Discrete characterization of cross-sections of molecular surfaces

Abstract

The three-dimensional shape of molecules can be described by appropriately chosen, formal molecular surfaces, such as electronic isodensity contours, molecular electrostatic isopotential contours, and similar functions. The characterization of the shape of such molecular surfaces, in particular, their changes along reaction paths, is of importance in several areas of theoretical and applied chemistry, biochemistry, and pharmacology. Molecular shapes are often represented in terms of cross-sections of such molecular surfaces, by appropriately chosen planes. Thus, the characterization of 3D shape is transformed into the 2D problem of characterizing a number of plane curves, the cross-sections. The problem is further simplified if these continuous curves are characterized using the methods of discrete mathematics, which are often more suitable for computer applications. In this work we formalize a number of possible approaches to provide a discrete characterization of molecular surface cross-sections. The method is graph-theoretical in nature and it allows one to provide a concise description of the curves, and their changes with the rearrangements in the nuclear configuration. The vertices of the graph are the inflexion points of the cross-section and the edges are their mutual visibility relations. The procedure is easily programmable as an algorithm, permitting an automatic evaluation of the shape descriptors for a large number of cross-sections and molecules. This possibility is of importance with respect to molecular similarity studies of computer-assisted drug design. Several simple examples are chosen to illustrate the shape descriptors. The basic ideas are illustrated by detailed cross-sections of the electronic density for the molecule of water, as functions of changes in bond angle and bond lengths. Molecular electrostatic maps for molecules of biochemical interest (nucleotide bases) are analyzed with respect to shape similarities. Finally, some of the notions are generalized to deal with cross-sections of surfaces of hard-sphere molecular models (the so-called van der Waals surfaces).