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

TL;DR

This study analyzes driven polymer translocation through extended patterned pores in two dimensions, revealing a robust finite-size scaling form $\langle \tau \rangle \sim N^{\gamma} \mathcal{F}(L_p N^{\phi})$ with $\gamma \approx 3.0$ and $\phi \approx 1.5$ for both patterned and unpatterned pores. The results show a geometry–drive competition: in short cylindrical pores, increasing $L_p$ accelerates translocation, while in long conical pores the decreasing axial drive can slow progress, with patterning further modulating exit residence. The scaling function decays as a power law with geometry-dependent exponents ($p \approx 0.85$ for cylindrical, $p \approx 0.6$–$0.8$ for conical/patterned pores), and the exponent relation $\beta \approx \gamma - (\phi p)/2$ links $N$- and $L_p$-dependence. These findings provide design principles for nanopore transport and a quantitative framework for patterning-based discrimination in driven translocation, though extensions to 3D with hydrodynamics are needed for direct experimental comparison.

Abstract

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.

Figures (7)

• Figure 1: Top row shows a schematic diagram of pore-polymer set up for translocation in two-dimensions. The pore is 5 beads long with a half apex angle $\alpha$. The pore diameter on the cis end is $d=2.25\sigma$, whereas it is $y = d + 2L_p \tan \alpha$ on the trans end. The flexible polymer is made up of beads with diameter $\sigma=1$ connected with springs. The subsequent rows show patterned pores with different pore lengths, $L_{p}=$ 5, 10, and 15, respectively. The blue beads are attractive while the yellow and green beads are repulsive in interaction with polymer chain.
• Figure 2: Plots showing the variation of $\tau$ vs. $\alpha$ behavior, with the change of pore-length, $L_{p}$. Here, the polymer is flexible with length $N$ of 128 beads, and pore-length varies from 5$\sigma$ to 10$\sigma$.
• Figure 3: $\langle \tau \rangle$ as a function of $N$ (in log-log scale) for three different driving forces when the polymer is translocating from an extended cylindrical pore (i.e., $\alpha = 0$) of length (a) $L_p = 5\sigma$, and (b) $L_p = 10\sigma$. The slopes mentioned in the plot are the values of the translocation exponent, $\beta$, defined by $\langle \tau \rangle \sim N^{\beta}$, for different driving forces.
• Figure 4: $\langle \tau \rangle / N^{\gamma}$ as a function of $L_p N^{\phi}$ for an extended pore ($\alpha = 0$) for three different driving forces (a) 0.2, (b) 0.6, and (c) 1.0. The lines are the scaling function $\mathcal{F}(x)\sim 1/x^p$ with exponent $p=0.85$.The dashed line above (below) is for exponent value $p = 0.80 \ (p = 0.90)$
• Figure 5: Comparison of $\tau$ vs. $\alpha$ for three different pore sizes $L_{p}=$ 5, 10 and 15 for two different forces $f_{0}= 0.6$ and 1.0. The length $N$ of the flexible polymer chain is 128 beads.