A beta-shape from the voronoi diagram of atoms for protein structure analysis

Abstract

In this paper, we present a ss-shape and a ss-complex for a set of atoms with arbitrary sizes for a faster response to the topological queries among atoms. These concepts are the generalizations of the well-known a-shape and a-complex (and their weighted counterparts as well). To compute a ss-shape, we first compute the Voronoi diagram of atoms and then transform the Voronoi diagram to a quasi-triangulation which is the topological dual of the Voronoi diagram. Then, we compute a ss-complex from the quasi-triangulation by analyzing the valid intervals for each simplex in the quasi-triangulation. It is shown that a ss-complex can be computed in O(m) time in the worst case from the Voronoi diagram of atoms, where m, is the number of simplices in the quasi-triangulation. Then, a ss-shape for a particular ss consisting of k simplices can be located in O(log m + k) time in the worst case from the simplicies in the ss-complex sorted according to the interval values.