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, pp. 706-708
Section 22.1.3. Definitions of protein surface^{a}Department of Molecular Biophysics & Biochemistry, 266 Whitney Avenue, Yale University, PO Box 208114, New Haven, CT 06520, USA |
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. 22.1.3.1). 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.
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. 22.1.3.2).
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).
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. 22.1.3.2). 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 Chapter 22.2 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 22.1.4.1 and Fig. 22.1.3.2.
The area of the van der Waals surface will be calculated by the various area algorithms (see Section 22.2.2 in Chapter 22.2) when the probe radius is set to zero. This is a mathematical calculation only. There is no physical procedure that will measure van der Waals surface area directly. From a mathematical point of view, it is just the first of a set of solvent-accessible surfaces calculated with differing probe radii.
The solvent-accessible surface is convex and closed, with defined areas assignable to each individual atom (Lee & Richards, 1971). However, the individual calculated values vary in a complex fashion with variations in the radii of the probe and protein atoms. This radius is frequently, but not always, set at a value considered to represent a water molecule (1.4 Å). The total SAS area increases without bound as the size of the probe increases.
Like the solvent-accessible surface, the molecular surface is also closed, but it contains a mixture of convex and concave patches, the sum of the contact and re-entrant surfaces. The ratio of these two surfaces varies with probe radius. In the limit of infinite probe radius, the molecular surface becomes convex and attains a limiting minimum value (i.e. it becomes a convex hull, similar to the one described above). The molecular surface cannot be divided up and assigned unambiguously to individual atoms.
The contact surface is not closed. Instead, it is a series of convex patches on individual atoms, simply related to the solvent-accessible surface of the same atoms. In complementary fashion, the re-entrant surface is also not closed but is a series of concave patches that is part of the probe surface where it contacts two or three atoms simultaneously. At infinite probe radius, the re-entrant areas are plane surfaces, at which point the molecular surface becomes a convex surface. The re-entrant surface cannot be divided up and assigned unambiguously to individual atoms. Note that the molecular surface is simply the union of the contact and re-entrant surfaces, so in terms of area MS = CS + RS.
The detail provided by these surfaces will depend on the radius of the probe used for their construction.
One may argue that the behaviour of the rolling probe sphere does not accurately model real hydrogen-bonded water. Instead, its `rolling' more closely mimics the behaviour of a nonpolar solvent. An attempt has been made to incorporate more realistic hydrogen-bonding behaviour into the probe sphere, allowing for the definition of a hydration surface more closely linked to the behaviour of real water (Gerstein & Lynden-Bell, 1993c).
The definitions of accessible surface and molecular surface can be related back to the Voronoi construction. The molecular surface is similar to `time-averaging' the surface formed from the faces of Voronoi polyhedra (the Voronoi surface) over many water configurations, and the accessible surface is similar to averaging the Delaunay triangulation of the first layer of water molecules over many configurations.
There are a number of other definitions of protein surfaces that are unrelated to either the probe-sphere method or Voronoi polyhedra and provide complementary information (Kuhn et al., 1992; Leicester et al., 1988; Pattabiraman et al., 1995).
References
Connolly, M. L. (1991). Molecular interstitial skeleton. Comput. Chem. 15, 37–45.Finney, J. L., Gellatly, B. J., Golton, I. C. & Goodfellow, J. (1980). Solvent effects and polar interactions in the structural stability and dynamics of globular proteins. Biophys. J. 32, 17–33.
Gerstein, M. & Chothia, C. (1996). Packing at the protein–water interface. Proc. Natl Acad. Sci. USA, 93, 10167–10172.
Gerstein, M. & Lynden-Bell, R. M. (1993c). What is the natural boundary for a protein in solution? J. Mol. Biol. 230, 641–650.
Gerstein, M., Tsai, J. & Levitt, M. (1995). The volume of atoms on the protein surface: calculated from simulation, using Voronoi polyhedra. J. Mol. Biol. 249, 955–966.
Kuhn, L. A., Siani, M. A., Pique, M. E., Fisher, C. L., Getzoff, E. D. & Tainer, J. A. (1992). The interdependence of protein surface topography and bound water molecules revealed by surface accessibility and fractal density measures. J. Mol. Biol. 228, 13–22.
Lee, B. & Richards, F. M. (1971). The interpretation of protein structures: estimation of static accessibility. J. Mol. Biol. 55, 379–400.
Leicester, S. E., Finney, J. L. & Bywater, R. P. (1988). Description of molecular surface shape using Fourier descriptors. J. Mol. Graphics, 6, 104–108.
O'Rourke, J. (1994). Computational Geometry in C. Cambridge University Press.
Pattabiraman, N., Ward, K. B. & Fleming, P. J. (1995). Occluded molecular surface: analysis of protein packing. J. Mol. Recognit. 8, 334–344.
Richards, F. M. (1974). The interpretation of protein structures: total volume, group volume distributions and packing density. J. Mol. Biol. 82, 1–14.
Richards, F. M. (1977). Areas, volumes, packing, and protein structure. Annu. Rev. Biophys. Bioeng. 6, 151–176.