A Probabilistic Model For the Time to Unravel a Strand of DNA

A common model for the time σL (sec) taken by a DNA strand of length L (cm) to unravel is to assume that new points of unraveling occur along the strand as a Poisson process of rate λ 1/(cm x sec) in space-time and that the unraveling propagates at speed v/2 (cm/sec) in each direction until time σL. We solve the open problem to determine the distribution of σL by finding its Laplace transform and using it to show that as x = L2λ/v → ∞, σL is nearly a constant:σL=1λvlogL2λv12We also derive (modulo some small gaps) the more precise limiting asymptotic formula: for - ∞ < θ < ∞,PσL<1λvψ12[log(L2λv)]+θψ12[log(L2λv)]→e-e-θwhere ψ is defined by the equation: ψ(x) = log ψ(x)+x, x⩾1. These results are obtained by interchanging the role of space and time to uncover an underlying Markov process which can be studied in detail.