Identity of Distributions of Direct Uphill and Downhill Translocation Times for Particles Traversing Membrane Channels

We study the distribution of direct translocation times for particles passing through membrane channels connecting two reservoirs. The direct translocation time is a conditional first-passage time defined as the residence time of the particle in the channel while passing to the other side of the membrane directly, i.e., without returning to the reservoir from which it entered. We show that the distributions of direct translocation times are identical for translocation in both directions, independent of any asymmetry in the potential across the channel and, hence, the translocation probabilities.

Figure 1

Membrane channel and the potential of mean force, $U(x)$, along the channel axis; $\Delta U$ is the potential drop across the channel.

Figure 2

Auxiliary potential $V(x)$ is identical to $U(x)$ on the interval corresponding to the channel, $x_L\le x\le x_R$, and tends to infinity as $|x|\to \infty$.

Figure 3

Fragment of an equilibrium trajectory that goes from left to right and its mirror image that goes from right to left. The direct translocation times $t_{L\to R}$ and $t_{R\to L}$ are equal.