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

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(tf|x0) of the first passage time tf 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. Graphical abstract