1. Volume, Surface and their derivatives

The accessible surface of a protein is defined as the van der Waals envelope of the molecule expanded by the radius of the solvent sphere about each atom center [1] .

The vdW surface of a molecule (shown in red) is defined as the surface of the union of balls representing
all atoms, with radii set to the vdW radii.
The accessible surface of the same molecule (shown in green) is the surface generated by the center of a sphere rolling
on the vdW surface. The radius of this sphere is usually set to 1.4 Angstroms, the radius of a water molecule.
The molecular surface (shown in magenta) is the lower envelope generated by the rolling sphere.
It differs from the vdW surface in that some areas are inaccessible to the rolling sphere.

Computational methods that evaluate the area of this surface can be divided into approximate and exact methods. Most of the approximate methods rely on numerical integration, by representing the surface with a large number of dots [2,3]. Some of the approximations are analytical but treat multiple overlapping balls probabilistically or ignore them [4-6]. The first exact analytical methods for computing the accessible surface area were introduced by Connolly [7] and Richmond [8]. They have been improved in recent years, the focus being on computational efficiency [9,10] and stability [11,12]. The idea of using inclusion­exclusion to reduce intersections of five or more balls to linear combinations of at most four balls was introduced by Kratky [13] and Gibson and Scheraga [14]. Doing the reduction correctly remains however computationally expensive. The Alpha Shape theory solves this problem using Delaunay triangulations and their filtrations [15]. Alpha shapes have been used to compute the surface area and volume of proteins as well as for detecting and measuring cavities in proteins [16].

The distinction between approximate and exact computation also applies to existing methods for computing the derivatives of the surface area with respect to atomic coordinates. The computationally more efficient methods are the approximate methods implemented in the MSEED [17] and SASAD [18] programs. The computational efficiency of MSEED is a consequence of ignoring the contribution of internal atoms, at the cost of missing buried cavities. A numerical procedure for computing the accessible fraction of the circle of intersection between two surface spheres is implemented in SASAD. The analytical method for surface area calculation proposed by Richmond [8] also provides analytical gradients, and revised versions of this approach were implemented in molecular dynamics programs [10,19].

All existing methods for computing the derivatives are extensions of strategies used for computing the surface area, and therefore suffer the same stability problems. The Alpha Shapes software proposes a robust solution to the latter problem, by implementing arbitrary precision arithmetic to avoid numerical problems and systematically resolving all singularities without explicitly perturbing the positions of the sphere centers. The latter method is referred to as Simulation of Simplicity [20]. We have derived an extension of the Alpha Shapes method that includes the robust, exact, and analytical computation of the surface and volume of a protein, as well as their derivatives.

There is an inherent difficulty in using a potential based on surface area for energy minimization or molecular dynamics. Although the accessible surface area is continuous in the position of the atoms, its derivatives are not. Scheraga and co­workers have published a list of situations in which discontinuity in the derivative is observed [21]. We have re­examined this issue within the framework of the Alpha Shape method and have related discontinuities with combinatorial changes in the subcomplex of the Delaunay triangulation that is dual to the space­filling diagram. These combinatorial changes are resolved exactly in the process of building the Delaunay triangulation using the Simulation of Simplicity, and consequently all discontinuities in the area derivatives are resolved.

2. Protein Pockets

Active sites of proteins are often found in structural pockets or cavities.

A pocket is an empty concavity on the protein surface into which solvent can gain access. Such a concavity has one or more surfaces connecting their interior with the outside bulk solution. We call these surfaces the "mouths"of the pocket.

A cavity (or void) is an interior empty space of the protein structure that is not accessible to the solvent. It has no mouths. The positions of the pockets and cavities of a protein can be identified automatically by studying the difference between the Delaunay triangulation and the Alpha complex computed from the positions of all atoms in the proteins [16].

3. Example

A test case illustrating the Alpha Shape procedure for computing protein surface and volume, and detecting pockets, is available here.

References

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2. Shrake, A and Rupley, JA. Environment and exposure to solvent of protein atoms in lyzyzyme and insulin. J. Mol. Biol., 79, 351-371 (1973).

3. Legrand, SM and Merz, KM. Rapid approximation to molecular surface area via the use of Boolean logic and lookup tables. J. Comp. Chem., 14, 349-352 (1993).

4. Wodak, SJ and Janin, J. Analytical approximation to the accessible surface area of proteins. Proc. Natl. Acad. Sci. (USA), 77, 1736-1740 (1979).

5. Hasel, W, Hendrikson, TF and Still, WC. A rapid approximation to the solvent accessible surface areas of atoms. Tetrahed. Comp. Method., 1, 103-116 (1988).

6. Street, AG and Mayo, SL. Pairwise calculation of protein solvent-accessible surface areas. Folding & Design, 3, 253-258 (1998).

7. Connolly, M. Analytical molecular surface calculation. J. Appl. Cryst., 16, 548-558 (1983).

8. Richmond, TJ. Solvent accessible surface area and excluded volume in proteins. J. Mol. Biol.,178, 63-89 (1984).

9. Von Freyberg, B, Richmond, TJ and Braun, W. Surface area included in energy refinements of proteins: a comparative study on atomic solvation parameters. J. Mol. Biol., 233, 275-292 (1993).

10. Fraczkiewicz, R. and Braun, W. Exact and efficient analytical calculation of the accessible surface area and their gradient for macromolecules. J. Comput. Chem., 19, 319-333 (1998).

11. Eisenhaber, F and Argos, P. Improved strategy in analytic surface calculation for molecular systems: handling of singularities and computational efficiency. J. Comput. Chem., 14, 1272-1280 (1993).

12. Gogonea, V and Osawa, E. An improved algorithm for the analytical computation of solvent-excluded volume. The treatment of singularities in solvent accessible surface area and volume functions. J. Comput. Chem., 7, 817-842 (1995).

13. Kratky, KW. The area of the intersection of n equal cicular disks. J. Phys. A: Math. Gen., 11, 1017-1024 (1978).

14. Gibson, KD and Scheraga, HA. Exact calculation of the volume and surface area of fused hard sphere molecule with unequal atomic radii. Mol. Phys., 62, 1247-1265 (1987).

15. Edelsbrunner, H. The union of balls and its dual shape. Discrete Comput. Geom., 13, 415-440 (1995).

16. Liang, J, Edelsbrunner, H, Fu, P, Sudhakar, PV and Subramaniam, S. Analytical shape computation of macromolecules I and II. Proteins: Struct. Funct. Genet., 33, 1-17 and 18-29 (1998).

17. Perrot, G, Cheng, B, Gibson, KD, Vila, J, Palmer, KA, Nayeem, A, Maigret, B and Scheraga, HA. MSEED: a program for the rapid analytical determination of accessible surface-areas and their derivatives. J. Comput. Chem., 13, 1-11 (1992).

18. Sridharan, S, Nicholls, A and Sharp, KA. A rapid method for calculating derivatives of solvent accessible surface areas of molecules. J. Comput. Chem., 16, 1038-1044 (1994).

19. Wesson, L and Eisenberg, D. Atomic solvation parameters applied to molecular dynamics of proteins in solution. Protein Sci., 1, 227-235 (1992).

20. Edelsbrunner, H and Mucke, EP. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graphics, 9, 66-104 (1990).

21. Wawak, RJ, Gibson, KD and Scheraga, HA. Gradient discontinuities in calculations involving molecular surface area. J. Math. Chem., 15, 207-232 (1994).