Topology of Molecular Shape and Chirality

Abstract

The three-dimensional shape of various functions, for example, of electron density contours describing the body of molecules, is an essential tool for the characterization of molecules in drug design. A detailed description of such shapes is possible by the symmetry-independent shape group method, developed recently. The shape group analysis of molecules is based on the initial generation of a molecular surface, such as a surface of a quantum chemical isodensity contour, a contour of the electrostatic potential, a van der Waals, or solvent accessibility, or a so-called union surface obtained for an enzyme cavity by the superposition of contour surfaces of several active drug molecules. The curvature properties and mutual interpenetrations of such contour surfaces define a family of topological objects, which are characterized by their homology groups of algebraic topology. These homology groups are the shape groups of the original contour surface, applicable for a precise shape characterization of arbitrary, asymmetric molecules. The shape group method (SGM) is combined with a novel representation of molecular chirality properties, using a simple algebraic test for chirality, based on a recent finding in a branch of topology, called knot theory.