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

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 $\langle \tau_\text{LD} \rangle \sim N^\alpha/V$ is observed, with $\alpha \sim 1.40 - 1.48$ depending on the nanopore size. Fitting the mean first passage time given by the Fokker-Planck theory, $\langle \tau_\text{FP}\rangle$,to simulation results helps gain insights into the diffusivity $k_\text{FP}$ used in the theory. We report a scaling of $k_\text{FP}\sim N^\beta$. The $r_p-$dependent values of the exponent $\beta$ significantly deviate from the Rouse theory prediction of $\beta = -1$ for center-of-mass diffusivity of a polymer chain.