Interconversion between 3D molecular representations: some macromolecular applications of spherical harmonic-Bessel expansions about an arbitrary center

An accurate interconversion of three-dimensional molecular spatial informat ion between two alternative orthogonal representations is demonstrated. The se alternative descriptions are useful for exhaustive six-dimensional grid searches of empirical energy functions and for estimating the contents of a symmetric units in sparsely packed, non-centrosymmetric crystalline arrays. To illustrate the fidelity of this interconversion, it is implemented to p erform an exhaustive grid search in six-dimensional relative rotational and translational space (phase space) with the spherical harmonic-Bessel repre sentation. This search is an extension of Crowther's crystallographic fast rotational overlap function from Patterson space to direct, three-dimension al 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. T he 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 mol ecular shape accurately, if this shape is contained entirely within the sph erical zone of expansion about each grid point. This arbitrariness for the choice of origin, and the accuracy of interconversion between three-dimensi onal spherical harmonic-Bessel and Fourier representations, suggest a metho d for describing the contents of non-centrosymmetric, sparsely packed cryst als. Incomplete Fourier representations (diffraction amplitudes) provide co mplex spherical harmonic-Bessel coefficients for a crystal packed with symm etry-enforced, non-centrosymmetrically arranged, non-overlapping spherical expansion zones. (C) 1999 Elsevier Science Ltd. All rights reserved.