PHYSICAL REVIEW E 86, 061915 (2012) Entropy hysteresis and nonequilibrium thermodynamic efficiency of ion conduction in a voltage-gated potassium ion channel Biswajit Das, Kinshuk Banerjee, and Gautam Gangopadhyay * S. N. Bose National Centre For Basic Sciences Block JD, Sector III, Salt Lake, Kolkata 700098, India (Received 14 July 2012; revised manuscript received 16 October 2012; published 27 December 2012) 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. DOI: 10.1103/PhysRevE.86.061915 PACS number(s): 87.16.Vy, 05.70.Ln, 05.10.Gg, 05.90.+m I. INTRODUCTION The study of ion channels plays an important role in understanding the propagation of nerve impulse and a wide variety of phenomena associated with excitable tissue of neural as well as non-neural nature [1–4]. Ion channels maintain a controlled exchange of ions between the cells and the extracellular medium through ion-permeable pores with the rearrangement of the tertiary structure of channel proteins. A great deal of understanding about the function of ion channels owes its origin in the experiments using the voltage clamp method [3–11]. In a traditional voltage clamp technique, ion flow across a cell membrane is measured as electric current, while the membrane voltage is held under experimental control with a feedback circuit [3–8]. Current due to a single ion channel can also be measured using a patch clamp experiment based on a similar principle [4]. Recently, nonequilibrium response spectroscopy [12,13] has added a new dimension in the field of ion channel experiments using the oscillating-voltage protocol. This technique has been used for the selection of an appropriate Markov model from various possible schemes of ion channel kinetics [12–16]. From kinetic studies, it has been found qualitatively that the oscillating voltage drives the ion channel out of equilibrium and resists the system to relax back to equilibrium [12–15]. The oscillating-voltage protocol [16] thus offers an opportunity to explore nonequilibrium response properties of the ion channel such as hysteresis [17] at nonequilibrium steady state (NESS). Hysteresis has a long history [17] in its wide manifestation in various magnetic [18,19] and other condensed-matter systems [20,21] as well as in biological processes [16,22,23]. In voltage-gated ion channels, hysteresis can occur when the time period of the oscillating external voltage is comparable to the characteristic relaxation time of the conformational transitions between conducting and nonconducting states [23–26]. The channel hysteresis has biological relevance; for * gautam@bose.res.in example, it plays an important functional role in regulating physiological phenomena and is also a governing factor in maintaining the action of a neuron pacemaker [25]. A detailed theoretical description of hysteresis in ion channel for oscillating voltage was given by Pustovoit et al. [26] by considering a simple two-state model. Recently, Ander- sson described the hysteresis of ionic conductance [27] for oscillating voltage by considering the analysis of Pustovoit et al. [26] and then they have extended the study of the channel gating schemes for multiple states with independent as well as cooperative gating. Their studies [26,27] reveal that the probability-voltage as well as the current-voltage hysteresis is dynamic in nature. The hysteresis loop area vanishes at the low and high frequency limits of the external oscillating voltage due to the wide time scale separation. Now, particularly for time-dependent external voltage, the system can go arbitrarily far away from equilibrium. Hence, along with the kinetic properties, the nonequilibrium thermodynamic features of the ion channel must also be explored. In this perspective, we have raised the following questions. (i) Are these low and high frequency limiting situations equivalent from the thermodynamic viewpoint or does the vanishing of an out-of-equilibrium phenomenon like hysteresis ensure that the system is at thermodynamic equilibrium? (ii) At NESS, how is the supplied energy utilized for the production of ionic current? To address the above issues coherently, we present a detailed nonequilibrium thermodynamic analysis of a voltage-gated Shaker potassium ion channel. The ion channel kinetics is described by a master equation constructed on the basis of the best-suited Markov process proposed in an experimental work [14]. Starting from a model consisting of five states, we have discussed about how the stochastic conformational states are connected with the essential features of a traditional Hodgkin-Huxley equation at a constant voltage. We then have explored the nonequilibrium thermodynamic features due to oscillating voltage. The paper is organized as follows. In Sec. II A, we describe the kinetic scheme which is efficient to describe the ion channel kinetics both for constant and oscillating voltages [14]. For 061915-1 1539-3755/2012/86(6)/061915(10) ©2012 American Physical Society