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Activation kinetics of T-type calcium channel by
a path probability approximation
Mahmut Ozer
Department of Electrical and Electronics Engineering, Engineering Faculty, Zonguldak Karaelmas University, 67100 Zonguldak, Turkey Corresponding Author: mahmutozer2002@yahoo.com
Received 8 March 2004; accepted 24 March 2004 DOI: 10.1097/01.wnr.0000128429.81961.0a
We previously formulated dynamics of ion channel gates by the path probability method. In this study, we apply that theoretical approach to derive the activation rate kinetics of T-type calcium channel in thalamic relay neurons. We derive explicit expressions of the forward and backward rate constants and show that the proposed rate constants accurately capture form of the empirical time constant, and that they also provide its saturation to a
constant value at depolarized membrane potentials. We also com-pare our derivations with linear and nonlinear thermodynamic models of rate kinetics obtained from the same calcium channel, and show that it is possible to capture saturation of the time constant for the depolarized membrane potentials by the only
proposed rate constants. NeuroReport 15:1451^1455 c 2004
Lippincott Williams & Wilkins.
Key words: Activation gate; Ion channel; Path probability model; T-type calcium channel; Thermodynamic model
INTRODUCTION
Ion channels are key molecules for cellular regulation and communication. They couple biomolecular events to electric signals. Voltage-gated ion channels open and close in a stochastic manner dependent on the transmembrane vol-tage. They 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.
Voltage-gated ion channels are formed by pore-like proteins whose functions are dictated by their possible conformations. They have some charged regions to be stimulated by membrane potential. During the last few years, there have been enormous strides in our under-standing of structure-function relationships in the ion channels. In a recent breakthrough, the molecular structure of the Streptomyces lividans potassium channel has been determined from crystallographic analyses [1]. 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 [2]. 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 [3,4]. These results showed 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. In the accompanying paper [4], 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. Although the result does not provide a definitive answer on how a change in the membrane potential opens and closes the channel [5] 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 [6], it 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 [7]. In this context, different approaches were proposed to address the mechan-ism of voltage sensing and gating in these channels. Yang et al. [8] proposed a statistical mechanical model for ion channels in the presence of electric fields. They obtained the maximum fractions of open potassium and sodium chan-nels by solving a self-consistent non-linear algebraic equation under a mean-field approximation, but formulated their model just for static electric field and not for time-varying electric fields. Roux [9] developed a rigorous statistical mechanical formulation of the equilibrium prop-erties of selective ion channels incorporating the influence of the membrane potential, multiple occupancy and saturation effects. Lee and Sung [10] presented a stochastic model to describe the coupled behaviors of ion transport and channel conformation under an applied transmembrane potential. 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 the cluster
variation method [11]. Then, we proposed a method, which combines statistical equilibrium theory and the thermo-dynamics of irreversible processes for the study of the relaxation phenomena in the activation and inactivation gates of ion channels [12]. In a recent study, we formulated dynamics of the voltage-gated ion channels by the path probability method of the non-equilibrium statistical phy-sics [13].
In this study, we formulate the kinetic equation for the time-dependent open-state probability of the gate with three parameters explicitly, which is also convenient for fitting experimental data. We applied a theoretical approach, developed earlier [13], to derive the activation rate kinetics of T-type calcium channel from thalamic relay neurons by using empirically derived equations by Huguenard and McCormick [14]. We also compare our derivations with the linear and non-linear thermodynamic models of the rate kinetics obtained from the same calcium data by Destexhe and Huguenard [15].
MATERIALS AND METHODS
In our previous papers [11,13], we considered a simple two-state gate system in which the conformational change consists of the movement of gating particles. At any moment, the particles are in one of two positions, 1 and 2, which are associated with closed and open states, respec-tively. The positions 1 and 2 correspond to two wells in an energy profile, and there is a single energy barrier between
them. The internal variables are indicated by x1and x2. x1
with energy E1is the fraction of gating particles in position 1
and x2while energy E2is the fraction of gating particles in
position 2. Since the positions 1 and 2 are associated with
closed and open states, respectively x1and x2correspond to
the probabilities of being closed and open states for the gate.
Hence x1+ x2¼1. In order to study the dynamics of such a
system, first we studied the steady-state behavior of the system, especially to obtain the steady state equation for the gate. This equation was calculated by means of the cluster variation method (CVM), and we found the following equation for the probability of being open state for the gate
x2as follows [11]:
x2¼ 1
1 þ exp bze½ 0ðV VHÞ ð1Þ
where b ¼ 1=kBT, kBis the Boltzmann constant, and T is the
absolute temperature. z is the number of charges on the
particle in the gate, e0is the elementary electronic charge, V
is the membrane potential and VH is the voltage at which
half of the gates are open.
In order to construct an equation describing dynamic behavior of the gate, we used the path probability method (PPM) in our accompanying paper [13]. In our model, we used the first recipe introduced by Kikuchi [16] and obtained the dynamic equation for the probability of being open state as follows [13]:
dx2
dt ¼
k
Zðe2x1 e1x2Þ ð2Þ
e1and e2are defined in our previous study [11] and defined
as follows:
e1¼ exp½bðe1þ ze0VÞ; e2¼ exp½be2 ð3Þ
In eqn 2, Z is the partition function of the system and defined as follows:
Z ¼X
2 i¼1
e1 ð4Þ
We can express Eqn (2) in a similar form with that of Hodgkin-Huxley [17] as follows:
dx2
dt ¼ a V; bð Þ 1 xð 2Þ b V; bð Þx2 ð5Þ
where a V; bð Þ ¼ ke2Z1and b V; bð Þ ¼ ke1Z1are the
voltage-and temperature-dependent forward voltage-and backward rate functions, respectively.
In this paper, we express a V; bð Þ and b V; bð Þ in explicit
forms by using the definitions of functions Z and ei as
follows: aðV; bÞ ¼ ke2Z1¼ k= 1 þ exp bze ½ 0ðV VHÞ ¼ ka0 V; b ð Þ ð6Þ bðV; bÞ ¼ ke1Z1¼ k= 1 þ exp bze ½ 0ðV VHÞ ¼ kb0ðV; bÞ ð7Þ
The relationship between a0 and b0 can be obtained as
a0ðV; bÞ þ b0ðV; bÞ ¼ 1 from eqns 6 and 7. Therefore we
obtain a relation of k ¼ a V; bð Þ þ b V; bð Þ. This result leads to
k ¼ 1= in the sense of H-H formalism, where t represents time constant.
RESULTS
We apply a theoretical approach, developed earlier [13] and described briefly in Materials and Methods, to derive the activation rate kinetics of T-type calcium channel from thalamic relay neurons. Huguenard and McCormick [14] developed empirical mathematical equations that describe the voltage-dependent kinetics of the activation gate of T-type calcium channel in the thalamic relay neurons. We use these empirical equations to parameterize our model. Then, we derive explicit expressions of its forward and backward rate constants. The steady-state activation relation was fit by Boltzmann function, leading to the following function [14]:
m1¼ 1
1 þ e Vþ57ð Þ=6:2 ð8Þ
They also fitted the voltage-dependence of time constant to the experimental data by multiexponential functions as follows:
m¼ 0:612 þ 1
e Vþ132ð Þ=16:7þ eðVþ16:8Þ=18:2 ð9Þ
In order to parameterize our model, we obtained the
parameters VHand 1=bze0 from eqn 8 as 57 and 6.2 mV,
respectively, then we determined the parameter k using the relationship k ¼ 1= from eqn 9.
First, we examined effect of temperature on the steady-state activation using eqn 1 and eqns 6 and 7 separately with above values of the parameters for different values of the temperature. In the calculations with eqns 6 and 7,
temperature effect is just involved in a0(V,b) and b0(V,b).
We observed that a0 and b0 exactly captured effect of the
temperature on the steady-state activation obtained by eqn 1, and that the parameter k was just dependent on the
membrane potential (not shown). Therefore, we tried to derive explicit function of the parameter k depending on the membrane potential by using its empirical values with the
relationship k ¼ 1=. Acceptable fits to the empirical eqn 9 with the relation k¼1/t were obtained for a sum of the Gaussion functions given by
k ¼X 3 i¼1 Aiexp V Bi Ci 2 " # ð10Þ This procedure led to the following parameters for a membrane potential range from 120 mV to 60 mV:
A1¼2.972 ms1, B1¼145.2 mV, C1¼75.62 mV, A2¼3.682 ms1,
B2¼229.6 mV, C2¼72.17 mV, A3¼1.172 ms1, B3¼26.27 mV,
C3¼52.17 mV. We represent the rate constant given by eqn 10
with these parameter values as k1. Activation time constants
for empirical model given by eqn 9 and our model, we call
path probability model, obtained by eqns 6 and 7 with k¼k1
are shown in Fig. 1a. Figure 1a shows that the path
probability model with k1 could capture the form of the
voltage dependence of the activation time constant. In particular, it could fit the saturating value of the time constant at depolarized potential levels. However, the fit to the time constant was poor for a membrane potential range
Fig. 1. Activation time constant of the empirical and path probability
models for the T-type calcium channel. (a) a¼f(k1), b¼f(k1). (b) a¼f(k2),
b¼f(k2). (c) a¼f(k1), b¼f(k2).
Fig. 2. Forward rate constant of the empirical and path probability
models with k¼k1and k¼k2for theT-type calcium channel.
Fig. 3. Backward rate constant of the empirical and path probability
between around 85 mV and 50 mV. Therefore, we examined the poor fit in this specific range based on the
activation rate constants with k1. We observed that the
forward rate constant of the path probability model with k1
exactly matched to that of empirical model as seen in Fig. 2. However, the backward rate constant of the path probability model captured the behavior of the empirical backward rate constant only in a same range of the membrane potential as in the case of the time constant, as shown in Fig. 3. Consequently this poor fit to the backward rate constant resulted in a poor fit to the time constant for a specific range of membrane potential.
Since the parameter k1 results in a poor fit to the time
constant in a specific range of the membrane potential, we attempted to fit to rate constant k for a narrow poten-tial range from 120 mV to 20 mV. This procedure led
to the following parameters A1¼37.36 ms1, B1¼3475 mV,
C1¼1481 mV, A2¼1.217 ms1, B2¼24.67 mV, C2¼47.2 mV,
A3¼0.6496 ms1, B3¼149.7 mV, C3¼36.15 mV. We represent
this rate constant parameter as k2. Activation time constants
for the empirical model and path probability model
obtained by eqns 6 and 7 with k¼k2are shown in Fig. 1b.
Figure 1b shows that time constant of the path probability
model with k2 exactly matched to that of the empirical
model up to 20 mV. However, it was not possible to capture the saturation of the time constant to a constant value at
420 mV. Therefore, we tried to determine its origin based
on the activation rate constants with k2. In contradistinction
to k1, the backward rate constant of the path probability
model with k2exactly matched to that of empirical model as
shown in Fig. 3 while the forward rate constant of the model largely deviated from that of the empirical model at
420 mV and showed a bell-shaped curve, as shown in
Fig. 2. We concluded that this behavior of the forward rate
constant with k2 leaded to the time constant reaching
arbitrarily large values at 420 mV.
Our findings above on k1and k2suggested us to use two
different rate constants kiði ¼ 1; 2Þ in eqns 6 and 7 as follows:
aðV; bÞ ¼ k1a0ðV; bÞ
bðV; bÞ ¼ k2b0ðV; bÞ ð11Þ
Activation time constants for empirical model given by eqn 9 and path probability model obtained by eqn 11 are shown in Fig. 1c. Figure 1c shows that the path probability
model with a ¼ fðk1Þ and b ¼ fðk2Þ accurately captured the
form of the voltage dependence of the empirical activation time constant. It also provided the saturation of the time constant to a constant value at depolarized potential levels. We also studied the probability of being open state for the gate as a function of time or relaxation process that occurs from a state far from its steady-state value. The relaxation curves for several values of the membrane potential are plotted in Fig. 4. Different relaxation curves were obtained for 70, 60, 55, 50 and 45 mV membrane potential
levels with mð0Þ ¼ 1:0, a ¼ fðk1Þ and b ¼ fðk2Þ by solving
eqn 5. The relaxed values of the curves were the same with the steady-state activation values obtained using eqn 1 as 0.1094, 0.3813, 0.58, 0.7557 and 0.8739 corresponding to 70, 60, 55, 50 and 45 mV membrane potential levels, respectively. These results also indicate to consistency of the proposed activation rate constants of T-type calcium channel by the path probability method.
Finally, we compared our derivations with the linear and nonlinear thermodynamic models of the rate kinetics obtained from the same calcium channel by Destexhe and Huguenard [15]. We used the activation rate equations for the linear and nonlinear thermodynamic models in their eqns 19 and 20, and eqn 28, respectively. The time constants are obtained for a wide range of the potential and shown in Fig. 5. The discussion of Fig. 5 will be given in the next section.
DISCUSSION
We recently presented a new methodology to define the equilibrium value function in the kinetics of (in)activation gates based on the lowest approximation of the cluster variation method [11]. Then, we formulated dynamics of the voltage-gated ion channels by the path probability method of the non-equilibrium statistical physics [13]. In this study, we formulated the kinetic equation for the time-dependent
open-state probability of the gate with three parameters VH,
Fig. 4. Relaxation curves of theT-type calcium channel activation for
dif-ferent potential levels with m(0)¼1.0. Fig. 5. Activation time constant of the path probability (a¼f(k1) and
b¼f(k2)), linear and non-linear thermodynamic models for theT-type
z and k explicitly. The first two of these parameters are also involved in eqn 1 for the open-state probability for the gate in the steady state. Since the quantities directly observable by voltage-clamp experiments are the steady state (in)acti-vation and the in(acti(in)acti-vation) time-constant, these para-meters are also convenient for fitting experimental data.
Then we applied this theoretical framework to derive activation rate kinetics of the T-type calcium channel in thalamic relay neurons. We used empirical equations given by Huguenard and McCormick [14] to parameterize our model. In the case of the activation kinetics for T-type calcium channel, we found that the parameter k was dependent only on the membrane potential. Therefore, we attempted to derive its explicit expression depending on the membrane potential and obtained acceptable fits for a sum of Gaussian functions given by eqn 10. By fitting eqn 10 to the experimental data for a wide range of the potential from 120 mV to 60 mV, we obtained optimal values of the
parameters Ai;Bi;Ciði ¼ 1; 2; 3Þ. We denoted this rate
con-stant as k1. Although activation time constant obtained by
eqns 6 and 7 with a ¼ fðk1Þ and b ¼ fðk1Þ captured form of
the empirical time constant and its saturation to a constant value at depolarized potential levels, the fit was poor a membrane potential range between around 85 mV and 50 mV as shown in Fig. 1a. We examined this poor fit based on forward and backward rate constants and observed that the forward rate constant exactly matched to that of empirical model while the backward rate constant captured the behavior of the empirical constant only in a specific range of the membrane potential.
Since k1 could not accurately captured behavior of the
backward rate constant and the backward rate constant depends on the voltage in a narrow range, eqn 10 was fit for a narrow range from 120 mV and 20 mV, and a new set of
the parameters Ai;Bi;Ciði ¼ 1; 2; 3Þ was obtained. We
de-noted this rate constant as k2. In the case of a ¼ fðk2Þ and
b ¼fðk2Þ, time constant accurately matched to the empirical
time constant up to 20 mV. However, it reached arbitrarily large values at 420 mV as shown in Fig. 1b. We again examined this behavior based on the forward and backward rate constants and observed that the backward rate constant exactly matched to that of empirical model. While the forward rate constant largely deviated from that of
the empirical model. Consequently, we decided to use k1
for the forward rate constant and k2for the backward rate
constant, i.e. a ¼ k1a0and b ¼ k2b0. This approach accurately
captured the form of the empirical time constant. It also provided its saturation to a constant value at depolarized potential levels as shown in Fig. 1c.
The equilibrium values of the activation were also
obtained for self-consistency by solving eqn 5 with a ¼ k1a0
and b ¼ k2b0. We found that the activation curves relax to the
values determined by using the self-consistent equilibrium value function given by eqn 1 as shown in Fig. 4.
Finally, since the linear and non-linear thermodynamic models of the activation rate kinetics obtained from the same T-type calcium channel were given by Destexhe and Huguenard [15], we also compared our derivations with its linear and non-linear models as shown in Fig. 5. The time
constant of the nonlinear thermodynamic model produced a trajectory more consistent with that of the path probability model up to around 6 mV, but leaded to arbitrarily large values at above 6 mV. The activation time constant was also close to zero in the linear thermodynamic model for these potentials. However, it was possible to capture saturation of the activation time constant to a constant value for these depolarized membrane potentials in the path probability model. Since the time constant obtained by the path probability model accurately matched to that of the empirical model and leaded to saturation of it to a constant value for depolarized membrane potentials, we may also point out that the statistical treatment in this case is more powerful than the thermodynamic one.
CONCLUSION
The results of the present study indicate that our methodol-ogy suggests a general theoretical framework for ion channels. In our model, the parameter k in the rate eqns 6 and 7 plays an important role as a scale factor for the forward and backward rate constants. It is also very crucial for deriving optimal rate constant expressions.
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Acknowledgements: I would like to thank Professor Lyle J. Graham of UFR Biomedicale de l’Universite¤ Rene Descartes for the helpful suggestions related to this paper.
