Pattern formation through genetic drift at expanding population fronts

TL;DR

This paper studies how genetic drift influences pattern formation at the front of expanding populations, revealing fractal sector boundaries and modeling their dynamics as Brownian motions to understand spatial genetic diversity.

Contribution

It introduces a model linking sector boundary dynamics to Brownian motion, providing a unified understanding of pattern formation in expanding populations.

Findings

Sector boundaries behave as time-changed Brownian motions.

A mapping between circular and linear geometries explains sector statistics.

The model predicts fractal sector boundary patterns.

Abstract

We investigate the nature of genetic drift acting at the leading edge of range expansions, building on recent results in [Hallatschek et al., Proc.\ Natl.\ Acad.\ Sci., \textbf{104}(50): 19926 - 19930 (2007)]. A well mixed population of two fluorescently labeled microbial species is grown in a circular geometry. As the population expands, a coarsening process driven by genetic drift gives rise to sectoring patterns with fractal boundaries, which show a non-trivial asymptotic distribution. Using simplified lattice based Monte Carlo simulations as a generic caricature of the above experiment, we present detailed numerical results to establish a model for sector boundaries as time changed Brownian motions. This is used to derive a general one-to-one mapping of sector statistics between circular and linear geometries, which leads to a full understanding of the sectoring patterns in terms of annihilating diffusions.