Aspects of direct phasing in femtosecond nanocrystallography

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.

An outlook on using serial femtosecond crystallography in drug discovery

Introduction: X-ray crystallography has made important contributions to modern drug development but its application to many important drug targets has been extremely challenging. The recent emergence of X-ray free electron lasers (XFELs) and advancements in serial femtosecond crystallography (SFX) have offered new opportunities to overcome limitations of traditional crystallography to accelerate the structure-based drug discovery (SBDD) process. Areas covered: In this review, the authors describe the general principles of X-ray generation and the main properties of XFEL beams, outline details of SFX data collection and processing, and summarize the progress in the development of associated instrumentation for sample delivery and X-ray detection. An overview of the SFX applications to various important drug targets such as membrane proteins is also provided. Expert opinion: While SFX has already made clear advancements toward the understanding of the structure and dynamics of several major drug targets, its robust application in SBDD still needs further developments of new high-throughput techniques for sample production, automation of crystal delivery and data collection, as well as for processing and storage of large amounts of data. The expansion of the available XFEL beamtime is a key to the success of SFX in SBDD.

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.

Fourier phase retrieval with a single mask by Douglas-Rachford algorithms

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization.

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.

Serial femtosecond crystallography: the first five years

Protein crystallography using synchrotron radiation sources has had a tremendous impact on biology, having yielded the structures of thousands of proteins and given detailed insight into their mechanisms. However, the technique is limited by the requirement for macroscopic crystals, which can be difficult to obtain, as well as by the often severe radiation damage caused in diffraction experiments, in particular when using tiny crystals. To slow radiation damage, data collection is typically performed at cryogenic temperatures. With the advent of free-electron lasers (FELs) capable of delivering extremely intense femtosecond X-ray pulses, this situation appears to be remedied, allowing the structure determination of undamaged macromolecules using either macroscopic or microscopic crystals. The latter are exposed to the FEL beam in random orientations and their diffraction data are collected at cryogenic or room temperature in a serial fashion, since each crystal is destroyed upon a single exposure. The new approaches required for crystal growth and delivery, and for diffraction data analysis, including de novo phasing, are reviewed. The opportunities and challenges of SFX are described, including applications such as time-resolved measurements and the analysis of radiation damage-prone systems.

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.