Section 22.1.1.2.1. Volume in terms of Voronoi polyhedra: overview

Protein volume can be defined in a straightforward sense through a particular geometric construction called the Voronoi polyhedron. In essence, this construction provides a useful way of partitioning space amongst a collection of atoms. Each atom is surrounded by a single convex polyhedron and allocated the space within it (Fig. 22.1.1.1). The faces of Voronoi polyhedra are formed by constructing dividing planes perpendicular to vectors connecting atoms, and the edges of the polyhedra result from the intersection of these planes.

Figure 22.1.1.1 | top | pdf | The Voronoi construction in two and three dimensions. Representative Voronoi polyhedra from 1CSE (subtilisin) are shown. (a) Six polyhedra around the atoms in a Phe ring. (b) A single polyhedron around the side-chain hydroxyl oxygen (OG) of a serine. (c) A schematic showing the construction of a Voronoi polyhedron in two dimensions. The broken lines indicate planes that were initially included in the polyhedron but then removed by the `chopping-down' procedure (see Fig. 22.1.1.4).

Voronoi polyhedra were originally developed by Voronoi (1908) nearly a century ago. Bernal & Finney (1967) used them to study the structure of liquids in the 1960s. However, despite the general utility of these polyhedra, their application to proteins was limited by a serious methodological difficulty. While the Voronoi construction is based on partitioning space amongst a collection of `equal' points, all protein atoms are not equal. Some are clearly larger than others. In 1974, a solution was found to this problem (Richards, 1974), and since then Voronoi polyhedra have been applied to proteins.