  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.
