International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 |
International Tables for Crystallography (2012). Vol. F, ch. 22.1, p. 706
Section 22.1.3.2. Definitions of surface in terms of Voronoi polyhedra (the convex hull)^{a}Department of Molecular Biophysics & Biochemistry, 266 Whitney Avenue, Yale University, PO Box 208114, New Haven, CT 06520, USA |
More fundamentally, however, the `problem of the protein surface' indicates how closely linked the definitions of surface and volume are and how the definition of one, in a sense, defines the other. That is, the two-dimensional (2D) surface of an object can be defined as the boundary between two 3D volumes. More specifically, the polyhedral faces defining the Voronoi volume of a collection of atoms also define their surface. The surface of a protein consists of the union of (connected) polyhedra faces. Each face in this surface is shared by one solvent atom and one protein atom (Fig. 22.1.3.2).
Another somewhat related definition is the convex hull, the smallest convex polyhedron that encloses all the atom centres (Fig. 22.1.3.2). This is important in computer-graphics applications and as an intermediary in many geometric constructions related to proteins (Connolly, 1991; O'Rourke, 1994). The convex hull is a subset of the Delaunay triangulation of the surface atoms. It is quickly located by the following procedure (Connolly, 1991): Find the atom farthest from the molecular centre. Then choose two of its neighbours (as determined by the Delaunay triangulation) such that a plane through these three atoms has all the remaining atoms of the molecule on one side of it (the `plane test'). This is the first triangle in the convex hull. Then one can choose a fourth atom connected to at least two of the three in the triangle and repeat the plane test, and by iteratively repeating this procedure, one can `sweep' across the surface of the molecule and define the whole convex hull.
Other parts of the Delaunay triangulation can define additional surfaces. The part of the triangulation connecting the first layer of water molecules defines a surface, as does the part joining the second layer. The second layer of water molecules, in fact, has been suggested on physical grounds to be the natural boundary for a protein in solution (Gerstein & Lynden-Bell, 1993c). Protein surfaces defined in terms of the convex hull or water layers tend to be `smoother' than those based on Voronoi faces, omitting deep grooves and clefts (see Fig. 22.1.3.2).
References
Connolly, M. L. (1991). Molecular interstitial skeleton. Comput. Chem. 15, 37–45.Gerstein, M. & Lynden-Bell, R. M. (1993c). What is the natural boundary for a protein in solution? J. Mol. Biol. 230, 641–650.
O'Rourke, J. (1994). Computational Geometry in C. Cambridge University Press.