A Rigorous Approach to Derive Analytical Expressions in Coarse-Grained Force Fields

Abstract

Analytical expressions for the energy terms in coarse-grained force fields are usually imported from the all-atom force fields. However, coarse-grained force fields have a different origin because they stem from the potentials of mean force of the systems of interest in which the degrees of freedom are averaged out. In our earlier work [1] we developed a rigorous approach to derive coarse-grained force fields by factorization of the potentials of mean force into Kubo's cluster-cumulant functions. In this work, we extend this methodology to derive physics-based functional forms of the effective energy terms. The all-atom potential energy is expressed in terms of the squares of interatomic distances, which (and, thereby, also the energy) are then expanded into the Taylor series in the rotation angles of the extended units about the virtual-bond axes. By using the Kubo cumulant expansion [2] and averaging over the rotation angles, the combinations of variables which the effective energy terms depend on, and approximate energy expressions can be determined. In particular, the expressions for the ‘torsional’ potentials must include the virtual-bond-valence angles to be meaningful. As an example, new valence-torsional potentials for the united-residue UNRES force field for proteins and the united-residue NARES-2P force field for nucleic acids developed in our laboratory are presented. As opposed to the torsional-only potentials, the new potentials possess minima at the alpha-helical and not only extended structures of polypeptide chains, in agreement with the statistics derived from the Protein Data Bank. The performance of the new potentials in the simulations of protein folding and DNA hybridization in comparison with the old potentials will be discussed.[1] Liwo, A., et al., J. Chem. Phys., 115, 2323-2347 (2001).[2] Kubo, R., J. Phys. Soc. Japan, 17, 1100-1120 (1962).