Branching processes and bacterial growth

TL;DR

This paper develops a detailed mathematical model of bacterial growth using branching processes, incorporating cell size, growth rate variability, and different bacterial types, and connects it to growth-fragmentation equations.

Contribution

It introduces a novel branching process model with multiple bacterial types and variable offspring, extending previous models and establishing key properties like the many-to-one formula.

Findings

Model is well-defined and mathematically rigorous.

The mean empirical measure satisfies a growth-fragmentation PDE.

Includes new biological features like cell aging and variable offspring number.

Abstract

We model the growth of a cell population using a piecewise deterministic Markov branching tree. In this model, each cell splits into two offspring at a division rate $B(x)$, which depends on its size $x$. The size of each cell increases exponentially over time, with a growth rate that varies for each individual. Expanding upon the model studied in \cite{hof}, we introduce a scenario with two types of bacteria: those with a young pole and those with an old pole. Additionally, we account for the possibility that a bacterium may not always divide into exactly two offspring. We will demonstrate that our branching process is well-defined and that it satisfies a many-to-one formula. Furthermore, we establish that the mean empirical measure of the model adheres to a growth-fragmentation equation when structured by size, growth rate, and type as state variables.