In the infinite dilution limit, assuming that the macromolecule has one preferential conformation, the scattering intensity of a sample solution containing the macromolecule is proportional to the scattering intensity of a single macromolecule of scattering density (
r), surrounded by a solvent of average electron density . The scattering intensity of this macromolecule at a given value of the wavevector norm
q is the spherical average of the scattering intensity on the sphere of radius
q in reciprocal space (
Equation 1).
where A is the excess form factor of the system and are respectively the form factor of the solute
in vacuo and of the solvent-excluded-volume. is the excess form factor of the hydration shell.
We perform the spherical averaging using the cubature formulae (30 ). Following the effective-atomic-scattering-form-factor method (25 ), the solute and the solvent-excluded-volume's form factors are computed as a sum over all N non-Hydrogen atoms of the solute (
Equations 2 and
3).
where
fj is the atomic form factor
in vacuo (19 ), computed as a sum of a constant and four gaussians whose parameters depend on the atom type, is the solvent volume displaced by atom
j. is an overall expansion factor, as defined in (19 ), with C
1 being the ratio of the adjusted and computed average atomic radii (default value = 1.0).
Programs such as AquaSol (28 (
link)) [or 3D-RISM/Amber (29 )] compute solvent density maps around the solute, based on the physical interactions within the system. The method used in AquaSol is based on the Poisson–Boltzmann formalism, where the solvent is no longer described as a continuum dielectric medium but rather as an assembly of self-orienting dipoles of variable density on a grid. It was shown that the resulting water distribution is in good agreement with experimental data and with the chemical nature of the atoms exposed to the solvent, both at the atomic and residue-level (26 (
link)). These maps are typically cubic grids of given size and resolution (a), where each grid point r is associated to a given density value . Basically, in such maps, one expects a density of 0 inside the solute, and 1 (in units of bulk density ) in the bulk region of the solvent, i.e
. far from the solute. At the boundary between the solute and bulk region, the density is determined by the physico–chemical nature of the environment.
We compute the form factor for the hydration shell as in
Equation 4.
The sum runs over all points with nonzero density. In practice, to reduce computation time, grid points with a density close to 1 (i.e
. typically within 1.10
−4) are removed from the sum. On Urate Oxidase (example mentioned below), allowing a tolerance of 1.10
−6 slowed down the computation by a factor of three and did not affect the resulting profile: the same fitting parameters were found, and the goodness-of-fit (cf
Equation 6) was similar (1.688 versus 1.691).
Besides the solute and solvent, another possible contributor to the SAXS profile is the ion atmosphere surrounding the solute. AquaSol (28 (
link)) computes the density maps of free cations and anions, and, in principle, these maps could be used to compute the excess form factors of ions. However, at physiological concentrations (200 mM NaCl) the ratio of the fugacities of ions and water is < 0.5%. At this stage, the contribution of ions was not implemented into AquaSAXS, except in the form of explicitly bound and fixed ions. Nevertheless, the presence of free ions can indirectly affect the solvent density in the hydration shell (screening effect), so the user is prompted for the ionic strength of the solution.
Alternatively, the hydration shell's form factor can be computed as in FoXS (20 (
link)), following
Equation 5.
where is the fraction of solvent accessible surface of the atom
j (31 (
link)) and is the water form factor. is a scale factor used to adjust the hydration shell's contribution (default value = 1.0).
The computed profile is fitted to a given experimental SAXS profile (with experimental error ) by minimizing the goodness-of-fit function with respect to three adjustable parameters:
C1,
C2 and
C (
Equation 6).
C1 and
C2 values are scanned within a given range ( , and ), in steps of 0.0055, 0.014 and 0.04,
resp., and for each pair, a linear-least-squares minimization is performed to adjust the scaling constant C. The pair leading to the minimal
χ is kept to compute the returned profile.
Poitevin F., Orland H., Doniach S., Koehl P, & Delarue M. (2011). AquaSAXS: a web server for computation and fitting of SAXS profiles with non-uniformally hydrated atomic models. Nucleic Acids Research, 39(Web Server issue), W184-W189.