Section 22.1.2.2.6. Analytic surface calculations and the Gauss–Bonnet theorem

An analytical method was also proposed for calculating approximate accessible areas (Wodak & Janin, 1980). It assumed random distributions of neighbouring atoms, but this can be a sufficient approximation when calculating the area of an entire molecule. The areas of spherical and toroidal pieces of surface can be calculated exactly by analytic and differential geometry (Richmond, 1984; Connolly, 1983). An advantage of analytical expressions over the prior numerical approximations is that analytical derivatives of the areas can be calculated, albeit with significant difficulty. This then provides the opportunity to optimize atomic positions with respect to surface area. Pseudo-energy functions that approximate the hydrophobic contribution to free energy with a term proportional to the accessible surface area (Richards, 1977) can therefore be incorporated in energy-minimization programs. Although rigorous, these methods are computationally cumbersome and are not used in all energy-minimization routines. Incorporation of solvent effects may become more universal with the Gaussian atom approximations discussed below.