Channel transport: gating, geometry, and heterogeneous diffusion

Abstract

Channel-mediated transport is ubiquitous in biology. A series of works by different theoreticians have sought to determine how the diffusive flux through a channel depends on (a) stochastic gating, (b) channel geometry, and (c) heterogeneous diffusion. In this paper, we derive an explicit estimate for the diffusive flux through a channel that accounts for these three factors. We show that our estimate is exact in certain parameter regimes. We further use stochastic simulations to confirm that our estimate remains accurate across a very broad range of parameters. Our estimate differs from some results in the physics literature.

Figures (4)

• Figure 1: Diffusing particles can pass from the left (or "west") bulk region through a channel to the right (or "east") bulk region. Particles have diffusivities $D_{\mathrm{w}}$, $D_{\mathrm{c}}$, and $D_{\mathrm{e}}$ in the west bulk, channel, and east bulk. The west entrance to the channel is stochastically gated, which means that it randomly opens and closes. Particles can freely enter and exit the west side of the channel when the gate is open (left diagram), whereas particles can neither enter nor exit through the west side of the channel when the gate is closed (right diagram).
• Figure 2: Comparison of the limiting splitting probabilities in \ref{['eq:fastchannelswitching']}-\ref{['eq:fastbulkswitching']}.
• Figure 3: Comparison of the short channel formula in \ref{['eq:shortchannelformula']} (curves) to stochastic simulation (markers) for different values of the open probability $p_0$. In Panel A, $\Gamma$ is the unit disk. In Panel B, $\Gamma$ is the unit square.
• Figure 4: Comparison of $\mathcal{P}_{*}$ in \ref{['eq:Pmain']} (black curves) to stochastic simulations (markers).