Section 22.1.1.3. Definitions of protein surface

When one is carrying out the Voronoi procedure, if a particular atom does not have enough neighbours the `polyhedron' formed around it will not be closed, but rather will have an open, concave shape. As it is not often possible to place enough water molecules in an X-ray crystal structure to cover all the surface atoms, these `open polyhedra' occur frequently on the protein surface (Fig. 22.1.1.6). Furthermore, even when it is possible to define a closed polyhedron on the surface, it will often be distended and too large. This is the problem of the protein surface in relation to the Voronoi construction.

Figure 22.1.1.6 | top | pdf | The problem of the protein surface. This figure shows the difficulty in constructing Voronoi polyhedra for atoms on the protein surface. If all the water molecules near the surface are not resolved in a crystal structure, one often does not have enough neighbours to define a closed polyhedron. This figure should be compared with Fig. 22.1.1.1, illustrating the basic Voronoi construction. The two figures are the same except that in this figure, some of the atoms on the left are missing, giving the central atom an open polyhedron. 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).

There are a number of practical techniques for dealing with this problem. First, one can use very high resolution protein crystal structures, which have many solvent atoms positioned (Gerstein & Chothia, 1996). Alternatively, one can make up the positions of missing solvent molecules. These can be placed either according to a regular grid-like arrangement or, more realistically, according to the results of molecular simulation (Finney et al., 1980; Gerstein et al., 1995; Richards, 1974).

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. 22.1.1.7).

Figure 22.1.1.7 | 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). (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. 22.1.1.7). 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. 22.1.1.7).

In the absence of solvent molecules to define Voronoi polyhedra, one can define the protein surface in terms of the position of a hypothetical solvent, often called the probe sphere, that `rolls' around the surface (Richards, 1977) (Fig. 22.1.1.7). The surface of the probe is imagined to be maintained at a tangent to the van der Waals surface of the model.

Various algorithms are used to cause the probe to visit all possible points of contact with the model. The locus of either the centre of the probe or the tangent point to the model is recorded. Either through exact analytical functions or numerical approximations of adjustable accuracy, the algorithms provide an estimate of the area of the resulting surface. (See Section 22.1.2 for a more extensive discussion of the definition, calculation and use of areas.)

Depending on the probe size and whether its centre or point of tangency is used to define the surface, one arrives at a number of commonly used definitions, summarized in Table 22.1.1.2 and Fig. 22.1.1.7.