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

Ab initio reconstruction from one-dimensional crystal diffraction data

Filamentary and rod-like assemblies are ubiquitous in biological systems, and single such assemblies can form one-dimensional (1D) crystals. New, intense X-ray sources, such as X-ray free-electron lasers, make it feasible to measure diffraction data from single 1D crystals. Such experiments would present some advantages, since cylindrical averaging of the diffraction data in conventional fiber diffraction analysis is avoided, there is coherent signal amplification relative to single-particle imaging, and the diffraction data are oversampled compared with those from a 3D crystal so that the phase problem is better determined than for a 3D crystal [Millane (2017). Acta Cryst. A73, 140-150]. Phasing of 1D crystal diffraction data is examined, by simulation, using an iterative projection algorithm. Ab initio phasing is feasible with realistic noise levels and little envelope information is required if a shrink-wrap algorithm is also incorporated. Some practical aspects of the proposed experiments are explored.

Macromolecular phasing using diffraction from multiple crystal forms

A phasing algorithm for protein crystallography using diffraction data from multiple crystal forms is proposed. The algorithm is evaluated by simulation, and practical aspects and potential for ab initio phasing are discussed.

A phasing algorithm for macromol­ecular crystallography is proposed that utilizes diffraction data from multiple crystal forms – crystals of the same mol­ecule with different unit-cell packings (different unit-cell parameters or space-group symmetries). The approach is based on the method of iterated projections, starting with no initial phase information. The practicality of the method is demonstrated by simulation using known structures that exist in multiple crystal forms, assuming some information on the mol­ecular envelope and positional relationships between the mol­ecules in the different unit cells. With incorporation of new or existing methods for determination of these parameters, the approach has potential as a method for ab initio phasing.

The phase problem for two-dimensional crystals. II. Simulations

Phasing of diffraction data from two-dimensional crystals using only minimal molecular envelope information is investigated by simulation. Two-dimensional crystals are an attractive target for studying membrane proteins using X-ray free-electron lasers, particularly for dynamic studies at room temperature. Simulations using an iterative projection algorithm show that phasing is feasible with fairly minimal molecular envelope information, supporting recent uniqueness results for this problem [Arnal & Millane (2017).Acta Cryst.A73, 438–448]. The effects of noise and likely requirements for structure determination using X-ray free-electron laser sources are investigated.