Molecular Surfaces
7. Molecular Volume and Protein Packing

Michael L. Connolly

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The van der Waals volume of a molecule is the volume of the union of overlapping spheres. This problem in solid geometry has been the subject of methods both numerical (Rowlinson, 1963; Pavani and Ranghino, 1982; Gavezzotti, 1983) and analytic (Richmond, 1984; Kang, Nemethy and Scheraga, 1987; Gibson, and Scheraga, 1987a; Gibson, and Scheraga, 1987b; Gibson, and Scheraga, 1988; Guerrero-Ruiz, Ocadiz-Ramirez and Garduno-Juarez, 1991; Petitjean, 1994). These methods either point out that overlap volumes of four or more atoms do not occur in real molecules, or else point out a theorem from mathematics that higher-order unions and intersections can always be reduced to lower-order unions and intersections (Edelsbrunner, 1987). In addition, some people have gone on to compute the derivatives of the van der Waals volume with respect to the atomic coordinates (Kundrot, Ponder and Richards, 1991; Gogonea and Osawa, 1994b; Gogonea and Osawa, 1995).

Although Richmond (1984) has defined the solvent-excluded volume to mean the volume contained within the accessible surface, that is the union of the expanded atom spheres, more commonly the term has been used to denote the volume inside the molecular surface (Rellick and Becktel, 1995). This solvent-excluded volume is made up of the van der Waals volume plus the interstitial volume. In the picture below, the crambin protein is sectioned. The van der Waals volume is blue, and the interstitial volume is red. The outer molecular surface is yellow.

Grid-based methods for computing this kind of solvent-excluded volume have been developed by (Pavlov and Fedorov, 1983), (MŸller, 1983) and (Higo and Go) 1989. Also see PROVE (PRotein Volume Evaluator) at the Unite Conformation de Macromolecules Biologiques, l'Universite Libre de Bruxelles. An analytic algorithm has been developed by (Connolly, 1985b; Connolly, 1994). It cuts the solvent-excluded volume into non-overlapping pieces. Below is an exploded view of the decomposition of an asparagine amino acid.

Water-sized and smaller probe spheres can also be (computationally) rolled around inside protein molecules to define and identify protein cavities, tunnels, and probably sites for internal waters. Richards (1979) studied the connectivity of protein packing defects in order to identify possible channels from the surface into the interior. Connolly (1986c) cataloged water-sized packing defects in the static crystal structures of a sample of proteins. Rashin, Iofin and Honig (1986) identified packing defects, and also modeled water molecules inside of them if there were hydrophilic side chains surrounding the cavity. Internal waters and cavities have been studied by (Williams, Goodfellow and Thornton, 1994), who found that "the total volume of a protein's large cavities is proportional to its molecular weight and is not dependent on structural class." Packing defects and tunnels in myoglobin have been studied with respect to how a xenon atom might move between the solvent and the interior (Tilton, Singh, Weiner, Connolly, Kuntz, Kollman, Max and Case, 1986). A fast program for computing tunnels and cavities has been developed (Voorintholt, Kosters, Vegter, Vriend and Hol, 1989) and can be used in conjunction with Gerrit Vriend's WHATIF program. Philippe Alard and Shoshana Wodak (1991) have developed a method to identify cavities in van der Waals and accessible volumes. Levitt and Banaszak (1992) have written a computer program called POCKET for locating protein cavities. A method for finding void regions in proteins has been developed by Kleywegt and Jones (1994). A good review of work on the packing in protein interiors is given by (Baldwin and Matthews, 1994). Protein cavities and internal waters have also been studied by Hubbard, Gross and Argos (1994) and Hubbard and Argos (1995). Matthews, Morton and Dahlquist (1995) have used NMR to detect water molecules within nonpolar protein cavities. Michael Prisant's ray tracing software can identify cavities and internal waters. Also see the section on solvation and hydrophobicity.

Computer-science approaches to the problem of finding protein voids have been developed by Delaney (1992) and Edelsbrunner, Facello, Fu and Liang (1995). Wodak and Janin (1981) have used surface area measurement to identify structural domains in proteins. Another computational-geometry approach to studying the packing of protein interiors has been Voronoi polyhedra (Voronoi, 1907). Examples of two-dimensional Voronoi diagrams can be found at Carnegie-Mellonand SUNY Stony Brook. There is also a list server for Voronoi and computational geometry applications. The Voronoi polyhedra partition the solvent-excluded volume of the protein into polyhedra, with one protein atom at the center of each polyhedron (Richards, 1974; Finney, 1978; Gellatly and Finney, 1982; David, 1988). An algorithm for constructing Voronoi polyhedra has been given by Brostow, Dussault and Fox (1978). The boundary between the protein and the solvent has been a special focus of this approach (Gerstein and Lynden-Bell, 1993; Gerstein, Tsai, Levitt, 1995). Voronoi polyhedra have even been applied to chromosomes! (Eils, Bertin, Saracoglu, Rinke, Schrock, et al., 1995). A related idea from computer science is the skeleton, also called the symmetric axis transform and the medial axis transform (Blum, 1967; Calabi and Hartnett, 1968; Nackman and Pizer, 1985). So far, it has receive only limited application to protein structure (Connolly, 1991; Sanner, 1992; Boissonnat, Devillers, Duquesne and Yvinec, 1994). Another related construction from computational geometry is Delaunay triangulation (Vaisman, Tropsha and Singh, 1995; Singh, Tropsha and Vaisman, 1996).

[ 1. Introduction ] [ 2. Physical Molecular Models ] [ 3. Electron Density Fitting ] [ 4. Molecular Graphics ] [ 5. Solvent-Accessible Surfaces ] [ 6. Molecular Surface Graphics ] [ *** 7. Molecular Volume and Protein Packing *** ] [ 8. Shapes of Small Molecules and Proteins ] [ 9. Structure-based Drug Design ] [ 10. Protein-Protein Interactions ] [ 11. Surface Biology, Chemistry and Physics ] [ 12. Bibliography ]

All material in ths article Copyright © 1996 by Michael L. Connolly



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