Can deleterious mutations surf deterministic population waves? A functional law of large numbers for a spatial model of Muller's ratchet

TL;DR

This paper models the spread of deleterious mutations in expanding populations using a spatial Muller's ratchet, deriving PDE limits and analyzing mutation surfing during population expansion.

Contribution

It introduces a scaling limit of the spatial Muller's ratchet model, connecting particle systems to PDEs, and analyzes mutation surfing in expanding populations.

Findings

Weak convergence to PDE system confirmed non-rigorous predictions.

Bounds established on mutation density ratios in monostable regimes.

Determined conditions under which deleterious mutations can surf population waves.

Abstract

The spatial Muller's ratchet is a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population. The model is an interacting particle system consisting of particles performing symmetric random walks that reproduce and die with rates that depend on the local number of particles. For each particle, we keep track of the number of deleterious mutations that it carries, and after each birth event, with some positive probability, the offspring particle can acquire an additional mutation that gives it a lower reproduction rate than its parent. We show that under an appropriate scaling, the process converges weakly to the solution of an infinite system of partial differential equations (PDEs), confirming non-rigorous computations of Foutel-Rodier and Etheridge. In the PDE limit, when the reaction term of the system of PDEs is monostable, we establish bounds on the ratio between the density of particles with a given number of mutations and the density of particles without mutations. If the reaction term satisfies a Fisher-KPP condition, we can also rigorously determine the spreading speed of the population into an empty habitat. Finally, by considering the PDE limit of a form of tracer dynamics, we answer the question of whether deleterious mutations can surf population waves in this setting.