Diffusion and binding of finite-size particles in confined geometries

Describing the diffusion of particles through crowded, confined environments with which they can interact is of considerable biological and technological interest. Under conditions where the confinement dimensions become comparable to the particle dimensions, steric interactions between particles, as well as particle-wall interactions, will play a crucial role in determining transport properties. To elucidate the effects of these interactions on particle transport, we consider the diffusion and binding of finite-size particles within a channel whose diameter is comparable to the size of the particles. Using a simple lattice model of this process, we calculate the steady-state current and density profiles of both bound and free particles in the channel. We show that the system can exhibit qualitatively different behavior depending on the ratio of the channel width to the particle size. We also perform simulations of this system and find excellent agreement with our analytic results.

Figure 1

Schematic illustration of the interior of the channel for a typical particle distribution of bound particles (shaded sites), unbound particles (cross-hatched sites), and unoccupied (white) sites.

Figure 2

Dimensionless bound particle density ${\lambda }_b\left(x\right)∕\Lambda$ for the no-hole case obtained from the simulations (points), as compared to the theoretical predictions both at the mean-field level (dashed lines) and including the correlations (solid lines). For all data shown, $N=5$, $p_{\text{on}}=0.01$, $p_{\text{off}}=0.001$, and $p_{\text{hop}}=1∕150$; for the simulations we also set ${\tilde{p}}_{\text{hop}}(j,j^{\prime })=p_{\text{hop}}$ for all $j,j^{\prime }$. The value of $p_{\text{enter}}$ for each simulation sets the left-end boundary condition ${\lambda }_b^{\left(0\right)}$ for the theoretical curves. The values of $p_{\text{enter}}$ and ${\lambda }_b^{\left(0\right)}$, respectively, are 0.006, 0.60 (top, squares); 0.02, 0.79 (top, triangles); 0.01, 0.70 (bottom, circles); and 0.5, 0.90 (bottom, diamonds). For the theoretical predictions that include the effects of correlations, all of the curves use the same fitting parameter, $ϵ=0.27$.

Figure 3

Dimensionless steady-state current $J_0∕k_{\text{hop}}$ obtained from the simulations (points) for various values of the left-end boundary condition ${\lambda }_b^{\left(0\right)}∕\Lambda$, as compared to the theoretical predictions both at the mean-field level (dashed lines) and including the correlations (solid lines). All parameter values are identical to those used in Fig. 2.

Figure 4

Dimensionless bound particle density ${\lambda }_b\left(x\right)∕\Lambda$ for the hole case obtained from the simulations (points), as compared to the mean-field theoretical prediction (solid lines). For all data shown, $N_H=1$; the remaining parameter values are identical to those used in Fig. 2. The values of $p_{\text{enter}}$, and the resultant values of ${\lambda }_b^{\left(0\right)}$, are, respectively, 0.004, 0.41 (circles); 0.008, 0.53 (squares); 0.02, 0.63 (diamonds); and 0.5, 0.72 (triangles).

Figure 5

Dimensionless steady-state current $J_0∕k_{\text{hop}}$ obtained from the simulations (points) for various values of the left-end boundary condition ${\lambda }_b^{\left(0\right)}∕\Lambda$, as compared to the mean-field theoretical predictions Eq. (33) (solid lines). All parameter values are identical to those used in Fig. 4.