Entropy hysteresis and nonequilibrium thermodynamic efficiency of ion conduction in a voltage-gated potassium ion channel

Abstract

Here we have studied the nonequilibrium thermodynamic response of a voltage-gated Shaker potassium ion channel using a stochastic master equation. For a constant external voltage, the system reaches equilibrium indicated by the vanishing total entropy production rate, whereas for oscillating voltage the current and entropy production rates show dynamic hysteretic behavior. Here we have shown quantitatively that although the hysteresis loop area vanishes in low and high frequency domains of the external voltage, they are thermodynamically distinguishable. In the very low frequency domain, the system remains close to equilibrium, whereas at high frequencies it goes to a nonequilibrium steady state (NESS) associated with a finite value of dissipation function. At NESS, the efficiency of the ion conduction can also be related with the nonlinear dependence of the dissipation function on the power of the external field. Another intriguing aspect is that, at the high frequency limit, the total entropy production rate oscillates at NESS with half of the time period of the external voltage.

Dynamic memory of a single voltage-gated potassium ion channel: A stochastic nonequilibrium thermodynamic analysis

“…27 It is easy to see that, even in this case, the ionic current does not have this symmetry. One can easily verify that the same temporal symmetry relations given in Eq.…”

“…For k 1 (0) k −1 (0) ≫ 1, the ion-conducting state probability P ss 4 (t) dominates with P ss 0 (t), P ss 1 (t), P ss 2 (t) being negligibly small. 27 Then, from Eq. (20), we can write approximatelẏ…”

“…The details of their behavior, particularly the dynamic hysteresis, are already discussed elsewhere. 27 However, for the sake of completeness and readability, we provide some representative plots here in the long-time limit.…”

“…The state C 4 , with all the subunits in the active state, is taken as the ionconducting state. 22,27 For the dynamics of a single ion channel, the number of subunits in active state, m, is a fluctuating quantity. [33][34][35] Then, it is appropriate to describe the time evolution of the single channel as a Markov process.…”

“…Taking an extended version of the ion channel model, it has been found that, although current-voltage as well as total EPR-voltage hysteresis are dynamic in nature, the (average) total EPR increases with voltage frequency to saturation. 27 With this background in mind, it is therefore natural to ask the questions: (i) how does the vanishing hysteresis correspond with the finite dissipation? and more importantly, (ii) is the vanishing hysteresis in the limiting conditions a general feature, valid for all the periodic responses of the system in the long time limit?…”