Mutant number distribution in an exponentially growing population

TL;DR

This paper provides an explicit solution to a classic model of mutation emergence in exponentially growing bacterial populations, offering exact formulas for mutant distribution and connecting to recent stochastic results.

Contribution

It introduces an explicit analytical solution for the mutant number distribution in a deterministic exponential growth model with stochastic mutations, extending previous stochastic models.

Findings

Exact generating function for total mutants

Simple expression for zero mutants probability

Recursion formula for mutant count probability

Abstract

We present an explicit solution to a classic model of cell-population growth introduced by Luria and Delbrueck 70 years ago to study the emergence of mutations in bacterial populations. In this model a wild-type population is assumed to grow exponentially in a deterministic fashion. Proportional to the wild-type population size, mutants arrive randomly and initiate new sub-populations of mutants that grows stochastically according to a supercritical birth and death process. We give an exact expression for the generating function of the total number of mutants at a given wild type population size. We present a simple expression for the probability of finding no mutants, and a recursion formula for the probability of finding a given number of mutants. In the "large population-small mutation"-limit we recover recent results of Kessler and Levin for a fully stochastic version of the process.