Stochastic dynamics of electrical membrane with voltage-dependent ion channel fluctuations

¹ Department of Applied Mathematics, University of Washington - Seattle, WA 98195-3925, USA
² College of Mathematics, Jilin University - Changchun, Jilin 130012, PRC
³ Department of Mathematics, Zhejiang Normal University - Jinhua, Zhejiang 321004, PRC
⁴ School of Mathematical Sciences, Peking University - Beijing 100871, PRC

Received: 30 October 2013
Accepted: 28 March 2014

Abstract

A Brownian-ratchet–like stochastic theory for the electrochemical membrane system of Hodgkin-Huxley (HH) is developed. The system is characterized by a continuous variable , representing mobile membrane charge density, and a discrete variable K_t representing ion channel conformational dynamics. A Nernst-Planck-Nyquist-Johnson–type equilibrium is obtained when multiple conducting ions have a common reversal potential. Detailed balance yields a previously unknown relation between the channel switching rates and membrane capacitance, bypassing an Eyring-type explicit treatment of gating charge kinetics. From a molecular structural standpoint, the membrane charge Q_m is a more natural dynamic variable than the potential V_m; our formalism treats Q_m-dependent conformational transition rates as intrinsic parameters. Therefore, in principle,  vs. V_m is experimental-protocol–dependent, e.g., different from voltage or charge clamping measurements. For constant membrane capacitance per unit area C_m and neglecting the membrane potential induced by gating charges, , and HH's formalism is recovered. The presence of two types of ions, with different channels and reversal potentials, gives rise to a nonequilibrium steady state with positive entropy production e_p. For rapidly fluctuating channels, an expression for e_p is obtained.