Solvent accessible surface area and excluded volume in proteins. Analytical equations for overlapping spheres and implications for the hydrophobic effect.

An analytical formula has been derived for the calculation of the solvent accessible surface area of a protein molecule or equivalently the surface area exterior to an arbitrary number of overlapping spheres. The directional derivative of this function with respect to atomic co-ordinates is provided to facilitate minimization procedures used with molecular docking algorithms and energy calculations. An analytical formula for the calculation of the volume enclosed within the accessible surface, the excluded volume, is also derived. Although the area function is not specific to the structures of proteins, the derivation was motivated by the need for a computationally feasible simulation of the hydrophobic effect in proteins. A computer program using the equations for area has been tested and has had limited application to the docking of protein alpha-helices. Possible relationships of the solvent excluded volume to hydrophobic interaction free energy and transfer free energy of solute molecules are derived from the statistical mechanics of solution.