Scaling solution in the large population limit of the general asymmetric stochastic Luria-Delbr\"uck evolution process

TL;DR

This paper develops a scaling solution for the asymmetric stochastic Luria-Delbrück process in large populations, connecting it to Levy stable distributions and highlighting the limitations of moment-based characterizations.

Contribution

It introduces a novel scaling solution for large population limits of the Luria-Delbrück process, linking it to Levy distributions and emphasizing ensemble differences.

Findings

Distribution moments are misleading for typical behavior.

The fixed population size ensemble differs from the fixed time ensemble.

The solution connects the process to Levy α-stable distributions.

Abstract

One of the most popular models for quantitatively understanding the emergence of drug resistance both in bacterial colonies and in malignant tumors was introduced long ago by Luria and Delbr\"uck. Here, individual resistant mutants emerge randomly during the birth events of an exponentially growing sensitive population. A most interesting limit of this process occurs when the population size $N$ is large and mutation rates are low, but not necessarily small compared to $1/N$. Here we provide a scaling solution valid in this limit, making contact with the theory of Levy $\alpha$-stable distributions, in particular one discussed long ago by Landau. One consequence of this association is that moments of the distribution are highly misleading as far as characterizing typical behavior. A key insight that enables our solution is that working in the fixed population size ensemble is not the same as working in a fixed time ensemble. Some of our results have been presented previously in shortened form.