Downloaded from https://journals.lww.com/neuroreport by BhDMf5ePHKav1zEoum1tQfN4a+kJLhEZgbsIHo4XMi0hCywCX1AWnYQp/IlQrHD3z484KQalTVETncwKJTq3oIukANfFw4Ygi6Bwd5rp328= on 08/14/2020 Downloadedfrom https://journals.lww.com/neuroreportby BhDMf5ePHKav1zEoum1tQfN4a+kJLhEZgbsIHo4XMi0hCywCX1AWnYQp/IlQrHD3z484KQalTVETncwKJTq3oIukANfFw4Ygi6Bwd5rp328=on 08/14/2020
An improved non-linear thermodynamic model of
voltage-dependent ionic currents
Mahmut Ozer
CADepartment of Electrical and Electronics Engineering, Engineering Faculty, Zonguldak Karaelmas University, 67100 Zonguldak, Turkey
CACorresponding Author: mahmutozer2002@yahoo.com
Received 31 May 2004; accepted 5 July 2004
Thermodynamic models of ionic currents are used to deduce exact functional form of rate constant. It is ¢rst assumed that free en-ergy depends linearly on voltage. A major criticism of this approach is that time constant can reach arbitrarily small values resulting in an aberrant behavior. Recently, non-linear e¡ects of electric ¢eld on the free energy were considered to solve this problem based onT-type calcium channel. In this study, we show that the current
model approximately captures the voltage-dependence of the time constant only in a speci¢c range of voltage and does not provide saturation of the time constant outside this range. Then, we pro-pose an improved non-linear thermodynamic model and illustrate its applicability based on T-type calcium channel. NeuroReport 15:1953^1957 c 2004 Lippincott Williams & Wilkins.
Key words: Ion channel; Rate constant; Thermodynamic model; T-type calcium channel
INTRODUCTION
Voltage-gated ion channels are formed by pore-like proteins whose functions are dictated by their possible conforma-tions. They include charged regions which make their structure susceptible to the membrane potential. Hodgkin and Huxley (H-H) [1] provided the first quantitative description of the voltage-dependent gating of the 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.
During the last few years, there have been enormous strides in our understanding of structure-function relation-ships in the ion channels. In a recent breakthrough, the molecular structure of the Streptomyces lividans potassium channel has been determined from crystallographic analyses [2]. It is also expected that crystal structures of other ion channel will usher us into a new era in ion channel studies, where predicting the function of channels from their atomic structures will become the main quest [3]. In this context, one of the most exciting recent developments in the ion channel gating is the determination of an X-ray crystal structure of a voltage-gated potassium channel [4,5]. It was found that the voltage sensors, called paddles, are attached to the central ion-conducting pore by flexible hinges and apparently move in response to the membrane potential changes by carrying their positive charges across the membrane. It was also shown that the voltage sensor paddles are positioned inside the membrane, near the intracellular surface, when the channel is closed, and that the paddles move a large distance across the membrane from inside to outside when the channel opens [5]. Although the result does not provide a definitive answer
on how a change in the membrane potential opens and closes the channel [6], and additional structures will be necessary to clarify how the voltage sensors pack against the pore in the membrane in the closed and open conformations [7], this finding directly challenges previous models.
Parallel to these landmark experimental findings, there have been also important advances in computational biophysics. There is a need to develop models that can relate the structural parameters of the channels to experi-mental data and thereby build a theoretical framework that can explain different sets of observations [8]. In this context, thermodynamic models were proposed to deduce exact functional form of voltage-dependence of rate constants, which are used to fit voltage-clamp experiments in order to be parameterized. Thermodynamic models assume that a free energy barrier is associated with transitions by using an analogy with the theory of reaction rates [9,10]. In the linear thermodynamic models, the effect of the electric field is assumed to be linearly related to the free energy [10,11]. A major criticism of this thermodynamic model is that the time constant can reach arbitrarily small values, despite the fact that conformational changes require a certain minimum amount of time to occur [12]. This problem was attempted to be overcome by introducing a minimal rate [13], forcing the rate constant to saturate [12] or using additional voltage-independent rate-limiting transitions with Markov repre-sentation [14,15]. Recently, 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 [14]. They also illustrated its applicability by modeling the voltage dependence of T-type calcium currents from thalamic relay neurons [14].
The purpose of this paper is to illustrate the inadequacy of the standard form of the non-linear thermodynamic model proposed by Destexhe and Huguenard [14] and to introduce an improved form for its rate constants based on the T-type calcium channel. In this context, we show that the non-linear model approximately captures the voltage-depen-dence of the time constant only in a specific range of the voltage and does not provide saturation of the time constant to a constant value outside this range. Then, we also show that the proposed model more accurately captures the voltage-dependence of the time constant in this range, and also provides saturation of the time constant to a constant value outside this range based on T-type calcium channel.
MATERIALS AND METHODS
Thermodynamic models assume that the gating of ion channels operates through conformational changes of the ion channel gates. Therefore, the transitions between two states of the channel correspond to the conformational change of the ion channel protein. We consider a transition between an initial (1) and a final (2) state with a voltage-dependent rate constant k(V). According to the theory of reaction rates [9,10], the rate of transition k(V) between the two states depends exponentially on the free energy barrier between them as follows:
kðVÞ ¼ k0eDGðVÞ=RT ð1Þ
where k0is a constant and DG(V) is the free energy barrier
given by
DGðVÞ ¼ GðVÞ G1ðVÞ ð2Þ
where G*(V) is the free energy of an activated complex and G1(V) is the free energy of the initial (1) state. The
equilibrium distribution between these states are deter-mined by using the relative values of the free energy of the initial and final states, G1(V) and G2(V). However,
the kinetics of the transition is determined by the size of the free energy barrier DG(V). The free energy of a given state i can be written as a Taylor series expansion of the form [14–16]:
GiðVÞ ¼ Aiþ BiV þ CiV2þ . . . : ð3Þ where the constant Aicorresponds to the free energy that is
independent of the electric field, the linear term BiV
corresponds to the interaction between electric field with isolated charges and rigid dipoles [11,17–19]. The linear term results if the conformational change accompanies the translation of a freely moving charge inside the structure of the channel [15]. Non-linear terms describe effects such as electronic polarization and pressure induced by membrane potential V [11,18,19], as well as mechanical constraints in the movement of charges due to the structure of the ion channel protein [14].
The rate constant k(V) can be expressed in a general form by substituting eqns (2) and (3) into eqn (1) as follows:
kðVÞ ¼ k0e ðA þBVþCV2þ...:ÞðA 1þB1VþC1V2þ...:Þ ½ ¼ k0eðaþbVþcV 2þ...Þ=RT ð4Þ where a ¼ A A 1, b ¼ B B1, c ¼ C C1. . .. Applying eqn (4) to forward and backward reactions of the particular case of a reversible open-closed transition, we get the
following general expression for the voltage-dependence of rate constants as follows:
aðVÞ ¼ a0eða1þb1Vþc1V
2þ...:Þ=RT
bðVÞ ¼ b0eða2þb2Vþc2V
2þ...:Þ=RT ð5Þ
where a and b are the forward and backward rate constants, respectively, and a1, b1, c1, a2, b2, c2,y, are constants specific
of this transition. Since the three different conformations considered here might have different distributions of charges, these constants are not necessarily interrelated [15]. This general form for the voltage dependence of the rate constants is called non-linear thermodynamic model [14,15]. The linear thermodynamic model, which is the simplest approximation, considers only first-order terms in V for the rate constants given by eqn (5). The non-linear thermodynamic model considers different complexity by including quadratic, cubic or higher-order terms in V for the rate constants. In this case, the selection of the complexity is based on the acceptable fits.
RESULTS AND DISCUSSION
Destexhe and Huguenard [14] proposed the non-linear thermodynamic models of voltage-dependent currents leading to a physically plausible solution to a major criticism of the linear thermodynamic model, which is that the time constant can reach arbitrarily small values. They illustrated its applicability to fit experimental data by considering the case of the T-type calcium current in thalamic relay neurons, and performed the fitting simulta-neously on two data sets, time constants and steady-state values, to determine the exact functional form of the rate constants am, bm for the activation and ah, bh for the
inactivation of T-type calcium channel. The steady-state activation and inactivation relations were fit by Boltzmann function, leading to the following functions [14,20]:
m1¼ 1
1 þ eðVþ57Þ=6:2 ð6Þ
h1¼ 1
1 þ eðVþ81Þ=4 ð7Þ
They also fitted multiexponential functions to the experi-mental data to represent the voltage-dependence of time constants, leading to the following expression for activation [14,20]:
m¼ 0:612
þ 1= expððV þ 132Þ=16:7Þ þ expððV þ 16:8Þ=18:2Þ ð8Þ and for inactivation [14,20]:
h¼ 28 þ exp ðV þ 22Þ=10:5 ; V 81 exp ðV þ 467Þ=66:6 ; Vo 81 ð9Þ Eqns (6)–(9) are also called an empirical model for the T-type calcium channel. Destexhe and Huguenard [14] considered non-linear thermodynamic models (eqn (5)) of different complexity to fit the voltage-clamp data of the T-current and obtained acceptable fits for a cubic expansion of the rate constants given in their eqn (28). We obtained activation and inactivation time constants of the empirical model by eqns (8) and (9), respectively, and of the non-linear thermodynamic models by substituting their eqn (28) into ¼1=ða þ bÞ. The results are shown in Fig. 1a,b for the
activation and inactivation time constants, respectively. The activation time constant of the non-linear thermodynamic model produced approximately a consistent voltage-depen-dent trajectory with that of empirical model up to around 0 mV, but it reached arbitrarily large values from that potential level as seen in Fig. 1a. However, activation time constant of the empirical model saturated to a constant value of around 0.6 ms for these depolarized membrane potentials. In the case of inactivation time constant, the non-linear thermodynamic model resulted in large deviations in the time constant up to 81 mV, while it produced a more consistent trajectory with that of empirical model up to around 6 mV. However, it could not capture the saturation of inactivation time constant to a constant value, and it begun to fall down toward zero from the latter potential level.
Although Destexhe and Huguenard [14] suggested that the rate constants, derived from non-linear thermodynamic model and fitted to the experimental data, captured the
saturation of the time constants to a minimal value, we showed that the activation time constant lead to arbitrarily large values and inactivation time constant was close to zero for the depolarized membrane potential levels as seen in Fig. 1a,b, respectively. They also concluded that non-linear expansions of higher order provided quantitatively but not qualitatively better fits [14]. These results motivated us to reconsider the rate function form for the non-linear thermodynamic model given by eqn (5). By considering more complex expansions of the free energy and preserving the non-linear terms, we proposed that the following equation for the rate function could provide the best fits of the non-linear thermodynamic model:
aðVÞ ¼X n i¼1 a0;iexp aa;i RT V Va;i 2 h i bðVÞ ¼X n i¼1
b0;iexp ab;i
RT V Vb;i
2
h i ð10Þ
In the following, this functional form for the voltage dependence of the rate constants will be called the improved non-linear thermodynamic model. In fact, eqn (10) is a sum of Gaussion distribution. Considering that the rate constant k(V) involves the sum of many exponentials k0;iexp DG iðVÞ=RT, eqn (10) can be easily derived. In this derivation, we assumed that n distinct transitions deter-mined by different energy barriers results in one transition given by eqn (1). An important property obtained from the detailed studies of a simple protein is that proteins exist in a huge number of quasidegenerate microscopic substates, corresponding to a single macroscopic conformation [21]. Therefore, this assumption takes this property into account. We also assumed that approximating the derivation by a quadratic expansion would be enough to capture the voltage-dependence of the rate constants since it involves the sum of many quadratic expansions. Therefore, in the simplest case of n¼1, it provides the quadratic expansion of eqn (5). In other cases, it takes into account more sophisticated effects of the membrane potential on the channel structure by taking the huge number of almost degenerate conformational substates into account.
We fitted eqn (10) to values of the empirical model of T-type calcium channel to estimate parameter values of the rate constants. Acceptable fits were obtained with n¼4 for the activation of T-type calcium channel. Estimated para-meter values are given in Table 1 and Table 2 for the forward and backward rate constants, respectively. The activation time constant of the improved non-linear thermodynamic model was obtained by using the proposed rate functions and shown in Fig. 1a with the empirical one. The activation time constant of the improved model accurately matched to that of empirical model. It also provided saturation of activation time constant to a constant value, which was the primary goal of deriving the non-linear thermodynamic model.
Fig. 1. Activation and inactivation time constants of theT-type calcium
channel for di¡erent models. (a) Activation time constant of the empiri-cal, non-linear and improved non-linear thermodynamic models. (b) Inac-tivation time constant of the empirical, linear and improved non-linear thermodynamic models.
Table 1. Estimated parameter values of forward rate constant of the
activation.
i¼1 i¼2 i¼3 i¼4
a0,Ims1 1.584 0.1309 0.1312 0.007484
Va,imV 60.96 53.74 12.34 110.8
Finally, we fitted eqn (10) to values of the empirical model of T-type calcium channel to estimate parameter values of the inactivation rate constants. Acceptable fits were obtained with n¼3 for forward rate constant and n¼4 for backward rate constant. Estimated parameter values are given in Table 3 and Table 4 for the inactivation forward and backward rate constants, respectively. The inactivation time constant of the improved non-linear thermodynamic model was obtained by using the proposed rate functions and shown in Fig. 1b with the empirical one. The inactivation time constant of the improved model also accurately matched to that of empirical model. Although non-linear thermodynamic model resulted in large deviations in the inactivation time constant up to 81 mV, the improved model refined these large deviations as shown in Fig. 1b. The improved model produced a much more consistent trajectory with that of empirical model than non-linear thermodynamic model up to around 6 mV. In addition, it also provided saturation of the inactivation time constant to a constant value of 28 ms for the depolarized voltage levels. When the applied voltages are relatively small, i.e. in the low-field limit, the contribution of the higher-order terms may be negligible [11,16,19]. Approximating the form of the rate constant by a second- or third-order polynomial relies on the low-field limit [14]. Considering a quadratic expan-sion, the proposed form of the rate constant in the present study also overlaps with the hypothesis that the voltage is not too large. However, while the non-linear thermody-namic model of T-type calcium channel approximately captures the voltage-dependence of time constant only in a specific range of the membrane potential, the proposed model removes this limitation due to contribution of a summation factor as shown in Fig. 1a,b. The proposed model also captures the voltage-dependence of the time constants more accurately than the conventional non-linear thermodynamic model in this specific range. Destexhe and Huguenard [14] indicated that one must consider more
complex expansion of the free energy, which takes into account more sophisticated effects of the voltage on the channel structure for very large voltages outside this specific range. In this context, we think that the proposed model might address to this consideration.
In the improved non-linear thermodynamic model, the number of the substates, n, depended on the range of applied voltages and temperatures used in the experiment is needed for agreement with single-channel recordings. However, at present, such data are not available for the T-type calcium channel. Therefore, the improved non-linear thermodynamic model for T-type calcium channel constitu-tes more acceptable representation than the current non-linear thermodynamic model. On the other hand, we recently proposed a new approach to address the mechan-ism of voltage gating in the ion channel gates based on the lowest approximation of the cluster variation method and the path probability method of statistical physics [22,23]. Then, we applied this theoretical framework to derive the activation kinetics of T-type calcium channel from thalamic relay neurons and showed that the proposed rate constants accurately captured form of the empirical time constant [24]. By comparing the activation time constant of the path probability model shown in Fig. 1c of the latest study [24] with that of the improved non-linear thermodynamic model shown in the Fig. 1a of the present paper, we conclude that the path probability model still constitute the most acceptable biophysical representation for the T-type calcium channel.
REFERENCES
1. Hodgkin AL and Huxley AF. A quantitative description of membrane currents and its application to conduction and excitation in nerve. J Physiol Lond 1952; 117:500–544.
2. Doyle DA, Cabral JM, Pfyetzner RA, Kuo A, Gulbis JM, Cohen SL et al. The structure of the potassium channel: molecular basis of K+
conduction and selectivity. Science 1998; 280:69–77.
3. Chung SH and Kuyucak S. Recent advances in ion channel research. Biochim Biophys Acta 2002; 1565:267–286.
4. Jiang Y, Lee A, Chen J, Ruta V, Cadene M, Chait BT and MacKinnon R. X-ray structure of a voltage-dependent K+channel. Nature 2003; 423:33–41.
5. Jiang Y, Ruta V, Chen J, Lee A and MacKinnon R. The principle of gating charge movement in a voltage-dependent K+ channel. Nature 2003;
423:42–48.
6. Woolley GA and Lougheed T. Modeling ion channel regulation. Curr Opin Chem Biol 2003; 7:710–714.
7. MacKinnon R. Potassium channels. FEBS Lett 2003; 555:62–65. 8. Chung SH, Allen TW, Hoyles M and Kuyucak S. Permeation of ions
across the potassium channel: Brownian dynamic studies. Biophys J 1999; 77:2517–2533.
9. Eyring H. The activated complex in chemical reactions. J Chem Phys 1935; 3:107–115.
10. Johnson FH, Eyring H and Stover BJ. The Theory of Rate Processes in Biology and Medicine. New York: John Wiley and Sons; 1974.
11. Andersen O and Koeppe RE II. Molecular determinants of channel function. Physiol Rev 1992; 72:S89–S158.
12. Willms AR, Baro DJ, Harris-Warrick RM and Guckenheimer J. An improved parameter estimation method for Hodgkin-Huxley models. J Comput Neurosci 1999; 6:145–168.
13. Borg-Graham LJ. Modeling the non-linear conductances of excitable membranes. In: Wheal H and Chad J (eds). Cellular and Molecular Neurobiology: A Practical Approach. Oxford: Oxford University Press; 1991, pp. 247–275.
14. Destexhe A and Huguenard JR. Non-linear thermodynamic models of voltage-dependent currents. J Comput Neurosci 2000; 9:259–270. 15. Destexhe A and Huguenard J. Which formalism to use for modeling
voltage-dependent conductances? In: De Schutter E (ed.). Computational
Table 2. Estimated parameter values of backward rate constant of the
activation.
i¼1 i¼2 i¼3 i¼4
b0,ims1 190.6 0.07389 0.2103 0.0551
Vb,imV 230.7 120.5 122.1 63.12
ab,iJmV2 2.6789 15.9048 2.2227 6.1855
Table 3. Estimated parameter values of forward rate constant of the
inactivation.
i¼1 i¼2 i¼3
a0,ims1 0.004367 0.002905 0.003907
Vb,imV 128.4 118.8 100.3
aa,iJmV2 53.1037 20.5738 5.47856
Table 4. Estimated parameter values of backward rate constant of the
inactivation.
i¼1 i¼2 i¼3 i¼4
b0,ims1 0.03501 0.001271 0.01883 0.01561
Vb,imV 56.57 82.26 10.73 43.18
Neuroscience: Realistic Modeling for Experimentalists. New York: CRC Press; 2001, pp. 129–157.
16. Destexhe A, Mainen ZF and Sejnowski TJ. Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formulation. J Comput Neurosci 1994; 1:195–230. 17. Tsien RW and Noble D. A transition state theory approach to the kinetics
of conductances in excitable membranes. J Membr Biol 1969; 1:248–273. 18. Hill TL and Chen YD. On the theory of ion transport across nerve
membranes. VI. Free energy and activation free energies of conformational change. Proc Natl Acad Sci USA 1972; 69:1723–1726. 19. Stevens CF. Interactions between intrinsic membrane protein and electric
field. Biophys J 1978; 22:295–306.
20. Huguenard JR and McCormick DA. Simulation of the currents involved in rhythmic oscillations in thalamic relay neurons. J Neurophysiol 1992; 68:1373–1383.
21. Goychuk I and Hanggi P. Ion channel gating: a first-passage time analysis of the Kramer type. Proc Natl Acad Sci 2002; 99:3552–3556.
22. Ozer M and Erdem R. A new methodology to define the equilibrium value function in the kinetics of (in)activation gates. Neuroreport 2003; 14:1071–1073.
23. Ozer M, Erdem R and Provaznik I. A new approach to define dynamics of the ion channel gates. Neuroreport 2004; 15:335–338.
24. Ozer M. Activation kinetics of T-type calcium channel by a path probability approximation. Neuroreport 2004; 15:1451–1455.
