Phase retrieval for multiple objects from their averaged diffraction.

The problem of reconstructing multiple objects from the average of their diffracted intensities is investigated. Reconstruction feasibility (uniqueness) depends on the number of objects, their support shapes and dimensionality, and an appropriately calculated constraint ratio. For objects with sufficiently different supports, and a favorable constraint ratio, the reconstruction problem has a unique solution. For objects with identical supports, there can be multiple solutions, even with a favorable constraint ratio. However, positivity of the objects and noncentrosymmetry of the support reduce the number of multiple solutions, and a unique solution may exist with a favorable constraint ratio. An iterative projection based algorithm to reconstruct the individual objects is described. The efficacy of the reconstruction algorithm and the uniqueness results are demonstrated by simulation.

Diffraction by nanocrystals II.

Nanocrystals with more than one molecule in the unit cell will generally crystallize with incomplete unit cells on the crystal surface. Previous results show that the ensemble-averaged diffraction by such crystals consists of a usual Bragg component and two other Bragg-like components due to the incomplete unit cells. Using an intrinsic flexibility in the definition of the incomplete-unit-cell part of a crystal, the problem is formulated such that the magnitude of the Bragg-like components is minimized, which leads to a simpler and more useful interpretation of the diffraction. Simulations show the nature of the relative magnitudes of the diffraction components in different regions of reciprocal space and the effect of crystal faceting.

Diffraction by nanocrystals.

X-ray femtosecond nanocrystallography is a new, potentially powerful technique for imaging biological macromolecules that uses ensemble-averaged measurements of diffraction of x-ray free-electron laser pulses from nanocrytalline specimens. Nanocrystals have some diffraction characteristics that are distinct from those of macroscopic crystals, due to the presence of different kinds of unit cell in the crystal and of truncated unit cells on the crystal surface. Expressions are derived for diffraction by nanocrystals with variable and incomplete unit cells, averaged over a distribution of crystal sizes and shapes. The diffraction contains differently modulated Bragg components that are due to interference effects within and between the full and incomplete unit cells. Estimates are obtained for the relative magnitudes of the components. The nature of the diffraction is illustrated by two-dimensional simulations. Implications for molecular imaging are discussed.

Ab initio phasing of the diffraction of crystals with translational disorder.

To date X-ray protein crystallography is the most successful technique available for the determination of high-resolution 3D structures of biological molecules and their complexes. In X-ray protein crystallography the structure of a protein is refined against the set of observed Bragg reflections from a protein crystal. The resolution of the refined protein structure is limited by the highest angle at which Bragg reflections can be observed. In addition, the Bragg reflections alone are typically insufficient (by a factor of two) to determine the structure ab initio, and so prior information is required. Crystals formed from an imperfect packing of the protein molecules may also exhibit continuous diffraction between and beyond these Bragg reflections. When this is due to random displacements of the molecules from each crystal lattice site, the continuous diffraction provides the necessary information to determine the protein structure without prior knowledge, to a resolution that is not limited by the angular extent of the observed Bragg reflections but instead by that of the diffraction as a whole. This article presents an iterative projection algorithm that simultaneously uses the continuous diffraction as well as the Bragg reflections for the determination of protein structures. The viability of this method is demonstrated on simulated crystal diffraction.

Flow-aligned, single-shot fiber diffraction using a femtosecond X-ray free-electron laser.

A major goal for X-ray free-electron laser (XFEL) based science is to elucidate structures of biological molecules without the need for crystals. Filament systems may provide some of the first single macromolecular structures elucidated by XFEL radiation, since they contain one-dimensional translational symmetry and thereby occupy the diffraction intensity region between the extremes of crystals and single molecules. Here, we demonstrate flow alignment of as few as 100 filaments (Escherichia coli pili, F-actin, and amyloid fibrils), which when intersected by femtosecond X-ray pulses result in diffraction patterns similar to those obtained from classical fiber diffraction studies. We also determine that F-actin can be flow-aligned to a disorientation of approximately 5 degrees. Using this XFEL-based technique, we determine that gelsolin amyloids are comprised of stacked ß-strands running perpendicular to the filament axis, and that a range of order from fibrillar to crystalline is discernable for individual α-synuclein amyloids.

Aspects of direct phasing in femtosecond nanocrystallography.

X-ray free-electron laser diffraction patterns from protein nanocrystals provide information on the diffracted amplitudes between the Bragg reflections, offering the possibility of direct phase retrieval without the use of ancillary experimental data. Proposals for implementing direct phase retrieval are reviewed. These approaches are limited by the signal-to-noise levels in the data and the presence of different and incomplete unit cells in the nanocrystals. The effects of low signal to noise can be ameliorated by appropriate selection of the intensity data samples that are used. The effects of incomplete unit cells may be small in some cases, and a unique solution is likely if there are four or fewer molecular orientations in the unit cell.

Direct phasing in femtosecond nanocrystallography. I. Diffraction characteristics.

X-ray free-electron lasers solve a number of difficulties in protein crystallography by providing intense but ultra-short pulses of X-rays, allowing collection of useful diffraction data from nanocrystals. Whereas the diffraction from large crystals corresponds only to samples of the Fourier amplitude of the molecular transform at the Bragg peaks, diffraction from very small crystals allows measurement of the diffraction amplitudes between the Bragg samples. Although highly attenuated, these additional samples offer the possibility of iterative phase retrieval without the use of ancillary experimental data [Spence et al. (2011). Opt. Express, 19, 2866-2873]. This first of a series of two papers examines in detail the characteristics of diffraction patterns from collections of nanocrystals, estimation of the molecular transform and the noise characteristics of the measurements. The second paper [Chen et al. (2014). Acta Cryst. A70, 154-161] examines iterative phase-retrieval methods for reconstructing molecular structures in the presence of the variable noise levels in such data.

Direct phasing in femtosecond nanocrystallography. II. Phase retrieval.

X-ray free-electron laser diffraction patterns from protein nanocrystals provide information on the diffracted amplitudes between the Bragg reflections, offering the possibility of direct phase retrieval without the use of ancillary experimental diffraction data [Spence et al. (2011). Opt. Express, 19, 2866-2873]. The estimated continuous transform is highly noisy however [Chen et al. (2014). Acta Cryst. A70, 143-153]. This second of a series of two papers describes a data-selection strategy to ameliorate the effects of the high noise levels and the subsequent use of iterative phase-retrieval algorithms to reconstruct the electron density. Simulation results show that employing such a strategy increases the noise levels that can be tolerated.