Diffuse scattering resulting from macromolecular frustration

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.

Stabilization of Polar Nanoregions in Pb-free Ferroelectrics

The formation of polar nanoregions through solid-solution additions is known to enhance significantly the functional properties of ferroelectric materials. Despite considerable progress in characterizing the microscopic behavior of polar nanoregions (PNR), understanding their real-space atomic structure and dynamics of their formation remains a considerable challenge. Here, using the method of dynamic pair distribution function, we provide direct insights into the role of solid-solution additions towards the stabilization of polar nanoregions in the Pb-free ferroelectric of Ba(Zr,Ti)O_{3}. It is shown that for an optimum level of substitution of Ti by larger Zr ions, the dynamics of atomic displacements for ferroelectric polarization are slowed sufficiently below THz frequencies, which leads to increased local correlation among dipoles within PNRs. The dynamic pair distribution function technique demonstrates a unique capability to obtain insights into locally correlated atomic dynamics in disordered materials, including new Pb-free ferroelectrics, which is necessary to understand and control their functional properties.

Continuous diffraction of molecules and disordered molecular crystals

The intensities of far-field diffraction patterns of orientationally aligned molecules obey Wilson statistics, whether those molecules are in isolation (giving rise to a continuous diffraction pattern) or arranged in a crystal (giving rise to Bragg peaks). Ensembles of molecules in several orientations, but uncorrelated in position, give rise to the incoherent sum of the diffraction from those objects, modifying the statistics in a similar way as crystal twinning modifies the distribution of Bragg intensities. This situation arises in the continuous diffraction of laser-aligned molecules or translationally disordered molecular crystals. This paper develops the analysis of the intensity statistics of such continuous diffraction to obtain parameters such as scaling, beam coherence and the number of contributing independent object orientations. When measured, continuous molecular diffraction is generally weak and accompanied by a background that far exceeds the strength of the signal. Instead of just relying upon the smallest measured intensities or their mean value to guide the subtraction of the background, it is shown how all measured values can be utilized to estimate the background, noise and signal, by employing a modified 'noisy Wilson' distribution that explicitly includes the background. Parameters relating to the background and signal quantities can be estimated from the moments of the measured intensities. The analysis method is demonstrated on previously published continuous diffraction data measured from crystals of photosystem II [Ayyer et al. (2016 ▸), Nature, 530, 202-206].

X-ray Scattering Studies of Protein Structural Dynamics

X-ray scattering is uniquely suited to the study of disordered systems and thus has the potential to provide insight into dynamic processes where diffraction methods fail. In particular, while X-ray crystallography has been a staple of structural biology for more than half a century and will continue to remain so, a major limitation of this technique has been the lack of dynamic information. Solution X-ray scattering has become an invaluable tool in structural and mechanistic studies of biological macromolecules where large conformational changes are involved. Such systems include allosteric enzymes that play key roles in directing metabolic fluxes of biochemical pathways, as well as large, assembly-line type enzymes that synthesize secondary metabolites with pharmaceutical applications. Furthermore, crystallography has the potential to provide information on protein dynamics via the diffuse scattering patterns that are overlaid with Bragg diffraction. Historically, these patterns have been very difficult to interpret, but recent advances in X-ray detection have led to a renewed interest in diffuse scattering analysis as a way to probe correlated motions. Here, we will review X-ray scattering theory and highlight recent advances in scattering-based investigations of protein solutions and crystals, with a particular focus on complex enzymes.

Mesoscale Polarization by Geometric Frustration in Columnar Supramolecular Crystals

Columnar supramolecular phases with polarization along the columnar axis have potential for the development of ultrahigh‐density memories as every single column might function as a memory element. By investigating structure and disorder for four columnar benzene‐1,3,5‐trisamides by total X‐ray scattering and DFT calculations, we demonstrate that the column orientation, and thus the columnar dipole moment, is receptive to geometric frustration if the columns aggregate in a hexagonal rod packing. The frustration suppresses conventional antiferroelectric order and heightens the sensitivity towards collective intercolumnar packing effects. The latter finding allows for the building up of mesoscale domains with spontaneous polarization. Our results suggest how the complex interplay between steric and electrostatic interactions is influenced by a straightforward chemical design of the molecular synthons to create spontaneous polarization and to adjust mesoscale domain size.

Measuring and modeling diffuse scattering in protein X-ray crystallography

X-ray diffraction has the potential to provide rich information about the structural dynamics of macromolecules. To realize this potential, both Bragg scattering, which is currently used to derive macromolecular structures, and diffuse scattering, which reports on correlations in charge density variations, must be measured. Until now, measurement of diffuse scattering from protein crystals has been scarce because of the extra effort of collecting diffuse data. Here, we present 3D measurements of diffuse intensity collected from crystals of the enzymes cyclophilin A and trypsin. The measurements were obtained from the same X-ray diffraction images as the Bragg data, using best practices for standard data collection. To model the underlying dynamics in a practical way that could be used during structure refinement, we tested translation-libration-screw (TLS), liquid-like motions (LLM), and coarse-grained normal-modes (NM) models of protein motions. The LLM model provides a global picture of motions and was refined against the diffuse data, whereas the TLS and NM models provide more detailed and distinct descriptions of atom displacements, and only used information from the Bragg data. Whereas different TLS groupings yielded similar Bragg intensities, they yielded different diffuse intensities, none of which agreed well with the data. In contrast, both the LLM and NM models agreed substantially with the diffuse data. These results demonstrate a realistic path to increase the number of diffuse datasets available to the wider biosciences community and indicate that dynamics-inspired NM structural models can simultaneously agree with both Bragg and diffuse scattering.

Macromolecular diffractive imaging using imperfect crystals

The three-dimensional structures of macromolecules and their complexes are predominantly elucidated by X-ray protein crystallography. A major limitation is access to high-quality crystals, to ensure X-ray diffraction extends to sufficiently large scattering angles and hence yields sufficiently high-resolution information that the crystal structure can be solved. The observation that crystals with shrunken unit-cell volumes and tighter macromolecular packing often produce higher-resolution Bragg peaks^1,2 hints that crystallographic resolution for some macromolecules may be limited not by their heterogeneity but rather by a deviation of strict positional ordering of the crystalline lattice. Such displacements of molecules from the ideal lattice give rise to a continuous diffraction pattern, equal to the incoherent sum of diffraction from rigid single molecular complexes aligned along several discrete crystallographic orientations and hence with an increased information content³. Although such continuous diffraction patterns have long been observed—and are of interest as a source of information about the dynamics of proteins⁴ —they have not been used for structure determination. Here we show for crystals of the integral membrane protein complex photosystem II that lattice disorder increases the information content and the resolution of the diffraction pattern well beyond the 4.5 Å limit of measurable Bragg peaks, which allows us to directly phase⁵ the pattern. With the molecular envelope conventionally determined at 4.5 Å as a constraint, we then obtain a static image of the photosystem II dimer at 3.5 Å resolution. This result shows that continuous diffraction can be used to overcome long-supposed resolution limits of macromolecular crystallography, with a method that puts great value in commonly encountered imperfect crystals and opens up the possibility for model-free phasing^6,7.

Acoustic vibrations contribute to the diffuse scatter produced by ribosome crystals

The diffuse scattering pattern produced by frozen crystals of the 70S ribosome from Thermus thermophilus is as highly structured as it would be if it resulted entirely from domain-scale motions within these particles. However, the qualitative properties of the scattering pattern suggest that acoustic displacements of the crystal lattice make a major contribution to it.

Experimental uncertainties of three-dimensional pair distribution function investigations exemplified on the diffuse scattering from a tris-tert-butyl-1,3,5-benzene tricarboxamide single crystal

Diffuse scattering from a substitutionally disordered tris-tert-butyl-1,3,5-benzene tricarboxamide single crystal is analyzed with the three-dimensional difference pair distribution function (3D-ΔPDF) method. The real structure of the crystal is shown to consist of infinite polar molecular stacks along thecaxis, which are laterally packed in a hexagonal fashion. The orientation of the stacks is disordered, but neighboring stacks strongly prefer antiparallel arrangements. Quantitative orientational pair correlation coefficients are determined for all lateral pairs separated by less than 100 Å. A careful analysis of the factors influencing the accuracy of the 3D-ΔPDF refinement is presented. It is shown that the effect of statistical errors is small compared to systematic errors coming from diffraction geometry distortions, reciprocal space resolution or incompletely corrected background. Various strategies for identifying and decreasing systematic errors are discussed. The impact of the systematic errors on the uncertainty of the results is not just specific for 3D-ΔPDF investigations but also relevant for other quantitative diffuse scattering modeling techniques.

Diffuse scattering and partial disorder in complex structures

The study of single-crystal diffuse scattering (SCDS) goes back almost to the beginnings of X-ray crystallography. Because SCDS arises from two-body correlations, it contains information about local (short-range) ordering in the sample, information which is often crucial in the attempt to relate structure to function. This review discusses the state of the field, including detectors and data collection and the modelling of SCDS using Monte Carlo and ab initio techniques. High-quality, three-dimensional volumes of SCDS data can now be collected at synchrotron light sources, allowing ever more detailed and quantitative analyses to be undertaken, and opening the way to approaches such as three-dimensional pair distribution function studies (3D-PDF) and automated refinement of a disorder model, powerful techniques that require large volumes of low-noise data.

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.

Understanding the structure details when drying hydrate crystals of pharmaceuticals – interpretations from diffuse scattering and inter-modulation satellites of a partially dehydrated crystal

Simplified models for the crystal lattice of the sesquihydrate form of the hemi-sulfate salt of (5S,6S,9R)-5-amino-6-(2,3-difluorophenyl)-6,7,8,9-tetrahydro-5H-cyclohepta[b]pyridin-9-yl 4-(2-oxo-2,3-dihydro-1H-imidazol[4,5b]pyridin-1-yl)-1-piperidine carboxylate (BMS-927711, C28H29F2N6O3+) are used to calculate diffuse diffraction features in order to develop a mechanistic understanding of the dehydration process with respect to disruption of the lattice, since a Bragg model cannot be established. The model demonstrates that what we observe when the water leaves the crystal is partial transformation from the parent form to a child form (a new form, less hydrated and structurally related to the parent). Yet this `dried' structure is not a pure phase. It consists of semi-random layers of both child, parent and an interfacial layer which has a modulated structure that represents a transitory phase. Understanding the fact that a single `dried' crystal can have the disordered layer structure described as well as understanding mechanistic relationships between the phases involved can have implications in understanding the effect of common large scale bulk drying procedures. During the development of BMS-927711, difficulties did arise during characterization of the dried bulk when using only routine solid-state analysis. The material is now better understood from this diffraction study. The diffraction experiments also reveal intermodulation satellites, which upon interpretation yield even more structural information about the crystal transformation. The model suggests the mechanism of transformation is laminar in which layers of the crystal are driven to approach a stableB-centered supercell phase of lower water content.

Continuous X-ray diffractive field in protein nanocrystallography

The recent development of X-ray free-electron laser sources has created new opportunities for the structural analysis of protein nanocrystals. The extremely small sizes of the crystals, as well as imperfections of the crystal structure, result in an interference phenomenon in the diffraction pattern. With decreasing crystallite size the structural imperfections play a role in the formation of the diffraction pattern that is comparable in importance to the size effects and should be taken into account during the data analysis and structure reconstruction processes. There now exists a need to develop new methods of protein structure determination that do not depend on the availability of good-quality crystals and that can treat proteins under conditions close to the active form. This paper demonstrates an approach that is specifically tailored to nanocrystalline samples and offers a unique crystallographic solution.