The vertebrate muscle superlattice: discovery, consequences, and link to geometric frustration.

Early x-ray diffraction studies of muscle revealed spacings larger than the basic thick filament lattice spacing and led to a number of speculations on the mutual rotations of the filaments in the myosin lattice. The nature of the arrangements of the filaments was resolved by John Squire and Pradeep Luther using careful electron microscopy and image analysis. The intriguing disorder in the rotations, that they termed the myosin superlattice, remained a curiosity, until work with Rick Millane and colleagues showed a connection to "geometric frustration," a well-known phenomenon in statistical and condensed matter physics. In this review, we describe how this connection gives a satisfying physical basis for the myosin superlattice, and how recent work has shown relationships to muscle mechanical behaviour.

Iterative projection algorithms for ab initio phasing in virus crystallography.

Iterative projection algorithms are proposed as a tool for ab initio phasing in virus crystallography. The good global convergence properties of these algorithms, coupled with the spherical shape and high structural redundancy of icosahedral viruses, allows high resolution phases to be determined with no initial phase information. This approach is demonstrated by determining the electron density of a virus crystal with 5-fold non-crystallographic symmetry, starting with only a spherical shell envelope. The electron density obtained is sufficiently accurate for model building. The results indicate that iterative projection algorithms should be routinely applicable in virus crystallography, without the need for ancillary phase information.

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 a frustrated system: the triangular Ising antiferromagnet.

Expressions are derived for diffraction by the triangular Ising antiferromagnet, a disordered lattice system consisting of two kinds of scatterer and exhibiting geometric frustration. Analysis of the expressions shows characteristics of the diffraction patterns, including the presence of Bragg and diffuse diffraction, superlattice reflections, and their behavior with temperature. These characteristics are illustrated by numerical simulations. The results have application to diffraction imaging of disordered 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.

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 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.

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.

Prior estimate-based compressed sensing in parallel MRI.

Two improved compressed sensing (CS)-based image reconstruction methods for MRI are proposed: prior estimate-based compressed sensing (PECS) and sensitivity encoding-based compressed sensing (SENSECS). PECS allows prior knowledge of the underlying image to be intrinsically incorporated in the image recovery process, extending the use of data sorting as first proposed by Adluru and DiBella (Int J Biomed Imaging 2008: 341648). It does so by rearranging the elements in the underlying image based on the magnitude information gathered from a prior image estimate, so that the underlying image can be recovered in a new form that exhibits a higher level of sparsity. SENSECS is an application of PECS in parallel imaging. In SENSECS, image reconstruction is carried out in two stages: SENSE and PECS, with the SENSE reconstruction being used as a image prior estimate in the following PECS reconstruction. SENSECS bypasses the conflict of sampling pattern design in directly applying CS recovery in multicoil data sets and exploits the complementary characteristics of SENSE-type and CS-type reconstructions, hence achieving better image reconstructions than using SENSE or CS alone. The characteristics of PECS and SENSECS are investigated using experimental data.

Macromolecular phasing using diffraction from multiple crystal forms.

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.

Geometric frustration in the myosin superlattice of vertebrate muscle.

Geometric frustration results from an incompatibility between minimum energy arrangements and the geometry of a system, and gives rise to interesting and novel phenomena. Here, we report geometric frustration in a native biological macromolecular system---vertebrate muscle. We analyse the disorder in the myosin filament rotations in the myofibrils of vertebrate striated (skeletal and cardiac) muscle, as seen in thin-section electron micrographs, and show that the distribution of rotations corresponds to an archetypical geometrically frustrated system---the triangular Ising antiferromagnet. Spatial correlations are evident out to at least six lattice spacings. The results demonstrate that geometric frustration can drive the development of structure in complex biological systems, and may have implications for the nature of the actin--myosin interactions involved in muscle contraction. Identification of the distribution of myosin filament rotations with an Ising model allows the extensive results on the latter to be applied to this system. It shows how local interactions (between adjacent myosin filaments) can determine long-range order and, conversely, how observations of long-range order (such as patterns seen in electron micrographs) can be used to estimate the energetics of these local interactions. Furthermore, since diffraction by a disordered system is a function of the second-order statistics, the derived correlations allow more accurate diffraction calculations, which can aid in interpretation of X-ray diffraction data from muscle specimens for structural analysis.

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.

Femtosecond X-ray coherent diffraction of aligned amyloid fibrils on low background graphene.

Here we present a new approach to diffraction imaging of amyloid fibrils, combining a free-standing graphene support and single nanofocused X-ray pulses of femtosecond duration from an X-ray free-electron laser. Due to the very low background scattering from the graphene support and mutual alignment of filaments, diffraction from tobacco mosaic virus (TMV) filaments and amyloid protofibrils is obtained to 2.7 Å and 2.4 Å resolution in single diffraction patterns, respectively. Some TMV diffraction patterns exhibit asymmetry that indicates the presence of a limited number of axial rotations in the XFEL focus. Signal-to-noise levels from individual diffraction patterns are enhanced using computational alignment and merging, giving patterns that are superior to those obtainable from synchrotron radiation sources. We anticipate that our approach will be a starting point for further investigations into unsolved structures of filaments and other weakly scattering objects.

The phase problem for one-dimensional crystals.

The phase problem for diffraction amplitudes measured from a one-dimensional crystal is examined. In the absence of any a priori information, the solution to this problem is shown to be unique up to a parameterized, low-dimensional set of solutions. Minimal additional a priori information is expected to render the solution unique. The effects of additional information such as positivity, molecular envelope and helical symmetry on uniqueness are characterized. The results are pertinent to structural studies of polymeric and rod-like biomolecular assemblies that form one-dimensional, rather than three-dimensional, crystals. This shows the potential for ab initio phasing of diffraction data from single such assemblies measured using new X-ray free-electron laser sources. Such an approach would circumvent the complicated inversion of cylindrically averaged diffraction that is necessary in traditional X-ray fibre diffraction analysis.

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.

Analysis of XFEL serial diffraction data from individual crystalline fibrils.

Serial diffraction data collected at the Linac Coherent Light Source from crystalline amyloid fibrils delivered in a liquid jet show that the fibrils are well oriented in the jet. At low fibril concentrations, diffraction patterns are recorded from single fibrils; these patterns are weak and contain only a few reflections. Methods are developed for determining the orientation of patterns in reciprocal space and merging them in three dimensions. This allows the individual structure amplitudes to be calculated, thus overcoming the limitations of orientation and cylindrical averaging in conventional fibre diffraction analysis. The advantages of this technique should allow structural studies of fibrous systems in biology that are inaccessible using existing techniques.

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.

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.

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.