Classification of continuous diffraction patterns: A numerical study

Single-Image Super-Resolution Improvement of X-ray Single-Particle Diffraction Images Using a Convolutional Neural Network

Femtosecond X-ray pulse lasers are promising probes for the elucidation of the multiconformational states of biomolecules because they enable snapshots of single biomolecules to be observed as coherent diffraction images. Multi-image processing using an X-ray free-electron laser has proven to be a successful structural analysis method for viruses. However, the performance of single-particle analysis (SPA) for flexible biomolecules with sizes ≤100 nm remains difficult. Owing to the multiconformational states of biomolecules and noisy character of diffraction images, diffraction image improvement by multi-image processing is often ineffective for such molecules. Herein, a single-image super-resolution (SR) model was constructed using an SR convolutional neural network (SRCNN). Data preparation was performed in silico to consider the actual observation situation with unknown molecular orientations and the fluctuation of molecular structure and incident X-ray intensity. It was demonstrated that the trained SRCNN model improved the single-particle diffraction image quality, corresponding to an observed image with an incident X-ray intensity (approximately three to seven times higher than the original X-ray intensity), while retaining the individuality of the diffraction images. The feasibility of SPA for flexible biomolecules with sizes ≤100 nm was dramatically increased by introducing the SRCNN improvement at the beginning of the various structural analysis schemes.

Start-to-end simulation of single-particle imaging using ultra-short pulses at the European X-ray Free-Electron Laser

Single-particle imaging with X-ray free-electron lasers (XFELs) has the potential to provide structural information at atomic resolution for non-crystalline biomolecules. This potential exists because ultra-short intense pulses can produce interpretable diffraction data notwithstanding radiation damage. This paper explores the impact of pulse duration on the interpretability of diffraction data using comprehensive and realistic simulations of an imaging experiment at the European X-ray Free-Electron Laser. It is found that the optimal pulse duration for molecules with a few thousand atoms at 5 keV lies between 3 and 9 fs.

A minimal view of single-particle imaging with X-ray lasers

The ability to serially interrogate single biomolecules with femtosecond X-ray pulses from free-electron lasers has ushered in the possibility of determining the three-dimensional structure of biomolecules without crystallization. However, the complexity of imaging a sample's structure from very many of its noisy and incomplete diffraction data can be daunting. In this review, we introduce a simple analogue of this imaging workflow, use it to describe a structure reconstruction algorithm based on the expectation maximization principle, and consider the effects of extraneous noise. Such a minimal model can aid experiment and algorithm design in future studies.

Effect of two-particle correlations on x-ray coherent diffractive imaging studies performed with continuum models

Coherent diffraction imaging (CDI) of single molecules at atomic resolution is a major goal for the x-ray free-electron lasers (XFELs). However, during an imaging pulse, the fast laser-induced ionization may strongly affect the recorded diffraction pattern of the irradiated sample. The radiation tolerance of the imaged molecule should then be investigated a priori with a dedicated simulation tool. The continuum approach is a powerful tool for modeling the evolution of irradiated large systems consisting of more than a few hundred thousand atoms. However, this method follows the evolution of average single-particle densities, and the experimentally recorded intensities reflect the spatial two-particle correlations. The information on these correlations is then inherently not accessible within the continuum approach. In this paper we analyze this limitation of continuum models and discuss the applicability of continuum models for imaging studies. We derive a formula to calculate scattered intensities (including both elastic and inelastic scattering) from the estimates obtained with a single-particle continuum model under conditions typical for CDI studies with XFELs. We demonstrate through numerical simulations that it describes the scattered signal with good accuracy. Two-particle correlation effects manifest themselves only in the region of low momentum transfers, together with the effects of the finite size of the sample. We also show that inelastic scattering on bound electrons can have a significant impact on the measured intensities: it contributes to the background that reduces the contrast of the recorded image. This effect is even more pronounced at larger momentum transfers. Therefore, whereas inelastic scattering can be neglected for nanocrystals, where Bragg scattering dominates, and in experiments imaging single objects at low resolution, it should be taken into account when planning atomic resolution imaging of nonperiodic samples. Finally, we show the effect of the electronic damage on the recorded total signal. Progressing damage does not change the positions of intensity peaks that correspond to the (fixed) positions of imaged ions. It only changes the contrast between intensity minima and maxima, which reduces the image contrast. Our results have implications for imaging-oriented studies of radiation damage performed with continuum models, as they define the limits of applicability of these models for CDI simulations.

Classifying and assembling two-dimensional X-ray laser diffraction patterns of a single particle to reconstruct the three-dimensional diffraction intensity function: resolution limit due to the quantum noise

A new two-step algorithm is developed for reconstructing the three-dimensional diffraction intensity of a globular biological macromolecule from many experimentally measured quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The first step is classification of the two-dimensional patterns into groups according to the similarity of direction of the incident X-rays with respect to the molecule and an averaging within each group to reduce the noise. The second step is detection of common intersecting circles between the signal-enhanced two-dimensional patterns to identify their mutual location in the three-dimensional wavenumber space. The newly developed algorithm enables one to detect a signal for classification in noisy experimental photon-count data with as low as ~0.1 photons per effective pixel. The wavenumber of such a limiting pixel determines the attainable structural resolution. From this fact, the resolution limit due to the quantum noise attainable by this new method of analysis as well as two important experimental parameters, the number of two-dimensional patterns to be measured (the load for the detector) and the number of pairs of two-dimensional patterns to be analysed (the load for the computer), are derived as a function of the incident X-ray intensity and quantities characterizing the target molecule.

Phase retrieval from single biomolecule diffraction pattern

In this paper, we propose the SPR (sparse phase retrieval) method, which is a new phase retrieval method for coherent x-ray diffraction imaging (CXDI). Conventional phase retrieval methods effectively solve the problem for high signal-to-noise ratio measurements, but would not be sufficient for single biomolecular imaging which is expected to be realized with femto-second x-ray free electron laser pulses. The SPR method is based on the Bayesian statistics. It does not need to set the object boundary constraint that is required by the commonly used hybrid input-output (HIO) method, instead a prior distribution is defined with an exponential distribution and used for the estimation. Simulation results demonstrate that the proposed method reconstructs the electron density under a noisy condition even some central pixels are masked.

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.

In-plane rotation classification for coherent X-ray imaging of single biomolecules

We report a new classification scheme with computation complexity well within the capacity of a PC for coherent X-ray imaging of single biomolecules. In contrast to current methods, which are based on data from large scattering angles, we propose to classify the orientations of the biomolecule using data from small angle scattering, where the signals are relatively strong. Further we integrate data to form radial and azimuthal distributions of the scattering pattern to reduce the variance caused by the shot noise. Classification based on these two distributions are shown to successfully recognize not only the patterns from molecules of the same orientation but also those that differ by an in-plane rotation.

Sacrificial tamper slows down sample explosion in FLASH diffraction experiments

Intense and ultrashort x-ray pulses from free-electron lasers open up the possibility for near-atomic resolution imaging without the need for crystallization. Such experiments require high photon fluences and pulses shorter than the time to destroy the sample. We describe results with a new femtosecond pump-probe diffraction technique employing coherent 0.1 keV x rays from the FLASH soft x-ray free-electron laser. We show that the lifetime of a nanostructured sample can be extended to several picoseconds by a tamper layer to dampen and quench the sample explosion, making <1 nm resolution imaging feasible.

Structural variability and the incoherent addition of scattered intensities in single-particle diffraction

X-ray lasers may allow structural studies on single particles and biomolecules without crystalline periodicity in the samples. We examine here the effect of sample dynamics as a source of structural heterogeneity on the resolution of the reconstructed image of a small protein molecule. Structures from molecular-dynamics simulations of lysozyme were sampled and aligned. These structures were then used to calculate diffraction patterns corresponding to different dynamic states. The patterns were incoherently summed and the resulting data set was phased using the oversampling method. Reconstructed images of hydrated and dehydrated lysozyme gave resolutions of 3.7 A and 7.6 A , respectively. These are significantly worse than the root-mean-square deviation of the hydrated ( 2.7 A for all atoms and 1.45 A for C-alpha positions) or dehydrated ( 3.7 A for all atoms and 2.5 A for C-alpha positions) structures. The noise introduced by structural dynamics and incoherent addition of dissimilar structures restricts the maximum resolution to be expected from direct image reconstruction of dynamic systems. A way of potentially reducing this effect is by grouping dynamic structures into distinct structural substates and solving them separately.

Reconstruction algorithm for single-particle diffraction imaging experiments

We introduce the EMC algorithm for reconstructing a particle's three-dimensional (3D) diffraction intensity from very many photon shot-noise limited two-dimensional measurements, when the particle orientation in each measurement is unknown. The algorithm combines a maximization step (M) of the intensity's likelihood function, with expansion (E) and compression (C) steps that map the 3D intensity model to a redundant tomographic representation and back again. After a few iterations of the EMC update rule, the reconstructed intensity is given to the difference-map algorithm for reconstruction of the particle contrast. We demonstrate reconstructions with simulated data and investigate the effects of particle complexity, number of measurements, and the number of photons per measurement. The relatively transparent scaling behavior of our algorithm provides an estimate of the data processing resources required for future single-particle imaging experiments.

Coherent-pulse 2D crystallography using a free-electron laser x-ray source

Coherent diffractive imaging for the reconstruction of a two-dimensional (2D) finite crystal structure with a single pulse train of free-electron laser radiation at 7.97 nm wavelength is demonstrated. This measurement shows an advance on traditional coherent imaging techniques by applying it to a periodic structure. It is also significant that this approach paves the way for the imaging of the class of specimens which readily form 2D, but not three-dimensional crystals. We show that the structure is reconstructed to the detected resolution, given an adequate signal-to-noise ratio.

Feasibility of imaging living cells at subnanometer resolutions by ultrafast X-ray diffraction

Detailed structural investigations on living cells are problematic because existing structural methods cannot reach high resolutions on non-reproducible objects. Illumination with an ultrashort and extremely bright X-ray pulse can outrun key damage processes over a very short period. This can be exploited to extend the diffraction signal to the highest possible resolution in flash diffraction experiments. Here we present an analysis of the interaction of a very intense and very short X-ray pulse with a living cell, using a non-equilibrium population kinetics plasma code with radiation transfer. Each element in the evolving plasma is modeled by numerous states to monitor changes in the atomic populations as a function of pulse length, wavelength, and fluence. The model treats photoionization, impact ionization, Auger decay, recombination, and inverse bremsstrahlung by solving rate equations in a self-consistent manner and describes hydrodynamic expansion through the ion sound speed. The results show that subnanometer resolutions could be reached on micron-sized cells in a diffraction-limited geometry at wavelengths between 0·75 and 1·5 nm and at fluences of 1011–1012 photons μm−2in less than 10 fs. Subnanometer resolutions could also be achieved with harder X-rays at higher fluences. We discuss experimental and computational strategies to obtain depth information about the object in flash diffraction experiments.