Continuous Diffraction Patterns Research Articles

Single Particle Study by X-Ray Diffraction: Crystallographic Approach

An increase in the power of X-ray sources, in particular, the commissioning of X-ray free electron lasers, opens up the possibility of recording the diffraction by single macromolecular biological particles. These opportunities open the way to weakening, and, ideally, removal, of the main limitation of X-ray structure analysis, namely, the need to prepare the sample in the single-crystal form. However, the possibility of the practical recording of diffraction by a single particle is currently limited to a very low-resolution zone, what is one of the main obstacles in the development of this approach. This paper discusses the similarities and differences in the study of crystal samples and single particles. It is shown that the problem of the determination of the structure of a single particle can be formulated as a standard problem of biological crystallography, namely, as the problem of retrieval the electron density distribution in some unit cell from the magnitudes of its Fourier coefficients. This makes it possible to apply the entire range of the methods of biological crystallography to the study of isolated particles. At the same time, the possibility of recording continuous diffraction pattern for a single particle (as opposed to a discrete set of Bragg reflections in the case of a crystal) significantly increases the amount of information derived from the experiment. The analytical properties of the continuous diffraction pattern create a potential opportunity both to restore the structure factors phases (lost in the diffraction experiment), and to extrapolate the experimentally observed pattern onto a wider area, which allows to increase the resolution of Fourier syntheses for the electron density distribution.

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].

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 peaks1,2 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 phasing6,7.

Controlled angular redirection of light via nanoimprinted disordered gratings

Enhanced control of diffraction through transparent substrates is achieved via disordered gratings in a silica sol-gel film. Tailoring the degree of disorder allows tuning of the diffractive behavior from discrete orders into broad distributions over large angular range. Gratings of optical quality are formed by silica sol-gel nanoimprint lithography and an optical setup for the measurement of continuous diffraction patterns is presented. Sound agreement is found between measurements and simulation, validating both the approach for redirection of light and the fabrication process. The disordered gratings are presented in the context of improved interior daylighting and may furthermore be suited to a wide variety of applications where controlled angular redirection of light is desired.

Continuous diffraction patterns from circular arrays of carbon nanotubes

We report the remarkable diffraction effects produced from circular patterned arrays of multiwalled carbon nanotubes (MWCNTs). Highly ordered circular arrays of multiwalled carbon nanotubes (with inter-nanotube spacings of 633 nm) display optical dispersion effects similar to compact discs. These arrays display remarkable diffraction patterns in the far field which are spatially continuous. High quality diffraction patterns were obtained experimentally which are in excellent agreement with the theoretical calculations. The achieved continuous diffraction patterns pave the way towards the utilization of engineered carbon nanotube arrays in applications like three dimensional holograms.

Emerging opportunities in structural biology with X-ray free-electron lasers.

X-ray free-electron lasers (X-FELs) produce X-ray pulses with extremely brilliant peak intensity and ultrashort pulse duration. It has been proposed that radiation damage can be 'outrun' by using an ultra intense and short X-FEL pulse that passes a biological sample before the onset of significant radiation damage. The concept of 'diffraction-before-destruction' has been demonstrated recently at the Linac Coherent Light Source, the first operational hard X-ray FEL, for protein nanocrystals and giant virus particles. The continuous diffraction patterns from single particles allow solving the classical 'phase problem' by the oversampling method with iterative algorithms. If enough data are collected from many identical copies of a (biological) particle, its three-dimensional structure can be reconstructed. We review the current status and future prospects of serial femtosecond crystallography (SFX) and single-particle coherent diffraction imaging (CDI) with X-FELs.

Noise-robust coherent diffractive imaging with a single diffraction pattern

The resolution of single-shot coherent diffractive imaging at X-ray free-electron laser facilities is limited by the low signal-to-noise level of diffraction data at high scattering angles. The iterative reconstruction methods, which phase a continuous diffraction pattern to produce an image, must be able to extract information from these weak signals to obtain the best quality images. Here we show how to modify iterative reconstruction methods to improve tolerance to noise. The method is demonstrated with the hybrid input-output method on both simulated data and single-shot diffraction patterns taken at the Linac Coherent Light Source.

Common arc method for diffraction pattern orientation

Very short pulses of X-ray free-electron lasers opened the way to obtaining diffraction signal from single particles beyond the radiation dose limit. For three-dimensional structure reconstruction many patterns are recorded in the object's unknown orientation. A method is described for the orientation of continuous diffraction patterns of non-periodic objects, utilizing intensity correlations in the curved intersections of the corresponding Ewald spheres, and hence named the common arc orientation method. The present implementation of the algorithm optionally takes into account Friedel's law, handles missing data and is capable of determining the point group of symmetric objects. Its performance is demonstrated on simulated diffraction data sets and verification of the results indicates a high orientation accuracy even at low signal levels. The common arc method fills a gap in the wide palette of orientation methods.

Single mimivirus particles intercepted and imaged with an X-ray laser

X-ray lasers offer new capabilities in understanding the structure of biological systems, complex materials and matter under extreme conditions. Very short and extremely bright, coherent X-ray pulses can be used to outrun key damage processes and obtain a single diffraction pattern from a large macromolecule, a virus or a cell before the sample explodes and turns into plasma. The continuous diffraction pattern of non-crystalline objects permits oversampling and direct phase retrieval. Here we show that high-quality diffraction data can be obtained with a single X-ray pulse from a non-crystalline biological sample, a single mimivirus particle, which was injected into the pulsed beam of a hard-X-ray free-electron laser, the Linac Coherent Light Source. Calculations indicate that the energy deposited into the virus by the pulse heated the particle to over 100,000 K after the pulse had left the sample. The reconstructed exit wavefront (image) yielded 32-nm full-period resolution in a single exposure and showed no measurable damage. The reconstruction indicates inhomogeneous arrangement of dense material inside the virion. We expect that significantly higher resolutions will be achieved in such experiments with shorter and brighter photon pulses focused to a smaller area. The resolution in such experiments can be further extended for samples available in multiple identical copies.

Hawk: the image reconstruction package for coherent X-ray diffractive imaging

The past few years have seen a tremendous growth in the field of coherent X-ray diffractive imaging, in large part due to X-ray free-electron lasers which provide a peak brilliance billions of times higher than that of synchrotrons. However, this rapid development in terms of hardware has not been matched on the software side. The release ofHawkis intended to close this gap. To the authors' knowledgeHawkis the first publicly available and fully open source software program for reconstructing images from continuous diffraction patterns. The software handles all steps leading from a raw diffraction pattern to a reconstructed two-dimensional image including geometry determination, background correction, masking and phasing. It also includes preliminary three-dimensional support and support for graphics processing units using the Compute Unified Device Architecture, which speeds up processing by orders of magnitude compared to a single central processing unit.Hawkimplements numerous algorithms and is easily extended. This, in combination with its open-source licence, provides a platform for other groups to test, develop and distribute their own algorithms.Hawkis available under the GNU General Public License from http://xray.bmc.uu.se/hawk.

Reconstruction from a single diffraction pattern of azimuthally projected electron density of molecules aligned parallel to a single axis

Diffraction from the individual molecules of a molecular beam, aligned parallel to a single axis by a strong electric field or other means, has been proposed as a means of structure determination of individual molecules. As in fiber diffraction, all the information extractable is contained in a diffraction pattern from incidence of the diffracting beam normal to the molecular alignment axis. The limited size of the object results in continuous diffraction patterns characterized by neither Bragg spots nor layer lines. Equations relating the scattered amplitudes to the molecular electron density may be conveniently formulated in terms of cylindrical harmonics. For simulated diffraction patterns from short C nanotubes aligned along their axes, iterative solution of the equation for the zeroth-order cylindrical harmonic and its inverse with appropriate constraints in real and reciprocal space enables the phasing of the measured amplitudes, and hence a reconstruction of the azimuthal projection of the molecule.

Classification of continuous diffraction patterns: A numerical study

The very intense and short pulses of future X-ray free electron lasers may allow the atomic resolution imaging of small, non-periodic objects. Preliminary estimates show that images obtained from single pulses do not contain statistically enough photons to allow successful reconstruction. Therefore multiple exposures of randomly oriented identical replicas have to be taken and the individual images have to be classified according to the object’s orientation. The classification has been analytically treated by Huldt et al. [Huldt, G., Szoke, A., Hajdu, J., 2003. J. Struct. Biol. 144, 219.]. In this paper we extend the analytical results with numerical model calculations. This allows us to simulate realistic situations, which we will face in real experiments. We find significant deviations from the analytical expectations, even in the ideal case of spherical particles with random atomic distributions. We introduce a new norm for the individual scattering patterns and describe a criterion to select images belonging to similar orientation, which makes the classification more reliable in practice. We also discuss the effects of particle shape and size, partial orientational ordering, the measurement’s resolution and the charge error caused by the Coulomb explosion.

Three-dimensional mapping of a deformation field inside a nanocrystal

Coherent X-ray diffraction imaging is a rapidly advancing form of microscopy: diffraction patterns, measured using the latest third-generation synchrotron radiation sources, can be inverted to obtain full three-dimensional images of the interior density within nanocrystals. Diffraction from an ideal crystal lattice results in an identical copy of this continuous diffraction pattern at every Bragg peak. This symmetry is broken by the presence of strain fields, which arise from the epitaxial contact forces that are inevitable whenever nanocrystals are prepared on a substrate. When strain is present, the diffraction copies at different Bragg peaks are no longer identical and contain additional information, appearing as broken local inversion symmetry about each Bragg point. Here we show that one such pattern can nevertheless be inverted to obtain a 'complex' crystal density, whose phase encodes a projection of the lattice deformation. A lead nanocrystal was crystallized in ultrahigh vacuum from a droplet on a silica substrate and equilibrated close to its melting point. A three-dimensional image of the density, obtained by inversion of the coherent X-ray diffraction, shows the expected facetted morphology, but in addition reveals a real-space phase that is consistent with the three-dimensional evolution of a deformation field arising from interfacial contact forces. Quantitative three-dimensional imaging of lattice strain on the nanometre scale will have profound consequences for our fundamental understanding of grain interactions and defects in crystalline materials. Our method of measuring and inverting diffraction patterns from nanocrystals represents a vital step towards the ultimate goal of atomic resolution single-molecule imaging that is a prominent justification for development of X-ray free-electron lasers.

Three-dimensional data merging of randomly oriented continuous diffraction patterns

Three-dimensional data merging of randomly oriented continuous diffraction patterns

Phasing diffraction data from a stream of hydrated proteins

We consider the problem of phase determination for continuous diffraction patterns obtained from a beam of identical, aligned large molecules (such as proteins), each coated with a layer of water or vitreous ice. Many laser-aligned molecules are assumed to lie within a wide continuous x-ray beam at any instant. An iterative phasing method is developed to extract the common target structure in three dimensions from diffraction patterns of these doped ice balls. Several measurements of the diffraction intensity in reciprocal space are needed. We found iteratively two boundaries (supports) (between protein and ice and the outer iceball support) by using the charge-flipping and multiple hybrid input-output algorithms, working with multiple sets of measured data. The approach is applied to simulated data from hydrated lysozyme proteins generated by the serial crystallography method of laser-aligned protein-beam diffraction proposed by Spence and Doak [Phys. Rev. Lett. 92, 198102 (2004)]. We consider also the effect of empty ice balls on the patterns. The algorithm can also be used to align images with different randomly chosen origins, so that the same embedded subunits overlap.

Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations

The quadratic phase function is fundamental in describing and computing wave-propagation-related phenomena under the Fresnel approximation; it is also frequently used in many signal processing algorithms. This function has interesting properties and Fourier transform relations. For example, the Fourier transform of the sampled chirp is also a sampled chirp for some sampling rates. These properties are essential in interpreting the aliasing and its effects as a consequence of sampling of the quadratic phase function, and lead to interesting and efficient algorithms to simulate Fresnel diffraction. For example, it is possible to construct discrete Fourier transform (DFT)-based algorithms to compute exact continuous Fresnel diffraction patterns of continuous, not necessarily bandlimited, periodic masks at some specific distances.

Three-dimensional imaging of microstructure in Au nanocrystals.

X-ray diffraction using a coherent beam involves the mutual interference among all the extremities of small crystals. The continuous diffraction pattern so produced can be phased because it can be oversampled. We have thus obtained three-dimensional images of the interiors of Au nanocrystals that show 50 nm wide bands of contrast with [111] orientation that probably arise from internal twinning by dynamic recrystallization during their formation at high temperature.

Use of coherent X-ray diffraction to map strain fields in nanocrystals

A new method is proposed to analyze the internal strains inside nanometer-sized crystals. The method employs a coherent beam of X-rays to obtain a continuous diffraction pattern in the immediate vicinity of each of the Bragg reflections. The symmetric part of the diffraction is given by the Fourier transform of the crystal’s shape, while the asymmetric part can be associated with the strain. Iterative Fourier transform methods can then be applied to reveal the strain as a three-dimensional spatial image.

Molecular structures from femtosecond x-ray pulses

In this issue of PNAS Miao, Hodgson, and Sayre (1) validate a method of three-dimensional structure determination for individual biological macromolecules. Their proposal, based on suggestions by Hajdu and colleagues (2), is supported by careful computer simulation using plausible assumptions, and rests on recent advances in a number of related fields. In brief, they adopt the idea that femtosecond (fsec) x-ray pulses generated by free electron, laser-based x-ray sources will allow the two-dimensional (2D) diffraction patterns (DPs) of individual molecules to be recorded (2). Appropriate registration and averaging of these patterns will allow the full three-dimensional DP of the molecule to be assembled into an effectively continuous function. Previous work by Sayre and colleagues (3, 4) and others (5) has shown that, for objects that are everywhere positive and that are confined to an envelope of known size, the phases of the continuous diffraction pattern can be recovered. Thus, an image of the molecule can be formed by Fourier transformation of the phased diffraction pattern. The quality of the images produced in the simulation is quite high, and generated a burst of pleasure and optimism in this commentator.

Average unit-cell approach to diffraction on Thue–Morse sequence and decorated quasicrystals

The average unit cell approach has been successfully used to construct the probability distributions of atomic displacements from the reference lattice points and to calculate the diffraction patterns for two aperiodic structures — the Thue–Morse sequence and decorated Fibonacci chains. For the singular continuous diffraction pattern of the Thue–Morse sequence fractal exponents and multiplication factors have been found for scaling with the number of atoms. General method able to distinguish between various quasilattice decorations has been elaborated and tested for decorated Fibonacci chains.