Solution x-ray scattering and structure formation in protein dynamics

We propose a computationally effective approach that builds on Landau mean-field theory in combination with modern nonequilibrium statistical mechanics to model and interpret protein dynamics and structure formation in small- to wide-angle x-ray scattering (S/WAXS) experiments. We develop the methodology by analyzing experimental data in the case of Engrailed homeodomain protein as an example. We demonstrate how to interpret S/WAXS data qualitatively with a good precision and over an extended temperature range. We explain experimental observations in terms of protein phase structure, and we make predictions for future experiments and for how to analyze data at different ambient temperature values. We conclude that the approach we propose has the potential to become a highly accurate, computationally effective, and predictive tool for analyzing S/WAXS data. For this, we compare our results with those obtained previously in an all-atom molecular dynamics simulation.

Figure 1

Top: Comparison of bond $\left({\kappa }_i\right)$ angles in the first NMR entry of the PDB structure 2JWT (dashed red line) and in the ensuing multisoliton solution (continuous blue line). Bottom: Comparison of torsion $\left({\tau }_i\right)$ angles in the first NMR entry of the PDB structure 2JWT (dashed red line) and in the ensuing multisoliton solution (continuous blue line); note that the torsion angle is defined as $mod\left(2\pi \right)$ and thus the apparent differences at sites $i=24$ and $i=59$ are smaller than they appear.

Figure 2

The residue-wise distance between the $\mathrm{C}\alpha$ atoms in the PDB structure 2JWT (first entry) and the ensuing multisoliton; the gray area corresponds to a 0.2 Å zero point fluctuation distance.

Figure 3

The interlaced $\mathrm{C}\alpha$ traces of the PDB structure 2JWT (first entry) (in light green) and the ensuing multisoliton (in dark blue).

Figure 4

Comparison between simulated temperature dependence in $\alpha$-helical content and the corresponding experimental data observed using 222-nm CD spectra in wild-type Engrailed. The experimental data are adapted from Fig. 3 (bottom) of Ref. [50].

Figure 5

Distribution of the radii of gyration in mean-field model simulations at different Glauber temperature factor values.

Figure 6

Distribution of the radii of gyration, as recovered from the experimental S/WAXS data at different temperatures.

Figure 7

Comparison between the temperature dependence of radius of gyration in the mean-field model and in the S/WASX data. The error bars denote the one $\sigma$ of the Gaussian fits, in Figs. 5 and 6, respectively.

Figure 8

Distribution of the radii of gyration in the MD simulations.

Figure 9

Dependence of the ${\chi }^2$ values obtained by fitting mean-field model simulations, complemented with Pulchra all-atom reconstruction. The solid lines connecting circles are a guide for the eye. (Top curve ${20}^{\circ }$, second curve ${30}^{\circ }$, third curve ${40}^{\circ }$, and bottom curve ${55}^{\circ }$.) Also shown separately are ${\chi }^2$ values we obtain using a simplified $\mathrm{C}\alpha \text{−}\mathrm{C}\beta$ (mock alanine) structure in Pulchra.

Figure 10

Top: Fitting of the Landau mean-field model pools generated at $k\theta ={10}^{-9}$ with the experimental S/WAXS. Bottom: Same as in the top panel for the mixed-temperature MD pool with the experimental S/WAXS. Plots are shifted vertically to enhance visibility. The experimental errors are shown in a light color. The inset (second plot) in the top and bottom panels shows the difference between experimental and fitted profiles. The inset corresponds to 1% of the scattering intensity extrapolated at $q=0$ (top curves ${55}^{\circ }$, second curves ${40}^{\circ }$, third curves ${30}^{\circ }$, and bottom curves ${20}^{\circ })$.