Polymer translocation across an oscillating nanopore: study of several distribution functions of relevant Brownian functionals

Abstract

In this paper, we study polymer translocation dynamics across an oscillating nanopore by proposing and inspecting several probability distribution functions (PDFs) of relevant Brownian functionals which specify the translocation across the nanopore. We model such translocation process by an overdamped Langevin equation of collective variable x. We introduce several probability distribution functions (PDFs) to identify the translocation process. We consider elegant backward Fokker-Planck method to derive analytically closed form expressions of several PDFs associated with such stochastic process. For instance, an important quantity for translocation processes is the first passage time, i.e. the time the molecule takes to cross the nanopore with initial collective value of the molecule x0. We derive analytical expressions for: (i) the PDF P (t f |x0) of the first passage time t f which specify the lifetime of protein translocation process, (ii) the PDF P (A|x0) of the area A till the first passage time and it provides us numerous valuable information about the average size and reactivity of the process, and (iii) the PDF P (M ) associated with the maximum value of collective mode, M , of the translocation process before the first passage time. Our analysis is limited to a regime where both drift and diffusion have the same periodic time dependence with a constant ratio between them. We further confirm our analytical predictions by computing the same PDFs with direct numerical simulations of the corresponding Langevin equation. We obtain a very good agreement of our theoretical predictions with the numerically simulated results. Finally, several nontrivial scaling behaviour in the asymptotic limits for the above mentioned PDFs are predicted, which can be verified further from experimental observation.

Statistics of the first passage area functional for an Ornstein–Uhlenbeck process

“…This surprisingly simple result was first identified in [17]. For the second central moment 2 ( ) C α A , which is simply the variance, one can quickly derive the following expression based on (7,19):…”

“…To characterize the distribution more fully, it is useful to study the third and fourth central moments. For the former we have from (7,19):…”

“…polymer translocation [7] and barrierless reaction kinetics [8]. Conditioning on a fixed first passage time provides insight into interface fluctuations [9], laser cooling of atoms [10] and colloidal suspensions [11]; the associated techniques being developed explicitly in [12,13], along with studies of functional correlations in [14,15].…”

First-passage functionals for Ornstein–Uhlenbeck process with stochastic resetting

“…The other important FPBFs studied in the literature are area and extreme maximum which have applications in statistical physics and queuing theory [29][30][31]. The FPBF has also been calculated in the context of snowmelt dynamics [32], biopolymer translocation dynamics [33], DNA breathing dynamics [34] and barrierless reactions [35]. There has been a renewed interest for studying functionals in the presence of stochastic resetting.…”

Antiresonant driven systems for particle manipulation

“…Such a scenario can be realized by means of optical, electric or magnetic fields [17][18][19] or by forces resulting from the geometrical confinement of the particles. This is the case of the ondulatory motion of worm-like organisms [20,21], peristaltic pumping [22][23][24][25], fluctuating ion channels and pores [26][27][28][29][30][31][32][33][34][35][36][37] as well as synthetic soft micro-nanofluidic devices [38][39][40]. Knowing what the impact of higher order harmonics is on the response of the system is a question of great current interest.…”