Interconversion between 3D molecular representations: some macromolecular applications of spherical harmonic-Bessel...

An accurate interconversion of three-dimensional molecular spatial information between two alternative orthogonal representations is demonstrated. These alternative descriptions are useful for exhaustive six-dimensional grid searches of empirical energy functions and for estimating the contents of asymmetric units in sparsely packed, non-centrosymmetric crystalline arrays. To illustrate the fidelity of this interconversion, it is implemented to perform an exhaustive grid search in six-dimensional relative rotational and translational space (phase space) with the spherical harmonic-Bessel representation. This search is an extension of Crowther's crystallographic fast rotational overlap function from Patterson space to direct, three-dimensional space. The search uses a three-dimensional spherical harmonic expansion about a single center to represent a probe molecule's shape, charge and van der Waals parameters and numerous complete expansions about single centers to represent the complementary shape, electrostatic potential and van der Waals potential of several target sites about the complementary molecule. The rapid calculation of unique, complete expansions about numerous sites on a grid is made possible by application of the Fourier convolution theorem. Expansions from nearby grid points can be used to reconstruct the same molecular shape accurately, if this shape is contained entirely within the spherical zone of expansion about each grid point. This arbitrariness for the choice of origin, and the accuracy of interconversion between three-dimensional spherical harmonic-Bessel and Fourier representations, suggest a method for describing the contents of non-centrosymmetric, sparsely packed crystals. Incomplete Fourier representations (diffraction amplitudes) provide complex spherical harmonic-Bessel coefficients for a crystal packed with symmetry-enforced, non-centrosymmetrically arranged, non-overlapping spherical expansion zones.