Spheres Unions and Intersections and Some of their Applications in Molecular Modeling

Abstract

The geometrical and computational aspects of spheres unions and intersections are described. A practical analytical calculation of their surfaces and volumes is given in the general case: any number of intersecting spheres of any radii. Applications to trilateration and van der Waals surfaces and volumes calculation are considered. The results are compared to those of other algorithms, such as Monte Carlo methods, regular grid methods, or incomplete analytical algorithms. For molecular modeling, these latter algorithms are shown to give strongly overestimated values when the radii values are in the ranges recommended in the literature, while regular grid methods are shown to give a poor accuracy. Other concepts related to surfaces and volumes of unions of spheres are evoked, such as Connolly’s surfaces, accessible surface areas, and solvent-excluded volumes.KeywordsContact PointMonte Carlo CalculationRadical PlaneSpherical TriangleSphere IntersectionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.