Beta-decomposition for the volume and area of the union of three-dimensional balls and their offsets

Abstract

Given a set of spherical balls, called atoms, in three-dimensional space, its mass properties such as the volume and the boundary area of the union of the atoms are important for many disciplines, particularly for computational chemistry/biology and structural molecular biology. Despite many previous studies, this seemingly easy problem of computing mass properties has not been well-solved. If the mass properties of the union of the offset of the atoms are to be computed as well, the problem gets even harder. In this article, we propose algorithms that compute the mass properties of both the union of atoms and their offsets both correctly and efficiently. The proposed algorithms employ an approach, called the Beta-decomposition, based on the recent theory of the beta-complex. Given the beta-complex of an atom set, these algorithms decompose the target mass property into a set of primitives using the simplexes of the beta-complex. Then, the molecular mass property is computed by appropriately summing up the mass property corresponding to each simplex. The time complexity of the proposed algorithm is O(m) in the worst case where m is the number of simplexes in the beta-complex that can be efficiently computed from the Voronoi diagram of the atoms. It is known in R3 that m = O(n) on average for biomolecules and m = O(n2) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in R2 and extended to R3. The proposed algorithms were implemented into the software BetaMass and thoroughly tested using molecular structures available in the Protein Data Bank. BetaMass is freely available at the Voronoi Diagram Research Center web site.