Ejection dynamics of a ring polymer out of a nanochannel

We investigate the ejection dynamics of a ring polymer out of a cylindrical nanochannel using both theoretical analysis and three dimensional Langevin dynamics simulations. The ejection dynamics for ring polymers shows two regimes like for linear polymers, depending on the relative length of the chain compared with the channel. For long chains with length $N$ larger than the critical chain length $N_{c}$, at which the chain just fully occupies the nanochannel, the ejection for ring polymers is faster compared with linear chains of identical length due to a larger entropic pulling force; while for short chains ($N<N_c$), it takes longer time for ring polymers to eject out of the channel due to a longer distance to be diffused to reach the exit of the channel before experiencing the entropic pulling force. These results can help understand many biological processes, such as bacterial chromosome segregation.

💡 Research Summary

The paper presents a combined theoretical and computational study of how a ring‑shaped polymer ejects from a cylindrical nanochannel. Using a blob picture adapted for circular topology, the authors first derive scaling relations for the longitudinal extension of both linear and ring polymers confined in an infinitely long channel. For a linear chain the extension scales as (R_{k,l}\sim (A\sigma)^{5/3} N D^{-2/3}); for a ring polymer the same scaling holds but with a smaller prefactor, (R_{k,r}=0.561,R_{k,l}). This difference originates from the helix‑like arrangement of two strings of half‑diameter blobs that the ring adopts inside the tube.

A critical chain length (N_c) is defined as the length at which the polymer just fills the channel of height (h). Because of the different prefactors, the critical length for a ring polymer is about 1.78 times larger than that for a linear chain. The ejection dynamics therefore falls into two regimes.

1. Long chains ( (N>N_c) ) – Part of the polymer already protrudes from the channel, creating an entropic pulling force that drives the remaining segment out. The free‑energy gradient yields a pulling force (f_l = B_l k_BT/D) for linear chains and a larger force (f_r = 4\sqrt{2},B_r k_BT/D) for rings. Balancing this force with the viscous drag (\xi n \dot x) gives an ejection time (\tau_{\text{long}}\propto \xi h^2 D^{1/\nu}/(B k_BT)). Since the universal constants satisfy (B_l/B_r\approx0.315), a ring polymer ejects roughly 30 % faster than a linear polymer of the same length.

2. Short chains ( (N<N_c) ) – The whole polymer is initially confined, so it must first diffuse to the channel exit before any pulling force can act. The diffusion time dominates: (\tau_1\approx (h-R_k)^2/(2D_{\text{diff}})) with (D_{\text{diff}}=k_BT/(N\xi)). Because the ring’s longitudinal size (R_{k,r}) is smaller than (R_{k,l}), the distance ((h-R_{k,r})) is larger, leading to a longer diffusion stage and thus a longer total ejection time for rings in this regime.

The authors validate these predictions with three‑dimensional Langevin dynamics simulations of bead‑spring polymers (Lennard‑Jones repulsion + FENE bonds). They explore channel diameters (D=5,7,9) and heights (h=20.5,30.5,40.5) (in reduced units), averaging over 700 independent runs for each parameter set. The simulation data collapse onto the theoretical scaling laws: (\tau_{\text{long}}\sim h^2 D^{5/3}) for long chains, and the chain length at which the ejection time reaches its maximum, (N^), scales as (N^\propto h D^{2/3}). Moreover, the ratio of ejection times for long rings versus linear chains matches the predicted factor of ≈0.315.

Beyond the polymer physics, the work has clear biological relevance. Many bacterial chromosomes and viral genomes are circular DNA molecules that must be segregated or injected through confined pathways. The finding that long circular polymers experience a stronger entropic pulling force and therefore eject more rapidly than their linear counterparts provides a mechanistic explanation for efficient chromosome segregation in bacteria and rapid DNA delivery by bacteriophages.

In summary, the paper delivers a comprehensive picture of ring‑polymer ejection from nanochannels: it establishes the appropriate scaling framework, identifies the crossover between diffusion‑limited and force‑driven regimes, confirms the theory with extensive simulations, and connects the results to real‑world biological processes. The study thus advances our understanding of confined polymer dynamics and offers quantitative guidance for nanofluidic device design and the interpretation of DNA manipulation experiments.