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We present a novel rate theory based on the notions of splitting probability and mean first passage time to describe single-ion conduction in narrow, effectively one-dimensional membrane channels. In contrast to traditional approaches such as transition state theory or Kramers theory, transitions between different conduction states in our model are governed by rates which depend on the full geometry of the potential of mean force (PMF) resulting from the superposition of an equilibrium free energy profile and a transmembrane potential induced by a nonequilibrium constraint. If a detailed theoretical PMF is available (e.g., from atomistic molecular dynamics simulations), it can be used to compute characteristic conductance curves in the framework of our model, thereby bridging the gap between the atomistic and the mesoscopic level of description. Explicit analytic solutions for the rates, the ion flux, and the associated electric current can be obtained by approximating the actual PMF by a piecewise linear potential. As illustrative examples, we consider both a theoretical and an experimental application of the model. The theoretical example is based on a hypothetical channel with a fully symmetric sawtooth equilibrium PMF. For this system, we explore how changes in the spatial extent of the binding sites affect the rate of transport when a linear voltage ramp is applied. Already for the case of a single binding site, we find that there is an optimum size of the site which maximizes the current through the channel provided that the applied voltage exceeds a threshold value given by the binding energy of the site. The above optimization effect is shown to arise from the complex interplay between the channel structure and the applied electric field, expressed by a nonlinear dependence of the rates with respect to the linear size of the binding site. In studying the properties of current-voltage curves, we find a double crossover between sublinear and superlinear behaviors as the size of the binding site is varied. The ratio of unidirectional fluxes clearly deviates from the Ussing limit and can be characterized by a flux ratio exponent which decreases below unity as the binding site becomes wider. We also explore effects arising from changes in the ion bulk concentration under symmetric ionic conditions and the presence of additional binding sites in the hypothetical channel. As for the experimental application, we show that our rate theory is able to provide good fits to conductance data for sodium permeation through the gramicidin A channel. Possible extensions of the theory to treat the case of an asymmetric equilibrium PMF, fluctuations in the mean number of translocating ions, the case of fluctuating energy barriers, and multi-ion conductance are briefly discussed.

Original publication

DOI

10.1063/1.3077205

Type

Journal article

Journal

J Chem Phys

Publication Date

28/02/2009

Volume

130

Keywords

Biological Transport, Electrophysiology, Ion Channels, Kinetics, Membrane Potentials, Models, Biological