the case of biology, knowledge of the structures of biological molecules provides essential information that allows us to understand the biological functionality of biomolecules and biomolecular systems. This knowledge is used to understand the fundamental molecular basis of biological function and processes, disease processes, and is also important in rational, or structure-based, drug design.

X-ray crystallography involves irradiating a crystal specimen of the molecule under study with a beam of X-rays, and measuring the resulting pattern of diffracted X-rays. The data consisting of measured diffraction patterns is then inverted computationally to produce an image of the molecule. This is often referred to as computational imaging or computational microscopy. If both the phase and amplitude of the diffracted X- ray could be measured, then inversion of the data to produce the image would be straightforward. However, in practice, one can measure only the amplitude, but not the phase, of the diffracted X-rays. This results in the famous so-called “phase problem” in crystallography. A method of determining the phases must be devised before the structure can be calculated.

The phase problem in crystallography has been studied for over one hundred years, and a number of clever methods have been devised for determining the phases in order for structures to be calculated. However, the phase problem is still an active area of research as current phasing techniques have significant limitations, and also because of the emergence of new kinds of instrumentation, specimens, and diffraction experiments.

This thesis is concerned with the phase problem and phase retrieval algorithms for biological (macromolecular) crystallography that have arisen, in part, through the recent introduction of a new kind of X-ray source called an X-ray free-electron laser, and through new kinds of specimens that can be used with these sources.

The thesis is divided into six chapters. The first chapter provides background information on diffraction imaging, X-ray crystallography, the phase problem, phase retrieval algorithms, and X-ray free-electron lasers and serial femtosecond crystallography. Original material is contained in Chapters 2 through 5. Concluding remarks are made in Chapter 6.

Chapter 2 is concerned with properties of the phase problem for 3D crystals. New relationships are derived that more carefully formalise uniqueness for this problem, the problem for the case of an unknown molecular support is studied in detail and the theoretical results are supported by simulations, and the effects of crystallographic and noncrystallographic symmetry are elucidated.

Chapters 3 and 4 form the first main part of the thesis and consider the phase problem for 2D crystals, a new kind of specimen that has been investigated with X-ray free-electron lasers. The two chapters are presented as two published journal papers for which the candidate is the primary author. In Chapter 3, the fundamental uniqueness properties of the phase problem for 2D crystals are derived, the nature of the solution set is elucidated, and the effects of various kinds of a priori information are evaluated by simulation. Chapter 4 follows up the results in Chapter 3, using simulations to investigate practical aspects of ab initio phase retrieval for 2D crystals using minimal molecular envelope information, and considering the characteristics of data available from X-ray free-electron laser sources.

Chapter 5 forms the second main part of the thesis and develops a new kind of ab initio phasing technique called ab initio molecular replacement phasing. This method uses diffraction data from the same molecule crystallised in two or more crystal forms. Uniqueness of the solution for such a dataset is evaluated, and a suitable phase re- trieval algorithm is developed and tested by simulation using a small protein of known structure.

Chapter 6 contains a brief summary of the outcomes of the thesis and suggestions for future research.