Direct phasing of protein crystals with high solvent content

A deep learning solution for crystallographic structure determination

The general de novo solution of the crystallographic phase problem is difficult and only possible under certain conditions. This paper develops an initial pathway to a deep learning neural network approach for the phase problem in protein crystallography, based on a synthetic dataset of small fragments derived from a large well curated subset of solved structures in the Protein Data Bank (PDB). In particular, electron-density estimates of simple artificial systems are produced directly from corresponding Patterson maps using a convolutional neural network architecture as a proof of concept.

A general method for directly phasing diffraction data from high-solvent-content protein crystals

A procedure is described for direct phase determination in protein crystallography, applicable to crystals with high solvent content. The procedure requires only the diffraction data and an estimate of the solvent content as input. Direct phase determination is treated as a constraint satisfaction problem, in which an image is sought that is consistent with both the diffraction data and generic constraints on the density distribution in the crystal. The problem is solved using an iterative projection algorithm, the Difference Map algorithm, which has good global convergence properties, and can locate the correct solution without any initial phase information. Computational efficiency is improved by breaking the problem down into two stages; initial approximation of the molecular envelope at low resolution, followed by subsequent phase determination using all of the data. The molecular envelope is continually updated during the phase determination step. At both stages, the algorithm is initiated with many different and random phase sets, which are evolved subject to the constraints. A clustering procedure is used to identify consistent results across multiple runs, which are then averaged to generate consensus envelopes or phase sets. The emergence of highly consistent phase sets is diagnostic of success. The effectiveness of the procedure is demonstrated by application to 42 known structures of solvent fraction 0.60-0.85. The procedure works robustly at intermediate resolutions (1.9-3.5 Å) but is strongly dependent on crystal solvent content, only working routinely with solvent fractions greater than 0.70.

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.

Mask-based approach to phasing of single-particle diffraction data. II. Likelihood-based selection criteria

A new type of mask-selection criterion is suggested for mask-based phasing. In this phasing approach, a large number of connected molecular masks are randomly generated. Structure-factor phases corresponding to a trial mask are accepted as an admissible solution of the phase problem if the mask satisfies some specified selection rules that are key to success. The admissible phase sets are aligned and averaged to give a preliminary solution of the phase problem. The new selection rule is based on the likelihood of the generated mask. It is defined as the probability of reproducing the observed structure-factor magnitudes by placing atoms randomly into the mask. While the result of the direct comparison of mask structure-factor magnitudes with observed ones using a correlation coefficient is highly dominated by a few very strong low-resolution reflections, a new method gives higher weight to relatively weak high-resolution reflections that allows them to be phased accurately. This mask-based phasing procedure with likelihood-based selection has been applied to simulated single-particle diffraction data of the photosystem II monomer. The phase set obtained resulted in a 16 Å resolution Fourier synthesis (more than 4000 reflections) with 98% correlation with the exact phase set and 69% correlation for about 2000 reflections in the highest resolution shell (20–16 Å). This work also addresses another essential problem of phasing methods, namely adequate estimation of the resolution achieved. A model-trapping analysis of the phase sets obtained by the mask-based phasing procedure suggests that the widely used `50% shell correlation' criterion may be too optimistic in some cases.

Ab initio phasing of the diffraction of crystals with translational disorder

This article reports on the combined use of Bragg reflections and diffuse scatter for structure determination in crystallography.

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.

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.

Resolution Dependence of an Ab Initio Phasing Method in Protein X-ray Crystallography

For direct phasing of protein crystals, a method based on the hybrid-input-output (HIO) algorithm has been proposed and tested on a variety of structures. So far, however, the diffraction data have been limited to high-resolution ones, i.e., higher than 2 Å. In principle, the methodology can be applied to data of lower resolutions, which might be particularly useful for phasing membrane protein crystals. For resolutions higher than 3.5 Å, it seems the atomic structure is solvable. For data of lower resolutions, information of the secondary structures and the protein boundary can still be obtained. Examples are given to support the conclusions. Real experimental data are used. Two aspects of the observed data have been discussed: removal of the measured low-resolution reflections and involvement of the unmeasured high-resolution reflections. The ab initio phasing employs histogram matching for density modification. A question arises whether the reference histogram used should match the resolution of the diffraction data or not. It seems that there is an optimal histogram which is good to use for data at various resolutions.

Improving the convergence rate of a hybrid input-output phasing algorithm by varying the reflection data weight

In an iterative projection algorithm proposed for ab initio phasing, the error metrics typically exhibit little improvement until a sharp decrease takes place as the iteration converges to the correct high-resolution structure. Related to that is the small convergence probability for certain structures. As a remedy, a variable weighting scheme on the diffraction data is proposed. It focuses on phasing low- and medium-resolution data first. The weighting shifts to incorporate more high-resolution reflections when the iteration proceeds. It is found that the precipitous drop in error metrics is replaced by a less dramatic drop at an earlier stage of the iteration. It seems that once a good configuration is formed at medium resolution, convergence towards the correct high-resolution structure is almost guaranteed. The original problem of phasing all diffraction data at once is reduced to a much more manageable one due to the dramatically smaller number of reflections involved. As a result, the success rate is significantly enhanced and the speed of convergence is raised. This is illustrated by applying the new algorithm to several structures, some of which are very difficult to solve without data weighting.

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.

Structure determination based on continuous diffraction from macromolecular crystals

Bright and coherent X-ray sources, such free-electron lasers, have spurred large activities in developing new methods to obtain the structures of biological macromolecules. In particular, single-molecule diffraction is highly desired, as it would abolish the need for crystallization. It provides considerably more diffraction intensity information than needed to solve a structure, unlike crystal diffraction, which is usually insufficient for direct phasing. To overcome the challenge of weak scattering signals of single molecules, the direct phasing approaches in coherent diffractive imaging have been combined with crystals in several imaginative ways. One of these, using crystals with translational disorder, has been used to phase continuous femtosecond X-ray diffraction data from photosystem II complexes, offering a paradigm shift in crystallography.

Structure Determination Using X-Ray Free-Electron Laser Pulses

The intense X-ray pulses from free-electron lasers, of only femtoseconds duration, outrun most of the processes that lead to structural degradation in X-ray exposures of macromolecules. Using these sources it is therefore possible to increase the dose to macromolecular crystals by several orders of magnitude higher than usually tolerable in conventional measurements, allowing crystal size to be decreased dramatically in diffraction measurements and without the need to cool the sample. Such pulses lead to the eventual vaporization of the sample, which has required a measurement approach, called serial crystallography, of consolidating snapshot diffraction patterns of many individual crystals. This in turn has further separated the connection between dose and obtainable diffraction information, with the only requirement from a single pattern being that to give enough information to place it, in three-dimensional reciprocal space, in relation to other patterns. Millions of extremely weak patterns can be collected and combined in this way, requiring methods to rapidly replenish the sample into the beam while generating the lowest possible background . The method is suited to time-resolved measurements over timescales below 1 ps to several seconds, and opens new opportunities for phasing. Some straightforward considerations of achievable signal levels are discussed and compared with a wide variety of recent experiments carried out at XFEL, synchrotron, and even laboratory sources, to discuss the capabilities of these new approaches and give some perspectives on their further development.

Improving the efficiency of molecular replacement by utilizing a new iterative transform phasing algorithm

An iterative transform method proposed previously for direct phasing of high-solvent-content protein crystals is employed for enhancing the molecular-replacement (MR) algorithm in protein crystallography. Target structures that are resistant to conventional MR due to insufficient similarity between the template and target structures might be tractable with this modified phasing method. Trial calculations involving three different structures are described to test and illustrate the methodology. The relationship of the approach to PHENIX Phaser-MR and MR-Rosetta is discussed.

Uniqueness of the macromolecular crystallographic phase problem

Uniqueness of the phase problem in macromolecular crystallography, and its relationship to the case of single particle imaging, is considered. The crystallographic problem is characterized by a constraint ratio that depends only on the size and symmetry of the molecule and the unit cell. The results are used to evaluate the effect of various real-space constraints. The case of an unknown molecular envelope is considered in detail. The results indicate the quite wide circumstances under which ab initio phasing should be possible.

Iterative projection algorithms in protein crystallography. II. Application

Iterative projection algorithms (IPAs) are a promising tool for protein crystallographic phase determination. Although related to traditional density-modification algorithms, IPAs have better convergence properties, and, as a result, can effectively overcome the phase problem given modest levels of structural redundancy. This is illustrated by applying IPAs to determine the electron densities of two protein crystals with fourfold non-crystallographic symmetry, starting with only the experimental diffraction amplitudes, a low-resolution molecular envelope and the position of the non-crystallographic axes. The algorithm returns electron densities that are sufficiently accurate for model building, allowing automated recovery of the known structures. This study indicates that IPAs should find routine application in protein crystallography, being capable of reconstructing electron densities starting with very little initial phase information.