Romain D. Arnal

The phase problem for two‐dimensional crystals. I. Theory

Properties of the phase problem for two-dimensional crystals are examined. This problem is relevant to protein structure determination using diffraction from two-dimensional crystals that has been proposed using new X-ray free-electron laser sources. The problem is shown to be better determined than for conventional three-dimensional crystallography, but there are still a large number of solutions in the absence of additional a priori information. Molecular envelope information reduces the size of the solution set, and for an envelope that deviates sufficiently from the unit cell a unique solution is possible. The effects of various molecular surface features and incomplete data on uniqueness and prospects for ab initio phasing are assessed. Simulations of phase retrieval for two-dimensional crystal data are described in the second paper in this series.

Phase retrieval for multiple objects

The problem of reconstructing multiple objects from the average of their diffracted intensities is considered. Three cases of technical interest are studied. The first is where the incoherent average is measured over a single object that adopts a number of positions described by a symmetry group. The second is where the average is over a small number of distinct objects. The third is where the average is over a set of unit cells that can occur in an ensemble of nanocrystals as a result of different edge terminations. As a result of some redundancy in the multi-dimensional phase problem, a unique solution can be obtained for these problems under some circumstances. Uniqueness is characterised using the constraint ratio. Iterative projection algorithms can be adapted to accommodate these cases and example simulated reconstructions are presented.

Reconstruction of an object from diffraction intensities averaged over multiple object clusters

A projection operator is derived for use in iterative phase retrieval algorithms when the Fourier intensity data is an average over the intensity from multiple clusters of identical objects. The projection operator is a generalization of the magnitude projection for conventional phase retrieval for a single object, and is applicable when the relative orientations and positions of the objects within the clusters are known. Simulations demonstrate that an iterative projection algorithm equipped with this new projection operator can successfully reconstruct an object from the averaged Fourier intensities from multiple clusters, each containing multiple copies of the object.

Image reconstruction in serial femtosecond nanocrystallography using x-ray free-electron lasers

Serial femtosecond nanocrystallography (SFX) is a form of x-ray coherent diffraction imaging that utilises a stream of tiny nanocrystals of the biological assembly under study, in contrast to the larger crystals used in conventional x-ray crystallography using conventional x-ray synchrotron x-ray sources. Nanocrystallography utilises the extremely brief and intense x-ray pulses that are obtained from an x-ray free-electron laser (XFEL). A key advantage is that some biological macromolecules, such as membrane proteins for example, do not easily form large crystals, but spontaneously form nanocrystals. There is therefore an opportunity for structure determination for biological molecules that are inaccessible using conventional x-ray crystallography. Nanocrystallography introduces a number of interesting image reconstruction problems. Weak diffraction patterns are recorded from hundreds of thousands of nancocrystals in unknown orientations, and these data have to be assembled and merged into a 3D intensity dataset. The diffracted intensities can also be affected by the surface structure of the crystals that can contain incomplete unit cells. Furthermore, the small crystal size means that there is potentially access to diffraction information between the crystalline Bragg peaks. With this information, phase retrieval is possible without resorting to the collection of additional experimental data as is necessary in conventional protein crystallography. We report recent work on the diffraction characteristics of nanocrystals and the resulting reconstruction algorithms.

Molecular imaging with x-ray free-electron lasers

High resolution imaging of biological macromolecules using x-ray crystallography is a key component of modern molecular biology, the results of which are essential for understanding biological processes in health and disease, and for drug design. Macromolecular imaging is currently undergoing a revolution as a result of the recent availability of x-ray free-electron lasers (XFELs). XFELs produce extremely intense, ultra-short x-ray pulses which offer the possibility of imaging specimens that are different to the 3D crystals used in conventional x-ray crystallography. The application of XFEL imaging to nano-crystalline fibrous specimens - long, slender systems that are periodic in their axial direction exhibit partial lateral crystallinity - is investigated. It is shown that individual Fourier amplitudes can be measured from XFEL data from such specimens. It is demonstrated that the image reconstruction problem from diffraction data for specimens with reduced crystallinity, specifically 2D membranes, is achievable. Although such specimens are weakly diffracting, they potentially offer more information in their diffraction than do 3D crystals. Image reconstruction is demonstrated by simulation.

Extreme imaging: Macromolecular imaging using x-ray free-electron lasers

X-ray free-electron lasers are new x-ray sources that are revolutionising high resolution molecular imaging. These sources produce extremely bright and extremely brief x-ray pulses that can overcome limitations of crystal growth and radiation damage in traditional protein x-ray crystallography. They introduce a new paradigm in diffraction imaging in which the specimen is destroyed, but not before the diffraction data are collected, and that requires radically new forms of specimen delivery and data processing. An overview of biomolecular imaging using x-ray free-electron lasers is provided, as well as future prospects.

The phase problem with structured sampling

The phase problem is a main limitation in diffraction imaging. Uniqueness properties of the phase problem are important, and are well characterised for well-sampled Fourier amplitude data. Here we consider uniqueness of the phase problem is cases where the Fourier amplitude data are sampled at a sufficient density, but the sample locations are structured in a particular way that arises in imaging of 1D and 2D crystals. Uniqueness is characterised by a suitably defined constraint ratio. Simulations of phase retrieval using an iterative projection algorithm show the influence of the sampling structure both in terms of difficulty of reconstruction and the resulting reconstruction error, and its dependence on the constraint ratio. The results show that, compared to random sampling of the Fourier intensity, structured sampling leads to more convergence difficulties and increased reconstruction error at equivalent constraint ratios. The results have implications for ab initio phasing in imaging 1D and 2D crystals using x-ray free-electron lasers.

Uniqueness for ab initio phase retrieval in macromolecular X-ray crystallography

Protein X-ray crystallography, the determination of three-dimensional structures of protein molecules from crystal X-ray diffraction data, requires determining the phases of the diffraction signal, since only the amplitude is measured. This so-called phase problem is a key component of protein X-ray crystallography. Most current methods for phasing are experimentally based and utilise either additional diffraction data from modified crystals or knowledge of the structure of a related protein. Ab initio phasing refers to computational methods that utilise only the measured diffraction amplitudes and some a priori information on protein electron densities, and potentially offer a dramatic reduction in effort over experimental methods. A key question in ab initio phasing is whether the diffraction amplitudes and the a priori information are sufficient to uniquely define the phases. The question of uniqueness is examined here for the case of a priori information on the molecular envelope.