A generalized Langevin algorithm for studying permeation across biological ion channels

Abstract

We present a new leapfrog algorithm for the numerical solution of the generalized Langevin equation (GLE) in the case where the friction kernel is exponentially decaying. Like other leapfrog and Verlet algorithms, our algorithm is second order in velocity and third order in position. It is relatively easy to implement compared with other available algorithms, and would therefore make a good candidate for exploring the effects of finite memory time-scales in situations where modelling the precise functional form of the memory kernel was not important. We have tested this algorithm on a one-dimensional barrier crossing model, and found good asymptotic agreement with limits obtained using Brownian dynamics (BD) simulations, as well as with a theoretical asymptotic limit. We have also used the algorithm to perform a more sophisticated simulation of ion conduction through a KcsA channel. The results are a close match to corresponding results obtained using the Langevin equation, thereby helping to justify the use of Brownian dynamics in KcsA and other similar ion channels.