Analytical approximation to the accessible surface area of proteins

Abstract

We propose an analytical substitute to the geometrical construction that is commonly used in calculating the protein surface area that is accessible to the solvent. A statistical approach leads to an expression of accessible surface areas as a function of distances between pairs of atoms or of residues in the protein structure, assuming only that these atoms or residues are randomly distributed in space but not penetrating each other. This function gives good estimates of the accessible surface area and of the area buried in subunit contacts for a number of proteins. Its evaluation is very fast, and the function can be differentiated, which opens the way to new applications of accessibility measurements in the study of proteins. As an example, we show that the presence of domains is easily detected by an automatic procedure based on surface areas only. The concept of accessible surface area, first proposed by Lee and Richards (1), has found many applications in the study of proteins (2). The accessible surface area of a protein atom is defined as the area of the surface over which a water molecule can be placed while making van der Waals contact with this atom and not penetrating any other protein atom. A geometrical construction (Fig. 1) leads to algorithms that calculate accessible surface areas from atomic coordinates derived from x-ray studies (1, 3, 4). These areas are linearly correlated to the free energies of transfer from polar to nonpolar solvents, or hydrophobic free energies, of hydrocarbons (5-8). Measurements of accessible surface areas and of area changes occurring in various biochemical processes may therefore give insights into the role of the solvent and of hydrophobicity in these processes. For instance, the evaluation of the surface area change when proteins fold (9) or associate (10) shows that hydrophobicity is the major driving factor in folding and in polymerization.Because of the complexity of protein structures and of the many atoms present, the geometrical algorithms used in measuring accessible surface areas are costly in computer time.Moreover, it would be desirable to represent the surface area as an analytical function of the atomic coordinates because the function and its derivatives could be used in minimization procedures. An increase in computing efficiency can be achieved by simplifying the protein structure and representing each amino acid residue rather than each atom of the structure by a sphere (11). We have shown that good estimates of accessible surface areas are obtained in this way, and a systematic analysis of the trypsin-pancreatic trypsin inhibitor complex was made possible by the quickness of the calculation (12), even with the algorithm of Lee and Richards.We develop here an analytical approximation to the accessible surface area, expressed as a function of interatomic distances only. The approximation is based on a statistical approach assuming that atoms or amino acid residues are randomly distributed. Measurements of accessible surface areas and of are...

Factors affecting the ability of energy functions to discriminate correct from incorrect folds

“…C a centers are also treated as simple spheres but are only used for determining the burial of side-chains (Park & Levitt, 1996). The exposed surface area of the side-chain is determined using the approximate method of Wodak & Janin (1980).…”

Kinetic studies of the refolding of yeast phosphoglycerate kinase: Comparison with the isolated engineered domains

“…This might be related to a difference in stability among the domains of the protein and also by the importance of the interdomain interactions. Domains in proteins have been defined as structural units sometimes bearing a functional site (Rossmann & Liljas, 1974;Rose, 1979;Wodak & Janin, 1980). It was proposed by Wetlaufer (1973) that domains are independent folding units.…”

Fast accurate evaluation of protein solvent exposure

“…We chose to minimize ␦A protein buried rather than ␦A residue buried (the residue-by-residue deviation) to make sure that the total buried areas are as accurate as possible. Note also that by minimizing ␦A (10) , we minimized both the average and the spread of ␦A protein buried for the ten-protein set. For n ϭ 0, the Street and Mayo method without generic side-chains, we minimized ␦A (10) with respect to one parameter, the scaling factor s. For n ϭ 0, we also implemented the version of Street and Mayo's method with three different scaling factors: for i and j both core residues, for i and j both non-core residues and for i or j a core residue and the other a non-core residue.…”

“…For two spheres in contact, there is a simple analytical formula to calculate the buried area (and therefore the exposed area) using the two radii and the separation between the spheres. 10 For proteins, this formula was initially applied to each pair of atoms, and the total exposed and buried areas were estimated via a statistical combination. 10 Street and Mayo 1 significantly improved on this statistical combination of pair-atom areas by calculating pair-residue areas.…”

“…10 For proteins, this formula was initially applied to each pair of atoms, and the total exposed and buried areas were estimated via a statistical combination. 10 Street and Mayo 1 significantly improved on this statistical combination of pair-atom areas by calculating pair-residue areas. In the presence of the backbone, two residues were placed at positions i and j, first separately and then together, and the areas buried by the two side-chains were obtained.…”