Rapid calculation of the solution scattering profile from a macromolecule of known structure

If one expands the structure factor equation in spherical coordinates, rotational averaging of the molecular Fourier transform, which leads directly to the solution scattering profile, is greatly simplified. It becomes a projection in the polar and azimuthal angular variables. The profile is given by I(R) = 1/2 infinity sigma n = 0 n sigma m = 0 epsilon mNm,n magnitude of Gm,n(R) 2 where Gm,n(R) = sigma jfjYm,n(theta j, phi j)jn(2 pi rjR) The index j runs over all atoms; r, theta, phi are atomic coordinates and epsilon and N are constants; the Ym,n are complex spherical harmonics, and jn are spherical Bessel functions; R = 2 sin theta/lambda. The effects of solvent have been modeled by subtracting from each protein atom a properly weighted water. Hydrogens have been included by using scattering curves fj derived from the spherical averaging of protein atoms with their attached hydrogens. This approach may also be satisfactory for neutron scattering. Published scattering profiles for lysozyme and BPTI have been accurately matched in less than one-tenth the time required by other methods. Separate, adjustable temperature factors for the protein, solvent waters, and bound waters are used, and appear to be needed. In the case of BPTI, as suggested by NMR observations, the observed diffraction pattern was much better accounted for by including only 4 tightly bound waters rather than the roughly 60 seen by crystallography.