The energy barrier to the activated state for the S4 voltage sensor of a K channel is dependent on the electrostatic force between positively charged S4 residues and negatively charged groups on neighboring segments, the potential difference across the membrane, and the dielectric boundary force on the charged residues near the interface between the solvent and the low dielectric region of the membrane gating pore. The variation of the potential function with transverse displacement and rotation of the S4 sensor across the membrane may be derived from a solution of Poisson's equation for the electrostatic potential. By approximating the energy of an S4 sensor along a path between stationary states by a piecewise linear function of the transverse displacement, the dynamics of slow activation, in the millisecond range, may be described by the lowest frequency component of an analytical solution of interacting diffusion equations of Fokker-Planck type for resting and barrier regions. The solution of the Smoluchowski equations for an S4 sensor in an energy landscape with several barriers is in accord with an empirical master equation for multistep activation in a voltage-dependent K channel.

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.

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.

The dynamics of the open or closed state region of an ion channel may be described by a probability density p(x, t) which satisfies a Fokker-Planck equation. The closed state dwell-time distribution f c (t) derived from the Fokker-Planck equation with a nonlinear diffusion coefficient D(x) ∝ exp(−γx), γ > 0 and a linear ramp potential U c (x), is in good agreement with experimental data and it may be shown analytically that if γ is sufficiently large, f c (t) ∝ t −2−ν for intermediate times, where ν = U ′ c /γ ≈ −0.3 for a fast Cl channel. The solution of a master equation which approximates the Fokker-Planck equation exhibits an oscillation superimposed on the power law trend and can account for an empirical rate-amplitude correlation that applies to several ion channels.

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