Large biomolecules such as proteins perform a variety of functions within living organisms, including catalysis of metabolic reactions, DNA replication, and molecular transport. To fully understand protein function, it is important to determine protein structure.^[1] Currently, four major techniques are used for structural elucidation: X-ray crystallography, nuclear magnetic resonance (NMR), cryo-electron microscopy (cryo-EM), and SAXS. These techniques can help determine protein structure at either high or low resolution. SAXS is a promising technique that can rapidly provide global structural information, including protein size and shape.^{[2, 3]} In addition, SAXS is useful in characterizing the flexibility of a protein in solution, because the scattering profile is the average of the multiple protein conformations present in the population. Due to its low-resolution nature, SAXS is usually used to complement high-resolution techniques. In such applications, termed “ integrative structural biology” , ^[4] computer simulations play an important role in combining SAXS and other experimental data. This review will introduce various computational methods for integrative modeling, which are aimed at determining the structures of proteins and investigating their flexibility in solution. To demonstrate the advantages of this hybrid strategy, at the end of the review we present our recent study, in which we determined the structural ensemble of a protein complex formed between the tandem SH3 domains of c-Cbl-associated protein and the proline-rich loop in human vinculin.

In a SAXS experiment, a protein sample is illuminated by x-rays, and the scattering intensities are recorded by a detector as a function of the scattering angle (typically 0.1° to 10° ). Because the sample solution is isotropic, the two-dimensional image obtained can be converted into a one-dimensional profile I(q) (Fig. 1(a)). The magnitude of the scattering vector q is 4π sin (θ )/λ , where 2θ is the scattering angle and λ is the x-ray wavelength. The radius of gyration (R_g) of the protein can be estimated directly from small q values using Guinier approximation, , where I(0) is the forward scattering intensity. Using the program GNOM^[5] and employing indirect Fourier transform, the pair-distance distribution function (PDDF), which describes the paired set of distances between all electrons within the protein, can be obtained from the SAXS profile (Fig. 1(b)). The PDDF can provide global protein shape information. For instance, a globular protein has a roughly symmetric PDDF with a single peak, whereas a multi-domain or multi-subunit protein usually shows an extended tail at large r, with multiple shoulder peaks. Conformational changes in the protein may result in detectable changes in its PDDF. For proteins with a dominant conformation, the SAXS envelope of the protein can be calculated by ab initio shape determination using the DAMMIN^[6] or GASBOR^[7] programs. For flexible protein systems, SAXS data can be used to determine the structural ensemble of the protein, using algorithms such as the ensemble optimization method (EOM).^[8]

Fig. 1.

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	Fig. 1. Experimental SAXS data for the CAP-SH3ab:vin857 complex. (a) The SAXS scattering profile of the complex and (b) the corresponding PDDF curve.

To integrate experimental SAXS data into structural modeling, one must be able to compute the theoretical SAXS profiles of the protein conformations generated by the simulation. Several programs are available that can calculate a SAXS profile from a protein structure, ^{[9– 18]} and they employ different methods for spherical averaging, as well as treatment of the excluded volume and the hydration layer.

Because proteins are randomly oriented in solution, spherical averaging is required in the calculation of the theoretical SAXS profile. For spherical averaging computation, CRYSOL^[9] employs multipole expansion to accelerate calculation whereas fast-SAXS^[13] and FoXS^[15] use the Debye formula of pairwise inter-atomic distances. In CRYSOL, the atomic structure of the protein must be provided. To increase the computational efficiency, fast-SAXS defines a residue-level coarse-grained (CG) model of the protein, whereas FoXS accepts either an atomistic or a CG representation. Solute molecules are surrounded by a hydration layer and exclude a certain volume of bulk solvent, and both need to be taken into account. Programs like CRYSOL and FoXS treat the hydration layer implicitly, that is to say, the protein is assumed to be surrounded by a continuous envelope of adjustable density. The hydration layer can also be treated explicitly; for example, in Fast-SAXS, water molecules are introduced during the calculation. The excluded volume term is typically related to the shape of the molecule. CRYSOL calculates this portion of the scattering by assuming that the electron density of the excluded volume is equivalent to that of the bulk solvent. Because the total volume varies depending on a set of atomic radii values, accurate computation of the excluded volume is difficult. Therefore, CRYSOL and FoXS allow adjustment of the excluded volume of the protein for optimal fitting of the experimental SAXS profile.

Usually, the theoretical SAXS profile is fitted to the experimental profile by minimizing

where I_exp(q) and I(q) are the experimental and theoretical scattering profiles, respectively, σ (q) is the experimental error, M is the number of data points in the profile, and c is the scaling factor.

To integrate low-resolution SAXS data into structural modeling of proteins, two strategies are generally applied: refining while sampling and screening after sampling.^[19]

c-Cbl-associated protein (CAP) is an important cytoskeletal adaptor that functions in the regulation of adhesion turnover.^[42] It consists of a SoHo domain and three tandem SH3 domains. Human vinculin is comprised of a head domain and a tail domain connected by a proline-rich region (prr).^[43] It has been reported that binding of the first two SH3 domains of CAP (called CAP-SH3ab) to the prr of vinculin is responsible for localization of CAP for cell-extracellular matrix adhesion.^[44] The crystal structure of the isolated SH3a domain in complex with residues 870– 879 of vinculin (called prr2) has been solved to a resolution of 1.41 Å (pdb code 4LNP), along with that of the isolated SH3b domain in complex with residues 858– 867 of vinculin (called prr1) with a resolution of 1.00 Å (pdb code: 4LN2) (Fig. 2(a)). However, because the prr domain of vinculin is highly flexible, it is difficult to obtain the combined crystal structure of the tandem SH3ab complexed with the C-terminal prr of vinculin (residues 857– 879, denoted as vin₈₅₇). We determined a structural model of this complex by combining high-resolution structures of SH3a:prr2 and SH3b:prr1 with SAXS data for the complex via computational simulation using a structure-based model (SBM). A flow chart of the model building process is shown in Fig. 2.

Fig. 2.

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Fig. 2. A flow diagram of the process of building the model of the CAP-SH3ab:vin857 complex. (a) Crystal structures of SH3a:prr2 and SH3b:prr1. The SH3a and SH3b domains are shown in red and yellow, respectively, whereas prr1, prr2, and vin857 are shown in blue. (b) The initial structural model of the CAP-SH3ab:vin857 complex. (c) Pool of conformations of the complex generated by MD simulations, using an all-heavy-atom SBM. (d) The EOM ensemble selected from the conformation pool. (e) The SAXS curve calculated (red solid line) for the EOM ensemble with χ = 0.26, compared to the experimental SAXS profile (black solid line). All structures were created using VMD.[50]

The SAXS data for the CAP-SH3ab:vin₈₅₇ complex were collected at beamline 12ID-B of the Advanced Photon Sources (APS) at Argonne National Laboratory (Chicago, IL, USA) using a wavelength of 1.033 Å . The data were analyzed using the ATSAS package.^[21] The radius of gyration (R_g) of the protein, calculated from the PDDF (Fig. 1(b)), is approximately 22.5 Å .

An initial structure for the vin₈₅₇ peptide was first built by homology modeling, in which the structures of prr1 and prr2 were taken from the crystal structures of SH3b:prr1 and SH3a:prr2, respectively (Fig. 2(a)). It was then straightforward to dock the SH3a and SH3b domains on the vin₈₅₇ peptide, and an initial model of the CAP-SH3ab:vin₈₅₇ complex was obtained (Fig. 2(b)).

The configurational space of the complex was explored via CG MD simulations using an all-heavy-atom SBM (Fig. 2(c)).^[45] By loading the initial structure of the complex on the SMOG web server, ^[46] topology and coordinate files were generated for the MD simulations using the GROMACS-4.5.5 package.^[47] The native contacts were defined using a shadow algorithm.^[48] To facilitate sampling of possible conformations, all stabilizing contacts were removed from the topology file, except those present in the crystal structures SH3a:prr2 and SH3b:prr1. Two independent MD simulations with different initial atomic velocities were performed. Each simulation was coupled to a temperature bath of 50.0 in reduced units via Langevin dynamics. A time-step of 0.002 time units was used and the total simulation time was 10⁵ time units, which corresponds to approximately 500 microseconds.^[49]

The EOM was used to screen for conformations in the MD trajectories that fit the SAXS data for the complex. In the ensemble chosen from the first simulation (Fig. 2(d)), the R_g values were between 21.4 Å and 23.8 Å , with an average of 22.1 Å . The theoretical SAXS profile was concordant with the experimental curve, with χ = 0.26 (Fig. 2(e)). The ensemble from the second simulation (data not shown) contained conformations with R_g ranging from 21.1 Å to 24.2 Å (the average was also 22.1 Å ), with χ = 0.26 when fit with the experimental data. The two independent simulations can yield similar results via EOM analysis, which indicate the ensemble shown in Fig. 2(d) is reliable. The theoretical scattering profile of each conformation in the MD trajectories was computed using CRYSOL. Fitting with the experimental SAXS data, all χ values of theoretical profiles were > 1.0. This result suggests that the CAP-SH3ab:vin ₈₅₇ complex is flexible in solution, and that it would therefore be better represented as an ensemble instead of as a single dominant conformation.

Structural modeling methods that integrate SAXS data have been established, and this strategy shows promise for constructing structural models of proteins. One important issue for these screening-after-sampling approaches is to generate a pool of structures that adequately covers the configurational space of the protein. Therefore, the algorithms need to be improved to accelerate sampling efficiency. Although it is efficient to determine the structures of large multi-domain proteins or protein complexes using SAXS-integrated structural modeling methods, it should be noted that the resolution of SAXS data is inherently low, which may lead to ambiguous results. To tackle this over-fit problem, efforts should be made to improve the algorithms used in the screening step. Use of experimental data from other techniques, such as mass spectrometry and cross-link experiments, in addition to SAXS data may be helpful in distinguishing these ambiguous structural models. Future research should investigate combining these methods with other low-resolution datasets for validation.