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

Abstract

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.

Figure 1

Reweighting ensembles using SAXS data calculated using different values for the parameters that effect the contribution from for the hydration layer and displaced solvent. The grids show the results from the iBME ensemble optimization with different combinations of δρ and r₀. The top row (a–c) shows Hst5, the second row (d–f) shows Sic1, the third row (g–i) shows Tau, and the last row (j–l) shows results for TIA1. For each protein, we show in the first column (a, d, g, and j) $\text{ln}\left({\chi }_{\text{red}}^2\right)$, we show in the second column (b, e, h, and k) φ_eff, and we show in the third column (c, f, i, and l) $\gamma =\text{ln}\left(\frac{{\chi }_{\text{red}}^2}{{\phi }_{\text{eff}}}\right)$. White spots correspond to ensembles in which the iBME reweighting failed. To see this figure in color, go online.

Figure 2

Comparing ensembles relative to the optimum. For each protein (a: Hst5, b: Sic1, c: Tau and d: TIA1) we calculated the effective fraction of frames (shown here as φ_eff) between the weights obtained using the parameters in Table 1 and the weights obtained at all other combinations of r₀ and δρ. White spots correspond to ensembles in which the iBME reweighting failed. Purple spots correspond to the minima for γ. To see this figure in color, go online.

Figure 3

Effect of the δρ and r₀ parameters on reweighted probability distributions of R_g. We use Sic1 as an example and show p(R_g) from both the optimal (lowest γ) parameters (blue) as well as three other choices of r₀ and δρ in the low-γ region (orange, green, and red). The insert shows the parameters used in each case and the results of the reweighting on the R_g distribution. To see this figure in color, go online.

Figure 4

Effect of the prior distribution. (a) Distributions of R_g of α-Synuclein sampled with flexible-meccano (FM), a99SB-disp (disp), and a03ws. (b) Reweighted R_g distributions, either from the optimal (lowest γ) δρ and r₀ parameters for each ensemble (solid lines) or using the default values, we propose (δρ= 3.34 e/nm³ and r₀ = 1.681 Å; dotted lines). To see this figure in color, go online.

Figure 5

Reweighting α-Synuclein ensembles using SAXS data calculated using different values for the parameters that effect the contribution from the hydration layer and displaced solvent. The grids show the results from the iBME ensemble optimization with different combinations of δρ and r₀. The top row (a–c) shows the results from the flexible-meccano ensemble, the second row (d–f) shows the results using a99SB-disp as the prior, and the third row (g–i) shows the results from a03ws as the prior. For each ensemble we show in the first column (a, d, and g) $\text{ln}\left({\chi }_{\text{red}}^2\right)$, in the second column we show (b, e, and h) φ_eff, and in the third column (c, f, and i) we show $\gamma =\text{ln}\left(\frac{{\chi }_{\text{red}}^2}{{\phi }_{\text{eff}}}\right)$. White spots correspond to ensembles in which the iBME reweighting failed. Purple spots in the third column correspond to the minima for γ. To see this figure in color, go online.

Figure 6

Estimating ‹R_g› from experimental SAXS profiles of (a) Hst5, (b) Sic1, (c) α-Synuclein, (d) Tau, and (e) TIA1 using Guinier fitting and ensemble refinement. We used the Guinier approximation to estimate R_g by fitting from the lowest measured value of q (in the case of Hst5 we ignored the first 10 points due to noise) to different values of q_max, reporting the results as R_g vs. q_maxR_g (black circles). The horizontal black lines are the ensemble-averaged R_g calculated from the conformational ensembles (in the case of α-Synuclein, we used the flexible-meccano prior) with the chosen optimal r₀ and δρ parameters (Table 1).