Diffusive reaction dynamics on invariant free energy profiles

A fundamental problem in the analysis of protein folding and other complex reactions in which the entropy plays an important role is the determination of the activation free energy from experimental measurements or computer simulations. This article shows how to combine minimum-cut-based free energy profiles (F(C)), obtained from equilibrium molecular dynamics simulations, with conventional histogram-based free energy profiles (F(H)) to extract the coordinate-dependent diffusion coefficient on the F(C) (i.e., the method determines free energies and a diffusive preexponential factor along an appropriate reaction coordinate). The F(C), in contrast to the F(H), is shown to be invariant with respect to arbitrary transformations of the reaction coordinate, which makes possible partition of configuration space into basins in an invariant way. A "natural coordinate," for which F(H) and F(C) differ by a multiplicative constant (constant diffusion coefficient), is introduced. The approach is illustrated by a model one-dimensional system, the alanine dipeptide, and the folding reaction of a double beta-hairpin miniprotein. It is shown how the results can be used to test whether the putative reaction coordinate is a good reaction coordinate.