The Fourier transforms of curves and filaments and their application to low-resolution protein crystallography

Abstract

A numerical method for computing the Fourier transform of an arbitrary space curve is described. The method is applicable to all sufficiently smooth curves and relies on the local geometric parameters describing a curve. The numerical results for a helical curve are compared with the exact analytical theory for the transform of a helix. It is shown that the transform of a filamentary density distribution radially symmetric around a curve is equivalent to the transform of that curve scaled by an appropriately defined weight function. These filamentary density distributions in conjunction with the numerical transform evaluation method can be used for simulating low-resolution diffraction data for protein crystals. Crystallographic structure factors obtained from a filament model representing a simple three-helix-bundle protein are compared with those calculated from conventional coordinate models. At low resolution, the filamentary representation provides an excellent approximation of the structure factors obtained from the standard coordinate model, but requires far fewer independent parameters.