Euclidean Voronoi diagrams of 3D spheres and applications to protein structure analysis

Abstract

Despite its many important applications in various disciplines in sciences and engineering, the Euclidean Voronoi diagram for spheres in 3D space has not been studied as much as it deserves. In this paper, we present an algorithm to compute a Euclidean Voronoi diagram for 3D spheres and show how the diagram can be used in the analysis of protein structures.Given an initial Voronoi vertex, the presented edge-tracing algorithm follows Voronoi edges until the construction is completed in O(mn) time in the worst-case, where m and n are the numbers of edges and spheres, respectively.Once a Voronoi diagram for 3D atoms of a protein is computed, it is shown that the diagram can be used to efficiently and precisely analyze the spatial structure of the protein. It turns out that this capability of a Voronoi diagram can be crucial to solving several important problems remaining to be solved in structural biology.

Pocket extraction on proteins via the Voronoi diagram of spheres

“…However, the ordinary Voronoi diagram of atom centers or the power diagram does not properly provide such information and therefore a potential global search is necessary to guarantee a correct result. Similar observations can be easily found in several recent studies [11,12,18]. Geode et al, for example, described an efficient algorithm to calculate the precise volume and density of a protein using the Voronoi diagram of atoms [12].…”

“…Geode et al, for example, described an efficient algorithm to calculate the precise volume and density of a protein using the Voronoi diagram of atoms [12]. Kim et al introduced various problems such as computing the molecular surface of a protein and locating the interaction interface in a polymer using the Voronoi diagram of atoms [16,18,19,21,23].…”

Euclidean Voronoi Diagrams of 3D Spheres: Their Construction and Related Problems from Biochemistry

“…Recently, Kim et al reported on the details of an edge-tracing algorithm and its full implementation for constructing the whole Voronoi diagram with discussions on various applications including the analysis of protein structures [20,21,22,23]. They showed that the whole Voronoi diagram can be constructed in O(n 3 ) time in the worst-case.…”

“…In this paper, balls are used to denote generators and spheres are used to denote tangent spheres corresponding to Voronoi vertices. For further details, please refer to [20][21][22][23].…”

Convex Hull and Voronoi Diagram of Additively Weighted Points

“…Other algorithms have been recently designed and implemented in the planar case [4,5]. In R 3 , we are only aware of two prototype implementations, one by Will [6] for computing a single cell, and one by Kim et al [7] that computes the entire diagram. None of these implementations is provably robust.…”