Probability distributions for polymer translocation

Abstract

We study the passage (translocation) of a self-avoiding polymer through a membrane pore in two dimensions. In particular, we numerically measure the probability distribution Q(T) of the translocation time T, and the distribution P(s,t) of the translocation coordinate s at various times t. When scaled with the mean translocation time T , Q(T) becomes independent of polymer length, and decays exponentially for large T. The probability P(s,t) is well described by a Gaussian at short times, with a variance of s that grows subdiffusively as talpha with alpha approximately 0.8. For times exceeding T , P(s,t) of the polymers that have not yet finished their translocation has a nontrivial stable shape.

Fractional Brownian motion approach to polymer translocation: The governing equation of motion

“…The obtained solution for W (s,t) demonstrates two characteristic features of fBm, which agree favorably with the recent findings [12][13][14] as regards nondriven translocation dynamics: (i) A Gaussian distribution for the translocation coordinate s during the translocation process and (ii) a subdiffusive behavior for the variance of the translocation coordinate (t) = s 2 (t) − s(t) 2 . Moreover, the survival probability S(t,s 0 ) in the long-time limit has a stretched-exponential form in agreement with recent findings [29] (albeit in contrast to a popular opinion about its scaling behavior [28]).…”

“…An interesting aspect of translocation dynamics in a system with two adsorbing boundaries has been considered recently [12][13][14], suggesting that, for a sufficiently long-time interval, the normalized distribution…”

“…Based on numerical results [12,13], one has tried [28] recently to link translocation dynamics to self-affine processes undergoing anomalous diffusion in bounded domains within the context of fBm. The argumentation [28] relies on two crucial assumptions: (i) The PDF W (s,s 0 ,t) has a self-affine form W (s,s 0 ,t) = (1/t α/2 )F (s/t α/2 ,s 0 /t α/2 ), and (ii) the socalled survival probability S(s 0 ,t) = N 0 W (s,s 0 ,t) ds has a long-time scaling behavior S(s 0 ,t) ∼ t −θ with an exponent θ = 1 − α/2.…”

“…While this latter property may readily be verified in numeric experiments, the unambiguous determination of the precise functional shape of W (s,t) is rather difficult as a result of progressively deteriorating statistics of the distribution at late times. Meanwhile, several recent publications [12][13][14], devoted to the unbiased translocation dynamics of a Gaussian onedimensional (1D) chain [12] as well as to that of a twodimensional (2D) self-avoiding chain [13], have validated the subdiffusive behavior of s 2 (t) with an exponent α 0.8. Nonetheless, these new findings cast serious doubts as to whether the FFPE indeed provides an adequate description of nondriven translocation dynamics:…”

“…(i) It was found [12][13][14] by means of computer simulations that the probability distribution W (s,t) of the translocation coordinate s stays Gaussian for different time moments not exceeding the mean translocation time τ . At larger times, this distribution attains a more complex form.…”