100 years of Debye's scattering equation

Advances, Applications, and Perspectives in Small-Angle X-ray Scattering of RNA

RNAs exhibit a plethora of functions far beyond transmitting genetic information. Often, RNA functions are entailed in their structure, be it as a regulatory switch, protein binding site, or providing catalytic activity. Structural information is a prerequisite for a full understanding of RNA-regulatory mechanisms. Owing to the inherent dynamics, size, and instability of RNA, its structure determination remains challenging. Methods such as NMR spectroscopy, X-ray crystallography, and cryo-electron microscopy can provide high-resolution structures; however, their limitations make structure determination, even for small RNAs, cumbersome, if at all possible. Although at a low resolution, small-angle X-ray scattering (SAXS) has proven valuable in advancing structure determination of RNAs as a complementary method, which is also applicable to large-sized RNAs. Here, we review the technological and methodological advancements of RNA SAXS. We provide examples of the powerful inclusion of SAXS in structural biology and discuss possible future applications to large RNAs.

Multiphonon scattering formula of dynamic structure factors for classical Debye solids

A semianalytic formula of the dynamic structure factor S(k,ω) for classical Debye solids over the entire wave-number (k) and frequency (ω) range is constructed by taking into account multiphonon thermal diffuse scattering up to infinite order. The formula adopts Gaussian approximations to the spatial and time decay of the multiphonon part of the displacement correlation function. Numerical illustrations for isotropic polycrystals reveal that, as k increases, sharp peaks due to one-phonon normal scattering in the hydrodynamic regime (k→0) are replaced by diffuse spectra consisting of umklapp scattering and multiphonon continuum; approach toward the ideal-gas spectra in the large-k limit is proven from analytic properties of the multiphonon term. When k coincides with a Bragg reflection point, total thermal diffuse scattering S_{TDS}(k,ω) exhibits a 1/ω divergence as ω→0, which in turn gives rise to a logarithmic enhancement of the corresponding static structure factor S_{TDS}(k). Overall accuracy of the theory is confirmed through the exact zeroth-order frequency-moment sum rule between S_{TDS}(k,ω) and S_{TDS}(k); agreement with the second-order sum rule is shown to be satisfactory except for the vicinity of the Debye cutoff region.

Atomistic origin of nano-silver paracrystalline structure: molecular dynamics and x-ray diffraction studies

Classical molecular dynamics (MD) and x-ray diffraction (XRD) have been used to establish the origin of the paracrystalline structure of silver nanoparticles at the atomic scale. Models based on the face-centred cubic structure have been computer generated and their atomic arrangements have been optimized by the MD with the embedded-atom model (EAM) potential and its modified version (MEAM). The simulation results are compared with the experimental XRD data in reciprocal and real spaces, i.e. the structure factor and the pair distribution function. The applied approach returns the structural models, defined by the Cartesian coordinates of the constituent atoms. It has been found that most of the structural features of Ag nanoparticles are better reproduced by the MEAM. The presence of vacancy defects in the structure of the Ag nanoparticles has been considered and the average concentration of vacancies is estimated to be 3 at.%. The average nearest-neighbour Ag-Ag distances and the coordination numbers are determined and compared with the values predicted for the bulk Ag, demonstrating a different degree of structural disorder on the surface and in the core, compared to the bulk crystalline counterpart. It has been shown that the paracrystalline structure of the Ag nanoparticles has origin in the surface disorder and the disorder generated by the presence of the vacancy defects. Both sources lead to network distortion that propagates proportionally to the square root of the interatomic distances.

Computational diffraction reveals long-range strains, distortions and disorder in molecular dynamics simulations of irradiated single crystals

Atomic-scale simulations, and in particular molecular dynamics (MD), are key assets to model the behavior of the structure of materials under the action of external stimuli, say temperature, strain or stress, irradiation, etc. Despite the widespread use of MD in condensed matter science, some basic material characteristics remain difficult to determine. This is, for instance, the case for the long-range strain tensor, and its root-mean-squared fluctuations, in disordered materials. In this work, computational diffraction is introduced as a fast and reliable structural characterization tool of atomic-scale simulation cells in the case of irradiated single crystals. In contrast to direct-space methods, computational diffraction operates in the reciprocal space and is therefore highly sensitive to long-range spatial correlations. With the example of irradiated UO2 single crystals, it is demonstrated that the normal strains, shear strains and rotations, as well as their root-mean-squared fluctuations (microstrain) and the atomic disorder, are straightforwardly and unambiguously determined. The methodology presented here has been developed with efficiency in mind, in order to be able to provide simple and reliable characterizations either operating in real time, in parallel with other analysis tools, or operating on very large data sets.

Superhelical DNA liquid crystals from dendrimer-induced DNA compaction

Electrostatic compaction of double stranded DNA induced by a positively charged poly(amidoamine) (PAMAM) dendrimer of generation four (G4) was found to produce two unique types of DNA mesophases, in which the DNA bent into superhelices packed in a tetragonal or hexagonal lattice. The structure formed at a lower dendrimer charge density was three-dimensionally (3D) ordered, as characterized by the P4₁2₁2 space group with a 4₁ screw axis in a tetragonal arrangement, showing that the weakly bent DNA superhelices with a pitch length of ca. 5.0 nm possessed both identical handedness and phase conservation. The 3D ordered structure transformed into a 2D mesophase at a higher dendrimer charge density, wherein the strongly bent superhelices with a pitch length of ca. 4.0 nm organized in a hexagonal lattice without lateral coherence of helical trajectory. The counterion valency of the protonic acid that is used to charge the dendrimer was found to influence the phase diagram. Under a given dendrimer charge density, the complex with a multivalent acid-protonated dendrimer tended to form structures with less curved DNA, attesting that the driving force of charge matching was reduced by increasing the counterion valency of the dendrimer.

Unifying the concepts of scattering and structure factor in ordered and disordered samples

Scattering methods are widely used in many research areas to analyze and resolve material structures. Given its importance, a large number of textbooks are devoted to this topic. However, technical details in experiments and disconnection between explanations from different perspectives often confuse and frustrate beginner students and researchers. To create an effective learning path, the core concepts of scattering and structure factor are reviewed in this article in a self-contained way. Classical examples of scattering photography and intensity scanning are calculated. Sample CPU and GPU codes are provided to facilitate the understanding and application of these methods.

Combining scattering analysis and atomistic simulation of wood-water interactions

Molecular-scale interactions between water and cellulose microfibril bundles in plant cell walls are not fully understood, despite their crucial role for many applications of plant biomass. Recent advances in X-ray and neutron scattering analysis allow more accurate interpretation of experimental data from wood cell walls. At the same time, microfibril bundles including hemicelluloses and water can be modelled at atomistic resolution. Computing scattering patterns from atomistic models enables a new, complementary approach to decipher some of the most fundamental questions at this level of the hierarchical cell wall structure. This article introduces studies related to moisture behavior of wood with small/wide-angle X-ray/neutron scattering and atomistic simulations, recent attempts to combine these two approaches, and perspectives and open questions for future research using this powerful combination. Finally, we discuss the opportunities of the combined method in relation to applications of lignocellulosic materials.

Extending the Debye scattering equation for diffraction from a cylindrically averaged group of atoms: detecting molecular orientation at an interface

The Debye scattering equation is now over 100 years old and has been widely used to interpret diffraction patterns from randomly oriented groups of atoms. The present work develops and applies a related equation that calculates diffraction intensity from groups of atoms randomly oriented about a fixed axis, a scenario that occurs when molecules are oriented at an interface by the presentation of a binding motif as in antibody binding. Using an example biomolecule, the high level of sensitivity of the diffraction pattern to the orientation of the molecule and to the direction of the incident beam is shown. The use of the method is proposed not only for determining the orientation of molecules in biosensors and at membrane interfaces, but also for determining molecular conformation without the need for crystallization.

Structure analysis of supported disordered molybdenum oxides using pair distribution function analysis and automated cluster modelling

Molybdenum oxides and sulfides on various low-cost high-surface-area supports are excellent catalysts for several industrially relevant reactions. The surface layer structure of these materials is, however, difficult to characterize due to small and disordered MoO x domains. Here, it is shown how X-ray total scattering can be applied to gain insights into the structure through differential pair distribution function (d-PDF) analysis, where the scattering signal from the support material is subtracted to obtain structural information on the supported structure. MoO x catalysts supported on alumina nanoparticles and on zeolites are investigated, and it is shown that the structure of the hydrated molybdenum oxide layer is closely related to that of disordered and polydisperse polyoxometalates. By analysing the PDFs with a large number of automatically generated cluster structures, which are constructed in an iterative manner from known polyoxometalate clusters, information is derived on the structural motifs in supported MoO x .

Simulating the diffraction line profile from nanocrystalline powders using a spherical harmonics expansion

An accurate description of the diffraction line profile from nanocrystalline powders can be obtained by a spherical harmonics expansion of the profile function. The procedure outlined in this work is found to be computationally efficient and applicable to the line profile for any crystallite shape and size. Practical examples of the diffraction pattern peak profiles resulting from cubic crystallites between 1 and 100 nm in size are shown.