Stochastic diffusion model of multistep activation in a voltage-dependent K channel

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.