The effects of thermal disorder on the solution-scattering profiles of macromolecules

Fitting high-resolution electron density maps from atomic models to solution scattering data

Solution scattering techniques, such as small- and wide-angle X-ray scattering (SWAXS), provide valuable insights into the structure and dynamics of biological macromolecules in solution. In this study, we present an approach to accurately predict solution X-ray scattering profiles at wide angles from atomic models by generating high-resolution electron density maps. Our method accounts for the excluded volume of bulk solvent by calculating unique adjusted atomic volumes directly from the atomic coordinates. This approach eliminates the need for one of the free fitting parameters commonly used in existing algorithms, resulting in improved accuracy of the calculated SWAXS profile. An implicit model of the hydration shell is generated that uses the form factor of water. Two parameters, namely the bulk solvent density and the mean hydration shell contrast, are adjusted to best fit the data. Results using eight publicly available SWAXS profiles show high-quality fits to the data. In each case, the optimized parameter values show small adjustments demonstrating that the default values are close to the true solution. Disabling parameter optimization produces significantly more accurate predicted scattering profiles compared to the leading software. The algorithm is computationally efficient, comparable to the leading software and up to 10 times faster for large molecules. The algorithm is encoded in a command line script called denss.pdb2mrc.py and is available open source as part of the DENSS v1.7.0 software package. In addition to improving the ability to compare atomic models to experimental SWAXS data, these developments pave the way for increasing the accuracy of modeling algorithms using SWAXS data and decreasing the risk of overfitting.

Fitting high-resolution electron density maps from atomic models to solution scattering data

Solution scattering techniques, such as small and wide-angle X-ray scattering (SWAXS), provide valuable insights into the structure and dynamics of biological macromolecules in solution. In this study, we present an approach to accurately predict solution X-ray scattering profiles at wide angles from atomic models by generating high-resolution electron density maps. Our method accounts for the excluded volume of bulk solvent by calculating unique adjusted atomic volumes directly from the atomic coordinates. This approach eliminates the need for a free fitting parameter commonly used in existing algorithms, resulting in improved accuracy of the calculated SWAXS profile. An implicit model of the hydration shell is generated which uses the form factor of water. Two parameters, namely the bulk solvent density and the mean hydration shell contrast, are adjusted to best fit the data. Results using eight publicly available SWAXS profiles show high quality fits to the data. In each case, the optimized parameter values show small adjustments demonstrating that the default values are close to the true solution. Disabling parameter optimization results in a significant improvement of the calculated scattering profiles compared to the leading software. The algorithm is computationally efficient, showing more than tenfold reduction in execution time compared to the leading software. The algorithm is encoded in a command line script called denss.pdb2mrc.py and is available open source as part of the DENSS v1.7.0 software package (https://github.com/tdgrant1/denss). In addition to improving the ability to compare atomic models to experimental SWAXS data, these developments pave the way for increasing the accuracy of modeling algorithms utilizing SWAXS data while decreasing the risk of overfitting.

Structure and ensemble refinement against SAXS data: Combining MD simulations with Bayesian inference or with the maximum entropy principle

Small-angle X-ray scattering (SAXS) is a powerful method for tracking conformational transitions of proteins or soft-matter complexes in solution. However, the interpretation of the experimental data is challenged by the low spatial resolution and the low information content of the data, which lead to a high risk of overinterpreting the data. Here, we illustrate how SAXS data can be integrated into all-atom molecular dynamics (MD) simulation to derive atomic structures or heterogeneous ensembles that are compatible with the data. Besides providing atomistic insight, the MD simulation adds physicochemical information, as encoded in the MD force fields, which greatly reduces the risk of overinterpretation. We present an introduction into the theory of SAXS-driven MD simulations as implemented in GROMACS-SWAXS, a modified version of the GROMACS simulation software. We discuss SAXS-driven parallel-replica ensemble refinement with commitment to the maximum entropy principle as well as a Bayesian formulation of SAXS-driven structure refinement. Practical considerations for running and interpreting the simulations are presented. The methods are freely available via GitLab at https://gitlab.com/cbjh/gromacs-swaxs.

Refining conformational ensembles of flexible proteins against small-angle x-ray scattering data

Intrinsically disordered proteins and flexible regions in multidomain proteins display substantial conformational heterogeneity. Characterizing the conformational ensembles of these proteins in solution typically requires combining one or more biophysical techniques with computational modeling or simulations. Experimental data can either be used to assess the accuracy of a computational model or to refine the computational model to get a better agreement with the experimental data. In both cases, one generally needs a so-called forward model (i.e., an algorithm to calculate experimental observables from individual conformations or ensembles). In many cases, this involves one or more parameters that need to be set, and it is not always trivial to determine the optimal values or to understand the impact on the choice of parameters. For example, in the case of small-angle x-ray scattering (SAXS) experiments, many forward models include parameters that describe the contribution of the hydration layer and displaced solvent to the background-subtracted experimental data. Often, one also needs to fit a scale factor and a constant background for the SAXS data but across the entire ensemble. Here, we present a protocol to dissect the effect of the free parameters on the calculated SAXS intensities and to identify a reliable set of values. We have implemented this procedure in our Bayesian/maximum entropy framework for ensemble refinement and demonstrate the results on four intrinsically disordered proteins and a protein with three domains connected by flexible linkers. Our results show that the resulting ensembles can depend on the parameters used for solvent effects and suggest that these should be chosen carefully. We also find a set of parameters that work robustly across all proteins.

Evolving SAXS versatility: solution X-ray scattering for macromolecular architecture, functional landscapes, and integrative structural biology

Small-angle X-ray scattering (SAXS) has emerged as an enabling integrative technique for comprehensive analyses of macromolecular structures and interactions in solution. Over the past two decades, SAXS has become a mainstay of the structural biologist’s toolbox, supplying multiplexed measurements of molecular shape and dynamics that unveil biological function. Here, we discuss evolving SAXS theory, methods, and applications that extend the field of small-angle scattering beyond simple shape characterization. SAXS, coupled with size-exclusion chromatography (SEC-SAXS) and time-resolved (TR-SAXS) methods, is now providing high-resolution insight into macromolecular flexibility and ensembles, delineating biophysical landscapes, and facilitating high-throughput library screening to assess macromolecular properties and to create opportunities for drug discovery. Looking forward, we consider SAXS in the integrative era of hybrid structural biology methods, its potential for illuminating cellular supramolecular and mesoscale structures, and its capacity to complement high-throughput bioinformatics sequencing data. As advances in the field continue, we look forward to proliferating uses of SAXS based upon its abilities to robustly produce mechanistic insights for biology and medicine.

Predicting X-ray solution scattering from flexible macromolecules

Wide-angle X-ray solution scattering (WAXS) patterns contain substantial information about the structure and dynamics of a protein. Solution scattering from a rigid protein can be predicted from atomic coordinate sets to within experimental error. However, structural fluctuations of proteins in solution can lead to significant changes in the observed intensities. The magnitude and form of these changes contain information about the nature and spatial extent of structural fluctuations in the protein. Molecular dynamics (MD) simulations based on a crystal structure and selected force field generate models for protein internal motions, and here we demonstrate that they can be used to predict the impact of structural fluctuations on solution scattering data. In cases where the observed and calculated intensities correspond, we can conclude that the X-ray scattering provides direct experimental validation of the structural and MD results. In cases where calculated and observed intensities are at odds, the inconsistencies can be used to determine the origins of these discrepancies. They may be because of overestimates or underestimates of structural fluctuations in MD simulations, under-sampling of the structural ensemble in the simulations, errors in the structural model, or a mismatch between the experimental conditions and the parameters used in carrying out the MD simulation.

SAS profile correlations reveal SAS hierarchical nature and information content

In structural biology, Small-Angle Scattering experiments (SAS) are unique, because although they provide low resolution data, they can be performed in closer-to-native conditions than those arising in X-Ray crystallography. A number of questions on SAS, however, remain unsolved, particularly in the light of modelling ensembles of conformers in solution. In this article, we study the ensemble average and covariance of SAS profiles analytically. Using this ensemble covariance, we demonstrate the hierarchical nature of SAS profiles. Furthermore, we show that the information content is not uniform and reaches its maximum in the intermediate q range. The arguments are generalized using microsecond-scale molecular dynamics trajectories of the lysozyme and on an ensemble of the intrinsically disordered protein p15PAF. We show that for highly flexible systems, the SAS profile is a representation of the ensemble of conformers in solution, and not that of one conformer in particular.

100 years of Debye's scattering equation

Debye's scattering equation (DSE) has spanned a century of scientific development, from the dawn of quantum mechanics and the investigation of the structure of atoms and molecules to the era of nanotechnology, paving the way tototal scatteringmethods. The formulation offers the most accurate representation of the intensity scattered by randomly oriented atomic aggregates, constructed by superimposing the signal from each atomic distance in the molecule. The present paper reviews some of the milestone applications, from the interpretation of the intensity curves from gases and vapours, to aggregates of increasing size and more extended order. Important developments, aimed at mitigating the prohibitive computational complexity of the DSE, and state-of-the-art methods for the characterization of static and dynamic displacements are also discussed.

Structural Properties of Protein-Detergent Complexes from SAXS and MD Simulations

In experimental studies of solubilized membrane proteins, the detergent corona influences the protein behavior and the resulting measurement. Thus, combinations of experimental techniques with atomistic modeling have been used to resolve corona structural parameters and distributions. Here, we used small-angle X-ray scattering (SAXS) data and molecular dynamics simulations to study a model protein-detergent complex (PDC) consisting of aquaporin-0 and dodecyl-β-maltoside molecules (βDDM). The corona morphology of single snapshots was found to be rough, but it is smooth and compacted in 100-ns-scale ensemble averages. Individual snapshots therefore were unable to accurately represent the ensemble information as captured by experimental SAXS. Mimicking of annular lipids by detergent was also observed. SAXS prediction using different published methods was used to identify optimal βDDM numbers. Explicit-solvent methods predicted best agreement using 290-βDDM PDCs, but implicit-solvent methods gave unclear predictions due to overcompensation by free solvation-layer density parameters. Thus, ensemble-based approaches and physically motivated constraints will help to extract structural information from SAXS data.

Accurate small and wide angle x-ray scattering profiles from atomic models of proteins and nucleic acids

A new method is introduced to compute X-ray solution scattering profiles from atomic models of macromolecules. The three-dimensional version of the Reference Interaction Site Model (RISM) from liquid-state statistical mechanics is employed to compute the solvent distribution around the solute, including both water and ions. X-ray scattering profiles are computed from this distribution together with the solute geometry. We describe an efficient procedure for performing this calculation employing a Lebedev grid for the angular averaging. The intensity profiles (which involve no adjustable parameters) match experiment and molecular dynamics simulations up to wide angle for two proteins (lysozyme and myoglobin) in water, as well as the small-angle profiles for a dozen biomolecules taken from the BioIsis.net database. The RISM model is especially well-suited for studies of nucleic acids in salt solution. Use of fiber-diffraction models for the structure of duplex DNA in solution yields close agreement with the observed scattering profiles in both the small and wide angle scattering (SAXS and WAXS) regimes. In addition, computed profiles of anomalous SAXS signals (for Rb(+) and Sr(2+)) emphasize the ionic contribution to scattering and are in reasonable agreement with experiment. In cases where an absolute calibration of the experimental data at q = 0 is available, one can extract a count of the excess number of waters and ions; computed values depend on the closure that is assumed in the solution of the Ornstein-Zernike equations, with results from the Kovalenko-Hirata closure being closest to experiment for the cases studied here.

Interpretation of solution x-ray scattering by explicit-solvent molecular dynamics

Small- and wide-angle x-ray scattering (SWAXS) and molecular dynamics (MD) simulations are complementary approaches that probe conformational transitions of biomolecules in solution, even in a time-resolved manner. However, the structural interpretation of the scattering signals is challenging, while MD simulations frequently suffer from incomplete sampling or from a force-field bias. To combine the advantages of both techniques, we present a method that incorporates solution scattering data as a differentiable energetic restraint into explicit-solvent MD simulations, termed SWAXS-driven MD, with the aim to direct the simulation into conformations satisfying the experimental data. Because the calculations fully rely on explicit solvent, no fitting parameters associated with the solvation layer or excluded solvent are required, and the calculations remain valid at wide angles. The complementarity of SWAXS and MD is illustrated using three biological examples, namely a periplasmic binding protein, aspartate carbamoyltransferase, and a nuclear exportin. The examples suggest that SWAXS-driven MD is capable of refining structures against SWAXS data without foreknowledge of possible reaction paths. In turn, the SWAXS data accelerates conformational transitions in MD simulations and reduces the force-field bias.

WAXSiS: a web server for the calculation of SAXS/WAXS curves based on explicit-solvent molecular dynamics

Small- and wide-angle X-ray scattering (SWAXS) has evolved into a powerful tool to study biological macromolecules in solution. The interpretation of SWAXS curves requires their accurate predictions from structural models. Such predictions are complicated by scattering contributions from the hydration layer and by effects from thermal fluctuations. Here, we describe the new web server WAXSiS (WAXS in solvent) that computes SWAXS curves based on explicit-solvent all-atom molecular dynamics (MD) simulations (http://waxsis.uni-goettingen.de/). The MD simulations provide a realistic model for both the hydration layer and the excluded solvent, thereby avoiding any solvent-related fitting parameters, while naturally accounting for thermal fluctuations.

Emerging applications of small angle solution scattering in structural biology

Small angle solution X-ray and neutron scattering recently resurfaced as powerful tools to address an array of biological problems including folding, intrinsic disorder, conformational transitions, macromolecular crowding, and self or hetero-assembling of biomacromolecules. In addition, small angle solution scattering complements crystallography, nuclear magnetic resonance spectroscopy, and other structural methods to aid in the structure determinations of multidomain or multicomponent proteins or nucleoprotein assemblies. Neutron scattering with hydrogen/deuterium contrast variation, or X-ray scattering with sucrose contrast variation to a certain extent, is a convenient tool for characterizing the organizations of two-component systems such as a nucleoprotein or a lipid-protein assembly. Time-resolved small and wide-angle solution scattering to study biological processes in real time, and the use of localized heavy-atom labeling and anomalous solution scattering for applications as FRET-like molecular rulers, are amongst promising newer developments. Despite the challenges in data analysis and interpretation, these X-ray/neutron solution scattering based approaches hold great promise for understanding a wide variety of complex processes prevalent in the biological milieu.