Mean field mutation dynamics and the continuous Luria-Delbr\"uck distribution

TL;DR

This paper develops a continuous mutation model for the Luria-Delbrück distribution using nonlinear statistical physics tools, deriving differential equations and confirming results with numerical simulations.

Contribution

It introduces a novel continuous framework for the Luria-Delbrück model using generalized Fokker-Planck equations, linking classical formulations to modern mathematical physics.

Findings

Derivation of differential models from classical formulations

Solutions expressed as generalized Luria-Delbrück distributions

Numerical simulations confirm theoretical predictions

Abstract

The Luria-Delbr\"uck mutation model has a long history and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using some mathematical tools from nonlinear statistical physics. Starting from the classical formulations we derive the corresponding differential models and show that under a suitable mean field scaling they correspond to generalized Fokker-Planck equations for the mutants distribution whose solutions are given by the corresponding Luria-Delbr\"uck distribution. Numerical results confirming the theoretical analysis are also presented.