Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 22.1, p. 706   | 1 | 2 |

Section Definitions of surface in terms of Voronoi polyhedra (the convex hull)

M. Gersteina* and F. M. Richardsa

aDepartment of Molecular Biophysics & Biochemistry, 266 Whitney Avenue, Yale University, PO Box 208114, New Haven, CT 06520, USA
Correspondence e-mail: Definitions of surface in terms of Voronoi polyhedra (the convex hull)

| top | pdf |

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.[link].


Figure | top | pdf |

Definitions of the protein surface. (a) The classic definitions of protein surface in terms of the probe sphere, the accessible surface and the molecular surface. (This figure is adapted from Richards, 1977[link]). (b) Voronoi polyhedra and Delaunay triangulation can also be used to define a protein surface. In this schematic, the large spheres represent closely packed protein atoms and the smaller spheres represent the small loosely packed water molecules. The Delaunay triangulation is shown by dotted lines. Some parts of the triangulation can be used to define surfaces. The outermost part of the triangulation of just the protein atoms forms the convex hull. This is indicated by the thick line around the protein atoms. For the convex-hull construction, one imagines that the water is not present. This is highlighted by the thick dotted line, which shows how Delaunay triangulation of the surface atoms in the presence of the water diverges from the convex hull near a deep cleft. Another part of the triangulation, also indicated by thick black lines, connects the first layer of water molecules (those that touch protein atoms). A time-averaged version of this line approximates the accessible surface. Finally, the light thick lines show the Voronoi faces separating the protein surface atoms from the first layer of water molecules. Note how this corresponds approximately to the molecular surface (considering the water positions to be time-averaged). These correspondences between the accessible and molecular surfaces and time-averaged parts of the Voronoi construction are understandable in terms of which part of the probe sphere (centre or point of tangency) is used for the surface definition. The accessible surface is based on the position of the centre of the probe sphere while the molecular surface is based on the points of tangency between the probe sphere and the protein atoms, and these tangent points are similarly positioned to Voronoi faces, which bisect interatomic vectors between solvent and protein atoms.

Another somewhat related definition is the convex hull, the smallest convex polyhedron that encloses all the atom centres (Fig.[link]. This is important in computer-graphics applications and as an intermediary in many geometric constructions related to proteins (Connolly, 1991[link]; O'Rourke, 1994[link]). The convex hull is a subset of the Delaunay triangulation of the surface atoms. It is quickly located by the following procedure (Connolly, 1991[link]): 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[link]). 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.[link].


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.

to end of page
to top of page