Molecular Surfaces
8. Shapes of Small Molecules and Proteins

Michael L. Connolly

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Basically different methods have been used to study the shapes of small molecules and proteins. Tony Hopfinger has developed a method for small molecules that he calls "molecular shape analysis" and which involves comparing electrostatic fields (Hopfinger, 1980; Tokarski and Hopfinger, 1994). Stouch and Jurs (1986) measured small molecule similarity by superposing the two molecules and identifying the volumes that do not match using a spatial grid of points. Arteca and Mezey (1988) applied homology theory to study the shape of small molecules. They call their method "molecular shape descriptors" (Arteca, 1996). Meyer and Richards (1991) have developed a method to measure the similarity of molecular shape. Hermann and Herron (1991) have developed a pair of programs, OVID and SUPER, for studying small molecule shape. The latter program superposes two molecules. Masek, Merchant and Matthew (1993ab) have developed a method for measuring shape matching they call "molecular skins". Instead of maximizing the overlap of the whole interior volume, they consider only a shell or skin. It has particular advantage when attempting to match molecules of substantially different size. Bemis and Kuntz (1992) have developed methods for molecular shape description based upon considering distances between all possible triplets of atoms. Molecular shape has been used to screen databases of small molecules for the purpose of identifying molecules of possible pharmaceutical use (Good, Ewing, Gschwend and Kuntz, 1995). A fast method for shape characterization has been developed by Nilakantan, Bauman and Venkataraghavan (1993). Petitjean (1995) has defined a metric on the mathematical space of small molecule shapes based upon the volume of the symmetric difference when the molecules as superposed for maximum overlap. Blaney, Edge and Phippen (1995) have compared small molecules by comparing their electrostatic vector fields.

The shape of the protein as a whole have been studied several methods. Gates (1979) and Janin (1979) has studied the protein surface area to volume ratio. Peter Bladon (1989) has used the symmetries of the regular dodecahedron. Fourier analysis and spherical harmonics have been applied by Leicester, Finney and Bywater (1988, 1994), Max and Getzoff (1988), Max (1988), and Duncan and Olson (1993a). Malhotra, Tan and Harvey (1994) have combined spherical harmonics with electron microscopy and molecular mechanics.

The shapes of regions of the protein surface have been studied by different methods. Lee and Rose (1985) have made topographic maps of proteins. A topographic contouring method based upon depth below the convex hull has also been developed (Badel-Chagnon, Nessi, Buffat and Hazout, 1994). A method for measuring local curvature has been developed by (Connolly, 1986a). It is based upon centering a sphere at the protein surface and measuring the fraction of the sphere inside the solvent-excluded volume of the protein. If more than half of the sphere is inside the protein, the region is concave, if less than half of the sphere is inside the protein, the shape is convex. Either the solid sphere or the sphere surface may be used. A two-dimensional example is shown below.

The chymotrypsin surface below has been colored according to convexity or concavity. A sphere of radius six angstroms was centered at several points on each surface face, and the solid angle of the sphere lying inside the protein's molecular surface was computed and averaged over the face. Each face was colored according to where its average solid angle fell in the range between zero and four pi steradians. Convex regions are yellow, concave regions are blue, and regions of intermediate curvature are orange, red, and purple. The surface has been clipped, and the inner side of the molecular surface has been colored grey. The substrate-binding pocket is at the center of the image and can be seen to be blue.

This method has been further developed to distinguish between regions of equal curvature, but different shape (Connolly, 1992). Instead of simply measuring the amount of sphere area lying inside the protein's surface, one classifies its shape by comparing it to several thousand different template shapes.

One way to describe a protein surface is to identify the critical points, that is maxima and minima of some function akin to elevation that identifies knobs and holes (Fischer, Norel, Wolfson and Nussinov, 1993; Lin, Nussinov, Fischer and Wolfson, 1994). Other methods of studying protein topography have been developed by the Darmstadt group (Zachmann, Heiden, Schlenkrich and Brickmann, 1992; Zachmann, Kast, Sariban and Brickmann, 1993; Heiden and Brickmann 1994). Methods for measuring protein surface shape have been developed by Duncan and Olson (1993b).

Besides general methods for analysing protein surface shape, which apply also to convex and flat regions, there have also been developed specific methods for identifying clefts, crevices, and ligand-binding pockets in protein surfaces. Arthur Lesk has called this molecular speleology (1986). Voronoi polyhedra have been used to find clefts and possible drug binding sites in protein surfaces: (David, 1984; Boulu, Crippen, Barton, Kwon and Marletta, 1990; Crippen, 1991; Srivastava, and Crippen, 1993; Bradley and Crippen, 1993; Bradley, Richardson and Crippen, 1993). Blaney and Dixon (1991) describe the application of distance geometry in modeling receptor pockets. Some methods try to find the largest sphere that can be placed tangent to a particular surface atom without overlapping any other atom (Kuhn, Siani, Pique, Fisher, Getzoff and Tainer, 1992; Yeates, 1995). Parsons and Canny (1994) have written an interesting comparison of protein docking to robotics. Herbert Edelsbrunner (1993) and his colleagues at the University of Illinois and the National Center for Supercomputing Applications (NCSA) have applied methods from computational geometry and algebraic topology to characterizing protein surface concavities. Edelsbrunner and MŸcke (1994) have used a generalization of the convex hull called a 3D Alpha Shape to identify depressions and pockets on protein surfaces. These ideas have been applied to gramacidin A, HIV-I protease and the heme pocket of apomyoglobin (Edelsbrunner, Facello and Liang, 1995). Peters, Fauck and Fršmmel (1996) have developed an algorithm (APROPOS) for finding possible ligand-binding sites on protein surfaces using alpha shapes. They note that ligand-binding regions are concave, while protein-protein interfaces are generally rather flat.

[ 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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