It is widely believed that characterizing the geometry of a protein molecule is essential for understanding its folding process as well as its interactions with other biomolecules and small ligands. Among all geometric measures, volume and surface are probably the most fundamental property to study. Atomic volumes and surfaces have been used for quantifying packing inter-actions in proteins, as well as in energy functions for implicit solvent models. In the latter, the effects of water on the protein of interest are accounted for using continuum models for electrostatics, and a volume or surface term for hydrophobic effects. Inclusion of such potentials in a biomolecular simulation require fast algorithms for computing the volume or surface occupied by the protein, and its derivatives with respect to the atomic positions.

Computational methods that evaluate the solvent-excluded volume or surface of a molecule can be divided into approximate and exact methods. Most of the approximate methods rely on numerical integrations. The first analytical method to compute the volume of a molecule was based on the inclusion-exclusion principle, with the approximation that only intersections of up to three balls need to be considered. Including all possible orders of intersection remains computational difficult and expensive. The Alpha Shape theory solves this problem exactly by using Delaunay triangulations and their filtrations. The distinction between approximate and exact computation also applies to existing methods for computing the derivatives of the solvent-excluded volume and surface with respect to atomic coordinates. All these methods have to take a large number of singularities into account, where approximations are usually required. The Alpha Shape theory proposes a robust solution to this problem, by implementing arbitrary precision arithmetic to avoid numerical problems and systematically resolving all singularities without explicitly perturbing the positions of the ball centers. We have derived two theorems that provide the weighted volume derivative and the weighted area derivative of a space filling diagram, and implemented these theorem within the Alpha Shapes theory, to provide an efficient, robust, exact computation of the derivatives of volumes and surface.