Position-dependent friction in protein folding from a GLE derived with a non-stationary localized projection distribution

Abstract

The Generalized Langevin Equation (GLE) is an integro-differential equation of motion for a general observable of a many-body system and is rigorously derived using the projection operator formalism. In the standard derivation, a stationary canonical phase space distribution is used for the projection. Here, we derive a novel class of GLEs using a non-stationary projection distribution that constrains the initial ensemble to a hypersurface in phase space on which the observable has a fixed value $A_0$. As a result, all GLE parameters, and in particular the memory friction kernel, depend on $A_0$ and can be extracted from short simulations that do not sample the entire phase space. Applying this non-stationary GLE to protein folding trajectories, we find for villin and an $α$-helical poly-alanine segment that the total friction is higher in the folded state. These results demonstrate that observable-dependent friction effects are non-negligible and can be accounted for using non-stationary GLEs derived by constrained projection schemes.