International
Tables for
Crystallography
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, pp. 707-708   | 1 | 2 |

Section 22.1.3.3. Definitions of surface in terms of a probe sphere

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:  Mark.Gerstein@yale.edu

22.1.3.3. Definitions of surface in terms of a probe sphere

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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[link]) (Fig. 22.1.3.2)[link]. 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[link] 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[link].

22.1.3.3.1. van der Waals surface (VDWS)

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The area of the van der Waals surface will be calculated by the various area algorithms (see Section 22.2.2[link] 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.

22.1.3.3.2. Solvent-accessible surface (SAS)

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The solvent-accessible surface is convex and closed, with defined areas assignable to each individual atom (Lee & Richards, 1971[link]). 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.

22.1.3.3.3. Molecular surface as the sum of the contact and re-entrant surfaces (MS = CS + RS)

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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.

22.1.3.3.4. Further points

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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[link]).

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[link]; Leicester et al., 1988[link]; Pattabiraman et al., 1995[link]).

References

Gerstein, M. & Lynden-Bell, R. M. (1993c). What is the natural boundary for a protein in solution? J. Mol. Biol. 230, 641–650.
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.
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. (1977). Areas, volumes, packing, and protein structure. Annu. Rev. Biophys. Bioeng. 6, 151–176.








































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