The fundamental aim of structural analyses in biophysics is to reveal a mutual relation between a molecule’s dynamic structure and its physiological function. Small-angle X-ray scattering (SAXS) is an experimental technique for structural characterization of macromolecules in solution and enables time-resolved analysis of conformational changes under physiological conditions. As such experiments measure spatially averaged low-resolution scattering intensities only, the sparse information obtained is not sufficient to uniquely reconstruct a three-dimensional atomistic model. Here, we integrate the information from SAXS into molecular dynamics simulations using computationally efficient native structure-based models. Dynamically fitting an initial structure towards a scattering intensity, such simulations produce atomistic models in agreement with the target data. In this way, SAXS data can be rapidly interpreted while retaining physico-chemical knowledge and sampling power of the underlying force field. We demonstrate our method’s performance using the example of three protein systems. Simulations are faster than full molecular dynamics approaches by more than two orders of magnitude and consistently achieve comparable accuracy. Computational demands are reduced sufficiently to run the simulations on commodity desktop computers instead of high-performance computing systems. These results underline that scattering-guided structure-based simulations provide a suitable framework for rapid early-stage refinement of structures towards SAXS data with particular focus on minimal computational resources and time.
We investigated structural transitions in three two-state protein systems, where the target structure was initially known.
For the VHP-based polypeptide system (Fig 1A inset), proof-of-concept simulations aimed at both transitions from bent to elongated and elongated to bent conformation. Target scattering data were calculated via the Debye equation (Eq 1). We analyzed the backbone’s elongatedness by extracting distances between N-terminal and C-terminal Cα atoms for both free and scattering-guided SBM simulations. The distance distributions (Fig 1A) show a clear shift towards the target structure’s end-to-end distance. This confirms that, as intended, conformations which are not in accordance with the target curve are avoided in scattering-guided simulations. To show the degree of similarity between typical X-ray scattering patterns from the refinement and the target data, Debye intensities of simulated structures are illustrated in Fig 1B. The curves converge to a certain extent, but do not show perfect agreement. This is due to the fact that the refinement is not only steered by the scattering bias, but also by the physico-geometrical SBM, so that an equilibrium between these two contributions settles in. Computation time scaled as 1. 4 to 1 for scattering-guided and free SBM simulations. As the Debye summation is an O (N 2) problem (see Eq 1), the ratio of scattering-guided and free computation times will substantially increase with system size. In this context, rapid evaluation of SAXS profiles from structural models becomes even more important for dynamic refinement procedures such as scattering-guided biomolecular simulations. Time-dependent RMSD curves and bias potential are depicted in Fig 1C and Panel A of S4 Fig for elongated-to-bent and bent-to-elongated transition, respectively. Guiding the simulations towards the target scattering data obviously causes the structural transition to bidirectionally occur back and forth. Minimum target RMSDs are 0. 15 nm (Fig 1D) and 0. 19 nm (Panel B in S4 Fig), respectively. This implies that structure-opening conformational changes from rather compact to more spacious structures are more difficult to sample than structure-closing ones. Free SBM simulations of bent and elongated conformation yielded average Cα RMSDs of 0. 17 nm and 0. 18 nm, respectively. In light of this, scattering-guided SBM simulations were capable of reproducing each target structure with the method’s inherent best possible accuracy. Despite the drastic change in secondary structure, they could model the conformational transition in both directions properly and persistently sample physically reasonable structures near the target conformation. Result parameters are summarized in Table 1 along with the values from analogous explicit-solvent MD simulations. Considering computation times τ 0. 2 comp, the structure-based method turned out to be faster by two orders of magnitude than the full-MD approach in terms of wall-clock time. Detailed explicit-solvent MD results can be found in S5 and S6 Figs. The bias energy was analyzed as a function of the trajectory’s target RMSD. As displayed in Fig 4A, low bias potential is principally associated with low target RMSD. A Pearson correlation ρ of 0. 44 indicates that they are in fact positively correlated. However, we find a considerably spread ensemble of distinct structures at equal bias potential levels. With a bias potential of 0. 88 ε, the minimum target RMSD structure is not exactly in the energetic minimum. Results for the reverse transition are presented in Panels C and D in S4 Fig. Analogous explicit-solvent MD results can be found in Panels C and D in S5 and S6 Figs. To examine the influence of temperature and bias weight, we conducted grid-search variational studies. Results are depicted in Fig 5A and 5B for elongated-to-bent and bent-to-elongated transition, respectively. As soon as the initially increasing V XS av (k χ) drops down (T = 50,70) or plateaus (T = 90,110), RMSD target min converges towards the average value of related free simulations (see Table 1). Average χ2 dissimilarity of simulated scattering curves with respect to the target data minimizes accordingly. Near these turnaround points labeled k χ * hereafter, structure-based potential and scattering bias are assumed to be thoroughly balanced. This promotes rapid conformational transitions according to the target data in due consideration of the physico-geometrical model, but prevents the data from being overfitted. In SBMs, the bias potential has to be weighted in such a manner as to introduce a distinct competing minimum to the original single-basin energy funnel. We set k χ ∼ O (k χ *) to ensure occurrence of a clear transition, whilst modifying the underlying regular potential as little as possible. The elongated-to-bent transition yielded a smaller k χ * ≈ 1 · 10 - 8 ε (see Fig 5A) compared to the bent-to-elongated case with k χ * ≈ 5 · 10 - 8 ε (see Fig 5B). This behavior confirms our previous finding of structure-closing transitions to be favored over structure-opening ones. For both transitions, we find the effect of gradually increasing the coupling constant to be less pronounced at higher temperatures (Fig 5). It is conspicuous that, independent of kχ, all simulations at T = 110 could—at least temporarily—sample conformations near the target. The increased thermal energy allows to overcome potential barriers in the energy landscape, resulting in greater protein flexibility and sampling power. However, these thermal structural fluctuations by itself are isotropic in conformational space and not directed towards any particular conformation as is the case with a major scattering bias. In contrast, at lower temperatures T = 50 and 70, RMSD target av values almost double with kχ increasing up to the order of 10−8 ε, before they significantly decline as well. With less thermal energy being available and an increased coupling of simulations to scattering intensities attaching more relative importance to structural information from SAXS, this is in accordance with the expectations. Remember that the global change in orientation of secondary structure elements with respect to each other substantially affects the polypeptide’s overall shape. As a consequence, this system required large temperature (and bias weight) to ensure sufficient global conformational flexibility and stably reach a conformation near the target. Though a basic trend should be maintained, these findings cannot be directly translated to other protein systems. The optimal combination of temperature T and bias weight kχ depends on the individual system and should be determined by grid search or other systematic parameter optimization methods. In SBMs, the overall contact and dihedral energy is set equal to the number of atoms in the system [49]. This choice yields folding temperatures near 1 in the structure-based reduced units, corresponding to approximately 120 reduced GROMACS temperature units, and ensures a consistent parameterization. Thus, model-inherent absolute energies are highly system-specific and not comparable among different systems. Not only differ biomolecular systems in general and thus their respective absolute energies, but also the nature of their individual conformational transitions each associated with a specific energy barrier of different (unknown) height. Due to the high diversity among biomolecular systems, different systems require different bias weights and temperatures to suitably impact the underlying structure-based potential and provide sufficient thermal energy to induce or accelerate the conformational transition of interest. These parameters are not transferable and have to be determined separately for each system. Modeling the large-scale structural transition between open and closed conformation based on artificial difference data gives a theoretically constructed test example of a real protein movement. Simulations started from open and closed state and aimed at closed and open state, respectively. We computed artificial target difference data (Fig 2B) using the Debye equation on amino-acid level. RMSD and bias energy curves of open-to-closed and closed-to-open transition are shown in Fig 2C and Panel A in S7 Fig, respectively, illustrating how the structural similarity to initial and target state develops over the course of the simulations. Both refinements showed one clear transition from initial to target conformation in form of an immediate intercept of RMSD curves. VXS instantaneously minimized accordingly. Subsequently, the target RMSD curves proceeded near respective average free RMSD values. Best structures as measured by target RMSD and bias energy are shown in Fig 2D and Panel B in S7 Fig, respectively. Minimum target RMSD is 0. 14 nm for both directions of the conformational transition. Detailed results of analogous explicit-solvent MD simulations can be found in S8 and S9 Figs. All result parameters are summarized in Table 2. Considering computation times τ 0. 2 comp, structure-based refinements turned out to be faster by almost two orders of magnitude than the full-MD approach, while yielding more accurate structures in terms of minimum target RMSD. We analyzed the bias energy as a function of target RMSD (Fig 6A and Panel C in S7 Fig), which revealed positive Pearson correlations throughout. As a result of the almost instantaneous structural transitions, numerous similar conformations with small bias energy and target RMSD less than 0. 2 nm do exist, yielding a dense cluster of fluctuating points (RMSDtarget, VXS) in this area. This behavior disrupts a potential linear relationship between these quantities as assumed in the Pearson correlation analysis, causing rather small but certainly positive values for ρ. We conducted grid-search variational studies for both open-to-closed (see Fig 7A) and closed-to-open (see Fig 7B) transition. As indicated by the regions of undefined bias potential in Fig 7, simulations using a bias weight kχ greater than 10−8 ε apparently blew up. Depending on χ2, an immoderate bias weight may produce a very large bias potential. This generates an unacceptably large force, which eventually results in a failure of the integrator. For both directions of the conformational transition, the turnaround bias weight k χ * is in the order of 10−10 ε. At this point, the bias potential clearly exhibits its global minimum and average χ2 dissimilarity significantly drops down accordingly. In contrast to the VHP polypeptide, lower temperatures were sufficient to stably sample conformations near the target structure. This is due to the fact that the structural transition of AKE does not induce a drastic overall change in its molecular shape. Upon binding lysine, LAO protein (Fig 3A) experiences major structural change [56]. Modeling this domain motion based on artificial difference data gives another test case of a real protein movement. Starting from the crystal structure of the unliganded holo state, these simulations aimed at the unbound apo state and vice versa. Reference and target scattering were calculated from the crystal structures with CRYSOL [57] and thus implicitly include hydration shell contributions. We generated artificial difference data (Fig 3B) by subtracting the initial solution scattering from the target solution scattering. Time-dependent initial and target RMSD as well as bias potential are shown in Fig 3C and Panel A in S10 Fig for structure-based holo-to-apo and apo-to-holo simulations, respectively. Biasing simulations towards theoretical difference data resulted in the transitions to readily occur. The bias potential minimized almost instantaneously according to the trajectory’s convergence towards the target state. The final target RMSD of approx. 0. 2 nm was consistent with corresponding free simulations. For both directions of the conformational transition, best structures exhibit a minimum target RMSD of 0. 09 nm (Fig 3D and Panel B in S10 Fig). Structure-based refinements were capable of producing structures in full agreement with the target state. Provided equal computing resources, they required only a small fraction of computing time by comparison with analogous explicit-solvent MD simulations. Detailed full-MD results can be found in S11 and S12 Figs. Considering the holo-to-apo transition, the explicit-solvent refinement (tsim = 10 ns) lasted for 4 d 15 h 5 min 35 s, whereas the SBM run (tsim = 2000 arb) spanned 4 h 11 min 55 s. According to computation times τ 0. 2 comp related to the trajectory approaching a state with a target RMSD less than 0. 2 nm, the SBM proved to be up to ten times faster in terms of wall-clock time. All result parameters are summarized in Table 3. As in the other test systems, bias energy and target RMSD exhibit positive Pearson correlations throughout. According to Fig 8A and Panel C in S11 Fig, the structural diversity at equal bias potential levels is similar for SBM and explicit-solvent MD. Though the best structure cannot definitely be identified from a trajectory on the basis of VXS on its own, the bias potential can serve as a primary indicator for a simulation’s current state and eventual success or failure. Grid-search variational studies for both holo-to-apo and apo-to-holo transition revealed a similar behavior as for AKE. As highlighted by the regions of undefined bias potential in Fig 9, simulations applying a bias weight kχ greater than 6 ⋅ 10−8 ε failed due to excessively large scattering-related forces. For both directions of the structural transition, the turnaround bias weight k χ * is approx. 10−10 ε. Average χ2 dissimilarity clearly minimizes here (Fig 9A and 9B, bottom), whereas the bias potential does not have a distinct minimum as is the case for AKE but starts to monotonically increase as a function of kχ (Fig 9A and 9B, middle). Again, the evolution of minimum target RMSD indicates lower temperatures to be sufficient to stably reach the target conformation (Fig 9A and 9B, top). This is due to the fact that the conformational transition corresponds to a relative movement of subdomains in the structure so that the molecular shape does not experience a drastic overall change. To assess the method’s robustness towards errors in the scattering data, we conducted a structure-based refinement of LAO protein’s holo-to-apo transition towards noisy artificial difference data. Theoretical absolute scattering curves of reference and target structure were blurred according to a random Gaussian noise. For each q point, mean and standard deviation were modeled as the related clean intensity value and its square root, respectively. Details are described in S3 Appendix. We calculated noisy difference data by subtracting the blurred reference intensity from the blurred target intensity (Fig 10). Using usual error propagation, errors were calculated as the sum of the Gaussians’ absolute standard deviations and used to individually weight the q points in the simulation. We applied the same parameters as in the refinement towards clean data. Although the bias potential levels off at a considerably higher value, which is to be expected, the simulation could produce equal-quality structures and thus proved to be robust against errors, at least for the level of noise assumed here (see S13 Fig). Note once more that scattering-guided SBM simulations dispense with computationally expensive solvent effects. In view of these results, we did not find a need for explicit solvation in refinement simulations comprising small-angle scattering data up to a maximum momentum transfer of 5 nm−1. The fact that SBM simulations coupled to small-angle difference scattering data could reproduce each target state with high accuracy indicates that such curves hold sufficient information to guide the simulation towards the correct conformation, at least for the systems studied here. Regardless of their reduced level of complexity by comparison with explicit-solvent MD, scattering-guided SBM simulations produced equal-quality results in a small fraction of computing time. A fundamental paradigm in protein biophysics is the interdependency of macromolecular structure and function. In light of this, small-angle X-ray scattering has significantly gained in importance, especially for structural analyses of dissolved macromolecules. Accurate interpretation of resulting scattering intensities in terms of atomistic models is still a challenging task. By incorporating information from SAXS into structure-based models, we aimed at efficiently interpreting scattering data within computational simulations. Studying three different test systems, we have proven our method to be capable of effectively probing real protein transitions, based only on low-resolution scattering data. Giving results equivalent to those from analogous full-MD methods [20], scattering-guided SBM simulations could expedite interpretation of intensities from biological SAXS by about two orders of magnitude. Such simulations benefit from extensive sampling as a result of their intrinsically accelerated dynamics. They could rapidly generate structural ensembles in accordance with the input data and provide a valuable alternative for efficient refinement of atomistic structures against SAXS data. Thus, they are particularly suitable for initial high-throughput analyses and can easily perform on usual commodity hardware. If desired, the resultant structure can still be given a final polish within a regular MD force field. As a result of technical advances in light sources and detectors, the wide-angle regime encoding local structural fluctuations has become increasingly accessible in the experiment. So as to level up with experimental resolution, increasingly fine-grained modeling may then be indicated at the cost of leaping computational demands [5,21]. Finally, it is important to note that some systems cannot be analyzed straightforward using SAXS. In all test systems, structural transitions could be modeled by a collective movement along one effective degree of freedom, which influences the protein’s shape and thus the difference curve at q ≤ 2 nm−1 crucially. As a consequence, structural fits were unique. However, at higher q values, multiple candidate structures can generate interfering features in the difference profiles. For example, the structural change in the cytoplasmic portion of a sensor histidine kinase protein (PDB code 2C2A [58]) induces a Cα RMSD shift of 1. 25 nm. This conformational transition can effectively be described as a rotation of one subdomain around a helix bundle, but influences the overall molecular shape only marginally. Despite a substantial decrease in bias potential, refinements towards theoretical difference data did not converge to the target structure. This implies that structures exist, which adequately reproduce the difference data, but are not compatible with structural models obtained from crystallographic methods. These findings are in accordance with results presented in Ref [20] and due to a lack of information in the low-resolution experimental data, resulting in unaccomplishably high demands on the theoretical model. Having said this, the protocol for interpretation of SAXS data within SBMs established in this work can serve as a suitable starting point for further developments. These include e. g. expanding single-basin SBMs to multi-Gō models with several minima and testing other functional forms of the bias potential. Furthermore, we intend to directly interface the structure-based refinement framework with parameter optimization methods such as Bayesian inference. In addition, we see several possibilities to extend our hybrid framework to additionally account for information derived from other experimental techniques than SAXS. We plan to extend the framework by considering co-evolutionary contact information from biomolecular sequence data [36], distance and angle information from NMR spectroscopy, and cryo-EM density maps [6]. Whereas co-evolutionary information can be considered by additional potential terms similar to usual SBM native contacts, NMR distance and angle information can be accounted for by implementing suitable spatial restraints. Provided cryo-EM data, another energetic term can be introduced to bias the structure towards the electron density map based on a spatial overlap. Performing simulations with such a hybrid structure-based/biased/restraint force field, the system can relax into configurations that are consistent with all these contributions.
As a starting point, all-atom SBMs were constructed from the considered system’s initial structure with eSBMTools [30] to obtain suitable coordinate and topology files. Debye scattering terms are encoded as a special type of bonded interaction in the topology file [20]. Scattering topology as well as related extended coordinate file were constructed with gmx genrestr. This command creates half a matrix of virtual-site type-3 pairs, i. e. Debye terms, for the input coordinate file. Amino-acid scatterers centered on virtual interaction sites at the respective residue’s center of mass were used. All residues were considered. The resulting topology include file was added to the system’s topology directly after the atoms section. The corresponding atom type ‘MW’ was manually appended to the atom types table. If Debye scattering data were used as a target, the initial scattering was calculated using gmx waxsdebye. Suitably adjusted run parameters for the SBM refinement are listed in S4 Appendix. Temperatures T and bias weights kχ were set as described in Results and discussion. Finally, SBMs were preprocessed with gmx grompp and run with gmx mdrun. The set-up of explicit-solvent MD simulations followed the common steps of adding hydrogen atoms, choosing potential and water model, neutralizing electric charge by adding an appropriate number of ions, minimizing energy, and equilibrating temperature and pressure. We used the CHARMM27 force field [60], TIP3P water model [61], Verlet cut-off scheme, and a constant temperature of 300 K. Electrostatics were treated with the Particle Mesh Ewald method. Parrinello-Rahman pressure coupling and V-rescale temperature coupling were applied. To obtain coordinate and topology file, initial models were preprocessed and protonated with gmx pdb2gmx. A periodic cubic box exceeding twice the longest inter-protein distance was constructed with gmx editconf. The structure was initially energy-minimized using the GROMACS preprocessor gmx grompp and simulation command gmx mdrun. After solvation and electric-charge neutralization, the structure was energy-minimized again. Subsequently, systems were equilibrated in the canonical and isothermal-isobaric ensemble until temperature and pressure converged. Non-hydrogen atoms were position-restrained to their initial positions. A half-matrix of Debye terms was constructed with gmx genrestr for the NPT-equilibrated structure, including all residues and using amino-acid scatterers. This created the scattering topology, which was manually included into the system’s topology. The initial reference scattering was generated with gmx waxsdebye. After preprocessing with gmx grompp, the scattering-guided MD simulation was performed using the gmx mdrun command. Results are shown for coupling strengths kχ optimized via grid-search variational studies comprising 16 simulations in total for each system.
Proteins are the molecular nanomachines in biological cells and thus vital to any known form of life. From the evolutionary perspective, viable protein structure emerges on the basis of a ‘form-follows-function’ principle. A protein’s designated function is inextricably linked to dynamic conformational changes, which can be observed by small-angle X-ray scattering. Intensities from SAXS contain low-resolution information on the protein’s shape at different steps of its functional cycle. We are interested in directly getting an atomistic model of this encoded structure. One powerful approach is to include the experimental data into computational simulations of the protein’s function-related physical motions. We combine scattering intensities with coarse-grained native structure-based models. These models are computationally highly efficient yet describe the system’s dynamics realistically. Here, we present our method for rapid interpretation of scattering intensities from SAXS to derive structural models, using minimal computational resources and time.
molecular dynamics
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Rapid interpretation of small-angle X-ray scattering data