A probabilistic model for the time to unravel a strand of DNA

A common model for the time σ_L (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 σ_L. We solve the open problem to determine the distribution of σ_L by finding its Laplace transform and using it to show that as x = L²λ/v → ∞, σ_L is nearly a constant:σ_L= 1 λvlog L²λ v ^{1 2}We also derive (modulo some small gaps) the more precise limiting asymptotic formula: for - ∞ < θ < ∞,Pσ_L< 1 λvψ 1 2[log(L² λ v)]+ θ ψ ^{1 2}[log(L² λ v)]→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.