Polymer translocation through extended patterned pores in two dimensions: scaling of the total translocation time

We study the translocation of a flexible polymer through extended patterned pores using molecular dynamics (MD) simulations. We consider cylindrical and conical pore geometries that can be controlled by the angle of the pore apex $α$. We obtained the average translocation time $\langle τ\rangle$ for various chain lengths $N$ and the length of the pores $L_p$ for various values $α$ and found that $\langle τ\rangle$ scales as $\langle τ\rangle \sim N^γ\mathcal{F}\left( L_p N^ϕ\right)$ with exponents $γ= 3.00\pm0.05$ and $ϕ= 1.50\pm0.05$ for both patterned and unpatterned pores.

💡 Research Summary

In this work the authors investigate the driven translocation of a flexible polymer through extended nanopores with patterned interactions using two‑dimensional Langevin dynamics simulations. Two pore geometries are considered: a cylindrical pore (apex angle α = 0) and a conical pore (α > 0). Within each geometry the pore interior is divided into attractive and repulsive sections, giving three distinct patterns: (A) fully attractive, (B) attractive entrances and exits with a repulsive middle segment, and (C) attractive entrance with a repulsive exit. The entrance is always attractive to guarantee capture of the polymer, while the exit can be tuned to be either sticky or repulsive. An external pulling force mimicking an electrophoretic drive is applied inside the pore; its magnitude varies along the pore as f_ext(x)=f₀ d/(d+2x tanα), ensuring a constant total integrated force for a given f₀.

The polymer is modeled as a bead‑spring chain with Lennard‑Jones (LJ) excluded volume and harmonic bonds. All energies, lengths and masses are reduced (ε = σ = m = 1), and the temperature is set to k_B T = 1. The friction coefficient η = 1 and the bond spring constant K = 500. For each set of parameters the authors average over 1500–2000 successful translocation events, reporting standard errors.

First, the dependence of the mean translocation time ⟨τ⟩ on the pore length L_p is examined for a fixed polymer length N = 128, a driving force f₀ = 1.0, and purely attractive pore–polymer interactions. For cylindrical pores (α = 0) ⟨τ⟩ decreases markedly as L_p grows from 5σ to 10σ, reflecting the fact that a longer pore contains more driven monomers and reduces confinement. For conical pores the behaviour is non‑monotonic: at short L_p, increasing α widens the exit, providing entropic relief and lowering ⟨τ⟩; at larger L_p the same widening is accompanied by a progressive weakening of the local driving field (because f_ext decays with x), which creates a bottleneck and makes ⟨τ⟩ increase. This competition between entropic assistance and reduced drive is observed for all three pore patterns.

Next, the authors study how ⟨τ⟩ scales with polymer length N for two fixed pore lengths (L_p = 5σ and 10σ) and three driving forces (f₀ = 0.2, 0.6, 1.0). In log‑log plots they extract the exponent β defined by ⟨τ⟩ ∝ N^β. For weak driving (f₀ = 0.2) β≈1.32 (L_p = 5σ) and β≈1.68 (L_p = 10σ), both below the equilibrium expectation 1 + ν (ν = 0.75 in 2D). As the drive increases, β grows, reaching 1.80–1.95 for f₀ = 1.0, exceeding 1 + ν. The increase of β with both L_p and f₀ indicates that stronger forces and longer pores promote a more collective tension propagation along the chain, thereby slowing down the scaling relative to the simple Rouse picture.

To capture the joint influence of N and L_p the authors propose a finite‑size scaling form
 ⟨τ⟩ ∼ N^γ F(L_p N^ϕ) ,
where γ and ϕ are universal exponents and F is a scaling function. By collapsing data for all N (64 ≤ N ≤ 512), all L_p (5–10σ), all three pore patterns, both cylindrical and conical geometries, and all three driving forces, they find an excellent data collapse with γ = 3.00 ± 0.05 and ϕ = 1.50 ± 0.05. The master curve shows a power‑law decay F(x) ∼ x^–p for large arguments, consistent with the observation that longer pores (larger x) reduce the translocation time for cylindrical pores but increase it for conical pores when the driving field weakens. Moreover, the authors note an empirical relation β ≈ γ − (ϕ p)/2 linking the N‑only exponent β to the combined exponents, confirming internal consistency of the scaling picture.

Importantly, the same γ and ϕ values are obtained for all three interaction patterns (A, B, C) and for a range of apex angles α, demonstrating that the scaling law is robust against detailed pore chemistry. The only effect of patterning is to shift the prefactor of the scaling function, i.e., to make translocation faster (fully attractive) or slower (repulsive exit), but not to alter the underlying exponents.

The authors conclude that the translocation time of a driven polymer through an extended pore is governed primarily by a combined length scale L_p N^ϕ, with a universal exponent γ≈3. This finding implies that, for practical nanopore applications such as DNA sequencing, gene therapy delivery, or polymer filtration, one can predict and control translocation speeds simply by adjusting the ratio of pore length to polymer length, without needing to fine‑tune the detailed pattern of attractive or repulsive sites along the pore. The work thus provides a clear, quantitative framework for designing synthetic nanopores with desired transport characteristics.