Large-Scale Dynamical Models and Estimation for Permeation in Biological Membrane Ion Channels

Biological ion channels are water-filled angstrom-unit $(1\ { \hbox{angstrom}}\ {\hbox{ unit}}=10^{-10}\ {\hbox{m}})$ sized pores formed by proteins in the cell membrane. They are responsible for regulating the flow of ions into and out of a cell and hence they control all electrical activities in a cell. This paper deals with constructing large scale stochastic dynamical models for explaining ion permeation; that is, how individual ions interact with the protein atoms in an ion channel and travel through the channel. These permeation models capture the dynamics of the ions at a femto-second time scale and angstrom-unit spatial scale. We review large scale multiparticle simulation methods such as Brownian dynamics for modeling permeation. Then we present a novel multiparticle simulation methodology, which we call adaptive controlled Brownian dynamics, for estimating the force experienced by a permeating ion at each discrete position along the ion-conducting pathway. The profile of this force, commonly known as the potential of mean force, results from the electrostatic interactions between the ions in the conduit and all the charges carried by atoms forming the channel the protein, as well as the induced charges on the protein wall. We illustrate the use of adaptive controlled Brownian dynamics in gramicidin channels and shape estimation of sodium channels.