Free Energy and Diffusivity in the Fokker-Planck Theory of Polymer Translocation

TL;DR

This work revisits the Fokker-Planck framework for driven polymer translocation through nanopores by combining metadynamics-derived free-energy landscapes with an analytical FP form. It introduces an entropic correction through an effective coordination number $z_{\text{eff}}$ and calibrates a uniform diffusivity $k_{\text{FP}}$ by matching FP predictions to Langevin dynamics results, across varying $N$, $r_p$, and $V$. The key findings show that $\langle \tau \rangle$ scales as $\sim V^{-1}N^{\alpha}$ with $\alpha\approx 1.40-1.48$ for finite pores, while $k_{\text{FP}}$ scales as $N^{\beta}$ with $\beta$ in the range $-0.5$ to $-0.74$ depending on confinement, indicating confinement-enhanced diffusivity and a deviation from simple Rouse predictions. The combined metadynamics-theory approach extends FP predictions to longer polymers and diverse pore geometries, aligning with experimental trends and offering a concrete mechanism for diffusion along the translocation coordinate under confinement.

Abstract

We revisit the Fokker-Planck based theory of driven polymer translocation through a narrow nanopore. A bead-spring model of a uniformly charged polyelectrolyte chain translocating through a semi-implicit model of a nanopore embedded in a membrane are used to gain insights into the underlying free energy landscape and kinetics of translocation. The free energy landscape is predicted using metadynamics simulation, an enhanced sampling method. A direct comparison with the theoretical free energy formulation proposed in the literature allows us to introduce a modification related to the entropic contribution in the theory. Additional classical Langevin dynamics simulation runs are performed to obtain the translocation time distribution for polymers of lengths $N$ driven by voltages $V$ through nanopores of radii $r_p$. In agreement with earlier reports, a scaling of the mean translocation time $\langle τ_\text{LD} \rangle \sim N^α/V$ is observed, with $α\sim 1.40 - 1.48$ depending on the nanopore size. Fitting the mean first passage time given by the Fokker-Planck theory, $\langle τ_\text{FP}\rangle$,to simulation results helps gain insights into the diffusivity $k_\text{FP}$ used in the theory. We report a scaling of $k_\text{FP}\sim N^β$. The $r_p-$dependent values of the exponent $β$ significantly deviate from the Rouse theory prediction of $β= -1$ for center-of-mass diffusivity of a polymer chain.

Figures (6)

• Figure 1: Schematic of the simulation setup. $N$ beads (orange) of the polymer chain translocate through the semi-implicit nanopore of radius $r_p$ and length $L(=16)$ (blue) from donor to receiver compartment in the presence of externally applied force ($\textbf{F}^\text{ext}$).
• Figure 2: (a) Variation of average pore-polymer interactions energy $\epsilon_\text{FP}$ with $r_p$. (b) Comparison of free energy landscapes from theory ($F_\text{th}$) and metadynamics ($F_{\text{meta}}$) for $N=121$ and $r_p=1$. The dot-dashed line shows $F_\text{th}$ for the effective coordination number $z_\text{eff}=1$, while the solid black line shows the same for $z_\text{eff}=2.46$. $\epsilon_\text{FP} = 0.172$ in both.
• Figure 3: (a) Comparison of $F_{\text{meta}}$ for $N=$ 61, 81, 101, and 121 (colored lines) and $F_{\text{th}}$ (solid black line) for $N=121$ during the nanopore filling stage. Inset shows $F_{\text{meta}}$ and $F_{\text{th}}$ using rescaled reaction coordinates, highlighting the independence of the free energy barrier with respect to $N$. (b) The effective coordination number of polymer beads inside the nanopore $z_\text{eff}$ obtained using $\epsilon_\text{FP}$ values from figure \ref{['fig:epsilon_FE_comparision']}(a) (filled left-triangles), and using $\epsilon_\text{FP}=0$ (empty left-triangles).
• Figure 4: Variation of mean translocation time from Langevin dynamics simulations $\langle \tau_{\text{LD}} \rangle$ (yellow diamonds) and mean first passage time from Fokker-Planck formalism with input free energy from (a) metadynamics simulations $\langle \tau_\text{meta} \rangle$ (blue left triangles) and (b) theoretical formulation $\langle \tau_\text{th} \rangle$ (green left triangles), with external trans-membrane voltage $V$.
• Figure 5: (a) Power law dependence of $\langle \tau \rangle$ on $N$ with the scaling exponent $\alpha=1.43$ for $rp=1$ and $V=86.7$mV. Inset shows a magnified view of the data for smaller $N$. (b) Values of $\alpha$ for different $r_p$ and $V$. $\alpha$ increases with increasing $r_p$, and is independent of $V$. Data for $r_p \rightarrow \infty$ is shown in grey color.