Evaluating the applicability of the Fokker-Planck equation in polymer translocation: A Brownian dynamics study

Abstract

Brownian dynamics (BD) simulations are used to study the translocation dynamics of a coarse-grained polymer through a cylindrical nanopore. We consider the case of short polymers, with a polymer length, N, in the range N = 21-61. The rate of translocation is controlled by a tunable friction coefficient, γ0p, for monomers inside the nanopore. In the case of unforced translocation, the mean translocation time scales with polymer length as <τ1> ∼ (N - Np)(α), where Np is the average number of monomers in the nanopore. The exponent approaches the value α = 2 when the pore friction is sufficiently high, in accord with the prediction for the case of the quasi-static regime where pore friction dominates. In the case of forced translocation, the polymer chain is stretched and compressed on the cis and trans sides, respectively, for low γ0p. However, the chain approaches conformational quasi-equilibrium for sufficiently large γ0p. In this limit the observed scaling of <τ1> with driving force and chain length supports the Fokker-Planck (FP) prediction that <τ> ∝ N/fd for sufficiently strong driving force. Monte Carlo simulations are used to calculate translocation free energy functions for the system. The free energies are used with the FP equation to calculate translocation time distributions. At sufficiently high γ0p, the predicted distributions are in excellent agreement with those calculated from the BD simulations. Thus, the FP equation provides a valid description of translocation dynamics for sufficiently high pore friction for the range of polymer lengths considered here. Increasing N will require a corresponding increase in pore friction to maintain the validity of the FP approach. Outside the regime of low N and high pore friction, the polymer is out of equilibrium, and the FP approach is not valid.

Dynamics of Polymer Translocation: A Short Review with an Introduction of Weakly-Driven Regime

“…Notice, however, that the scaling τ∼N2 was observed in Reference [31,32] in the opposite limit of low viscosity, where the polymer conformational relaxation is rapid. The same scaling is also observed in the limit of high imposed pore friction [36]; see also earlier works [23,24]. For practical purposes, one may add the solvent viscosity and the pore friction as extra parameters (possibly their combination with the chain length) to determine the appropriate regime.…”

Polymer translocation: the first two decades and the recent diversification

“…† More signicantly, numerics and 2D FB simulations showed that the distribution of translocation times has an exponential decay x exp(Àt/s) rather than the power-law behaviour predicted by the fractional FokkerPlanck equation, compare also ref. 104. In addition, the probability distribution of the translocation co-ordinate m was found to be Gaussian, but with the anomalous time dependence (6) of the mean squared displacement.…”

Segregation of polymers under cylindrical confinement: effects of polymer topology and crowding

“…tions in our previous study of polymer segregation, 31 as well as in simulation studies of polymer translocation [36][37][38][39] and backfolding of confined polymers. 40 To implement the SCH method, we carry out many independent simulations, each of which employs a unique "window potential" of the form:…”