A Scaling Relation Between Accessible Phase Space Volume And Escape Rates For Random Walks In High-Dimensional Hierarchical Energy Landscapes - Implications For Protein Dynamics

Protein dynamics can be described as a random walk within the underlying many particles free energy landscapes, which, as first shown by Frauenfelder et al., exhibit a multi-tier hierarchy of barrier heights. Each of these tiers in this hierarchy are characterized by motions of characteristic length scales and time scales. As a simple model of such landscapes, we have considered random walks in high dimensional cubic grids where transition rates are determined by exponentially distributed random free energy barriers, governed by a ‘ruggedness parameter’ ΔF, and have shown that this model can reproduce anomalous diffusion exponents seen in molecular dynamics (MD) trajectories. Here we show that this ruggedness, further, serves as an order parameter for a scaling relation between the accessible phase space volume and the corresponding escape rates.This result quantifies how the ruggedness and the effective dimensionality of hierarchical free energy landscapes determine phase space exploration.