Sequential mutations in exponentially growing populations

TL;DR

This paper develops a general mathematical framework to predict the number and arrival time of cells with multiple mutations in exponentially growing populations, revealing universal distribution patterns.

Contribution

It introduces a multitype branching process model for sequential mutations, deriving universal distributions for mutation counts and timings across various mutation effects.

Findings

Number and timing of mutants follow Mittag-Leffler and logistic distributions.

Results apply to large times and small mutation rates regimes.

Framework aids in mutation rate inference and understanding evolutionary dynamics.

Abstract

Stochastic models of sequential mutation acquisition are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells are there with $n$ alterations, and how long will it take for these cells to appear. For exponentially growing populations, these questions have been tackled only in special cases so far. Here, within a multitype branching process framework, we consider a general mutational path where mutations may be advantageous, neutral or deleterious. In the biologically relevant limiting regimes of large times and small mutation rates, we derive probability distributions for the number, and arrival time, of cells with n mutations. Surprisingly, the two quantities respectively follow Mittag-Leffler and logistic distributions regardless of $n$ or the mutations' selective effects. Our results provide a rapid method to assess how altering the fundamental division, death, and mutation rates impacts the arrival time, and number, of mutant cells. We highlight consequences for mutation rate inference in fluctuation assays.