Algorithms for calculating excluded volume and its derivatives as a function of molecular conformation and their use in energy minimization

A numerical method for calculating the volume of a macromolecule and its first and second derivatives as a function of atomic coordinates is presented. For N atoms, the method requires about 0.3 N ln(N) seconds of CPU time on a VAX‐8800 to evaluate the volume and derivatives. As a test case, the method was used to evaluate a pressure‐volume energy term in energy minimizations of the protein lysozyme at 1000 atm (1 atm = 1.013 × 10⁵ Pa). R.m.s. gradients of 10⁻⁴ kcal/mol/Å were obtained at convergence. The calculated structures exhibited pressure‐induced changes which were qualitatively similar to the changes observed in the 1000 atm structure determined by X‐ray crystallography.