From Brownian Dynamics to Markov Chain: an Ion Channel Example

TL;DR

This paper develops a discrete Markov chain model for multi-ion channels by reducing continuous ion diffusion dynamics to state transitions, enabling analysis of ion permeation, escape times, and optimal channel geometry.

Contribution

It introduces a method to derive Markovian transition rates from continuous Fokker-Planck equations for ion channels, bridging detailed dynamics with simplified models.

Findings

The Markov model accurately predicts stationary state probabilities.

The approach matches Brownian dynamics simulation results.

Optimal channel geometry for maximum ion flux is identified.

Abstract

A discrete rate theory for general multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model the Markovian transition rates can be determined. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed.