Shape analysis of molecular surfaces

The description of molecular shape is important in the analysis of protein-protein and protein-ligand interactions. We describe volumetric and surface-based techniques for computing shape properties of molecular surfaces. The surface is defined as an isocontour of an approximate electron density function. Each technique can compute several scalar and vector surface properties such as the Gaussian and mean curvature, principal curvatures, and principal curvature directions. Shape properties are derived from the eigenvalues and eigenvectors of a 3 by 3 matrix for each surface point. In the volumetric approach, the matrix is the second derivative of an approximate electron density function. In the surface-based approach, the matrix is the approximate gradient of the surface normal. Derivatives are computed by convolving the density or the surface normals with the derivatives of a Gaussian function. The variance of the Gaussian determines the effective length scale at which the surface is analyzed. Scalar surface properties are displayed as colored dots or shaded triangles, and vector properties are displayed as line segments from each surface point. This report describes the implementation of these procedures and their use in computing the shape properties of Cu-Zu superoxide dismutase.