SoftWAXS: a computational tool for modeling wide-angle X-ray solution scattering from biomolecules

Abstract

This paper describes a computational approach to estimating wide-angle X-ray solution scattering (WAXS) from proteins, which has been implemented in a computer program called SoftWAXS. The accuracy and efficiency of SoftWAXS are analyzed for analytically solvable model problems as well as for proteins. Key features of the approach include a numerical procedure for performing the required spherical averaging and explicit representation of the solute-solvent boundary and the surface of the hydration layer. These features allow the Fourier transform of the excluded volume and hydration layer to be computed directly and with high accuracy. This approach will allow future investigation of different treatments of the electron density in the hydration shell. Numerical results illustrate the differences between this approach to modeling the excluded volume and a widely used model that treats the excluded-volume function as a sum of Gaussians representing the individual atomic excluded volumes. Comparison of the results obtained here with those from explicit-solvent molecular dynamics clarifies shortcomings inherent to the representation of solvent as a time-averaged electron-density profile. In addition, an assessment is made of how the calculated scattering patterns depend on input parameters such as the solute-atom radii, the width of the hydration shell and the hydration-layer contrast. These results suggest that obtaining predictive calculations of high-resolution WAXS patterns may require sophisticated treatments of solvent.

Figure 1

Two approaches to modeling the excluded volume using atomic or group contributions. (a) The sum-of-excluded-volume functions. (b) The union-of-excluded-volume functions. For clarity, in these figures the atomic contributions have been modeled as hard spheres rather than Gaussians.

Figure 2

(a) Illustration of the atoms that define the line segment along which we plot the sum-of-excluded-volume function. (b) Sum of excluded volumes along the indicated line segment. Distances are in Å. The atomic excluded-volume functions are defined as in CRYSOL (Svergun et al., 1995 ▶). Along the plotted axis, the ASP66:CA atom is at 10⁵ Å on this line and the ALA122:N atom is at 44.0 Å.

Figure 3

The SoftWAXS hierarchical-cube approach to estimating excluded-volume and hydration-layer scattering. (a) Definition of the excluded volume and hydration layer. (b) An illustration of the recursive octree decomposition.

Figure 4

Calculations of the atomic form-factor contribution to scattering for lysozyme [PDB accession code 6lyz (Diamond, 1974 ▶)]. Calculations have been performed using the Debye formula, CRYSOL and SoftWAXS using 144, 400 or 900 quadrature points to evaluate the spherical averages. The curve for SoftWAXS using 900 quadrature points is indistinguishable on this scale from the curve obtained using the Debye formula. The scattering angle is in Å. Units on the ordinate axis are arbitrary intensity units.

Figure 5

Excluded-volume and hydration-layer contributions to scattering for lysozyme. Calculations have been performed with SoftWAXS using 900 quadrature points to evaluate the spherical averages. The results using an octree depth of 5 clearly show artifacts at wide angles ( Å) due to the use of coarser cubes. Units on the ordinate axis are arbitrary intensity units. Excluded-volume contributions to are as follows: for 5-level octree, ; for 6-level octree, ; for CRYSOL, .

Figure 6

Calculated scattering of lysozyme, using either the union-of-hard-spheres approach (with the surface defined as the solvent-excluded surface using a probe sphere of radius 1.4 Å) or the sum-of-Gaussians method (Svergun et al., 1995 ▶). The hard-sphere radii have been set to the optimal radii for fitting to lysozyme experimental data. The volumes for the atoms are: C 44.60, O 7.24, N 1.44, H 4.19 Å. Using this approach the excluded-volume contribution is . See §4.3.4 for more details. The sum-of-Gaussians calculation employed the same atomic excluded volumes as used in CRYSOL: C 16.44, O 9.13, N 2.49, H 5.15 Å (Fraser et al., 1978; Svergun et al., 1995 ▶). The excluded-volume contribution is .

Figure 7

Calculation of the excluded-volume scattering due to three spheres using analytical methods and the numerical methods in SoftWAXS. The scattering intensities calculated using 400-point and 900-point numerical quadrature for spherical averaging are indistinguishable at this scale. Using an octree depth of 5, the SoftWAXS calculation is essentially indistinguishable from the analytical solution except at the minimum at  Å, where the semilog scale provides adequate resolution to observe the minute discrepancy.

Figure 8

Calculations of WAXS patterns are sensitive to both hydration-layer thickness and contrast for  Å. For larger the parameters, when varied within reasonable limits, have relatively little effect on the pattern. In each plot the products of hydration-layer density (relative to bulk water) and thickness are the same for the two patterns. (a)  e Å⁻², (b)  e Å⁻², (c)  e Å⁻², (d)  e Å⁻².

Figure 9

A comparison of lysozyme scattering from experiment, CRYSOL and SoftWAXS. (a) Scattering calculated using CRYSOL [both with and without fitting to experiment (Svergun et al., 1995 ▶)]. (b) Scattering patterns calculated using SoftWAXS while varying the C, O and N radii; see text for details. For comparison, the experimental data have also been plotted (dashed line). (c) Scattering from experiment and the optimal radii.

Figure 10

A comparison of cytochrome c scattering from experiment, CRYSOL and SoftWAXS. (a) Scattering calculated using CRYSOL [both with and without fitting to experiment (Svergun et al., 1995 ▶)]. (b) Scattering patterns calculated using SoftWAXS while varying the C, O and N radii; see text for details. For comparison, the experimental data have also been plotted (dashed line). (c) Scattering from experiment and the optimal radii.