Reversible Transitions in a Fluctuation Assay Modify the Tail of Luria–Delbrück Distribution

Abstract

We consider a fluctuation test experiment in which cell colonies were grown from a single cell until they reach a given population size and were then exposed to treatment. While they grow, the cells may, with a low probability, acquire resistance to treatment and pass it on to their offspring. Unlike the classical Luria–Delbrück fluctuation test, and motivated by recent work on drug-resistance acquisition in cancer/microbial cells, we allowed the resistant cell state to switch back to a drug-sensitive state. This modification did not affect the central part of the Luria–Delbrück distribution of the number of resistant survivors: the previously developed approximation by the Landau probability density function applied. However, the right tail of the modified distribution deviated from the power law decay of the Landau distribution. Here, we demonstrate that the correction factor was equal to the Landau cumulative distribution function. We interpreted the appearance of the Landau laws from the standpoint of singular perturbation theory and used the asymptotic matching principle to construct uniformly valid approximations. Additionally, we describe the corrections to the distribution tails in populations initially consisting of multiple sensitive cells, a mixture of sensitive and resistant cells, and a cell with a randomly drawn state.