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

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 Kt 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 Qm is a more natural dynamic variable than the potential Vm; our formalism treats Qm-dependent conformational transition rates as intrinsic parameters. Therefore, in principle, vs. Vm is experimental-protocol–dependent, e.g., different from voltage or charge clamping measurements. For constant membrane capacitance per unit area Cm 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 ep. For rapidly fluctuating channels, an expression for ep is obtained.