Efficient minimization of angle-dependent potentials for polypeptides in internal coordinates

Angular potentials play an important role in the refinement of protein structures through angle-dependent restraints (e.g., those determined by cross-correlated relaxations, residual dipolar couplings, and hydrogen bonds). Analytic derivatives of such angular potentials with respect to the dihedral angles of proteins would be useful for optimizing such restraints and other types of angular potentials (i.e., such as we are now introducing into protein structure prediction) but have not been described. In this article, analytic derivatives are calculated for four types of angular potentials and integrated with the efficient recursive derivative calculation methods of Gō and coworkers. The formulas are implemented in publicly available software and illustrated by refining a low-resolution protein structure with idealized vector-angle, dipolar-coupling, and hydrogen-bond restraints. The method is now being used routinely to optimize hydrogen-bonding potentials in ROSETTA.