Direct determination of reaction paths and stationary points on potential of mean force surfaces

A simulation approach is introduced for directly determining reaction paths and stationary points on potential of mean force (PMF) surfaces associated with molecular events that occur in complex environments. The nudged elastic band approach was employed to search for steepest descent paths on the PMF surface using the relevant PMF derivatives from a series of local simulations. The steepest descent path on the PMF surface corresponds to the minimum PMF path (i.e. the path with the lowest local PMF barrier), which contains important information about stationary points (e.g. saddle points) on the PMF surface, which in turn can provide useful insights into the thermodynamics and kinetics for the process of interest. By working with the PMF defined in a low-dimensional sub-space rather than a potential energy function of full molecular dimensionality, the main features of the process under study are concisely represented and the orthogonal degrees of freedom are adequately sampled with the appropriate canonical distribution at the desired temperature (e.g. 300 K). Therefore, minimum PMF paths carry statistically meaningful mechanistic information and are complementary to reaction paths of full molecular dimensionality proposed in previous studies. The NEB based path optimization method is direct in the sense that no information regarding the global PMF surface is necessary for the determination of the local reaction path and stationary points along this path. Since only low-dimensional quantities (paths) are searched for, the PMF-path method is expected to scale better in terms of dimension of the PMF sub-space than those aims to fully explore multi-dimensional PMF surfaces. Test applications on simple molecular systems, the alanine di-peptide in vacuum and in solution and a micro-solvated proton-wire, indicate that reliable PMF paths can be determined for both conformational isomerization and chemical reaction processes. However, highly accurate PMF derivatives are required for determining more quantitative observables, such as the free energy profile along the minimum PMF path. Therefore, effective numerical algorithms for calculating local PMF derivatives and systematic protocols for defining the relevant sub-space are the main focus in the near future. Finally, we emphasize that the minimum PMF path defined here includes thermal (e.g. entropic) effects associated with the orthogonal degrees of freedom, but finite kinetic energies associated with the PMF degrees of freedom are not included; this can be improved by adopting a different definition of the reaction path, such as the maximum flux path, on the PMF surface, or thermally sampling all degrees of freedom orthogonal to the one-dimensional path.