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 ℝ(3) that m = O(n) on average for biomolecules and m = O(n(2)) in the worst case for general spheres where n is the number of atoms. The theory is first introduced in ℝ(2) and extended to ℝ(3). 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.

BetaVoid: Molecular voids via beta-complexes and Voronoi diagrams

“… and , and for a more intuitive explanation, see Ref. . To better understand the relationship between the Voronoi diagram, its quasi‐triangulation, and the beta‐complex in two‐dimensional space, readers are recommended to use the BetaConcept program available at VDRC.…”

“…The method we describe in this article uses the idea initially proposed by Kim and Sugihara with the geometrical property computation adopting the beta‐decomposition algorithm reported in Ref. . We introduce geometric modeling, a core method in Computer‐Aided Design (CAD) which deals with the unambiguous representation of shapes and algorithms to analyze and manipulate objects .…”

Tunnels and Voids in Molecules via Voronoi Diagram

“…It is easy to see that the boundary of a void can be found by searching the atoms that correspond to the vertices in V C new and this property can be used in the computation of mass property. We developed a different yet more efficient and better approach called the Beta-decomposition which was reported very recently [28]. The idea of the Beta-decomposition is as follows.…”

BetaMol: A Molecular Modeling, Analysis and Visualization Software Based on the Beta-Complex and the Quasi-Triangulation

“…The proximity information is stored in the topology of the quasi-triangulation and thus its computation is the most important precondition for many applications running on the BetaMol such as the computation of the Connolly surface (28), (29) , the docking simulation (35), (36) , the computation of the molecular volume (37) , the computation of the molecular sphericity (38) , etc.…”