Transitions and Metastable States  Biophysical Substrate of Adaptation

Kinetics of Two-State Ion Channels

We describe here the simplest possible model of a voltage-dependent ion channel based on equilibrium biophysics.

A dipole moment or gating charge on the S4 transmembrane segment in each domain of the ion channel protein biases the protein to flip between states at different voltages. Under the influence of a transmembrane voltage, the charged protein residues move across the membrane, changing the tertiary structure of the ion channel protein to allow selected species of ions to pass through the pore region linking the four protein domains  [Yang and Horn (1995),Yang et al. (1996),French et al. (1996),Larsson et al. (1996),Aggarwal and MacKinnon (1996),Seoh et al. (1996)].

The probability of an ion channel being in any given state at physical equilibrium is given by its Gibbs free energy, which is a function of the internal electrostatic and van der Waals interactions between protein residues, the entropy of protein folding, and the external electric field. For state $i$ with free energy $E_i$, this probability is given by the ratio of exponential terms: 

$$p(E_i) = { \exp( - E_i/ kT)\over \sum_{j} \exp( - E_j /kT)},$$ (ix)

where $T$ is the absolute temperature (in Kelvin), $k$ is Boltzmann's constant, and the sum in the denominator is over all accessible states.

Following the nomenclature of Almers (1978) and Hille (1992), the energy difference between two states can be divided into a conformational energy change $W$ of the protein and an energy change $-z_g V$ due to the movement of an effective gating charge $z_g$ across the membrane potential $V$.The effective charge depends on the position of positive charges in the S4 segment, the distance these move through the membrane, and the structure of the electric field within the membrane.

The simplest and most generally applicable model of a voltage-gated ion channel has a single open and closed state, as illustrated in the one-dimensional `reaction-coordinate' system of Fig. 11).

 

Figure 11: The simplest model of an voltage-dependent ion channel has two states, open and closed. Since such channels carry a charge, changing the voltage shifts the relative energies of the two states, biasing the probability for the channel to dwell in the open versus the closed state.

 

The reaction coordinate is simply a measure of the protein's folded shape as it changes from closed to open state or vice versa.

When the two states are populated with equal probability (dashed-dotted line), the equilibrium rate $r$ of transitions from the closed to open state is simply the Kramer's escape rate across the activation energy barrier 

$$r \propto \sqrt{ E''(x_{\text{min}}) \vert E''(x_{\text{max}})\vert} \; \exp (- E_0 /k T),$$ (x)

where $x$ is the reaction coordinate, $E''(x_{\text{min}})$ and $\vert E''(x_{\text{max}})\vert$are the curvatures of the solid curve in fig. 8 at the minimum and maximum, and $E_0$ is the height of the central peak above baseline.

For simplicity, we assume that the energy barrier is symmetric and that the electric field is linear across the membrane. As in a see-saw, changing the transmembrane potential $V$ shifts the Gibbs free energies of the open and closed states by opposite but equal amounts. The total energy change in the presence of an applied field is $\Delta E = W - z_g V$.

 The forward transition rate $\alpha(V)$ from closed to open state will be:
$$\begin{align*}\alpha(V) = & r \; \exp \Biggl[ - \biggl( {(W - z_g V) \over 2 k... ...exp \Biggl[ + \biggl( {(W - z_g V) \over 2 kT } \biggr) \Biggl]. \end{align*}$$
Here the rate $r$ denotes the Kramer's escape rate when the two states are equally balanced, as given by eq. x.

At the macroscopic scale of ion flow across thousands of channels, $m(V)$ represents the fraction of channels of one type that are open at a given time. The quantity $m(V)$ obeys the first-order differential equation 

$${d m \over d t} = \alpha(V) (1-m) - \beta(V) m.$$ (xi)

In the Hodgkin-Huxley equations [Hodgkin and Huxley (1952)], this is the equation for a single gating particle.

For each steady state transmembrane voltage $V$, the channel will be in the open state with a probability $m_\infty(V)$ as a function of voltage and in the closed state with probability $(1 - m_\infty(V))$. As long as the ion channel possesses only two states that are in energetic equilibrium, $m_\infty(V)$ must be an S-shaped function of voltage given by the Boltzmann equation: 

$$m_\infty(V) = 1/\{1 + \exp[ - s (V - V_{1 \over 2})]\},$$ (xii)

where $s = z_g / (k T)$ and $V_{1 \over 2} = W/z_g$. This is simply the solution of eq. xi when $dm/dt = 0$.$V_{1 \over 2}$ is the midpoint voltage of half-activation, and $s$ is the slope of the S-shaped function at its maximum inflection.

We can, of course, rewrite eq. xi in terms of $m_\infty(V)$ as 

$$\tau(V) {d m \over d t} = m_\infty(V) - m,$$ (xii)

with $\tau(V) = {[\alpha(V) + \beta(V)]}^{-1}$,which is the equation given in the text.

   
 Transitions and Metastable States  Biophysical Substrate of Adaptation
3/7/1998