Calculation of accurate small angle X-ray scattering curves from coarse-grained protein models

Binding structures of SERF1a with NT17-polyQ peptides of huntingtin exon 1 revealed by SEC-SWAXS, NMR and molecular simulation

The aberrant fibrillization of huntingtin exon 1 (Httex1) characterized by an expanded polyglutamine (polyQ) tract is a defining feature of Huntington's disease, a neurodegenerative disorder. Recent investigations underscore the involvement of a small EDRK-rich factor 1a (SERF1a) in promoting Httex1 fibrillization through interactions with its N terminus. By establishing an integrated approach with size-exclusion-column-based small- and wide-angle X-ray scattering (SEC-SWAXS), NMR, and molecular simulations using Rosetta, the analysis here reveals a tight binding of two NT17 fragments of Httex1 (comprising the initial 17 amino acids at the N terminus) to the N-terminal region of SERF1a. In contrast, examination of the complex structure of SERF1a with a coiled NT17-polyQ peptide (33 amino acids in total) indicates sparse contacts of the NT17 and polyQ segments with the N-terminal side of SERF1a. Furthermore, the integrated SEC-SWAXS and molecular-simulation analysis suggests that the coiled NT17 segment can transform into a helical conformation when associated with a polyQ segment exhibiting high helical content. Intriguingly, NT17-polyQ peptides with enhanced secondary structures display diminished interactions with SERF1a. This insight into the conformation-dependent binding of NT17 provides clues to a catalytic association mechanism underlying SERF1a's facilitation of Httext1 fibrillization.

Are Protein Conformational Ensembles in Agreement with Experimental Data? A Geometrical Interpretation of the Problem

The conformational variability of biological macromolecules can play an important role in their biological function. Therefore, understanding conformational variability is expected to be key for predicting the behavior of a particular molecule in the context of organism-wide studies. Several experimental methods have been developed and deployed for accessing this information, and computational methods are continuously updated for the profitable integration of different experimental sources. The outcome of this endeavor is conformational ensembles, which may vary significantly in properties and composition when different ensemble reconstruction methods are used, and this raises the issue of comparing the predicted ensembles against experimental data. In this article, we discuss a geometrical formulation to provide a framework for understanding the agreement of an ensemble prediction to the experimental observations.

Sulfated disaccharide protects membrane and DNA damages from arginine-rich dipeptide repeats in ALS

Hexanucleotide repeat expansion in C9ORF72 (C9) is the most prevalent mutation among amyotrophic lateral sclerosis (ALS) patients. The patients carry over ~30 to hundreds or thousands of repeats translated to dipeptide repeats (DPRs) where poly-glycine-arginine (GR) and poly-proline-arginine (PR) are most toxic. The structure-function relationship is still unknown. Here, we examined the minimal neurotoxic repeat number of poly-GR and found that extension of the repeat number led to a loose helical structure disrupting plasma and nuclear membrane. Poly-GR/PR bound to nucleotides and interfered with transcription. We screened and identified a sulfated disaccharide that bound to poly-GR/PR and rescued poly-GR/PR-induced toxicity in neuroblastoma and C9-ALS-iPSC-derived motor neurons. The compound rescued the shortened life span and defective locomotion in poly-GR/PR expressing Drosophila model and improved motor behavior in poly-GR-injected mouse model. Overall, our results reveal structural and toxicity mechanisms for poly-GR/PR and facilitate therapeutic development for C9-ALS.

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.

Accurate and Efficient SAXS/SANS Implementation Including Solvation Layer Effects Suitable for Molecular Simulations

Small-angle X-ray and neutron scattering (SAXS/SANS) provide valuable insights into the structure and dynamics of biomolecules in solution, complementing a wide range of structural techniques, including molecular dynamics simulations. As contrast-based methods, they are sensitive not only to structural properties but also to solvent-solute interactions. Their use in molecular dynamics simulations requires a forward model that should be as fast and accurate as possible. In this work, we demonstrate the feasibility of calculating SAXS and SANS intensities using a coarse-grained representation consisting of one bead per amino acid and three beads per nucleic acid, with form factors that can be corrected on the fly to account for solvation effects at no additional computational cost. By coupling this forward model with molecular dynamics simulations restrained with SAS data, it is possible to determine conformational ensembles or refine the structure and dynamics of proteins and nucleic acids in agreement with the experimental results. To assess the robustness of this approach, we applied it to gelsolin, for which we acquired SAXS data on its closed state, and to a UP1-microRNA complex, for which we used previously collected measurements. Our hybrid-resolution small-angle scattering (hySAS) implementation, being distributed in PLUMED, can be used with atomistic and coarse-grained simulations using diverse restraining strategies.

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.

Protein-induced transformation of unilamellar to multilamellar vesicles triggered by a polysaccharide

We report on the morphological transitions of didodecyldimethylammonium bromide (DDAB) cationic vesicles and hybrid DDAB/hyaluronic acid (HA) vesicles upon addition of BSA at pH 7 where BSA is overall negatively charged. Small angle neutron scattering (SANS) is used to extract the size distributions of the nanovesicles, the thickness of the DDAB bilayers and their lamellarity. Although the HA-decorated DDAB vesicles contain the negatively charged polysaccharide the interaction with BSA appears to be more intense in comparison to bare vesicles. Characteristic peaks in the SANS patterns indicate the presence of multilamellar interfaces while the formation of multilamellar vesicles induced by BSA depends on the amount of added HA. Consequently, higher lamellarities are observed at higher BSA contents. This work demonstrates a simple methodology to tune the encapsulation of globular proteins in vesicular nanoassemblies by affecting their lamellarity and has direct implications on the application of vesicles and liposomes in protein delivery.

From dilute to concentrated solutions of intrinsically disordered proteins: Interpretation and analysis of collected data

Intrinsically disordered proteins (IDPs) have a broad energy landscape and consequently sample many different conformations in solution. The innate flexibility of IDPs is exploited in their biological function, and in many instances allows a single IDP to regulate a range of processes in vivo. Due to their highly flexible nature, characterizing the structural properties of IDPs is not straightforward. Often solution-based methods such as Nuclear Magnetic Resonance (NMR), Förster Resonance Energy Transfer (FRET), and Small-Angle X-ray Scattering (SAXS) are used. SAXS is indeed a powerful technique to study the structural and conformational properties of IDPs in solution, and from the obtained SAXS spectra, information about the average size, shape, and extent of oligomerization can be determined. In this chapter, we will introduce model-free methods that can be used to interpret SAXS data and introduce methods that can be used to interpret SAXS data beyond analytical models, for example, by using atomistic and different levels of coarse-grained models in combination with molecular dynamics (MD) and Monte Carlo simulations.

Ab initio determination of the shape of membrane proteins in a nanodisc

New software, called Marbles, is introduced that employs SAXS intensities to predict the shape of membrane proteins embedded into membrane nanodiscs. To gain computational speed and efficient convergence, the strategy is based on a hybrid approach that allows one to account for the contribution of the nanodisc to the SAXS intensity through a semi-analytical model, while the embedded membrane protein is treated as a set of beads, similarly to as in well known ab initio methods. The reliability and flexibility of this approach is proved by benchmarking the code, implemented in C++ with a Python interface, on a toy model and two proteins with very different geometry and size.

Hybrid methods for combined experimental and computational determination of protein structure

Knowledge of protein structure is paramount to the understanding of biological function, developing new therapeutics, and making detailed mechanistic hypotheses. Therefore, methods to accurately elucidate three-dimensional structures of proteins are in high demand. While there are a few experimental techniques that can routinely provide high-resolution structures, such as x-ray crystallography, nuclear magnetic resonance (NMR), and cryo-EM, which have been developed to determine the structures of proteins, these techniques each have shortcomings and thus cannot be used in all cases. However, additionally, a large number of experimental techniques that provide some structural information, but not enough to assign atomic positions with high certainty have been developed. These methods offer sparse experimental data, which can also be noisy and inaccurate in some instances. In cases where it is not possible to determine the structure of a protein experimentally, computational structure prediction methods can be used as an alternative. Although computational methods can be performed without any experimental data in a large number of studies, inclusion of sparse experimental data into these prediction methods has yielded significant improvement. In this Perspective, we cover many of the successes of integrative modeling, computational modeling with experimental data, specifically for protein folding, protein-protein docking, and molecular dynamics simulations. We describe methods that incorporate sparse data from cryo-EM, NMR, mass spectrometry, electron paramagnetic resonance, small-angle x-ray scattering, Förster resonance energy transfer, and genetic sequence covariation. Finally, we highlight some of the major challenges in the field as well as possible future directions.

Lamellarity and size distributions in mixed DPPC/amphiphilic poly(2-oxazoline) gradient copolymer vesicles and their temperature response

Mixed liposomes of dipalmitoylphosphatidylcholine (DPPC) and gradient (pseudodiblock) poly(2-methyl-2-oxazoline)-grad-poly(2-phenyl-2-oxazoline) (MPOx) copolymers are investigated by small angle neutron scattering (SANS). All experimental data, from different phospholipid-copolymer compositions, concentrations and temperatures are fitted with one model. This model allows the determination of the separate contributions from vesicular populations of different lamellarity and size. MPOx copolymers are proved to modify both the size and lamellarity of DPPC liposomes. The gradient copolymer with higher hydrophilic content induces shrinkage of the uni- and bi-lamellar DPPC vesicles. The copolymer with lower hydrophilic content causes dramatic changes on the lamellarity of DPPC vesicles by the formation of hexa-lamellar vesicles. The tendency of multi-lamellar vesicles to transform into uni-lamellar ones as temperature increases is more pronounced in the presence of the copolymers. These findings may have direct implications on the drug loading and release properties of liposomes and their interactions with cells.

Estimating signal and noise of time-resolved X-ray solution scattering data at synchrotrons and XFELs

Elucidating the structural dynamics of small molecules and proteins in the liquid solution phase is essential to ensure a fundamental understanding of their reaction mechanisms. In this regard, time-resolved X-ray solution scattering (TRXSS), also known as time-resolved X-ray liquidography (TRXL), has been established as a powerful technique for obtaining the structural information of reaction intermediates and products in the liquid solution phase and is expected to be applied to a wider range of molecules in the future. A TRXL experiment is generally performed at the beamline of a synchrotron or an X-ray free-electron laser (XFEL) to provide intense and short X-ray pulses. Considering the limited opportunities to use these facilities, it is necessary to verify the plausibility of a target experiment prior to the actual experiment. For this purpose, a program has been developed, referred to as S-cube, which is short for a Solution Scattering Simulator. This code allows the routine estimation of the shape and signal-to-noise ratio (SNR) of TRXL data from known experimental parameters. Specifically, S-cube calculates the difference scattering curve and the associated quantum noise on the basis of the molecular structure of the target reactant and product, the target solvent, the energy of the pump laser pulse and the specifications of the beamline to be used. Employing a simplified form for the pair-distribution function required to calculate the solute-solvent cross term greatly increases the calculation speed as compared with a typical TRXL data analysis. Demonstrative applications of S-cube are presented, including the estimation of the expected TRXL data and SNR level for the future LCLS-II HE beamlines.

Building blocks for commodity augmented reality-based molecular visualization and modeling in web browsers

For years, immersive interfaces using virtual and augmented reality (AR) for molecular visualization and modeling have promised a revolution in the way how we teach, learn, communicate and work in chemistry, structural biology and related areas. However, most tools available today for immersive modeling require specialized hardware and software, and are costly and cumbersome to set up. These limitations prevent wide use of immersive technologies in education and research centers in a standardized form, which in turn prevents large-scale testing of the actual effects of such technologies on learning and thinking processes. Here, I discuss building blocks for creating marker-based AR applications that run as web pages on regular computers, and explore how they can be exploited to develop web content for handling virtual molecular systems in commodity AR with no more than a webcam- and internet-enabled computer. Examples span from displaying molecules, electron microscopy maps and molecular orbitals with minimal amounts of HTML code, to incorporation of molecular mechanics, real-time estimation of experimental observables and other interactive resources using JavaScript. These web apps provide virtual alternatives to physical, plastic-made molecular modeling kits, where the computer augments the experience with information about spatial interactions, reactivity, energetics, etc. The ideas and prototypes introduced here should serve as starting points for building active content that everybody can utilize online at minimal cost, providing novel interactive pedagogic material in such an open way that it could enable mass-testing of the effect of immersive technologies on chemistry education.

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.

Martini bead form factors for nucleic acids and their application in the refinement of protein–nucleic acid complexes against SAXS data

The use of small-angle X-ray scattering (SAXS) in combination with molecular dynamics simulation is hampered by its heavy computational cost. The calculation of SAXS from atomic structures can be speeded up by using a coarse-grain representation of the structure. Following the work of Niebling, Björling & Westenhoff [J. Appl. Cryst. (2014), 47, 1190–1198], the Martini bead form factors for nucleic acids have been derived and then implemented, together with those previously determined for proteins, in the publicly available PLUMED library. A hybrid multi-resolution strategy has also been implemented to perform SAXS restrained simulations at atomic resolution by calculating the virtual positions of the Martini beads on the fly and using them for the calculation of SAXS. The accuracy and efficiency of the method are demonstrated by refining the structure of two protein–nucleic acid complexes. Instrumental for this result is the use of metainference, which allows the consideration and alleviation of the approximations at play in the present SAXS calculations.

Evaluating Anti-CD32b F(ab) Conformation Using Molecular Dynamics and Small-Angle X-Ray Scattering

Complementary strategies of small-angle x-ray scattering (SAXS) and crystallographic analysis are often used to determine atomistic three-dimensional models of macromolecules and their variability in solution. This combination of techniques is particularly valuable when applied to macromolecular complexes to detect changes within the individual binding partners. Here, we determine the x-ray crystallographic structure of a F(ab) fragment in complex with CD32b, the only inhibitory Fc-γ receptor in humans, and compare the structure of the F(ab) from the crystal complex to SAXS data for the F(ab) alone in solution. We investigate changes in F(ab) structure by predicting theoretical scattering profiles for atomistic structures extracted from molecular dynamics (MD) simulations of the F(ab) and assessing the agreement of these structures to our experimental SAXS data. Through principal component analysis, we are able to extract principal motions observed during the MD trajectory and evaluate the influence of these motions on the agreement of structures to the F(ab) SAXS data. Changes in the F(ab) elbow angle were found to be important to reach agreement with the experimental data; however, further discrepancies were apparent between our F(ab) structure from the crystal complex and SAXS data. By analyzing multiple MD structures observed in similar regions of the principal component analysis, we were able to pinpoint these discrepancies to a specific loop region in the F(ab) heavy chain. This method, therefore, not only allows determination of global changes but also allows identification of localized motions important for determining the agreement between atomistic structures and SAXS data. In this particular case, the findings allowed us to discount the hypothesis that structural changes were induced upon complex formation, a significant find informing the drug development process. The methodology described here is generally applicable to deconvolute global and local changes of macromolecular structures and is well suited to other systems.

Hybrid Methods for Modeling Protein Structures Using Molecular Dynamics Simulations and Small-Angle X-Ray Scattering Data

Small-angle X-ray scattering (SAXS) is an efficient experimental tool to measure the overall shape of macromolecular structures in solution. However, due to the low resolution of SAXS data, high-resolution data obtained from X-ray crystallography or NMR and computational methods such as molecular dynamics (MD) simulations are complementary to SAXS data for understanding protein functions based on their structures at atomic resolution. Because MD simulations provide a physicochemically proper structural ensemble for flexible proteins in solution and a precise description of solvent effects, the hybrid analysis of SAXS and MD simulations is a promising method to estimate reasonable solution structures and structural ensembles in solution. Here, we review typical and useful in silico methods for modeling three dimensional protein structures, calculating theoretical SAXS profiles, and analyzing ensemble structures consistent with experimental SAXS profiles. We also review two examples of the hybrid analysis, termed MD-SAXS method in which MD simulations are carried out without any knowledge of experimental SAXS data, and the experimental SAXS data are used only to assess the consistency of the solution model from MD simulations with those observed in experiments. One example is an investigation of the intrinsic dynamics of EcoO109I using the computational method to obtain a theoretical profile from the trajectory of an MD simulation. The other example is a structural investigation of the vitamin D receptor ligand-binding domain using snapshots generated by MD simulations and assessment of the snapshots by experimental SAXS data.

Solution scattering approaches to dynamical ordering in biomolecular systems

Clarification of solution structure and its modulation in proteins and protein complexes is crucially important to understand dynamical ordering in macromolecular systems. Small-angle x-ray scattering (SAXS) and small-angle neutron scattering (SANS) are among the most powerful techniques to derive structural information. Recent progress in sample preparation, instruments and software analysis is opening up a new era for small-angle scattering. In this review, recent progress and trends of SAXS and SANS are introduced from the point of view of instrumentation and analysis, touching on general features and standard methods of small-angle scattering. This article is part of a Special Issue entitled "Biophysical Exploration of Dynamical Ordering of Biomolecular Systems" edited by Dr. Koichi Kato.

Conformational variability of the stationary phase survival protein E from Xylella fastidiosa revealed by X-ray crystallography, small-angle X-ray scattering studies, and normal mode analysis

Xylella fastidiosa is a xylem-limited bacterium that infects a wide variety of plants. Stationary phase survival protein E is classified as a nucleotidase, which is expressed when bacterial cells are in the stationary growth phase and subjected to environmental stresses. Here, we report four refined X-ray structures of this protein from X. fastidiosa in four different crystal forms in the presence and/or absence of the substrate 3'-AMP. In all chains, the conserved loop verified in family members assumes a closed conformation in either condition. Therefore, the enzymatic mechanism for the target protein might be different of its homologs. Two crystal forms exhibit two monomers whereas the other two show four monomers in the asymmetric unit. While the biological unit has been characterized as a tetramer, differences of their sizes and symmetry are remarkable. Four conformers identified by Small-Angle X-ray Scattering (SAXS) in a ligand-free solution are related to the low frequency normal modes of the crystallographic structures associated with rigid body-like protomer arrangements responsible for the longitudinal and symmetric adjustments between tetramers. When the substrate is present in solution, only two conformers are selected. The most prominent conformer for each case is associated to a normal mode able to elongate the protein by moving apart two dimers. To our knowledge, this work was the first investigation based on the normal modes that analyzed the quaternary structure variability for an enzyme of the SurE family followed by crystallography and SAXS validation. The combined results raise new directions to study allosteric features of XfSurE protein.

Pepsi-SAXS: an adaptive method for rapid and accurate computation of small-angle X-ray scattering profiles

A new method called Pepsi-SAXS is presented that calculates small-angle X-ray scattering profiles from atomistic models. The method is based on the multipole expansion scheme and is significantly faster compared with other tested methods. In particular, using the Nyquist-Shannon-Kotelnikov sampling theorem, the multipole expansion order is adapted to the size of the model and the resolution of the experimental data. It is argued that by using the adaptive expansion order, this method has the same quadratic dependence on the number of atoms in the model as the Debye-based approach, but with a much smaller prefactor in the computational complexity. The method has been systematically validated on a large set of over 50 models collected from the BioIsis and SASBDB databases. Using a laptop, it was demonstrated that Pepsi-SAXS is about seven, 29 and 36 times faster compared with CRYSOL, FoXS and the three-dimensional Zernike method in SAStbx, respectively, when tested on data from the BioIsis database, and is about five, 21 and 25 times faster compared with CRYSOL, FoXS and SAStbx, respectively, when tested on data from SASBDB. On average, Pepsi-SAXS demonstrates comparable accuracy in terms of χ² to CRYSOL and FoXS when tested on BioIsis and SASBDB profiles. Together with a small allowed variation of adjustable parameters, this demonstrates the effectiveness of the method. Pepsi-SAXS is available at http://team.inria.fr/nano-d/software/pepsi-saxs.

Quantitative evaluation of statistical errors in small-angle X-ray scattering measurements

A model is presented for the errors in small-angle X-ray scattering profiles that takes into account the physics of the measurement process. The model agrees quantitatively with the variations observed in experimental measurements and provides a straightforward prescription to add realistic errors to simulated scattering profiles.

A new model is proposed for the measurement errors incurred in typical small-angle X-ray scattering (SAXS) experiments, which takes into account the setup geometry and physics of the measurement process. The model accurately captures the experimentally determined errors from a large range of synchrotron and in-house anode-based measurements. Its most general formulation gives for the variance of the buffer-subtracted SAXS intensity σ²(q) = [I(q) + const.]/(kq), where I(q) is the scattering intensity as a function of the momentum transfer q; k and const. are fitting parameters that are characteristic of the experimental setup. The model gives a concrete procedure for calculating realistic measurement errors for simulated SAXS profiles. In addition, the results provide guidelines for optimizing SAXS measurements, which are in line with established procedures for SAXS experiments, and enable a quantitative evaluation of measurement errors.

Accurate optimization of amino acid form factors for computing small-angle X-ray scattering intensity of atomistic protein structures

Structure modelling via small-angle X-ray scattering (SAXS) data generally requires intensive computations of scattering intensity from any given biomolecular structure, where the accurate evaluation of SAXS profiles using coarse-grained (CG) methods is vital to improve computational efficiency. To date, most CG SAXS computing methods have been based on a single-bead-per-residue approximation but have neglected structural correlations between amino acids. To improve the accuracy of scattering calculations, accurate CG form factors of amino acids are now derived using a rigorous optimization strategy, termed electron-density matching (EDM), to best fit electron-density distributions of protein structures. This EDM method is compared with and tested against other CG SAXS computing methods, and the resulting CG SAXS profiles from EDM agree better with all-atom theoretical SAXS data. By including the protein hydration shell represented by explicit CG water molecules and the correction of protein excluded volume, the developed CG form factors also reproduce the selected experimental SAXS profiles with very small deviations. Taken together, these EDM-derived CG form factors present an accurate and efficient computational approach for SAXS computing, especially when higher molecular details (represented by the q range of the SAXS data) become necessary for effective structure modelling.

Functional Diversity of Cytotoxic tRNase/Immunity Protein Complexes from Burkholderia pseudomallei

Contact-dependent growth inhibition (CDI) is a widespread mechanism of inter-bacterial competition. CDI(+) bacteria deploy large CdiA effector proteins, which carry variable C-terminal toxin domains (CdiA-CT). CDI(+) cells also produce CdiI immunity proteins that specifically neutralize cognate CdiA-CT toxins to prevent auto-inhibition. Here, we present the crystal structure of the CdiA-CT/CdiI(E479) toxin/immunity protein complex from Burkholderia pseudomallei isolate E479. The CdiA-CT(E479) tRNase domain contains a core α/β-fold that is characteristic of PD(D/E)XK superfamily nucleases. Unexpectedly, the closest structural homolog of CdiA-CT(E479) is another CDI toxin domain from B. pseudomallei 1026b. Although unrelated in sequence, the two B. pseudomallei nuclease domains share similar folds and active-site architectures. By contrast, the CdiI(E479) and CdiI(1026b) immunity proteins share no significant sequence or structural homology. CdiA-CT(E479) and CdiA-CT(1026b) are both tRNases; however, each nuclease cleaves tRNA at a distinct position. We used a molecular docking approach to model each toxin bound to tRNA substrate. The resulting models fit into electron density envelopes generated by small-angle x-ray scattering analysis of catalytically inactive toxin domains bound stably to tRNA. CdiA-CT(E479) is the third CDI toxin found to have structural homology to the PD(D/E)XK superfamily. We propose that CDI systems exploit the inherent sequence variability and active-site plasticity of PD(D/E)XK nucleases to generate toxin diversity. These findings raise the possibility that many other uncharacterized CDI toxins may belong to the PD(D/E)XK superfamily.

BCL::SAXS: GPU accelerated Debye method for computation of small angle X-ray scattering profiles

Small angle X-ray scattering (SAXS) is an experimental technique used for structural characterization of macromolecules in solution. Here, we introduce BCL::SAXS--an algorithm designed to replicate SAXS profiles from rigid protein models at different levels of detail. We first show our derivation of BCL::SAXS and compare our results with the experimental scattering profile of hen egg white lysozyme. Using this protein we show how to generate SAXS profiles representing: (1) complete models, (2) models with approximated side chain coordinates, and (3) models with approximated side chain and loop region coordinates. We evaluated the ability of SAXS profiles to identify a correct protein topology from a non-redundant benchmark set of proteins. We find that complete SAXS profiles can be used to identify the correct protein by receiver operating characteristic (ROC) analysis with an area under the curve (AUC) > 99%. We show how our approximation of loop coordinates between secondary structure elements improves protein recognition by SAχS for protein models without loop regions and side chains. Agreement with SAXS data is a necessary but not sufficient condition for structure determination. We conclude that experimental SAXS data can be used as a filter to exclude protein models with large structural differences from the native.

Review of the fundamental theories behind small angle X-ray scattering, molecular dynamics simulations, and relevant integrated application

In this paper, the fundamental concepts and equations necessary for performing small angle X-ray scattering (SAXS) experiments, molecular dynamics (MD) simulations, and MD-SAXS analyses were reviewed. Furthermore, several key biological and non-biological applications for SAXS, MD, and MD-SAXS are presented in this review; however, this article does not cover all possible applications. SAXS is an experimental technique used for the analysis of a wide variety of biological and non-biological structures. SAXS utilizes spherical averaging to produce one- or two-dimensional intensity profiles, from which structural data may be extracted. MD simulation is a computer simulation technique that is used to model complex biological and non-biological systems at the atomic level. MD simulations apply classical Newtonian mechanics' equations of motion to perform force calculations and to predict the theoretical physical properties of the system. This review presents several applications that highlight the ability of both SAXS and MD to study protein folding and function in addition to non-biological applications, such as the study of mechanical, electrical, and structural properties of non-biological nanoparticles. Lastly, the potential benefits of combining SAXS and MD simulations for the study of both biological and non-biological systems are demonstrated through the presentation of several examples that combine the two techniques.

Methods for SAXS-based structure determination of biomolecular complexes

Measurements from small-angle X-ray scattering (SAXS) are highly informative to determine the structures of bimolecular complexes in solution. Here, current and recent SAXS-driven developments are described, with an emphasis on computational modeling. In particular, accurate methods to computing one theoretical scattering profile from a given structure model are discussed, with a key focus on structure factor coarse-graining and hydration contribution. Methods for reconstructing topological structures from an experimental SAXS profile are currently under active development. We report on several modeling tools designed for conformation generation that make use of either atomic-level or coarse-grained representations. Furthermore, since large, flexible biomolecules can adopt multiple well-defined conformations, a traditional single-conformation SAXS analysis is inappropriate, so we also discuss recent methods that utilize the concept of ensemble optimization, weighing in on the SAXS contributions of a heterogeneous mixture of conformations. These tools will ultimately posit the usefulness of SAXS data beyond a simple space-filling approach by providing a reliable structure characterization of biomolecular complexes under physiological conditions.

MARTINI bead form factors for the analysis of time-resolved X-ray scattering of proteins

Time-resolved small- and wide-angle X-ray scattering (SAXS and WAXS) methods probe the structural dynamics of proteins in solution. Although technologically advanced, these methods are in many cases limited by data interpretation. The calculation of X-ray scattering profiles is computationally demanding and poses a bottleneck for all SAXS/WAXS-assisted structural refinement and, in particular, for the analysis of time-resolved data. A way of speeding up these calculations is to represent biomolecules as collections of coarse-grained scatterers. Here, such coarse-graining schemes are presented and discussed and their accuracies examined. It is demonstrated that scattering factors coincident with the popular MARTINI coarse-graining scheme produce reliable difference scattering in the range 0 < q < 0.75 Å-1. The findings are promising for future attempts at X-ray scattering data analysis, and may help to bridge the gap between time-resolved experiments and their interpretation.

Accurate SAXS profile computation and its assessment by contrast variation experiments

A major challenge in structural biology is to characterize structures of proteins and their assemblies in solution. At low resolution, such a characterization may be achieved by small angle x-ray scattering (SAXS). Because SAXS analyses often require comparing profiles calculated from many atomic models against those determined by experiment, rapid and accurate profile computation from molecular structures is needed. We developed fast open-source x-ray scattering (FoXS) for profile computation. To match the experimental profile within the experimental noise, FoXS explicitly computes all interatomic distances and implicitly models the first hydration layer of the molecule. For assessing the accuracy of the modeled hydration layer, we performed contrast variation experiments for glucose isomerase and lysozyme, and found that FoXS can accurately represent density changes of this layer. The hydration layer model was also compared with a SAXS profile calculated for the explicit water molecules in the high-resolution structures of glucose isomerase and lysozyme. We tested FoXS on eleven protein, one DNA, and two RNA structures, revealing superior accuracy and speed versus CRYSOL, AquaSAXS, the Zernike polynomials-based method, and Fast-SAXS-pro. In addition, we demonstrated a significant correlation of the SAXS score with the accuracy of a structural model. Moreover, FoXS utility for analyzing heterogeneous samples was demonstrated for intrinsically flexible XLF-XRCC4 filaments and Ligase III-DNA complex. FoXS is extensively used as a standalone web server as a component of integrative structure determination by programs IMP, Chimera, and BILBOMD, as well as in other applications that require rapidly and accurately calculated SAXS profiles.

Advances in the determination of nucleic acid conformational ensembles

Conformational changes in nucleic acids play a key role in the way genetic information is stored, transferred, and processed in living cells. Here, we describe new approaches that employ a broad range of experimental data, including NMR-derived chemical shifts and residual dipolar couplings, small-angle X-ray scattering, and computational approaches such as molecular dynamics simulations to determine ensembles of DNA and RNA at atomic resolution. We review the complementary information that can be obtained from diverse sets of data and the various methods that have been developed to combine these data with computational methods to construct ensembles and assess their uncertainty. We conclude by surveying RNA and DNA ensembles determined using these methods, highlighting the unique physical and functional insights obtained so far.

Reconstruction of SAXS Profiles from Protein Structures

Small angle X-ray scattering (SAXS) is used for low resolution structural characterization of proteins often in combination with other experimental techniques. After briefly reviewing the theory of SAXS we discuss computational methods based on 1) the Debye equation and 2) Spherical Harmonics to compute intensity profiles from a particular macromolecular structure. Further, we review how these formulas are parameterized for solvent density and hydration shell adjustment. Finally we introduce our solution to compute SAXS profiles utilizing GPU acceleration.

Fast-SAXS-pro: a unified approach to computing SAXS profiles of DNA, RNA, protein, and their complexes

A generalized method, termed Fast-SAXS-pro, for computing small angle x-ray scattering (SAXS) profiles of proteins, nucleic acids, and their complexes is presented. First, effective coarse-grained structure factors of DNA nucleotides are derived using a simplified two-particle-per-nucleotide representation. Second, SAXS data of a 18-bp double-stranded DNA are measured and used for the calibration of the scattering contribution from excess electron density in the DNA solvation layer. Additional test on a 25-bp DNA duplex validates this SAXS computational method and suggests that DNA has a different contribution from its hydration surface to the total scattering compared to RNA and protein. To account for such a difference, a sigmoidal function is implemented for the treatment of non-uniform electron density across the surface of a protein/nucleic-acid complex. This treatment allows differential scattering from the solvation layer surrounding protein/nucleic-acid complexes. Finally, the applications of this Fast-SAXS-pro method are demonstrated for protein/DNA and protein/RNA complexes.

PHAISTOS: a framework for Markov chain Monte Carlo simulation and inference of protein structure

We present a new software framework for Markov chain Monte Carlo sampling for simulation, prediction, and inference of protein structure. The software package contains implementations of recent advances in Monte Carlo methodology, such as efficient local updates and sampling from probabilistic models of local protein structure. These models form a probabilistic alternative to the widely used fragment and rotamer libraries. Combined with an easily extendible software architecture, this makes PHAISTOS well suited for Bayesian inference of protein structure from sequence and/or experimental data. Currently, two force-fields are available within the framework: PROFASI and OPLS-AA/L, the latter including the generalized Born surface area solvent model. A flexible command-line and configuration-file interface allows users quickly to set up simulations with the desired configuration. PHAISTOS is released under the GNU General Public License v3.0. Source code and documentation are freely available from http://phaistos.sourceforge.net. The software is implemented in C++ and has been tested on Linux and OSX platforms.

How random are intrinsically disordered proteins? A small angle scattering perspective

While the crucial role of intrinsically disordered proteins (IDPs) in the cell cycle is now recognized, deciphering their molecular mode of action at the structural level still remains highly challenging and requires a combination of many biophysical approaches. Among them, small angle X-ray scattering (SAXS) has been extremely successful in the last decade and has become an indispensable technique for addressing many of the fundamental questions regarding the activities of IDPs. After introducing some experimental issues specific to IDPs and in relation to the latest technical developments, this article presents the interest of the theory of polymer physics to evaluate the flexibility of fully disordered proteins. The different strategies to obtain 3-dimensional models of IDPs, free in solution and associated in a complex, are then reviewed. Indeed, recent computational advances have made it possible to readily extract maximum information from the scattering curve with a special emphasis on highly flexible systems, such as multidomain proteins and IDPs. Furthermore, integrated computational approaches now enable the generation of ensembles of conformers to translate the unique flexible characteristics of IDPs by taking into consideration the constraints of more and more various complementary experiment. In particular, a combination of SAXS with high-resolution techniques, such as x-ray crystallography and NMR, allows us to provide reliable models and to gain unique structural insights about the protein over multiple structural scales. The latest neutron scattering experiments also promise new advances in the study of the conformational changes of macromolecules involving more complex systems.

Integrative structural modeling with small angle X-ray scattering profiles

Recent technological advances enabled high-throughput collection of Small Angle X-ray Scattering (SAXS) profiles of biological macromolecules. Thus, computational methods for integrating SAXS profiles into structural modeling are needed more than ever. Here, we review specifically the use of SAXS profiles for the structural modeling of proteins, nucleic acids, and their complexes. First, the approaches for computing theoretical SAXS profiles from structures are presented. Second, computational methods for predicting protein structures, dynamics of proteins in solution, and assembly structures are covered. Third, we discuss the use of SAXS profiles in integrative structure modeling approaches that depend simultaneously on several data types.

Computation of small-angle scattering profiles with three-dimensional Zernike polynomials

Small-angle X-ray scattering (SAXS) methods are extensively used for characterizing macromolecular structure and dynamics in solution. The computation of theoretical scattering profiles from three-dimensional models is crucial in order to test structural hypotheses. Here, a new approach is presented to efficiently compute SAXS profiles that are based on three-dimensional Zernike polynomial expansions. Comparison with existing methods and experimental data shows that the Zernike method can be used to effectively validate three-dimensional models against experimental data. For molecules with large cavities or complicated surfaces, the Zernike method more accurately accounts for the solvent contributions. The program is available as open-source software at http://sastbx.als.lbl.gov.

Determination of the structures of symmetric protein oligomers from NMR chemical shifts and residual dipolar couplings

Symmetric protein dimers, trimers, and higher-order cyclic oligomers play key roles in many biological processes. However, structural studies of oligomeric systems by solution NMR can be difficult due to slow tumbling of the system and the difficulty in identifying NOE interactions across protein interfaces. Here, we present an automated method (RosettaOligomers) for determining the solution structures of oligomeric systems using only chemical shifts, sparse NOEs, and domain orientation restraints from residual dipolar couplings (RDCs) without a need for a previously determined structure of the monomeric subunit. The method integrates previously developed Rosetta protocols for solving the structures of monomeric proteins using sparse NMR data and for predicting the structures of both nonintertwined and intertwined symmetric oligomers. We illustrated the performance of the method using a benchmark set of nine protein dimers, one trimer, and one tetramer with available experimental data and various interface topologies. The final converged structures are found to be in good agreement with both experimental data and previously published high-resolution structures. The new approach is more readily applicable to large oligomeric systems than conventional structure-determination protocols, which often require a large number of NOEs, and will likely become increasingly relevant as more high-molecular weight systems are studied by NMR.