Rapid calculation of first and second derivatives of conformational energy with respect to dihedral angles for proteins general recurrent equations

A fast algorithm to calculate first and second partial derivatives of conformational energy in proteins with respect to dihedral angles is described. The method is based on the evaluation of new recurrent equations which allow the calculation of the gradient and the Hessian of the conformational energy parallel to the calculation of the conformational energy and in approximately the same number of operations. The recurrent equations are derived by using the hierarchical tree structure of interaction sets in a polypeptide chain. In contrast to a previously published procedure the summation of the new recurrent equations need only a memory space proportional to n. The method is tested for a small sized protein, bovine pancreatic trypsin inhibitor. Potential applications of the method are the minimization of conformational energy and the normal mode analysis of fluctuations in proteins.