Voltage dependence of Hodgkin-Huxley rate functions for a multistage $K^{+}$ channel voltage sensor within a membrane

The activation of a ${\text{K}}^{+}$ channel sensor in two sequential stages during a voltage clamp may be described as the translocation of a Brownian particle in an energy landscape with two large barriers between states. A solution of the Smoluchowski equation for a square-well approximation to the potential function of the S4 voltage sensor satisfies a master equation and has two frequencies that may be determined from the forward and backward rate functions. When the higher-frequency terms have small amplitude, the solution reduces to the relaxation of a rate equation, where the derived two-state rate functions are dependent on the relative magnitude of the forward rates ($\alpha$ and $\gamma$) and the backward rates ($\beta$ and $\delta$) for each stage. In particular, the voltage dependence of the Hodgkin-Huxley rate functions for a ${\text{K}}^{+}$ channel may be derived by assuming that the rate functions of the first stage are large relative to those of the second stage—$\alpha \gg \gamma$ and $\beta \gg \delta$. For a Shaker IR ${\text{K}}^{+}$ channel, the first forward and backward transitions are rate limiting ($\alpha <\gamma$ and $\delta \ll \beta$), and for an activation process with either two or three stages, the derived two-state rate functions also have a voltage dependence that is of a similar form to that determined for the squid axon. The potential variation generated by the interaction between a two-stage ${\text{K}}^{+}$ ion channel and a noninactivating ${\text{Na}}^{+}$ ion channel is determined by the master equation for ${\text{K}}^{+}$ channel activation and the ionic current equation when the ${\text{Na}}^{+}$ channel activation time is small, and if $\beta \ll \delta$ and $\alpha \ll \gamma$, the system may exhibit a small amplitude oscillation between spikes, or mixed-mode oscillation, in which the slow closed state modulates the ${\text{K}}^{+}$ ion channel conductance in the membrane.

Figure 1

Two-stage activation model of a ${\mathrm{K}}^{+}$ ion channel voltage sensor, where the occupation probabilities of the closed states $n_1$, $n_2$ and the open state $n$ satisfy a master equation, and $\alpha ,\beta ,$ $\gamma$, and $\delta$ are voltage-dependent rate functions between states.

Figure 2

The derived rate functions ${\alpha }_{n,2}$ and${\beta }_{n,2}$ (solid line) in Eqs. (17) and (18) provide a good approximation to the HH rate functions (${\mathrm{ms}}^{-1}$) ${\alpha }_n=$ $0.01(V+50)/\{1-exp[-0.1(V+50)\left]\right\}$ and ${\beta }_n=0.125exp[-(V+60)/80]$ (dotted line) when the rate functions for two-stage activation (${\mathrm{ms}}^{-1}$) are $\alpha \left(V\right)=6.4exp[0.3(V-V_0)/25]$, $\beta \left(V\right)=17.6exp[-1.4(V-V_0)/25]$, $\gamma \left(V\right)=0.24exp$ $[0.345(V-V_0)/25]$, $\delta \left(V\right)=0.125exp[-0.312(V-V_0)/25]$, $V_0=-57.9$.

Figure 3

Survival probability of the state $n\left(t\right)$ during a voltage clamp for a two-stage activation model of a ${\mathrm{K}}^{+}$ ion channel (solid line) and for the HH squid axon model (dotted line) during (a) activation and (b) deactivation (see Fig. 2).

Figure 4

The derived rate functions ${\alpha }_{n,2}$ and ${\beta }_{n,2}$ (solid line) in Eqs. (15) and (16) are calculated for the rate functions determined experimentally for the Shaker ${\mathrm{K}}^{+}$ ion channel [11] $\alpha \left(V\right)=1.1exp(0.25V/25)$, $\beta \left(V\right)=0.37exp(-1.6V/25)$, $\gamma \left(V\right)=2.8exp(0.32V/25)$, $\delta \left(V\right)=0.021exp(-1.1V/25)$ (${\mathrm{ms}}^{-1}$), and may be approximated by the rate functions ${\alpha }_H=0.019(V+45.8)/\{1-exp[-0.15(V+45.8)]\}$ and ${\beta }_H=0.135exp[-0.052(V+45.8)]$ for single transition activation (dotted line).

Figure 5

Survival probability of the state $n\left(t\right)$ during a voltage clamp for a two-stage model of a ${\mathrm{K}}^{+}$ channel (solid line) and HH model (dotted line) during (a) activation and (b) deactivation, for the rate functions described in the legend of Fig. 4, and (c) activation when $\beta \left(V\right)$ and $\gamma \left(V\right)$ are increased by a factor of 3, and the rate functions ${\alpha }_H=0.02(V+45.8)/\{1-exp[-0.15(V+45.8)]\}$ and ${\beta }_H=0.12exp[-0.057(V+45.8)].$

Figure 6

Three-stage activation model of a ${\text{K}}^{+}$ ion channel voltage sensor, where the occupation probabilities of the closed states $n_1$, $n_2$, and $n_3$ and the open state $n$ satisfy a master equation, and ${\alpha }_i$, ${\beta }_i$ for $i=1$ to 3 are voltage-dependent rate functions between states.

Figure 7

The derived rate functions ${\alpha }_{n,2}\left(V\right)$ and ${\beta }_{n,2}\left(V\right)$ are calculated for a two-stage ${\text{K}}^{+}$ channel (solid line), where $\alpha \left(V\right)=1.1exp(0.25V/25)$, $\beta \left(V\right)=0.37$ $exp(-1.6V/25)$, $\gamma \left(V\right)=2.8exp(0.32V/25)$, $\delta \left(V\right)=0.021exp(-1.1V/25)$ (${\mathrm{ms}}^{-1}$), and ${\alpha }_{n,3}\left(V\right)$, ${\beta }_{n,3}\left(V\right)$ are calculated for a three-stage ${\text{K}}^{+}$ ion channel (dotted line), where the rate functions ${\alpha }_1\left(V\right)=1.1exp(0.25V/25)$, ${\alpha }_2\left(V\right)=24.0exp$ $(0.28V/25)$, ${\alpha }_3\left(V\right)=4.0exp(0.28V/25)$, ${\beta }_1\left(V\right)=1.17exp(-0.75V/25)$, ${\beta }_2\left(V\right)=7.0exp(-0.75V/25)$, ${\beta }_3\left(V\right)=0.021exp$ $(-1.1V/25)$.

Figure 8

Survival probability of the state $n\left(t\right)$ during a voltage clamp for a two-stage (solid line) and three-stage (dotted line) model of activation of a ${\text{K}}^{+}$ ion channel voltage sensor during (a) activation and (b) deactivation, for the rate functions described in the legend of Fig. 7.

Figure 9

The limit-cycle solution of Eqs. (50, 51, 52) (solid line) may be approximated by the solution of Eqs. (14), (17), (18), and (50) (dotted line) when $\beta \gg \delta$ and $\alpha \gg \gamma .$ The rate functions $\alpha \left(V\right)=6.4exp[0.3(V-V_0)/25]$, $\beta \left(V\right)=17.6exp[-1.4(V-V_0)/25]$, $\gamma \left(V\right)=0.24exp[0.345(V-V_0)/25]$, $\delta \left(V\right)=0.125exp$ $[-0.312(V-V_0)/25]$, ${\overline{g}}_{\mathrm{K}}=34$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_{\text{Na}}=15$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_L=0.1$ mS/${\mathrm{cm}}^2$, $V_{\mathrm{K}}=-72$ mV, $V_{\text{Na}}=55$ mV, $V_L=-49.4$ mV, $I=92$ $\mu A$/${\mathrm{cm}}^2$, $V_0=-57.9$, $z_m=1.3$, $V_m=58$.

Figure 10

The bifurcation diagram for Eqs. (50, 51, 52), when $\beta \gg \delta$ and $\alpha \gg \gamma$, represents the extremal values for the stable periodic solutions (thick solid line), unstable periodic solutions (thick dashed line), and the subcritical bifurcation points HB1 and HB2 at the intersection between stable stationary points (thin solid line) and unstable stationary points (thin dashed line)—see Fig. 9 for a limit-cycle solution.

Figure 11

The limit-cycle solution of Eqs. (50, 51, 52) (solid line) may be approximated by the solution of Eqs. (14), (50), (54), and (55) (dotted line) when $\beta \gg \delta$ and $\gamma \gg \alpha .$ The rate functions $\alpha \left(V\right)=1.1exp(0.25V/25)$, $\beta \left(V\right)=1.1exp(-1.6V/25)$, $\gamma \left(V\right)=8.4exp(0.32V/25)$, $\delta \left(V\right)=0.021exp(-1.1V/25)$, ${\overline{g}}_{\mathrm{K}}=17$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_{\text{Na}}=9$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_L=0.1$ mS/${\mathrm{cm}}^2$, $V_{\mathrm{K}}=-81$ mV, $V_{\text{Na}}=46$ mV, $V_L=-58.4$ mV, $I=13$ $\mu A$/${\mathrm{cm}}^2$, $z_m=1.5$, $V_m=51.9$.

Figure 12

The bifurcation diagram for Eqs. (50, 51, 52), when $\beta \gg \delta$ and $\gamma \gg \alpha$, where the subcritical (HB1) and supercritical (HB2) bifurcation points are at the junction between stable stationary points (thin solid line) and unstable stationary points (thin dashed line)—see Fig. 11 for a limit-cycle solution.

Figure 13

The bifurcation diagram for Eqs. (50) and (51), where $\beta \ll \delta$ and $\alpha \ll \gamma$, the slow variable $n_1$ is treated as a parameter in the ($V$, $n$) subsystem (red and blue line), and the $V$ vs. $n_1$ projection is computed from Eqs. (50, 51, 52) (solid line). For $n_1$ less than $0.48$, the stationary point is stable but for $n_1$ above this value, it is unstable. The rate functions are $\alpha \left(V\right)=0.17exp(0.5V/25)$, $\beta \left(V\right)=0.02exp(-V/25)$, $\gamma \left(V\right)=2.8exp(0.45V/25)$, $\delta \left(V\right)=0.44exp(-V/25)$, ${\alpha }_m\left(V\right)$, and ${\beta }_m\left(V\right)$ defined by Eq. (53), $V_m=45$, ${\overline{g}}_{\mathrm{K}}=36$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_{\text{Na}}=12$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_L=0.4$ mS/${\mathrm{cm}}^2$, $V_{\mathrm{K}}=-90$ mV, $V_{\text{Na}}=70$ mV, $V_L=-70$ mV, $I=236$ $\mu A$/${\mathrm{cm}}^2$.

Figure 14

When $\beta \ll \delta$ and $\alpha \ll \gamma$, Eqs. (50, 51, 52) may exhibit a small amplitude oscillation in $V$ that alternates with a cluster of spikes and is described as a bursting oscillation (see Fig. 13).

Figure 15

During a limit-cycle solution of Eqs. (56, 57, 58), the two-stage activation of ${\text{K}}^{+}$ channels, defined by Eqs. (51, 52) and $\beta \gg \delta$, $\alpha \gg \gamma$ (solid line), may be approximated by a rate equation, Eqs. (14), (17), and (18) (dotted line). The rate functions $\alpha \left(V\right)=6.4exp[0.3(V-V_0)/25]$, $\beta \left(V\right)=17.6exp[-1.4(V-V_0)/25]$, $\gamma \left(V\right)=0.24exp[0.345(V-V_0)/25]$, $\delta \left(V\right)=0.125exp$ $[-0.312(V-V_0)/25]$, ${\alpha }_m\left(V\right)$ and ${\beta }_m\left(V\right)$ defined by Eq. (53), $V_m=60$, ${\overline{g}}_{\mathrm{K}}=36$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_{\text{Na}}=120$ mS/${\mathrm{cm}}^2$, ${\overline{g}}_L=0.3$ mS/${\mathrm{cm}}^2$, $V_{\mathrm{K}}=-72$ mV, $V_{\text{Na}}=55$ mV, $V_L=-49.4$ mV, $I=42$ $\mu A$/${\mathrm{cm}}^2$, $V_0=-60$.

Figure 16

The bifurcation diagram for Eqs. (51) and (52) and Eqs. (56, 57, 58), when $\beta \gg \delta$ and $\alpha \gg \gamma$, represents the extremal values for the stable periodic solutions (thick solid line), unstable periodic solutions (thick dashed line), and the subcritical (HB1) and supercritical (HB2) bifurcation points, which occur at the intersection between stable stationary points (thin solid line) and unstable stationary points (thin dashed line)—see Fig. 15 for a limit-cycle solution.