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

Abstract

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.

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

“…16,18 A potential function for an alpha helical S4 sensor located within the gating pathway may be defined that is dependent on negatively charged amino acids on neighboring segments, and the potential difference across the membrane, 19 but also has contributions from the interaction energy between S4 charges and the charge at the dielectric boundaries induced by both positive and negative residues. 20,21 The calculated voltage dependence of the stationary distribution of the gating charge for a VSD model is in good agreement with experimental data from wild-type and charge-neutralized mutants of a K channel. 20 Assuming that the dielectric boundaries between the low dielectric region of the membrane and the solvent incorporate internal and external cavities formed from transmembrane helices, the variation of potential within the membrane and solvent may be determined from a numerical solution of the Poisson-Boltzmann equation.…”

“…Although low order terms give an approximate estimate for the potential ψ j , [19][20][21] the exact expression in Eq. (1) is a slowly converging series, in general and, therefore, terms of higher order are required to calculate the voltage dependence of the energy barriers of activation, and sufficient accuracy is obtained by including at least six terms in the expansion.…”

“…27 For a planar dielectric membrane between internal and external solvents, the electrostatic energy of the S4 residues within the membrane field U V (Z) = l Q lṼ F V (z l ), [20][21][22]31 whereṼ = V + V s is the potential difference across the gating pore, V is the potential difference across the membrane, V s is the surface charge potential,…”

“…24 An analytical solution of the Smoluchowski equation for a potential function determined by the electrostatic forces on an alpha helical S4 segment in an inhomogeneous dielectric medium may be approximated by a master equation that has a similar form to empirical models of K channel activation. 21 However, for the parameters adopted within the VSD model, the lowest frequency of the solution is not in accord with the voltage dependence of the activation time constant determined from a single exponential fit to the late phase of ionic or gating current for moderate depolarizing S1 S2 S3 S4 S5 S6 S1 S2 S3 S4 S5 S6 R1 R2 R3 R4 E283 E293 D316 lipid lipid pore Subunit Subunit FIG. 1.…”

Derivation of Hodgkin-Huxley equations for aNa+channel from a master equation for coupled activation and inactivation

“…The master equation may be derived from a Smoluchowski equation applied to the resting and barrier regions of an energy landscape for each of the S4 sensors in the domains DI to DIV [14,15]. The translocation of the S4 segment through the gating pore for Na+ (or K+) channels requires sufficient energy to overcome several barriers that are dependent on the Coulomb force between positively charged residues on the S4 sensor and negatively charged residues on neighboring helices, the dielectric boundary force, the electric field between internal and external aqueous crevices, and hydrophobic forces [16].…”