Flux through membrane channel: linear transport vs. single-molecule approaches

One of the most subtle steps in the single-molecule approach to the flux through the membrane channel, which uses the one-dimensional Smoluchowski equation, is to describe the molecule's "behavior" at the contacts between the channel openings and the bulk. Earlier, to handle this issue, we introduced the so-called "radiation boundary conditions" that account for the interplay between the two types of trajectories of the molecules starting at the openings, specifically, the ones that eventually return to the channel and the ones that escape to infinity. The latter trajectories represent the true translocation events on the condition that initially the molecule entered the channel from the opposite side. Here, we demonstrate that the single molecule approach based on the one-dimensional Smoluchowski equation with radiation boundary conditions leads to the same expression for the flux through the channel as the conventional approach based on the linear transport theory.