Copyright © 1999 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 76, Issue 4, 1918-1921, 1 April 1999
doi:10.1016/S0006-3495(99)77350-9
Biophysical Theory and Modeling
A Possible Resolution of the Gating Paradox
L.P. Endresen
, 
and J.S. Høye
Institutt for fysikk, Norges Teknisk Naturvitenskapelige Universitat, N-7034 Trondheim, Norway
Address reprint requests to Dr. Lars Petter Endresen, Institutt for fysikk, NTNU, N-7034 Trondheim, Norway. Tel.: 47-73-59-36-63; Fax: 47-73-59-33-72.
Abstract
We introduce a Markov model for the gating of membrane channels. The model features a possible solution to the so-called gating current paradox, namely that the bell-shaped curve that describes the voltage dependence of the kinetics is broader than expected from, and shifted relative to, the sigmoidal curve that describes the voltage dependence of the activation. The model also predicts some temperature dependence of this shift, but presence of the latter has not been tested experimentally so far.
Introduction
The gating of membrane channels is of vital importance for the electrophysiological activity of nerve, heart, and muscle. While some of these channels appear to have fractal-like gating (Liebovitch, 1995), most membrane channels do display activity that can be well approximated by a simple Markov process (Korn and Horn, 1988). However, Clay et al revealed a gating current paradox that has been difficult to explain with a standard type (Hille, 1992) Markov model. The paradox is that the bell-shaped curve that describes the voltage dependence of the kinetics is shifted significantly relative to the sigmoidal curve that describes the voltage dependence of the activation. The standard type model (Hille, 1992) does not allow such a shift. Also, the former curve is broader than the one predicted by the standard model.
Fig. 1Here we introduce a new Markov model that extends and generalizes the standard one. Our generalization consists of introducing an alternative route between the open and the closed positions of the gate. With two routes, or two membrane protein folding pathways, we are able to obtain results consistent with the observed ones. Thus such a model presents a possible resolution of the above paradox. A more complete resolution requires investigation of the detailed physical mechanism present in real membrane channels to see how they compare with the model. The idea with two routes, a rapid one and a slow one, is that the probability of choosing one or the other also depends upon the voltage through a Boltzmann factor. This will affect the kinetics, but not the equilibrium distribution (stationary state), and a relative shift of curves can take place.
The model
We imagine that a membrane channel has one open and one closed state, as in the simplest standard (Hille, 1992) Markov model for this problem. However, between these states we now assume that there exist two routes (i=1, 2). This gives,
where the rate constants
α1,
α2 and
β1,
β2, which are functions of voltage (but are constant at any given voltage), control the transitions between the closed (C) and the open (O) states of the gate. The
αi is the rate for a closed channel to open, and
βi the rate for an open channel to close. We introduce effective rate constants
α and
β,
where the probabilities
p1 and
p2 are related in a standard way to the difference
ΔGb in energy barriers that must be overcome for each of the two routes,
Let
x denote the average fraction of gates that are open or, equivalently, the probability that a given gate will be open, and let us imagine that a
Markov, 1906 model is suitable to describe the gating. One then has, as usual
where
Here
x∞ denotes the steady stationary state fraction of open gates and
τ the relaxation time. At equilibrium, the probability for a channel to be in the open state is
x∞, and the probability to be in the closed state is (
1−x∞)
. The ratio of these two probabilities is given by the Boltzmann distribution,
where
T is the absolute temperature,
k is Boltzmann's constant, and
ΔGx denotes the energy difference between the open and the closed positions. Thus,
At equilibrium, each of the forward reactions must occur just as frequently as each of the reverse reactions, giving
This is the principle of detailed balance, which is present in dynamical systems (reversible mechanics). As in the standard model the rates are then assumed to be
where
λi is assumed to be independent of
ΔGx. Thus the relaxation time (Eq.
(8)) can then be written as
Using Eqs.
(4), we obtain
where
To be more specific, the voltage dependences of
ΔGx and
ΔGb are needed. For the energy difference between the open state and the closed state we assume as usual,
where the term
qxvx is due to the difference in mechanical conformation energy between the two states;
qxv represents the electrical potential energy change associated with the redistribution of charge during the transition, and
sx is due to the difference in entropy between the two states. A similar expression can be assumed for the energy difference between the two barriers in routes 1 and 2,
Here
v is voltage, while
qx, vx, sx, qb, vb, and
sb are constants. However, the assumed voltage dependence in Eq.
(22) is in no way obvious, but we find it reasonable in the sense that the choice between the two routes may possibly depend upon the voltage in a way similar to the fraction
x∞ of open and closed gates.
One notes that the curve for the relaxation time τ has a shift in position due to the term γ. Inserted for the special case ΔGb=ΔGx the above yields
Here we find that the voltage dependence of the curve for the relaxation time (Eq.
(24)) is shifted by an amount
2γkT/qx relative to the steady-state activation curve (Eq.
(23)), which means that the magnitude of the shift depends upon temperature. With
ΔGb≠ ΔGx, Eq.
(24) becomes more complex, as follows from Eq.
(18), and the shape of the former curve is modified. This, however, is dealt with in the next section.
Results
We will now compare the model with the experimental results of Clay et al and show that it is consistent with the latter. Thus it presents a mechanism that represents a possible solution to the gating current paradox. The temperature dependence of the currents was not considered in those experiments, so here sx and sb can be incorporated into vx and vb. With the use of Eqs. (21), Eqs. (10) become
where
kx=
kT/qx and
kb=
kT/qb. These expressions were evaluated numerically, adjusting the parameters present to obtain a best possible fit to the experimental data. A least-squares fit weighting various points in accordance with experimental uncertainty was used. The results of this evaluation are shown in the figure below, where the data of
Clay et al are presented together with the curves given by Eqs.
(25) using the parameters shown in the figure legend.
However, the curves are not very sensitive to the values of these parameters except γ, i.e., the other parameters can be varied quite a bit and still give essentially the same curves. From these curves we find that the model is fully consistent with the experimental results within the uncertainties in the latter. Since the results of our proposed model for the gating heavily rely upon the assumption in Eq. (22), one can ask oneself whether other known models will fit experimental data in a similar way by adjusting parameters. As far as we can see, this is not possible, e.g., Clay et al tried to do so with the standard model, and as we find too, the obvious shift in the two curves can in no way be accounted for even with some asymmetry between α and β. That is, asymmetry can only produce a minor shift before the bell-shaped form of the curve for τ is lost. In this respect we did a standard statistical test evaluating the expression
where
n=
N−M is the number of degrees of freedom,
N is the total number of experimental points,
M is the number of adjustable parameters,
ti are the various theoretical values,
ei the experimental averages, and
σi the corresponding uncertainties of the latter. In our case with
M=
6 we find
χ2/n=
0.91 while the standard theory referred to above with
M=
4 yields
χ2/n=
16.8.Discussion
We have presented a Markov model that yields a possible solution to the gating current paradox announced by Clay et al. It gives a simple explanation of the voltage shift of the bell-shaped curve for the relaxation time relative to the steady-state activation curve. Also, the width and shape of the relaxation time curve can be modified in a way consistent with experiments. A novel feature of the present model is that the voltage shift is temperature-dependent. It is not clear whether such a temperature dependence can be observed experimentally.
Acknowledgments
Lars Petter Endresen thanks professor Jan Myrheim for illuminating discussions in connection with this work.
This work was supported by a fellowship from NTNU.
References
Clay et al., 1995 Clay, J.R., Ogbaghebriel, A., Paquette, T., Sasyniuk, B.I., and Shrier, A. (1995). A quantitative description of the E-4031-sensitive repolarization current in rabbit ventricular myocytes. Biophys. J. 69, 1830–1837. Abstract | | CrossRef | PubMed
Hille, 1992 Hille, B. (1992). Ionic channels of excitable membranes.. (Sunderland, MA: Sinauer Associates), . PubMed
Korn and Horn, 1988 Korn, S.J., and Horn, R. (1988). Statistical discrimination of fractal and Markov models of single-channel gating. Biophys. J. 54, 871–877. Abstract | | CrossRef | PubMed
Liebovitch, 1995 Liebovitch, L.S. (1995). Single channels: from Markovian to fractal models. In Cardiac Electrophysiology: From Cell to Bedside. Zipes, D.P., Jalife, J., eds. (Philadelphia: W. B. Saunders), pp. 293–304. PubMed
Markov, 1906 Markov, A.A. (1906). Extension de la loi de grands nombres aux événements dependants les uns de autres. Bulletin de La Société Physico-Mathématique de Kasan 15, 135–156. PubMed
Publication Information
Received: September 28, 1998
Revised: January 8, 1999
