Direct phasing in femtosecond nanocrystallography. II. Phase retrieval

Inter-Bragg crystallographic phase retrieval from shape transforms, stacking faults and substitutional disorder

One of the brilliant ideas of John Spence when he saw the first diffraction patterns from the Linac Coherent Light Source was that one could solve the crystallographic phase problem by utilising the intensities between Bragg peaks. Because these intensities are due to the Fourier transform of the shape of the crystal, the approach came to be known as "shape-transform phasing." Shape-transform phasing was developed over the next ten years and formed the basis for many other interesting ideas and pursuits. Here we describe the current best implementation of the original idea using a lattice occupancy formalism and show that certain types of crystal defects can also be modelled via this approach, allowing the molecular structure to be recovered from the additional information offered by the inter-Bragg intensities from these crystal defects.

Advances in methods for atomic resolution macromolecular structure determination

Recent technical advances have dramatically increased the power and scope of structural biology. New developments in high-resolution cryo-electron microscopy, serial X-ray crystallography, and electron diffraction have been especially transformative. Here we highlight some of the latest advances and current challenges at the frontiers of atomic resolution methods for elucidating the structures and dynamical properties of macromolecules and their complexes.

Shape transform phasing of edgy nanocrystals

Diffraction patterns from small protein crystals illuminated by highly coherent X-rays often contain measurable interference signals between Bragg peaks. This coherent `shape transform' signal introduces enough additional information to allow the molecular densities to be determined from the diffracted intensities directly, without prior information or resolution restrictions. However, the various correlations amongst molecular occupancies/vacancies at the crystal surface result in a subtle yet critical problem in shape transform phasing whereby the sublattices of symmetry-related molecules exhibit a form of partial coherence amongst lattice sites when an average is taken over many crystal patterns. Here an iterative phase retrieval algorithm is developed which is capable of treating this problem; it is demonstrated on simulated data.

Frontier methods in coherent X-ray diffraction for high-resolution structure determination

In 1912, Max von Laue and collaborators first observed diffraction spots from a millimeter-sized crystal of copper sulfate using an X-ray tube. Crystallography was born of this experiment, and since then, diffraction by both X-rays and electrons has revealed a myriad of inorganic and organic structures, including structures of complex protein assemblies. Advancements in X-ray sources have spurred a revolution in structure determination, facilitated by the development of new methods. This review explores some of the frontier methods that are shaping the future of X-ray diffraction, including coherent diffractive imaging, serial femtosecond X-ray crystallography and small-angle X-ray scattering. Collectively, these methods expand the current limits of structure determination in biological systems across multiple length and time scales.

XFELs open a new era in structural chemical biology

X-ray crystallography, the workhorse of structural biology, has been revolutionized by the advent of serial femtosecond crystallography using X-ray free electron lasers. Here, the fast pace and history of discoveries are discussed together with current challenges and the method’s great potential to make new structural discoveries, such as the ability to generate molecular movies of biomolecules at work.

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.

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.

Phasing coherently illuminated nanocrystals bounded by partial unit cells

With the use of highly coherent femtosecond X-ray pulses from a free-electron laser, it is possible to record protein nanocrystal diffraction patterns with far more information than is present in conventional crystallographic diffraction data. It has been suggested that diffraction phases may be retrieved from such data via iterative algorithms, without the use of a priori information and without restrictions on resolution. Here, we investigate the extension of this approach to nanocrystals with edge terminations that produce partial unit cells, and hence cannot be described by a common repeating unit cell. In this situation, the phase problem described in previous work must be reformulated. We demonstrate an approximate solution to this phase problem for crystals with random edge terminations.

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.