Evaluating diffusion resistance of a constriction in a membrane channel by the method of boundary homogenization

In this paper we analyze diffusive transport of noninteracting electrically uncharged solute molecules through a cylindrical membrane channel with a constriction located in the middle of the channel. The constriction is modeled by an infinitely thin partition with a circular hole in its center. The focus is on how the presence of the partition slows down the transport governed by the difference in the solute concentrations in the two reservoirs separated by the membrane. It is assumed that the solutions in both reservoirs are well stirred. To quantify the effect of the constriction we use the notion of diffusion resistance defined as the ratio of the concentration difference to the steady-state flux. We show that when the channel length exceeds its radius, the diffusion resistance is the sum of the diffusion resistance of the cylindrical channel without a partition and an additional diffusion resistance due to the presence of the partition. We derive an expression for the additional diffusion resistance as a function of the tube radius and that of the hole in the partition. The derivation involves the replacement of the nonpermeable partition with the hole by an effective uniform semipermeable partition with a properly chosen permeability. Such a replacement makes it possible to reduce the initial three-dimensional diffusion problem to a one-dimensional one that can be easily solved. To determine the permeability of the effective partition, we take advantage of the results found earlier for trapping of diffusing particles by inhomogeneous surfaces, which were obtained with the method of boundary homogenization. Brownian dynamics simulations are used to corroborate our approximate analytical results and to establish the range of their applicability.

Figure 1

Sketch of a cylindrical channel of length $L$ and radius $a$ with an infinitely thin partition, located in the middle of the channel; the partition has a hole of radius $b$, $b<a$, in its center.

Figure 2

Function $M\left(\xi \right)$, solid curve drawn according to Eq. (1.6), compared to $M\left(\xi \right)$ values, open circles, calculated using Eq. (3.3) with $\delta \tau$ obtained from Brownian dynamics simulations, $\delta \tau =\delta {\tau }_{\mathrm{sim}}$, for $D=1$, $a=1$, and $L=2$.