On Reaction Coordinate Optimality

The following question is addressed: how to establish that a constructed reaction coordinate is optimal, i.e., that it provides an accurate description of dynamics. It is shown that the reaction coordinate is optimal if its cut free energy profile, determined using length-weighted transitions, is constant, i.e., it is position and sampling interval independent. The observation leads to a number of interesting results. In particular, the equilibrium flux between two boundary states can be computed exactly as diffusion on a free energy profile associated with the coordinate. The mean square displacement, for the trajectory projected onto the coordinate, grows linear with time. That for the same trajectory projected onto a suboptimal coordinate grows slower than linear with time. The results are illustrated on a number of model systems, Sierpinski gasket, FIP35 protein, and beta3s peptide.