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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.

Methods are described for processing XFEL data from individual crystalline fibrils. The methods are applied to data collected at the Linac Coherent Light Source from an amyloid-forming oligopeptide from the adenovirus shaft., 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.

The application of a new generation of x-ray sources called X-ray Free Electron Lasers (XFELs) to diffractive imaging has allowed structural studies of specimens not previously accessible. Specimens of reduced crystallinity are of particular interest, including fibrous nano-crystals and single fibrous molecules. Diffractive imaging experiments using XFELs generate large datasets of diffraction frames from specimens with random, unknown orientations. The orientation of each diffraction frame needs to be determined from features in the pattern in order to register and merge the dataset for subsequent structural analysis. Certain sample delivery techniques simplify this process by limiting the range of orientations a specimen may take. In this paper we consider two sample delivery techniques: a liquid jet and a fixed target on a silicon wafer. Orientations determined from diffraction patterns from each delivery method are classified in order to investigate the type of orientation present. This information also helps to characterize the quality of sample preparations and provides feedback valuable for designing future experiments.

The application of high-powered X-ray Free Electron Lasers (XFELs) in Serial Femtosecond Crystallography (SFX) has led to an increasing number of crystal structure determinations. Achieving the same success for systems of diminished crystallinity, such as fibrous systems and single particles, is inhibited by low signal strength and difficulties in orientating the data. We present methods developed for analyzing serial diffraction data from fibrous systems. The data are processed to identify cases where a single XFEL pulse intersects a single fibril. The fibril orientations are then determined by analysis of detected features within the diffraction data. With the fibril orientation determined, serial diffraction data is merged in 3D reciprocal space, allowing the individual structure amplitudes to be calculated. This allows structural studies of previously inaccessible fibrous systems to be performed, and represents a step closer towards the long-term goal of imaging single particles.

X-ray free-electron lasers are new x-ray sources that are revolutionising high resolution molecular imaging. These sources produce extremely bright and extremely brief x-ray pulses that can overcome limitations of crystal growth and radiation damage in traditional protein x-ray crystallography. They introduce a new paradigm in diffraction imaging in which the specimen is destroyed, but not before the diffraction data are collected, and that requires radically new forms of specimen delivery and data processing. An overview of biomolecular imaging using x-ray free-electron lasers is provided, as well as future prospects.

Serial femtosecond nanocrystallography (SFX) is a form of x-ray coherent diffraction imaging that utilises a stream of tiny nanocrystals of the biological assembly under study, in contrast to the larger crystals used in conventional x-ray crystallography using conventional x-ray synchrotron x-ray sources. Nanocrystallography utilises the extremely brief and intense x-ray pulses that are obtained from an x-ray free-electron laser (XFEL). A key advantage is that some biological macromolecules, such as membrane proteins for example, do not easily form large crystals, but spontaneously form nanocrystals. There is therefore an opportunity for structure determination for biological molecules that are inaccessible using conventional x-ray crystallography. Nanocrystallography introduces a number of interesting image reconstruction problems. Weak diffraction patterns are recorded from hundreds of thousands of nancocrystals in unknown orientations, and these data have to be assembled and merged into a 3D intensity dataset. The diffracted intensities can also be affected by the surface structure of the crystals that can contain incomplete unit cells. Furthermore, the small crystal size means that there is potentially access to diffraction information between the crystalline Bragg peaks. With this information, phase retrieval is possible without resorting to the collection of additional experimental data as is necessary in conventional protein crystallography. We report recent work on the diffraction characteristics of nanocrystals and the resulting reconstruction algorithms.

A number of aspects of the two-point correlation behavior of the nearest-neighbor, triangular Ising antiferromagnet are studied using a combination of numerical evaluation of exact expressions and Monte Carlo simulation. Existing asymptotic results for on-axis correlations at finite temperatures are evaluated and shown to be of limited accuracy. The sublattice structure of the off-axis correlation function is clarified, and rotational invariance is studied as a function of temperature. Separations and temperatures for which the correlation function is significant are identified, and a simple functional expression is developed that allows accurate calculation of the correlation function in this region.

Muscular force is generated by molecular interactions between the contractile proteins actin and myosin. The myosin filaments in the sarcomere of vertebrate muscle pack on a triangular array into which the actin filaments are interdigitated. High resolution studies of the actin-myosin interactions are performed by x-ray fiber diffraction analysis of whole muscle fibers. In most vertebrate muscles however, the myosin filaments pack in a so-called “superlattice” arrangement that involves a semi-random distribution of two filament rotations. The unknown effects of this disorder on diffraction by muscle fibres have so far prevented a rigorous analysis of x-ray fiber diffraction patterns in terms of the structure of this complex system. We report a quantitative model of the disorder and its incorporation into calculations of x-ray fiber diffraction patterns from model structures. This allows rapid calculation of the diffraction and does not involve numerical averaging over the disorder. Calculations show that the disorder modulates the Bragg reflections in diffraction patterns and introduces diffuse diffraction. The results of this analysis will allow the effects of the disorder to be included in muscle structure refinement programs, allowing more accurate structure determination from x-ray fiber diffraction data.

An algorithm is presented for processing and analysis of differential interference contrast (DIC) microscopy images of the fovea to study the cone mosaic. The algorithm automatically locates the cones and their boundaries in such images and is assessed by comparison with results from manual analysis. Additional algorithms are presented that analyze the cone positions to extract information on cone neighbor relationships as well as the short-range order and domain structure of the mosaic. The methods are applied to DIC images of the human fovea.