Effects of multiple occupancy and interparticle interactions on selective transport through narrow channels: theory versus experiment

Abstract

Many biological and artificial transport channels function without direct input of metabolic energy during a transport event and without structural rearrangements involving transitions from a closed to an open state. Nevertheless, such channels are able to maintain efficient and selective transport. It has been proposed that attractive interactions between the transported molecules and the channel can increase the transport efficiency and that the selectivity of such channels can be based on the strength of the interaction of the specifically transported molecules with the channel. Herein, we study the transport through narrow channels in a framework of a general kinetic theory, which naturally incorporates multiparticle occupancy of the channel and non-single-file transport. We study how the transport efficiency and the probability of translocation through the channel are affected by interparticle interactions in the confined space inside the channel, and establish conditions for selective transport. We compare the predictions of the model with the available experimental data and find good semiquantitative agreement. Finally, we discuss applications of the theory to the design of artificial nanomolecular sieves.

Figure 1

Schematic diagram of transport through a channel. (A) Schematic illustration of the transport through a narrow channel. (B) Kinetic diagram of a one-site channel. (C) Kinetic diagram of a two-site channel.

Figure 2

Kinetic diagrams of transport through a channel of an arbitrary length. (A) Symmetric channel consisting of N positions (sites). The particles enter the channel at a site M with an average rate J. (B) Equivalent energetic diagram in the case when the exit rates are determined by the interaction (binding energy) with the channel. The exit rates at the channel ends are given by Arrhenius-Boltzmann factors of the energy barriers at the exits, E_→ and E_←: r_→ ∼ exp(−E_→/kT) and r_← ∼ exp(−E_←/kT). (C) Equivalent geometry of the channel in the case when the exit rates are due to spatial bottlenecks at the channel ends.

Figure 3

Efficiency of transport through a channel of an arbitrary length. (A) Transport efficiency as a function of the exit rate for J/r = 0.01, n_m = 1 for different entrance sites M. (Solid line) M = 1, N = 10; (shaded line) M = 4, N = 40. Corresponding dashed lines show the probability of a particle to traverse the channel; it is identical to a single particle transport efficiency in the limit J → 0 (see text). (B) Transport efficiency as a function of channel length N, for the optimal value of exit rate r_o = (Jr/(N− 1))^1/2, M = 1, J/r = 0.01, n_m = 1. (Solid line) J/r = 0.01, (dashed line) J/r = 0.1. (C) Transmitted flux J_out/J_out^∞ (see Eqs. 8 and 16) as a function of the normalized incoming flux J/r; (solid line) r_o/r = 0.01; (dashed line) r_o/r = 1; M = 1, and n_m = 1. Note that the transmitted flux saturates to a constant value J_out^∞ in the jammed regime. (D) Optimal exit rate as a function of the channel length N for M = 1, (solid line) J/r = 0.01, n_m = 1. (Dashed line) Same for J/r = 0.1.

Figure 4

Occupancy of the channel at the jamming transition. (A) Occupied fraction of the channel at the jamming transition, r_o = r_max, as a function of the channel length N, for different values of the incoming flux J/r. It shows that the channel can be occupied to a considerable degree—up to half of the available sites—before the jamming becomes significant. (B) Densities at the entrance site 1 (solid line) and exit site N (dashed line) as a function of the incoming flux J/r for r_o/r = 0.1, N = 5, and n_m = 1. Density at the entrance site saturates to 1, which causes the saturation of the transmitted flux. Density at the exit site stays low even in the regime when the transmitted flux through the pore saturates.

Figure 5

Flux through nanochannels: comparison with experiment. (A) Flux through the nanochannel as a function of the outside concentration of the transported ssDNA. (Black dots) Experimental data from Kohli et al. (16) for a nanochannel without trapping inside. (Black line) Theoretical fit from Eq. 8 with n_m = 6, Z = 1, D_in/D_out = 0.42, N = L/(2S). (Red dots) Experimental data from Kohli et al. (16) for a nanochannel with ssDNA hairpins grafted inside the channel, which are complementary to the transported ssDNA. (Red line) Theoretical prediction of Eq. 8 with n_m = 3, D_in/D_out = 0.0042, Z = 0.00007, and N = L/(2S). (B) Reduction of the flux through the channel as a function of the number of mismatches between transported ssDNA and the ssDNA hairpins grafted inside, relative to the flux of the perfect complement ssDNA measured at the feed ssDNA concentration 9 μM. (Dots) Experimental data from Kohli et al. (16) for a single mismatch at the edge of the transported DNA segment; (square) single mismatch in the middle of the transported ssDNA segment; (line) theoretical model. Same parameter values as used in panel A; see text.

Figure 6

Three-dimensional diffusion outside the channel. Schematic illustration of the three-dimensional diffusion outside the channel and one-dimensional diffusion inside. See text in Appendix.