Age-dependent Branching Processes and Applications to the Luria-Delbr\"uck Experiment

TL;DR

This paper develops a mathematical model for microbial populations with age-dependent and asymmetric cell division, providing tools to estimate mutation rates based on probability distributions of mutants.

Contribution

It introduces a novel probability generating function approach to model age-dependent asymmetric cell division and mutant distribution in microbial populations.

Findings

Derived a probability distribution for population size with age-dependent division

Established asymptotic growth rates for such populations

Provided a method to estimate mutation rates using the distribution

Abstract

Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbr\"uck that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur many generations before there is a selective advantage, and thus organisms follow Darwinian evolution instead of Lamarckian. Since then, new areas of research involving microbial evolution have spawned as a result of their experiment. Determining the mutation rate of a cell is one such area. Probability distributions that determine the number of mutants in a large population have been derived by D. E. Lea, C. A. Coulson, and J. B. S. Haldane. However, not much work has been done when time of cell division is dependent on the cell age, and even less so when cell division is asymmetric, which is the case in most microbial populations. Using probability generating function methods, we rigorously construct a probability distribution for the cell population size given a life-span distribution for both mother and daughter cells, and then determine its asymptotic growth rate. We use this to construct a probability distribution for the number of mutants in a large cell population, which can be used with likelihood methods to estimate the cell mutation rate.