Barrier crossing with concentration boundary conditions in biological channels and chemical reactions

Abstract

Ions move into biological cells through pores in proteins called ionic channels, driven by gradients of potential and concentration imposed across the channel, impeded by potential barriers and friction within the pore. It is tempting to apply to channels the chemical theory of barrier crossing, but important issues must first be solved: Concentration boundary conditions must be used and flux must be predicted for applied potentials of all sizes and for barriers of all shapes, in particular, for low barriers. We use a macroscopic analysis to describe the flux as a convolution integral of a mathematically defined adjoint function, a Green’s function. It so happens that the adjoint function also describes the first-passage time of a single particle moving between boundary conditions independent of concentration. The (experimentally observable) flux is computed from analytical formulas, from simulations of discrete random walks, and from simulations of the Langevin or reduced Langevin equations, with indistinguishable results. If the potential barrier has a single, large, parabolic peak, away from either boundary, an approximate expression reminiscent of Kramers’ formula can be used to determine the flux. The fluxes predicted can be compared with measurements of current through single channels under a wide range of experimental conditions.