DNA breathing dynamics: Analytic results for distribution functions of relevant Brownian functionals

TL;DR

This paper derives analytical distribution functions for various Brownian functionals related to DNA bubble dynamics, providing insights into bubble lifetime, reactivity, and size fluctuations, supported by numerical simulations.

Contribution

It introduces new analytical expressions for distributions of bubble lifetime, area, maximum size, and their joint probabilities in DNA breathing dynamics using advanced mathematical methods.

Findings

Analytical formulas for first-passage time distribution

Distribution of bubble area before reclosure

Joint distribution of maximum size and occurrence time

Abstract

We investigate DNA breathing dynamics by suggesting and examining several different Brownian functionals associated with bubble lifetime and reactivity. Bubble dynamics is described as an overdamped random walk in the number of broken base pairs. The walk takes place on the Poland-Scheraga free energy landscape. We suggest several probability distribution functions that characterize the breathing process, and adopt the recently studied backward Fokker-Planck method and the path decomposition method as elegant and flexible tools for deriving these distributions. In particular, for a bubble of an initial size $x_0$, we derive analytical expressions for (i) the distribution $P(t_f|x_0)$ of the first-passage time $t_f$, characterizing the bubble lifetime, (ii) the distribution $P(A|x_0)$ of the area $A$ till the first-passage time, providing information about the effective reactivity of the bubble to processes within the DNA, (iii) the distribution $P(M)$ of the maximum bubble size $M$ attained before the first-passage time, and (iv) the joint probability distribution $P(M,t_m)$ of the maximum bubble size $M$ and the time $t_m$ of its occurrence before the first-passage time. These distributions are analyzed in the limit of small and large bubble sizes. We supplement our analytical predictions with direct numerical simulations of the related Langevin equation, and obtain a very good agreement in the appropriate limits. The nontrivial scaling behavior of the various quantities analyzed here can, in principle, be explored experimentally.