Small-angle Scattering Data Research Articles

Combined Scheme of Reconstruction of the Particle Size Distribution Function Using Small-Angle Scattering Data

To analyze polydisperse systems of nanoparticles, the particle size distribution function can be determined from small-angle X-ray and neutron scattering data using some algorithms. The corresponding least squares problem is an ill-posed problem; correspondingly, a solution can strongly depend on the initial approximation and the parameters of search algorithms. A procedure based on solving a system of linear equations with regularization gives a stable solution, but it is not free of artifacts. Other algorithms that seek the distribution function both in an analytical form and in the form of a nonparametric histogram provide solutions that can depend on the search parameters and the initial approximation. In this work, a combined scheme of using these algorithms has been proposed to increase the stability of the solution and, thereby, its reliability.

Small Angle Scattering Data Analysis Assisted by Machine Learning Methods

ABSTRACTSmall angle scattering (SAS) is a widely used technique for characterizing structures of wide ranges of materials. For such wide ranges of applications of SAS, there exist a large number of ways to model the scattering data. While such analysis models are often available from various suites of SAS data analysis software packages, selecting the right model to start with poses a big challenge for beginners to SAS data analysis. Here, we present machine learning (ML) methods that can assist users by suggesting scattering models for data analysis. A series of one-dimensional scattering curves have been generated by using different models to train the algorithms. The performance of the ML method is studied for various types of ML algorithms, resolution of the dataset, and the number of the dataset. The degree of similarities among selected scattering models is presented in terms of the confusion matrix. The scattering model suggestions with prediction scores provide a list of scattering models that are likely to succeed. Therefore, if implemented with extensive libraries of scattering models, this method can speed up the data analysis workflow by reducing search spaces for appropriate scattering models.

SasPDF: pair distribution function analysis of nanoparticle assemblies from small-angle scattering data.

sasPDF, a method for characterizing the structure of nanoparticle assemblies (NPAs), is presented. The method is an extension of the atomic pair distribution function (PDF) analysis to the small-angle scattering (SAS) regime. The PDFgetS3 software package for computing the PDF from SAS data is also presented. An application of the sasPDF method to characterize structures of representative NPA samples with different levels of structural order is then demonstrated. The sasPDF method quantitatively yields information such as structure, disorder and crystallite sizes of ordered NPA samples. The method was also used to successfully model the data from a disordered NPA sample. The sasPDF method offers the possibility of more quantitative characterizations of NPA structures for a wide class of samples.

Classifying and analyzing small-angle scattering data using weightedknearest neighbors machine learning techniques

A consistent challenge for both new and expert practitioners of small-angle scattering (SAS) lies in determining how to analyze the data, given the limited information content of said data and the large number of models that can be employed. Machine learning (ML) methods are powerful tools for classifying data that have found diverse applications in many fields of science. Here, ML methods are applied to the problem of classifying SAS data for the most appropriate model to use for data analysis. The approach employed is built around the method of weightedknearest neighbors (wKNN), and utilizes a subset of the models implemented in theSasViewpackage (https://www.sasview.org/) for generating a well defined set of training and testing data. The prediction rate of the wKNN method implemented here using a subset ofSasViewmodels is reasonably good for many of the models, but has difficulty with others, notably those based on spherical structures. A novel expansion of the wKNN method was also developed, which uses Gaussian processes to produce local surrogate models for the classification, and this significantly improves the classification accuracy. Further, by integrating a stochastic gradient descent method during post-processing, it is possible to leverage the local surrogate model both to classify the SAS data with high accuracy and to predict the structural parameters that best describe the data. The linking of data classification and model fitting has the potential to facilitate the translation of measured data into results for both novice and expert practitioners of SAS.

A robust expectation-maximization method for the interpretation of small-angle scattering data from dense nanoparticle samples

The local monodisperse approximation (LMA) is a two-parameter model commonly employed for the retrieval of size distributions from the small-angle scattering (SAS) patterns obtained from dense nanoparticle samples (e.g. dry powders and concentrated solutions). This work features a novel implementation of the LMA model resolution for the inverse scattering problem. The method is based on the expectation-maximization iterative algorithm and is free of any fine-tuning of model parameters. The application of this method to SAS data acquired under laboratory conditions from dense nanoparticle samples is shown to provide good results.

BEES: Bayesian Ensemble Estimation from SAS

Many biomolecular complexes exist in a flexible ensemble of states in solution that is necessary to perform their biological function. Small-angle scattering (SAS) measurements are a popular method for characterizing these flexible molecules because of their relative ease of use and their ability to simultaneously probe the full ensemble of states. However, SAS data is typically low dimensional and difficult to interpret without the assistance of additional structural models. In theory, experimental SAS curves can be reconstituted from a linear combination of theoretical models, although this procedure carries a significant risk of overfitting the inherently low-dimensional SAS data. Previously, we developed a Bayesian-based method for fitting ensembles of model structures to experimental SAS data that rigorously avoids overfitting. However, we have found that these methods can be difficult to incorporate into typical SAS modeling workflows, especially for users that are not experts in computational modeling. To this end, we present the Bayesian Ensemble Estimation from SAS (BEES) program. Two forks of BEES are available, the primary one existing as a module for the SASSIE web server and a developmental version that is a stand-alone Python program. BEES allows users to exhaustively sample ensemble models constructed from a library of theoretical states and to interactively analyze and compare each model’s performance. The fitting routine also allows for secondary data sets to be supplied, thereby simultaneously fitting models to both SAS data as well as orthogonal information. The flexible ensemble of K63-linked ubiquitin trimers is presented as an example of BEES’ capabilities.

Reliable Biomolecular Structural Modelling with Small-Angle Scattering

The small-angle scattering (SAS) of X-rays or neutrons from biomolecules in solution can provide (1) unique structural insights, (2) information on the range of potential conformations present, (3) and powerful restraints for integrative/hybrid structural modelling (IHM). The rotational and ensemble averaging of the signal presents multiple challenges to the reliable utilisation of SAS data: (1) demonstrating SAS profile represents the species of interest, (2) reliably evaluating model fits, and (3) over-interpretation and over-fitting of data. A community driven project has been underway for more than a decade now to define the information that should be made available regarding sample quality, data acquisition and reduction, presentation of scattering data and validation, and modelling for biomolecular SAS experiments. These guidelines (Trewhella et al. (2017) Acta D73, 710) provide a reporting framework so that “readers can independently assess the quality of the data and the basis for any interpretations presented” concerning the results of bimolecular SAS experiments, including the generation or testing of 3D models. The validation task forces for SAS (Trewhella et al. (2013) Structure 21, 875) and for IHM (Sali et al. (2015) Structure 23, 1156) of the world-wide Protein Data Bank are additionally addressing the complex issues concerning the archiving and validation of IHM models that depend upon computational methods and disparate data from multiple techniques (Burley et al. (2017) Structure 25, 1317). Finally, the question of how many parameters are justified by the model evidence inherent in the SAS profile is an important question to answer when modelling conformational ensembles (Potrzebowski, Trewhella and Andr⌈ (2018) in review PLOS Computational Biology). This presentation will address the recommended reporting framework and its rationale, the reliable incorporation of SAS data in IHM, and the strengths and limits of SAS in ensemble modelling.

Bayesian inference of protein conformational ensembles from limited structural data.

Many proteins consist of folded domains connected by regions with higher flexibility. The details of the resulting conformational ensemble play a central role in controlling interactions between domains and with binding partners. Small-Angle Scattering (SAS) is well-suited to study the conformational states adopted by proteins in solution. However, analysis is complicated by the limited information content in SAS data and care must be taken to avoid constructing overly complex ensemble models and fitting to noise in the experimental data. To address these challenges, we developed a method based on Bayesian statistics that infers conformational ensembles from a structural library generated by all-atom Monte Carlo simulations. The first stage of the method involves a fast model selection based on variational Bayesian inference that maximizes the model evidence of the selected ensemble. This is followed by a complete Bayesian inference of population weights in the selected ensemble. Experiments with simulated ensembles demonstrate that model evidence is capable of identifying the correct ensemble and that correct number of ensemble members can be recovered up to high level of noise. Using experimental data, we demonstrate how the method can be extended to include data from Nuclear Magnetic Resonance (NMR) and structural energies of conformers extracted from the all-atom energy functions. We show that the data from SAXS, NMR chemical shifts and energies calculated from conformers can work synergistically to improve the definition of the conformational ensemble.

A Study of Beryllium-Based Materials and Comparison of Their X-Ray Homogeneities According to Small-Angle Scattering Data

A criterion for estimating the homogeneity of materials for refractive X-ray optics from small-angle X-ray scattering data is proposed. A material of new type—nanoberyllium—is considered. A number of samples of beryllium materials, including commercial products, have been analyzed. The X-ray homogeneity of the materials is determined, as well as the influence of technological processes.

Analysis of small-angle scattering data using model fitting and Bayesian regularization

The structure of macromolecules can be studied by small-angle scattering (SAS), but as this is an ill-posed problem, prior knowledge about the sample must be included in the analysis. Regularization methods are used for this purpose, as already implemented in indirect Fourier transformation and bead-modeling-based analysis of SAS data, but not yet in the analysis of SAS data with analytical form factors. To fill this gap, a Bayesian regularization method was implemented, where the prior information was quantified as probability distributions for the model parameters and included via a functional S. The quantity Q = χ2 + αS was then minimized and the value of the regularization parameter α determined by probability maximization. The method was tested on small-angle X-ray scattering data from a sample of nanodiscs and a sample of micelles. The parameters refined with the Bayesian regularization method were closer to the prior values as compared with conventional χ2 minimization. Moreover, the errors on the refined parameters were generally smaller, owing to the inclusion of prior information. The Bayesian method stabilized the refined values of the fitted model upon addition of noise and can thus be used to retrieve information from data with low signal-to-noise ratio without risk of overfitting. Finally, the method provides a measure for the information content in data, N g, which represents the effective number of retrievable parameters, taking into account the imposed prior knowledge as well as the noise level in data.

Recent advances in the UltraScan SOlution MOdeller(US-SOMO) hydrodynamic and small-angle scattering data analysis and simulation suite.

The UltraScan SOlution MOdeller (US-SOMO) is a comprehensive, public domain, open-source suite of computer programs centred on hydrodynamic modelling and small-angle scattering (SAS) data analysis and simulation. We describe here the advances that have been implemented since its last official release (#3087, 2017), which are available from release #3141 for Windows, Linux and Mac operating systems. A major effort has been the transition from the legacy Qt3 cross platform software development and user interface library to the modern Qt5 release. Apart from improved graphical support, this has allowed the direct implementation of the newest, almost two-orders of magnitude faster version of the ZENO hydrodynamic computation algorithm for all operating systems. Coupled with the SoMo-generated bead models with overlaps, ZENO provides the most accurate translational friction computations from atomic-level structures available (Rocco and Byron Eur Biophys J 44:417-431, 2015a), with computational times comparable with or faster than those of other methods. In addition, it has allowed us to introduce the direct representation of each atom in a structure as a (hydrated) bead, opening interesting new modelling possibilities. In the small-angle scattering (SAS) part of the suite, an indirect Fourier transform Bayesian algorithm has been implemented for the computation of the pairwise distance distribution function from SAS data. Finally, the SAS HPLC module, recently upgraded with improved baseline correction and Gaussian decomposition of not baseline-resolved peaks and with advanced statistical evaluation tools (Brookes et al. J Appl Cryst 49:1827-1841, 2016), now allows automatic top-peak frame selection and averaging.

Ambiguities and completeness of SAS data analysis: investigations of apoferritin by SAXS/SANS EID and SEC-SAXS methods

The method of small angle scattering (SAS) is widely used in the field of biophysical research of proteins in aqueous solutions. Obtaining low-resolution structure of proteins is still a highly valuable method despite the advances in high-resolution methods such as X-ray diffraction, cryo-EM etc. SAS offers the unique possibility to obtain structural information under conditions close to those of functional assays, i.e. in solution, without different additives, in the mg/mL concentration range. SAS method has a long history, but there are still many uncertainties related to data treatment. We compared 1D SAS profiles of apoferritin obtained by X-ray diffraction (XRD) and SAS methods. It is shown that SAS curves for X-ray diffraction crystallographic structure of apoferritin differ more significantly than it might be expected due to the resolution of the SAS instrument. Extrapolation to infinite dilution (EID) method does not sufficiently exclude dimerization and oligomerization effects and therefore could not guarantee total absence of dimers account in the final SAS curve. In this study, we show that EID SAXS, EID SANS and SEC-SAXS methods give complementary results and when they are used all together, it allows obtaining the most accurate results and high confidence from SAS data analysis of proteins.

Small Angle Scattering and Structural Biology: Data Quality and Model Validation.

This chapter provides a brief review of the current state-of-the-art in small-angle scattering (SAS) from biomolecules in solution in regard to: (1) sample preparation and instrumentation, (2) data reduction and analysis, and (3) three-dimensional structural modelling and validation. In this context, areas of ongoing research in regard to the interpretation of SAS data will be discussed with a particular focus on structural modelling using computational methods and data from different experimental techniques, including SAS (hybrid methods). Finally, progress made in establishing community accepted publication guidelines and a standard reporting framework that includes SAS data deposition in a public data bank will be described. Importantly, SAS data with associated meta-data can now be held in a format that supports exchange between data archives and seamless interoperability with the world-wide Protein Data Bank (wwPDB). Biomolecular SAS is thus well positioned to contribute to an envisioned federation of data archives in support of hybrid structural biology.

Characterization of Monoclonal Antibody-Protein Antigen Complexes Using Small-Angle Scattering and Molecular Modeling.

The determination of monoclonal antibody interactions with protein antigens in solution can lead to important insights guiding physical characterization and molecular engineering of therapeutic targets. We used small-angle scattering (SAS) combined with size-exclusion multi-angle light scattering high-performance liquid chromatography to obtain monodisperse samples with defined stoichiometry to study an anti-streptavidin monoclonal antibody interacting with tetrameric streptavidin. Ensembles of structures with both monodentate and bidentate antibody–antigen complexes were generated using molecular docking protocols and molecular simulations. By comparing theoretical SAS profiles to the experimental data it was determined that the primary component(s) were compact monodentate and/or bidentate complexes. SAS profiles of extended monodentate complexes were not consistent with the experimental data. These results highlight the capability for determining the shape of monoclonal antibody–antigen complexes in solution using SAS data and physics-based molecular modeling.

2017 publication guidelines for structural modelling of small-angle scattering data from biomolecules in solution: an update.

In 2012, preliminary guidelines were published addressing sample quality, data acquisition and reduction, presentation of scattering data and validation, and modelling for biomolecular small-angle scattering (SAS) experiments. Biomolecular SAS has since continued to grow and authors have increasingly adopted the preliminary guidelines. In parallel, integrative/hybrid determination of biomolecular structures is a rapidly growing field that is expanding the scope of structural biology. For SAS to contribute maximally to this field, it is essential to ensure open access to the information required for evaluation of the quality of SAS samples and data, as well as the validity of SAS-based structural models. To this end, the preliminary guidelines for data presentation in a publication are reviewed and updated, and the deposition of data and associated models in a public archive is recommended. These guidelines and recommendations have been prepared in consultation with the members of the International Union of Crystallography (IUCr) Small-Angle Scattering and Journals Commissions, the Worldwide Protein Data Bank (wwPDB) Small-Angle Scattering Validation Task Force and additional experts in the field.

Reduction of small-angle scattering profiles to finite sets of structural invariants.

This paper shows how small-angle scattering (SAS) curves can be decomposed in a simple sum using a set of invariant parameters called Kn which are related to the shape of the object of study. These Kn, together with a radius R, give a complete theoretical description of the SAS curve. Adding an overall constant, these parameters are easily fitted against experimental data giving a concise comprehensive description of the data. The pair distance distribution function is also entirely described by this invariant set and the Dmax parameter can be measured. In addition to the understanding they bring, these invariants can be used to reliably estimate structural moments beyond the radius of gyration, thereby rigorously expanding the actual set of model-free quantities one can extract from experimental SAS data, and possibly paving the way to designing new shape reconstruction strategies.

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake.

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, correlations can be built in the mass fractal amplitudes, which explains the decay of the scattering intensity I(q) ∼ qDs-4, with 1 < Ds < 2 being the fractal dimension of the perimeter. The curve I(q)q4-Ds is found to be log-periodic in the fractal region with a period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of the sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of the Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

Hybrid Applications of Solution Scattering to Aid Structural Biology.

Biomolecular applications of solution X-ray and neutron scattering (SAXS and SANS, respectively) started in late 1960s - early 1970s but were relatively limited in their ability to provide a detailed structural picture and lagged behind what became the two primary methods of experimental structural biology - X-ray crystallography and NMR. However, improvements in both data analysis and instrumentation led to an explosive growth in the number of studies that used small-angle scattering (SAS) for investigation of macromolecular structure, often in combination with other biophysical techniques. Such hybrid applications are nowadays quickly becoming a norm whenever scattering data are used for two reasons. First, it is generally accepted that SAS data on their own cannot lead to a uniquely defined high-resolution structural model, creating a need for supplementing them with information from complementary techniques. Second, solution scattering data are frequently applied in situations when a method such NMR or X-ray crystallography cannot provide a satisfactory structural picture, which makes these additional restraints highly desirable. Maturation of the hybrid bio-SAS approaches brings to light new questions including completeness of the conformational space sampling, model validation, and data compatibility.

Convex and concave successions of power-law decays in small-angle scattering

The small-angle scattering (SAS) structure factor from a new model of a 3D deterministic fractal in which the relative positions and the number of structural units vary with fractal iteration number is calculated. It is shown that, depending on the relative positions of scattering units inside the fractal, we can obtain various types of power-law successions, such as: convex/concave - when the absolute value of the scattering exponent of the first power-law decay is higher/smaller than that of the subsequent power- law decay, or any combination of them (i.e. convex-concave or concave-convex). The obtained results can explain experimental SAS (neutron or X-rays) data which are characterized by a succession of power-law decays of arbitrary length.

Extension of the sasCIF format and its applications for data processing and deposition.

Recent advances in small-angle scattering (SAS) experimental facilities and data analysis methods have prompted a dramatic increase in the number of users and of projects conducted, causing an upsurge in the number of objects studied, experimental data available and structural models generated. To organize the data and models and make them accessible to the community, the Task Forces on SAS and hybrid methods for the International Union of Crystallography and the Worldwide Protein Data Bank envisage developing a federated approach to SAS data and model archiving. Within the framework of this approach, the existing databases may exchange information and provide independent but synchronized entries to users. At present, ways of exchanging information between the various SAS databases are not established, leading to possible duplication and incompatibility of entries, and limiting the opportunities for data-driven research for SAS users. In this work, a solution is developed to resolve these issues and provide a universal exchange format for the community, based on the use of the widely adopted crystallographic information framework (CIF). The previous version of the sasCIF format, implemented as an extension of the core CIF dictionary, has been available since 2000 to facilitate SAS data exchange between laboratories. The sasCIF format has now been extended to describe comprehensively the necessary experimental information, results and models, including relevant metadata for SAS data analysis and for deposition into a database. Processing tools for these files (sasCIFtools) have been developed, and these are available both as standalone open-source programs and integrated into the SAS Biological Data Bank, allowing the export and import of data entries as sasCIF files. Software modules to save the relevant information directly from beamline data-processing pipelines in sasCIF format are also developed. This update of sasCIF and the relevant tools are an important step in the standardization of the way SAS data are presented and exchanged, to make the results easily accessible to users and to promote further the application of SAS in the structural biology community.