Relaxation Phenomena in the Activation and Inactivation Gates of Ionic Channels
Mahmut ¨Ozer
Department of Electrical and Electronics Engineering, Zonguldak Karaelmas University, 67100 Zonguldak, Turkey
(Received December 10, 2002)
The dynamics of a voltage-gated ionic channel is modeled by the conventional Hodgkin-Huxley mathematical formalism. In that formalism, the dynamics of the ionic channel activa-tion and inactivaactiva-tion gates is modeled by a first-order differential equaactiva-tion dependent on the gate variable and the membrane potential. In this study a method, which combines statistical equilibrium theory and the thermodynamics of irreversible processes, is proposed for the study of the relaxation phenomena in the activation and inactivation gates of ionic channels present in the excitable membranes of neurons. In order to study the relaxation phenomena, the as-sumption is made that the activation and inactivation gate order parameters can be treated as fluxes and forces, in the sense of Onsager’s theory of irreversible thermodynamics. The kinetic equations are solved by using the Runge-Kutta method, in order to study the relax-ation of the order parameters. It is found that the kinetic equrelax-ations are characterized by two relaxation times. The kinetic coefficients that relate the fluxes to the forces are determined. Furthermore, it is shown that the obtained relaxation times have the same results as those obtained by using the Hodgkin-Huxley model. These results therefore indicate the validity of the proposed approach.
PACS. 87.10.+e – General, theoretical, and mathematical biophysics. I. Introduction
Excitable membranes play a fundamental role in neural information processing. The mod-eling of a realistic neuronal structure is an important tool for neuroscientists trying to understand neuronal functions. Modern neurobiology methods have been significantly influenced by Hodgkin and Huxley. They derived mathematical equations describing two types of voltage-gated ionic channels in giant squid axons [1]. Neuroscientists have usually used the Hodgkin-Huxley (H-H) model of coupled four-dimensional nonlinear differential equations for the modeling of other neuronal structures [2]. Their mathematical model has been particularly useful in clarifying the time-dependent behavior of nerve excitations [3]. Therefore Hodgkin-Huxley’s mathematical for-malism is still used to describe the dynamics of voltage-gated ionic channels [4-7]. Although Clay [8] has proposed an improved model for the membrane potential of the giant squid axon, the H-H model remains as a paradigm for a conductance based model of neurons.
Various dynamical properties of the H-H model have been extensively studied in terms of their biological implications. Holden and Yoda [9] showed, by using the H-H model, that the specific channel density can act as a bifurcation parameter and can control the excitability and autorhythmicity of excitable membranes. They also investigated the effect of the channel
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density on the stability of the membrane potential and the response to applied currents using numerical solutions for the H-H equations [10]. Rinzel and Miller [11] argued that the conventional approach, in which it is accepted that the qualitative properties of H-H model can be reduced to a two dimensional flow, such as for the Fitzhugh-Nagumo model is not always valid. Aihara and Matsumoto [12] examined numerical solutions to the H-H equations for an intact squid axon bathed in potassium-rich sea water with an externally applied inward current. They showed that the equations have a complex bifurcation structure including, in addition to stable steady-states, a stable limit cycle, two unstable equilibrium points, and one asymptotically stable equilibrium point. Laboriau [13] has studied the H-H model in terms of the Hopf bifurcation, which is critical in locating regions of bistability. Meunier [14] has studied two different two-dimensional reductions of the H-H equations, and showed that the reductions display the same qualitative bifurcation scheme as the original equations but overestimate the current range where periodic emission occurs. Guckenheimer and Labouriau [15] have systematically analyzed how an axon changes its type of response from repetitive firing to single action potentials in the context of the H-H equations, and have illustrated how the dynamical behavior of the H-H model changes as a function of two of the system parameters. Kaplan et al. [16] have analyzed the subthreshold dynamics of periodically stimulated giant squid axons, showing that the subthreshold responses play a crucial role in generating complex aperiodic sequences of intermixed subthreshold responses and action potentials. In a recent study Doi and Kumagai [17] showed the existence of chaotic attractors in a modified H-H model. ¨Ozer and Erdem [18] proposed a new model to describe the dynamic behavior of ionic channel activation gates in ionic channels in terms of their internal energy by using the path probability method, which is extensively used in nonequilibrium statistical physics.
In the present study, a method that combines statistical equilibrium theory and the thermo-dynamics of irreversible processes is proposed for the study of the relaxation phenomena in the activation and inactivation gates of ionic channels present in the excitable membranes of neurons. We also obtain the relaxation times by using Onsager’s theory of irreversible thermodynamics.
This paper is organized as follows: In Section 2, the H-H mathematical model of an ionic channel is discussed briefly. In Section 3, we define an ionic channel gate model and the equilibrium properties are given. Section 4 contains the derivation of the entropy production and the kinetic equations using Onsager’s theory of irreversible thermodynamics. The results and a discussion are presented in Section 5, and Section 6 is devoted to a conclusion.
II. Hodgkin-Huxley mathematical model of an ionic channel
In the H-H mathematical model, each ionic channel is assumed to have one or more independent gates. A gate exists in just two states, closed or open. In order for the ionic channel to be open, all of its gates must be in the open state. With these assumptions, the conductance of an ionic channel through a population of identical ionic channels is defined by H-H as follows [1]:
GX(V; t) = gXmp(V; t)hp(V; t); (1)
where m and h show the voltage-dependent probability of being in an open state for the activation and inactivation gates respectively, gX is the maximal conductance of the ionic channel when all
of the gates are in the open state, p is the number of activation gates, and q is the number of inactivation gates.
The activation and inactivation gates open and close over time in response to the membrane potential according to the first-order differential equations
dm
dt = ®m(V )(1¡ m) ¡ ¯m(V )m; (2)
dh
dt = ®h(V )(1¡ h) ¡ ¯h(V )h: (3)
Eqs. (2) and (3) state that the closed activation gate (1¡ m) and the inactivation gate (1 ¡ h) open at a rate of ®m(V ) and ®h(V ), respectively; the open activation gate m and the inactivation
gate h close at a rate of ¯m(V ) and ¯h(V ), respectively. The rate functions ®(V ) and ¯(V )
are dependent on the potential across the membrane. The forms of the rate functions are usually determined by using a mix of theoretical and empirical considerations [19]. Eqs. (2) and (3) may be written as [19]: dm dt = m1(V )¡ m ¿m(V ) ; (4) dh dt = h1(V )¡ h ¿h(V ) ; (5)
where m1(V ) and h1(V ) are steady-state activation (i:e: steady-state open gate fraction for activation) and inactivation (i:e: steady-state open gate fraction for inactivation), respectively, since m and h will become asymptotically close to these values if the voltage is held constant for a sufficient duration; ¿m(V ) and ¿h(V ) show the voltage-dependent activation and inactivation
time constants, so that the time course for approaching these equilibrium values is described by a simple exponential with these time constants, and may be written as:
m1(V ) = ®m(V ) ®m(V ) + ¯m(V ) ; (6a) h1(V ) = ®h(V ) ®h(V ) + ¯h(V ) ; (6b) ¿m(V ) = 1 ®m(V ) + ¯m(V ) ; (7a) ¿h(V ) = 1 ®h(V ) + ¯h(V ) : (7b)
III. An ionic channel gate model and static properties in the molecular field approximation An ionic channel gate exists in just two states: closed (C) and open (O). The gate can change from one state to the other at random. The changes are stochastic events — that is to
say they occur at random in the time domain — and so we can describe their timing only in probabilistic terms. There is some structure or property of the gate that is concerned with the transition between these two states, and the word gate is used to describe this concept [20]. Gating is the process whereby the gate is opened and closed. There may be a number of different closed and open states, so the gating processes may involve a number of different sequential or alternative transitions from one gate state to another.
We consider a simple two-state gate system in which the conformational change consists of the movement of a single gating particle. At any moment this particle is in one of two positions, 1 or 2, which are associated with the closed and open states respectively. Positions 1 and 2 correspond to two wells in an energy profile, and there is a single energy barrier between them.
In a population of p identical activation gates and q identical inactivation gates, in a channel in which the activation and inactivation gates are independent, the internal variables will be indicated by x1 and x2 for the activation and by x3 and x4 for the inactivation, which are also
called the state or point variables. x1 and x3 are the fractions of gating particles in position 1,
with energy "1and "3 for the activation and inactivation gates, respectively, and x2 and x4 are the
fractions of gating particles in position 2, with energy "2and "4 for the activation and inactivation
gates, respectively, so that x1+ x2 = 1 and x3+ x4= 1. The weight factor can be expressed
in terms of the internal variables as:
= p!
(x1p)!(x2p)!
q! (x3q)!(x4q)!
: (8)
A simple expression for the internal energy of such a system, in the presence of a membrane potential, is written as:
E = p 2 X i=1 xi"i+ ze0px1V + q 4 X i=3 xi"i+ ze0qx3V; (9)
where z is the number of charges on each particle in the activation or inactivation gate, e0 is the
elementary electronic charge and V is the potential difference, also called the membrane potential. In the molecular field approximation, the entropy and the Helmholtz free energy of the system are given by SE = kBln =¡kBp 2 X i=1 xiln xi¡ kBq 4 X i=3 xiln xi; (10) F = E¡ T SE = p 2 X i=1 xi"i+ ze0px1V + q 4 X i=3 xi"i+ ze0qx3V +T kB Ã p 2 X i=1 xiln xi+ q 4 X i=3 xiln xi ! ; (11)
where kB is the Boltzmann constant, and T is the absolute temperature.
In our case x2 and x4 represent m and h and x1 and x3 represent (1¡ m) and (1 ¡ h),
respectively. By substituting m and h in Eq. (11) we get
F = p["1(1¡ m) + "2m] + ze0p(1¡ m)V + q["3(1¡ h) + "4h] + ze0q(1¡ h)V
+kBT p[(1¡ m) ln(1 ¡ m) + m ln m] + kBT q[(1¡ h) ln(1 ¡ h) + h ln h]:
The equilibrium values of the activation and inactivation order parameters are found by the con-ditions: @F @m = 0; (13a) @F @h = 0: (13b)
One can easily find the following set of self-consistent equations from Eqs. (12) and (13) by introducing a half-(in)activation voltage VH:
m = 1
1 + e¡¯ze0(V¡VH); (14)
h = 1
1 + e¡¯ze0(V¡VH); (15)
where ¯ = 1=kBT ., VH in Eq. (14) equals ¡("1 ¡ "2)=ze0 and VH in Eq. (15) equals
¡("3¡ "4)=ze0. The VH define a voltage at which half the gates are open.
IV. Kinetic equations and relaxation times for the activation and inactivation gates
In this section, we use the Onsager reciprocity theorem (ORT) [21-22] to obtain the ki-netic equations for the activation and inactivation gates and their relaxation times. It has been successfully applied to many transport or irreversible processes, such as steady-state interface mo-tion during phase transformamo-tion in a two-component system [23-24], transport in inhomogeneous media [25], gyrothermal effect with polyatomic gases [26], and spin-1 Ising system [27].
In the thermodynamics of irreversible processes, when the system is not in equilibrium, the derivatives of the Helmholtz free energy F of the system with respect to the order parameters are not equal to zero and can be regarded as generalized forces which cause changes in the parameters. According to the ORT, the relation between the time rate of change of deviations ®i
in the parameters, from their equilibrium values, and the generalized forces are written as Ji= d®i dt = X i;j °ijXj; (i; j = 1; 2; : : : ); (16)
where Ji are thermodynamic fluxes, Xj are the generalized forces and °ij is a matrix of kinetic
coefficients. Onsager’s reciprocity theorem requires that for °ij
°ij =§°ji; (17)
where the minus sign is taken when ®i and ®j have different parity, and the plus sign when ®i
and ®j have the same parity [27].
The generalized forces are obtained by differentiating the entropy production ¢SE with
respect to the deviations ®i as follows:
Xi = @(¢SE) @®i =¡X j ¯ij®j: (18)
One can write the following equation for the deviation of the entropy from its equilibrium value given in Eq. (18): ¢SE =¡ 1 2 X i;j ¯ij®i®j; (19)
where ¯ij denotes the entropy production coefficients defined by
¯ij =¡ µ @2SE @®i@®j ¶ eq = 1 T µ @2F @®i@®j ¶ eq ; (20)
where F is the Helmholtz free energy for the system given by Eqs. (11) and (12), and the subscript “eq” represents equilibrium.
In our case, we have two thermodynamic quantities, m for the activation and h for the inactivation, and two corresponding generalized forces, Xmand Xh. Deviations of m and h from
their equilibrium values are represented by (m¡ m0) and (h¡ h0), respectively. m0 and h0 are
the equilibrium values of m and h, respectively.
The entropy production for the system can be obtained by using Eq. (12) in Eqs. (19) and (20) as: ¢SE =¡ 1 2 £ A(m¡ m0)2+ 2B(m¡ m0)(h¡ h0) + C(h¡ h0)2 ¤ ; (21)
and the entropy production coefficients are A = 1 T µ @2F @m2 ¶ eq = kBp m0(1¡ m0) ; (22) B = 1 T µ @2F @m@h ¶ eq = 0; (23) C = 1 T µ @2F @h2 ¶ eq = kBq h0(1¡ h0) : (24)
The generalized forces Xm and Xh are found by substituting Eq. (21) into Eq. (18):
Xm=
@(¢SE)
@(m¡ m0)
=¡A(m ¡ m0)¡ B(h ¡ h0) =¡A(m ¡ m0) for B = 0; (25a)
Xh=
@(¢SE)
@(h¡ h0)
=¡B(m ¡ m0)¡ C(h ¡ h0) =¡C(m ¡ m0) for B = 0: (25b)
The linear relations between the fluxes and the generalized forces may be written in terms of the kinetic coefficients by substituting Eq. (25a) and (25b) into Eq. (16):
∙ Jm Jh ¸ = ∙ °m ° ° °h ¸ ∙ Xm Xh ¸ ; (26)
where a symmetric matrix of kinetic coefficients is introduced so as to satisfy the Onsager’s reciprocal relation. The off-diagonal kinetic coefficient ° is coupled with the activation and inactivation gate fluxes in Eq. (26). In our case, the gates are independent and have no interaction. Therefore, we take ° = 0. The rate equations are then obtained from the matrix of Eq. (26):
dm
dt =¡°mA(m¡ m0); (27a)
dh
dt =¡°hC(h¡ h0): (27b)
Assuming a form e¡t=¿ for both (m¡ m0) and (h¡ h0) for Eq. (27a) and (27b) one obtains the
following secular equation: ¯ ¯ ¯ ¯ (1=¿ )¡ °0 mA (1=¿ )0¡ °hC ¯ ¯ ¯ ¯ = 0: (28)
Two solutions of this equations are ¿1= 1 °mA ; (29a) ¿2= 1 °hC : (29b)
We have determined the kinetic coefficients °m and °h in Eq. (29a) and (29b) so as to
satisfy the time constants given by the H-H in Eq. (7): °m= ®m¯m kBp(®m+ ¯m) ; (30a) °h = ®h¯h kBq(®h+ ¯h) : (30b)
V. Results and discussion
In our calculations we consider the activation and inactivation gates of the sodium ionic channel for the giant squid axons. The activation and inactivation rate functions for the sodium ionic channel at 6.3±C are given as [28]:
®m(V ) = 0.1(40 + V ) 1¡ e¡(V +40)=10; ¯m(V ) = 0.108 e ¡V =18; (31) ®h(V ) = 0.0027 e¡V =20; ¯h(V ) = 1 1 + e¡(V +35)=10: (32)
The membrane potentials given in the rate functions above are the absolute potential according to the modern sign convention in contrast to the ones used by H-H, in which the potentials are expressed as the difference from the resting membrane potential.
FIG. 1. Equilibrium value curves for activation and inactivation gates of the sodium channel. See text for the definition of m and h.
First, sigmoidal-shaped curves are fitted to the steady-state activation (m1) and inactivation (h1) curves of the sodium ionic channel obtained by substituting Eqs. (31) and (32) into Eq. (6a) and (6b). The following parameter values are then estimated: [VH, kBT =ze0] = [-39.5819 mV,
9.5508 mV] and [-62.1965 mV, -7.0669 mV] for the activation and inactivation gates, respectively. The equilibrium value curves are plotted in Fig. 1 for the activation and inactivation by using Eqs. (14) and (15). The activation equilibrium value curve rises to 1 from 0 and the inactivation curve falls from 1 to 0 as the voltage is increased, as seen in Figure 1. Eqs. (14) and (15) are also known as the Boltzmann relation [20]. They give sigmoidal curves on a linear plot of m and h against V , as shown in Fig. 1. The half-activation voltage, VH, was used by Borg-Graham
[29] in the gating of ion channels to parameterize the Hodgkin-Huxley models. The forward and backward rate constants introduced by Borg-Graham resulted in the same self-consistent equation as Eqs. (14) and (15). In this respect, our methodology is consistent with the findings of the linear thermodynamic model. VH and z affect the steady-state activation/inactivation curves. Therefore
these parameters are also convenient for fitting experimental data [30]. Willms et al. [19] noted that the Boltzmann curve in particular has proved to be adequate for describing steady-state data from many currents. Our conclusions also support it. In this context, VH in Eqs. (14) and (15)
is equal to V2m in their Eq. (10) and have the same description. 1=¯ze0 corresponds to sm in
their Eq. (10), and it is called a slope parameter. The magnitude of 1=¯ze0 or sm determines the
steepness of the Boltzmann curve. The slope of the curve at VH is proportional to ¯ze0.
In order to determine the relaxation which occurs from states near equilibrium, it is necessary to solve Eqs. (27a) and (27b). We can solve these equations by using the Runge-Kutta method. For this purpose, we determine where the activation and the inactivation curves relax for different voltage values. Different activation curves against time are obtained for -60, -50, -39.5819, -30 and 10 mV membrane potential levels with m(0) = 1, by using Eqs. (22), (27a) and (30a), where m(0) represents the initial value of the activation. The results are shown in Fig. 2. The activation relaxes to different values, as can be seen in Fig. 2. Relaxed values of the activation curves are identical with the activation equilibrium values shown in Fig. 1: 0.105473, 0.251466, 0.5, 0.731698 and 0.956782, corresponding to -60, -50, -39.5819, -30 and 10 mV membrane potential levels, respectively. The different inactivation curves against time are then obtained for -80, -70,
FIG. 2. Sodium channel activation curves for different potential levels with m(0) = 1.0.
FIG. 3. Sodium channel inactivation curves for different potential levels with h(0) = 1.0.
FIG. 4. Sodium channel activation relaxation times for the H-H model and proposed approach.
FIG. 5. Sodium channel inactivation relaxation times for the H-H model and proposed approach.
-62.1965, -50 and -40 mV membrane potential levels with h(0) = 1, by using Eqs. (24), (27b), and (30b), where h(0) represents the initial value of the inactivation. The results are shown in Fig. 3. Relaxed inactivation curve values are the same as the inactivation equilibrium values shown in Fig. 1: 0.925482, 0.751052, 0.5, 0.15118 and 0.041451 corresponding to the -80, -70, -60 and -40 mV membrane potential levels, respectively.
The activation and inactivation order parameters m and h approach their equilibrium values m0 and h0, with two relaxation times, ¿1 and ¿2 given by Eq. (29a) and (29b), respectively. The
variations of the activation relaxation time with the voltage are obtained for Eq. (7a), and Eqs. (22), (29a), and (30a) separately, and are shown in Fig. 4. The variations of the inactivation relaxation time with the voltage are obtained for Eq. (7b), and Eqs. (24), (29b), and (30b) separately, and are shown in Fig. 5. We obtained the same relaxation times as given by the H-H model, as seen in Figs. 4 and 5. The activation gate relaxes to its equilibrium value much faster than the inactivation gate does, as is shown in Figs. 4 and 5. The kinetic coefficients °m and °h in Eq. (29a) and
FIG. 6. Variation of the kinetic coefficient °! with voltage.
FIG. 7. Variation of the kinetic coefficient °" with voltage.
(29b) are plotted on a semi-logarithmic scale against the voltage in Figs. 6 and 7, respectively. The kinetic coefficient curves have the same characteristics as the relaxation times, except for the magnitudes. Furthermore, it is seen that the kinetic coefficient °h is greater than °m in magnitude.
We determined the kinetic coefficients °m and °h in Eq. (29a) and (29b) so as to satisfy
the time constants given by the H-H in Eq. (7). The activation and inactivation curves obtained by solving Eq. (27a) and (27b), which involve the kinetic coefficients are shown to be relaxed to the same equilibrium values obtained from self-consistent Eqs. (14) and (15) for different voltage values. Thus the results support the validity of the proposed formalism. The rate equations in Eqs. (27a) and (27b) consist of equilibrium values and kinetic activation and inactivation coefficients. The quantities that are directly observable from the voltage-clamp experiments are the steady-state (equilibrium) activation and inactivation, and the activation and inactivation time constants which are given in Eqs. (4) and (5) [31]. Therefore, the proposed formalism can be used for the dynamic behavior analysis of different voltage-gated ionic currents which have voltage-clamp recordings.
Gating is a process involving a conformational state change in the channel protein [32]. Different conformational states may have different dipole moments, so the free energies of these conformational states will be influenced in different ways by the membrane electric potential [33]. In this context, the theory of electroconformational coupling (TEC) proposes that a transporter protein uses electric energy from the medium or high periodic fields that were generated by the membrane electric potential to drive a transport reaction away from equilibrium, in order to change the conformational state, because of an altered internal membrane energy state [34-35]. Therefore, further study using the TEC may provide additional insight into the detailed mechanism of the relaxation phenomena in the activation and inactivation gates of ionic channels.
VI. Conclusion
We have studied using Onsager’s reciprocity theorem, relaxation phenomena in the activation and inactivation gates of ionic channels present in excitable membranes. The equilibrium values of the activation and inactivation order parameters were obtained for self-consistency. We found that
the activation and inactivation curves relax to the values determined by using the self-consistent equilibrium value equations. It was found that the kinetic activation and inactivation equations are characterized by two relaxation times. Furthermore, we determined the kinetic coefficients, so as to satisfy the relaxation times given in the H-H model, and demonstrated that the obtained relaxation times have same results as are given by the H-H model.
Acknowledgements
I am indebted to Dr. R³za Erdem for fruitful discussions related to this paper. References
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