Shape group theory of van der Waals surfaces

In this article we present a method for the study of shapes of general, asymmetric van der Waals surfaces. The procedure is simple to apply and it consists of two steps. First, the surface is decomposed into spherical domains, according to the interpenetration of the van der Waals atomic spheres. Each domain defines a topological object that is either a 2-manifold or some truncated 2-manifold. Second, we compute the homology groups for all the objects into which the surface is divided. These groups are topological and homotopical invariants of the domains, hence they remain invariant to conformational changes that preserve the essential features of these domains of decomposition. In particular, these homology groups do not depend explicitly on the molecular symmetry. Major rearrangements of the nuclear configurations, however, do alter the decomposition into spherical domains, and the corresponding variation of the homology groups can be followed easily under conformational rearrangements. We discuss a partitioning of the metric internal configuration spaceM into shape regions of van der Waals surfaces, which allows one to identify those rearrangements which introduce an essential change in shape and to distinguish them from those which do not alter the fundamental shape of the molecular surface. The dependence of the shape group partitioning ofM on the symmetry under permutation of nuclear changes is discussed briefly, considering a simple illustrative example.