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A new approach to de¢ne dynamics of the ion
channel gates
Mahmut Ozer,
CARiza Erdem
1and Ivo Provaznik
2Department of Electrical and Electronics Engineering, Engineering Faculty, Zonguldak Karaelmas University, 67100 Zonguldak, Turkey;1Department of
Physics, Gaziosmanpasa University, 60250 Tokat, Turkey;2Department of Biomedical Engineering, Faculty of Electrical Engineering and Communication,
Brno University of Technology, 61200 Brno, Czech Republic
CACorresponding Author: mahmutozer2002@yahoo.com
Received 28 August 2003; accepted 13 October 2003 DOI: 10.1097/01.wnr.0000103272.91211.33
Voltage-gated ion channels are of great importance in the genera-tion and propagagenera-tion of electrical signals in the excitable cell mem-branes. How these channels respond to changes in the potential across the membrane has been a challenging problem, and di¡erent approaches have been proposed to address the mechanism of vol-tage sensing and gating in these channels. In this study, we attempt
a new approach by considering a simple two-state gate system and applying the path probability method to construct a nonequilibrium statistical mechanical model of the system. The model which is based on the principles of statistical physics provides a ¢rm physical basis for ion channel gating. NeuroReport 15:335^338 c 2004 Lippincott Williams & Wilkins.
Key words: (In)Activation gate; Ion channel; Path probability method; Statistical mechanics
INTRODUCTION
Ion channels provide conduction pathways for specific ions and thus constitute the fundamental elements for electric signalling in nerve, muscle and synapse [1]. Voltage-gated ion channels which form an important class of such channels are involved in the generation and propagation of electrical signals in the excitable cell membranes. The voltage-dependent gating of these channels between con-ducting and non-concon-ducting states is a major factor in controlling the transmembrane potential. The gating processes may involve a number of different sequential or alternative transitions from one channel state to another [2].
Hodgkin and Huxley (H-H) [3] provided the first quantitative description of the voltage-dependent gating of ion currents. In their formalism, voltage-dependent gating of the ion channel requires the movement of hypothetical gating particles able to sense the electric field across the membrane [4]. Their mathematical model has been particu-larly useful in clarifying the time-dependent behaviour of nerve excitation [5].
Voltage-gated ion channels are formed by pore-like proteins the functions of which are dictated by their possible conformations. They have some charged regions which are stimulated by membrane potential. The physical nature of the gating and its correlation with the charged group remain largely unknown [1]. In the conventional approach, it is assumed that the opening and closing of the channel take place by a conformational change of the protein. Suenaga et al. [6] proposed another possible mechanism and speculated from their observations that certain counterion
configurations correspond to the open state of the channel and the other configurations correspond to the closed state. Although it seems to be an attractive alternative to the conformational change model, such an open/close mechan-ism is difficult to verify experimentally.
In the energy-level models, it is assumed that the channel protein has a few conformational states which are separated by static energy barriers. It is also assumed that the passage of ions through the channel does not change the conforma-tional state of the channel protein and its energy barriers. A linear voltage-dependence of the free energy is usually assumed to be sufficient to describe the gating of ion channels [7–9]. In this context, thermodynamic models were attempted to provide a physical basis to parameterize the voltage dependence of the rate functions instead of using empirical functions. Thermodynamic models assume that a free-energy barrier is associated with the transition by using an analogy with the theory of reaction rates [10,11], and that the effect of the electrical field is linearly related to the free energy [12]. The major complication stems from this approach is that the time constant can reach arbitrarily small values. In a recent study, the non-linear effects of the electric field on the free energy were considered to solve this problem by including higher-order terms in the free-energy of a given state [13]. Lee and Sung [1] proposed a theoretical methodology to describe the stochastic behaviours of singly occupied ion channel by taking into account the interaction between ions and the associated channel conformational variable. Recently, we presented a new methodology to define the equilibrium value function in the kinetics of (in)activation gates based on the lowest approximation of
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the cluster variation method (CVM) and the static properties in the molecular field approximation [14].
Here we extend our previous work [14] and formulate a statistical mechanical model for the dynamics of ion channel gates. In order to construct an equation describing non-equilibrium behaviour for the gate, we use the path probability method which is a dynamical version of the CVM. The model which is constructed on the microscopic description of the ion channel gates is based on the principles of statistical physics and conceptually quite different from that of conventional models.
MATERIALS AND METHODS
We consider a simple two-state gate system in which the conformational change consists of the movement of a single gating particule. At any moment, this particule is in one of two positions, 1 and 2, which are associated with closed and open states, respectively. In Eyring rate theory terms, posi-tions 1 and 2 correspond to two wells in an energy profile, and there is a single energy barrier between them [2,15].
Given a population of p identical gates in a channel, the internal variables will be indicated by x1and x2for the gate.
x1 with energy E1 is the fractions of gating particle in
position 1 and x2 with energy E2 represents the fractions
of gating particle in position 2. Hence x1+ x2¼ 1. The
weight factor O can be expressed in terms of the internal variables as
O ¼p!=½ðx1pÞ!ðx2pÞ! ðeqn 1Þ A simple expression for the internal energy of such a system in the presence of a membrane potential is written as
E ¼ pX
2 i¼1
x1"iþ ze0px1V ðeqn 2Þ where z is the number of charges on the particle in the gate, e0is the elementary electronic charge and V is the membrane
potential.
We derived the equilibrium (steady-state) value function in our previous work [14]. In the present paper, the dynamic equation for our non-interactive two state gate model is derived by introducing the path probability rate coefficients which satisfy the detailed balancing relation. In order to construct an equation describing non-equilibrium behaviour for the gate, we use the path probability method (PPM) of Kikuchi [16]. The PPM which is the natural extension of the CVM into the time domain is a more general statistical-mechanical approach of formulating kinetic equations for a cooperative system. Kikuchi’s method gives a complete answer to the dynamical problem in the system.
The CVM proposes a general scheme for making approximate expressions for free energies and has been widely used to study thermal equilibrium states in a variety of systems. Although the CVM was originally developed to describe the configurational entropy of Ising lattices, it has transcended its original application and is now routinely applied to the study of numerous other systems. The main contribution of the CVM is the recipe of writing the entropy of a system in terms of the state variables (xi,i ¼ 1,2...) [17].
The state variables define the probability of occurrence of each configuration of a chosen cluster and specify a point on the free energy surface defined by the CVM [18]. After the entropy and the energy are written in terms of the state variables, the free energy is to be minimized with respect to the state variables to obtain the equilibrium state.
In PPM, a state of the system at t0 defined with the
variables xi(t0) is considered and the probability with which
the system changes toward a certain direction in a short time interval Dt is examined, as shown in Fig. 1. For this purpose, Kikuchi [16] introduced a number of parameters called the path parameters. The function which writes the probability of change in a short time interval Dt is called the path probability function (PPF). The basic procedure of the PPM is to maximize the PPF to obtain the most probable path [17]. In this respect, arrows in Fig. 1 represent fluctuated paths for the time interval Dt. The smooth curve in Fig. 1 represents the most probable path obtained by maximizing the PPF. Therefore, each consecutive state separated by Dt defined by the PPM is connected by the condition of the maximum of the PPF as the transition probability. In other words, the path parameters are deter-mined in such a way that the path probability is a maximum with respect to any variation of these parameters. The free energy always decreases following the most probable path which is for the maximum of the PPF. It is also shown that the physical meaning of log of the PPF is the increase of entropy [17]. Therefore, the PPF corresponds to the probability of finding a free energy of an ensemble in equilibrium.
The PPM has been successively applied to describe the non-equilibrium behaviour of a number of systems such as phonon and atomic diffusion systems [19] and spin-1 Ising model [20].
RESULTS
In the PPM, the rate of change of a state variable is written as
dxi=dt ¼ X
i6¼j
ðXji XyÞ ðeqn 3Þ
where Xij is the path probability rate coefficients for the
model to go from state i to j. The coefficients Xij are the
product of three factors: kij, the rate constants with
kij¼ kji, a temperature-dependent factor which guarantees
Xi (t)
Xi (t0)
t0 t0+∆t
t
Fig.1. Time dependence of the state variables xi(i ¼1,2,y) in an
irrever-sible process. Arrows represent £uctuated paths for the time interval Dt. The smooth curve represents the most probable path obtained by maximizing the path probability function.
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that the time-independent state is the equilibrium state, and a third factor which is the fraction of the system that is in the state i, e.g. xi. Detailed balancing
requires that
Xij¼ Xji ðeqn 4Þ
The following two options were introduced by Kikuchi [16] as follows:
Xij¼ kijZ1xiexpððb=pÞð@E=@xjÞÞ ðeqn 5aÞ Xij¼ kijZ1xiexpfðb=2pÞð@E=@xjÞ ð@E=@xjÞg ðeqn 5bÞ which both fulfill the necessary requirements expressed by eqn (4), Z is the partition function which was given explicitly in eqn (14) of our previous study [14] and b ¼ 1/ kBT, where kB is the Boltzmann constant and T is the
absolute temperature. In our model, we use the first option, eqn (5a), in order to derive the dynamic or rate equation. First and second options given by eqns (5a) and (5b) result in the same conclusion. By substituting eqn (2) into eqn (5a) we obtain X12¼ k12Z1x1expððb=pÞðdE=dx2ÞÞ ¼ k12Z1x1e2 ¼ k12Z1x1expðb"2Þ ðeqn 6aÞ X21¼ k21Z1x2expððb=pÞð@E=@x1ÞÞ ¼ k21Z1x2e1 ¼ k21Z1x1expðb½"1þ ze0VÞ ðeqn 6bÞ
Then the rate of change of the second state variable (x2) is
given as
dx2=dt ¼ X12 X21¼ k12Z1x1e2 k21Z1x2e1 ðeqn 7Þ or
dx2=dt ¼ X12 X21¼ ðk=ZÞðx1e2 x2e1Þ ðeqn 8Þ where k ¼ k12¼ k21 and Z ¼ (e1+ e2). Thus, the dynamic
equation for the probability of being open state for the gate in terms of symbols of the activation and inactivation gates, m and h, respectively is obtained:
dm=dt ¼ðk=ZÞ½ð1 mÞe2 me1 or dh=dt ¼ðk=ZÞ ½ð1 hÞe2 he1
ðeqn 9Þ By reorganizing eqn 9 we obtain
dm=dt ¼ ðk=ZÞ½ð1 mÞe2 me1
¼ ðk=ZÞ½e2 mðe1þ e2Þ ¼ ½e2=Z m
ðeqn 10Þ We showed that e2/Z equals the equilibrium value or
steady-state value of the activation gate for a given membrane potential and defined it in terms of VH and z
in our previous study [14]. VHdenotes a voltage at which
half of the gates are open. These parameters are also convenient for fitting experimental data [21]. Therefore eqn 10 can be written as follows:
dm=dt ¼ k½m1 m ðeqn 11Þ
We determined the k constant so as to satisfy the H-H mathematical formalism as follows:
k ¼ 1=m¼ ðamþ bmÞ ðeqn 12Þ
where amand bmare rate constants.
DISCUSSION
We proposed in our previous study [14] a new methodology to define the equilibrium (steady-state) value function in the kinetics of (in)activation gates. In the present paper, we extend our previous work [14] to describe the gate dynamics in the ion channel by using the path probability method. The dynamic equation for our non-interactive two-state gate model is derived by introducing the path probability rate coefficients which satisfy the detailed balancing relation.
The major complication stems from linear thermody-namic models is that the time constant can reach arbitrarily small values especially for depolarized membrane poten-tials. In a recent study, the non-linear effects of the electric field on the free energy were considered to solve this problem by including higher-order terms in the free-energy of a given state [13]. Assuming that the higher-order terms would also take into account the presence of mechanical constraints on the gating process, they concluded that non-linear thermodynamic models provide fits of comparable quality to empirical H-H models. We derived the dynamic equation for the probability of being open state for the gate and the determined k constant so as to satisfy the H-H mathematical formalism. Therefore our derivation does not cause any problems with the time constant. It also leads to the same formalism as that of H-H. In this context, it can be used as an attractive alternative to both linear and non-linear thermodynamic models. Since the time constants obtained by our approach exactly match those of H-H, we may point out that the statistical treatment in this case is more powerful than the thermodynamic one.
Although we compared our derivations with the H-H mathematical formalism as a reference, the recent gating current measurements and single-channel recordings re-vealed that the H-H mathematical formalism itself has some limitations. Recent data from Blum et al. [22] provided direct evidence that the opening of Nav1.9, a member of the
voltage-gated Na+ channel family, is mediated by ligand
binding rather than by voltage. A more serious objection to the macroscopic H-H equations is that they are not derived from a microscopic description of the neuronal membrane grounded on the opening and closing of single ion channels [23]. This emphasizes the difficulty in deriving a micro-scopic model from a macromicro-scopic description. However, the model proposed in this study is constructed based on the microscopic description of the ion channel gates. We use the principles of the statistical physics for deriving the dynamic equations. This approach is also conceptually quite different from that of the H-H.
We considered here a simple non-interactive two-state gate model, that is there is no explicit interactions between neighbouring channel states. Recent evidence also indicates that the coupling between activation and inactivation can no longer be neglected. In our approach, once the entropy and energy are written in terms of the state variables, the remaining part becomes a straightforward problem. On one hand it leads to the same formalism as that of the H-H with a constraint on k, on the other hand it enables us to generalize the model by incorporating the energy arising from the coupling or the mechanical constraints in eqn 2. Therefore, we may conclude that it provides a firm physical basis to the ion channel gating.
Voltage-gated ion channels fluctuate randomly between conformational states due to thermal agitation. Fluctuations
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A NEW APPROACH TO DEFINE DYNAMICS OF THE ION CHANNEL GATES NEUROREPORT
between conducting and non-conducting states result in noisy membrane currents and subthreshold voltage fluctua-tions which may contribute to variability in spike timing [24]. As we indicated in Materials and Methods, the basic procedure of the PPM is to maximize the PPF to obtain the most probable path. However, there are many other paths in addition to the most probable path which represent fluctuations in the PPF. Therefore, the PPM can treat not only the most probable motion of the system but also fluctuations from its most probable path owing to the variation principle of the PPM [25]. Consequently, our derivations also faciliate the investigation of fluctuations in the ion channel gates by making use of information of fluctuations in the PPF.
CONCLUSION
In the present study, we proposed a new approach to define dynamical behaviour of the ion channel gates by means of the path probability method which is natural extension of the cluster variation method into the time domain as a complementary work to our previous one [14]. Our approach provides a firm physical basis for the ion channel gating and does not cause any problems for the time constants with a constraint in which k constant is deter-mined so as to satisfy the H-H mathematical formalism. In this respect, we can propose it as an attractive alternative to the both linear and non-linear thermodynamic models. However how well it accounts for the experimental data can also be compared with the H-H, and linear and non-linear thermodynamic models without using any constraint on k. It should also be noted that the present model considers a simple non-interactive two-state gate model and does not take account of the presence of mechanical constraints on the gating process and interactions between neighbouring channel states. These questions will be addressed in the future publications. We will also investigate fluctuations in the ion channel gates by making use of information of fluctuations in the PPF as a new application of the PPM for the ion channel gates.
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