Reciprocal space representations of helical objects and their projection images for helices constructed from motifs without spherical symmetry

Formulas for predicting the 3-D reciprocal space cube and 2-D reciprocal space images that are slices of the cube (corresponding to real space images that are projections of the real space cube) are derived for the case when the motif of a helical object is not spherically symmetric. These formulas generalize much-used formulas due to Cochran et al. [The structure of synthetic polypeptides. I. The transform of atoms on a helix, Acta Crystallogr. 5 (1952) 581–586], for the case of a spherically symmetric motif. The new formulas allow control of the spatial resolution of the motif ranging from low-resolution spherically symmetric motifs as in Cochran et al. through moderate-resolution nonsymmetrical motifs to atomic-resolution nonsymmetrical motifs which can also be treated by the methods of Cochran et al. as a superposition of spherically symmetric atomic motifs. The ability to control resolution may be useful in reconstruction algorithms analogous to phase extension algorithms in X-ray crystallography.