Crowding effects in non-equilibrium transport through nano-channels

Transport through nano-channels plays an important role in many biological processes and industrial applications. Gaining insights into the functioning of biological transport processes and the design of man-made nano-devices requires an understanding of the basic physics of such transport. A simple exclusion process has proven to be very useful in ex- plaining the properties of several artificial and biological nano-channels. It is particularly useful for modeling the influence of inter-particle interactions on transport characteristics. In this paper, we explore several models of the exclusion process using a mean field approach and computer simulations. We examine the effects of crowding inside the channel and its immediate vicinity on the mean flux and the transport times of single molecules. Finally, we discuss the robustness of the theory’s predictions with respect to the crucial characteristics of the hindered diffusion in nano-channels that need to be included in the model.

💡 Research Summary

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The paper investigates how crowding inside and near nano‑channels influences both the average particle flux and the transport times of individual molecules. The authors employ a simple exclusion process—a one‑dimensional lattice where each site can accommodate up to m particles—to capture steric hindrance and inter‑particle interactions in a coarse‑grained manner. Particles enter the channel from the left with a constant external flux J and hop between neighboring sites with an internal diffusion rate r. Additional rates r_L and r_R describe hopping from the channel’s entrance and exit regions back into the bulk, while r_12 and r_21 model the specific dynamics at the immediate vicinity of the channel ends.

The analysis begins with the single‑particle case (no incoming flux). By solving the master equation in Laplace space, the authors obtain exact expressions for the forward and backward translocation probabilities (P→, P←) and for the mean first‑passage times (T→, T←). In an asymmetric channel (r_L ≠ r_R), P→ can take any value between 0 and 1, approaching unity when the exit rate to the right dominates. The mean escape time exhibits two regimes: when the left exit rate r_L is zero, the escape time grows exponentially with the binding energy (consistent with Kramers‑Arrhenius theory); when r_L > 0, the time saturates, reflecting the limiting role of the left‑hand “reflecting” boundary.

The study then extends to many‑particle conditions, where crowding becomes relevant. Using a mean‑field approximation, the steady‑state occupancy n_i^ss of each site is derived, leading to a closed‑form expression for the steady‑state rightward flux J→ = r_R n_N^ss. Remarkably, for a uniform channel the individual particle translocation probability remains identical to the single‑particle result, despite the presence of other particles. This indicates that crowding primarily reduces the available capacity (through the factor 1 – n_i/m) but does not alter the intrinsic probability of a particle that manages to enter the channel to exit on the right.

The transport efficiency, defined as Eff = J→/J, depends non‑linearly on the entrance/exit rates, channel length N, maximal occupancy m, and the internal diffusion rate r. When r_L and r_R are independent, the transmitted flux increases monotonically with each rate. However, if the two rates are linked by a common physical mechanism (as in a symmetric channel), the flux exhibits a maximum at an optimal value of the exit rate, suggesting that tuning the “trapping” strength at the channel ends can enhance selectivity.

A key contribution of the paper is the exploration of how the external flux J influences the internal diffusion rate r. Three scenarios are examined: (i) r independent of J (classical saturation behavior), (ii) r increasing with J (e.g., crowding reduces binding energy, facilitating diffusion), and (iii) r decreasing with J (e.g., conformational changes of the channel induced by high occupancy). In cases (ii) and (iii) the transmitted flux shows, respectively, a monotonic saturation curve and a non‑monotonic curve with a clear optimum at a finite J. Similar optimal behavior is observed when the exit rates themselves depend on J (linearly or quadratically). These findings mirror experimental observations in DNA nanopores, ion channels, and synthetic nano‑filters where high concentrations can either open or block the conduit.

The analytical results are validated against kinetic Monte‑Carlo simulations. The simulations reproduce the predicted translocation probabilities, mean first‑passage times, and flux‑versus‑flux curves with high accuracy, confirming the reliability of the mean‑field approach for the parameter regimes considered. Sensitivity analyses reveal that the core predictions (translocation probability, efficiency, optimal flux) are robust against variations in channel length, maximal occupancy, and asymmetry of the hopping rates.

In conclusion, the authors present a tractable yet physically insightful framework for non‑equilibrium transport through nano‑channels. By incorporating exclusion, site‑specific hopping rates, and flux‑dependent diffusion, the model captures essential features of crowding, asymmetry, and kinetic bottlenecks observed in both biological and engineered nano‑systems. The work provides quantitative guidance for designing artificial nano‑channels with desired selectivity and throughput, and offers a theoretical basis for interpreting single‑molecule experiments that probe transport dynamics under crowded conditions.