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

TL;DR

The spatial Muller's ratchet model, a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population, is considered, and the question of whether deleterious mutations can surf population waves in this setting is answered.

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.

Figures (2)

• Figure 1: Numerical simulation of the system of PDEs \ref{['Paper02_PDE_scaling_limit']} with Fisher–KPP type dynamics with parameters $m = 3$, $q_+(U) = 1$ and $q_-(U) = U$ for all $U \geq 0$, $\mu = 0.025$, and $s_k = 0.95^k$ for all $k \in \mathbb{N}_0$. The horizontal axis indicates spatial position $x$, and the vertical axis shows population density. The time the snapshot is taken is denoted by $T$. Colours indicate the density of population carrying different numbers of mutations, i.e. $u_k(T,\cdot)$ for $k \in \{0,1,\ldots,4\}$ as shown in the legend of the $T = 0$ figure. The simulation illustrates that the fractions of the population carrying positive numbers of mutations invade previously uncolonised habitat at the same speed as the fraction carrying no mutations. We emphasise that, as will be shown in Theorem \ref{['Paper02_thm_tracer_dynamics_no_gene_surfing']}, this phenomenon occurs due to a mutation--selection equilibrium rather than gene surfing of deleterious mutations.
• Figure 2: Numerical simulation of a population with Fisher–KPP type dynamics, governed by the systems of PDEs \ref{['Paper02_PDE_scaling_limit']} and \ref{['Paper02_PDE_system_labelled_particles']}, using the same parameters as in Figure \ref{['Paper02_FKPP_no_tracer']}. The horizontal axis indicates spatial position $x$, and the vertical axis shows population density. The time the snapshot is taken is denoted by $T$. Colours indicate the density of population carrying different numbers of mutations, i.e. $u_k(T,\cdot)$ for $k \in \{0,1,\ldots,4\}$ as shown in the legend of the $T = 0$ figure. Shaded regions indicate the labelled fraction of the population. At time $T = 0$, only the subpopulation without mutations is unlabelled. As shown in Theorem \ref{['Paper02_thm_tracer_dynamics_no_gene_surfing']}, the labelled population, descended from the initial mutant subpopulation, does not propagate with the expansion wave.

Theorems & Definitions (123)

• Theorem 2.1
• Remark 2.2
• Definition 2.3: Monostable and Fisher-KPP reaction terms
• Remark 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Proposition 3.1