Position-dependent stochastic diffusion model of ion channel gating

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

The activation of a 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 (α and γ) and the backward rates (β and δ) for each stage. In particular, the voltage dependence of the Hodgkin-Huxley rate functions for a K^{+} channel may be derived by assuming that the rate functions of the first stage are large relative to those of the second stage-α≫γ and β≫δ. For a Shaker IR K^{+} channel, the first forward and backward transitions are rate limiting (α<γ and δ≪β), 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 K^{+} ion channel and a noninactivating Na^{+} ion channel is determined by the master equation for K^{+} channel activation and the ionic current equation when the Na^{+} channel activation time is small, and if β≪δ and α≪γ, the system may exhibit a small amplitude oscillation between spikes, or mixed-mode oscillation, in which the slow closed state modulates the K^{+} ion channel conductance in the membrane.

Voltage dependence of a stochastic model of activation of an alpha helical S4 sensor in a K channel membrane

The voltage dependence of the ionic and gating currents of a K channel is dependent on the activation barriers of a voltage sensor with a potential function which may be derived from the principal electrostatic forces on an S4 segment in an inhomogeneous dielectric medium. By variation of the parameters of a voltage-sensing domain model, consistent with x-ray structures and biophysical data, the lowest frequency of the survival probability of each stationary state derived from a solution of the Smoluchowski equation provides a good fit to the voltage dependence of the slowest time constant of the ionic current in a depolarized membrane, and the gating current exhibits a rising phase that precedes an exponential relaxation. For each depolarizing potential, the calculated time dependence of the survival probabilities of the closed states of an alpha helical S4 sensor are in accord with an empirical model of the ionic and gating currents recorded during the activation process.