Trapping of particles diffusing in two dimensions by a hidden binding site

We study trapping of particles diffusing in a two-dimensional rectangular chamber by a binding site located at the end of a rectangular sleeve. To reach the site a particle first has to enter the sleeve. After that it has two options: to come back to the chamber or to diffuse to the site where it is trapped. The main result of the present work is a simple expression for the mean particle lifetime as a function of its starting position and geometric parameters of the system. This expression is obtained by an approximate reduction of the initial two-dimensional problem to the effective one-dimensional one which can be solved with relative ease. Our analytical predictions are tested against the results obtained from Brownian dynamics simulations. The test shows excellent agreement between the two for a wide range of the geometric parameters of the system.

Figure 1

Rectangular chamber of length $L$ and width $W$ with a sleeve of length $l$ and width $w\le W$ terminated by a binding site (a), and an approximate one-dimensional model of the particle dynamics in the system (b). The particle starting point is located at distance $x_0$ from the chamber wall opposite to the wall containing the sleeve entrance.

Figure 2

Rectangular chamber of length $L$ and width $W$ with an absorbing interval of width $w\le W$ located in the center of its right wall (a), and an approximate one-dimensional model of the particle dynamics in the system (b).

Figure 3

One-dimensional model used in our derivation of the expression for the mean particle lifetime ${\tau }_l\left(x_0\right)$ given in Eq. (1.1). The derivation exploits the steady-state picture, with the steady state maintained by a constant flux $j$ injected at the particle starting point $x_0$.

Figure 4

Numerical test of the $l$ dependence of the mean particle lifetime ${\tau }_l\left(x_0\right)$, Eq. (1.1). To test the $l$ dependence predicted by Eq. (1.1) we wrote this equation in the form given in Eq. (3.4). The straight lines in the figure are the linear dependence on $l/L$ in the right-hand side of Eq. (3.4) for three values of the slope, $w/W=0.25$, $0.5$, and $0.75$ from bottom to top. Symbols are the values of the expression on the left-hand side of Eq. (3.4) found for $x_0=L$ using the mean lifetimes ${\tau }_l^{\left(\mathrm{sim}\right)}\left(L\right)$ obtained from Brownian dynamics simulations and ${\tau }_0\left(L\right)$ given in Eq. (3.1).