The selective advantage of neighborhood-aware mutants in Moran process

TL;DR

The paper introduces replacers, a Moran-process phenotype whose offspring always replaces an individual of the opposite type when available, thereby reducing wasted reproduction. It derives fixation and elimination probabilities for replacers in well-mixed and one-dimensional lattice populations, showing neutral replacers have fixation probabilities of order $1/\sqrt{N}$ and exponentially small elimination probabilities, making them remarkably robust once established. For non-neutral mutants, replacers maintain advantageous fixation for $r>1$ and dramatically increase persistence against resident invasions for $r<1$, with the threshold for mutation-selection dominance tending toward $r^*=1/e$ as $N$ grows (and $r^*=1/2$ on cycles). The results generalize to spatial structures and connect to evolutionary games, offering a framework where neighborhood-aware strategies can dominate mutation-selection balance even at a cost, and suggesting broader implications for spatially structured evolution.

Abstract

Evolution occurs in populations of reproducing individuals. In stochastic descriptions of evolutionary dynamics, such as the Moran process, individuals are chosen randomly for birth and for death. If the same type is chosen for both steps, then the reproductive event is wasted, because the composition of the population remains unchanged. Here we introduce a new phenotype, which we call a replacer. Replacers are efficient competitors. When a replacer is chosen for reproduction, the offspring will always replace an individual of another type (if available). We determine the selective advantage of replacers in well-mixed populations and on one-dimensional lattices. We find that being a replacer substantially boosts the fixation probability of neutral and deleterious mutants. In particular, fixation probability of a single neutral replacer who invades a well-mixed population of size $N$ is of the order of $1/\sqrt N$ rather than the standard $1/N$. Even more importantly, replacers are much better protected against invasions once they have reached fixation. Therefore, replacers dominate the mutation selection equilibrium even if the phenotype of being a replacer comes at a substantial cost: curiously, for large population size and small mutation rate the relative fitness of a successful replacer can be as low as $1/e$.

Figures (8)

• Figure 1: Moran process with oblivious mutants and with replacers. In each step of the Moran Birth-death process, an offspring of a reproducing mutant (yellow circle) migrates along one of the red arrows to replace the individual at its other end. a, In the standard Moran process, a reproducing mutant is oblivious: when selected for reproduction, the offspring could migrate anywhere, possibly replacing another mutant. b, When the mutant is a replacer, it never "wastes its turn", by replacing a random resident rather than a random individual.
• Figure 2: Forward biases in the well-mixed populations. In the standard Moran process with $i$ oblivious mutants, the number of mutants is as likely to increase as it is to decrease, that is, the forward bias satisfies $\gamma_i=1$ (black, dashed). When the mutants are replacers, the forward bias in a population of size $N$ satisfifes $\gamma^R_i=\frac{N-1}{N-i}$, so it increases with the number $i$ of mutants, reaching up to $\gamma^R_{N-1}=N-1$. Here population sizes are $N\in\{4.5,\dots,10\}$ (colors) and the number of mutants $i$ varies from 1 to $N-1$.
• Figure 3: Fixation and elimination probabilities of neutral replacers.a, Neutral replacers have fixation probability of the order of $\rho^R(N)\sim 1/\sqrt N$, which is substantially higher than the fixation probability $\rho(N)=1/N$ of standard (oblivious) mutants. Moreover, replacers have exponentially small elimination probability of the order of $\psi^R(N)\sim 1/e^N$, which is substantially smaller than the elimination probability $\psi(N)=1/N$ of standard (oblivious) mutants. b, While all fixation and elimination probabilities tend to 0 as the population size $N$ grows large, they do so at very different rates. For oblivious mutants both the fixation and the elimination probability are equal to $\rho(N)=\psi(N)=1/N$, which serves as a natural baseline (blue). The fixation probability $\psi^R(N)$ of replacers is substantially larger (green), and their elimination probability $\psi^R(N)$ is substantially smaller (red).
• Figure 4: Fixation and elimination probabilities of non-neutral replacers.a, When the mutation is deleterious, the fixation probability is exponentially small regardless of whether the mutant is oblivious or a replacer. However, for replacers the base of the exponential is larger, so fixation is exponentially more likely. Perhaps more importantly, the elimination probability of replacers is also exponentially small, meaning that once established, the replacers are well protected against resident invasions. This strongly contrasts with a constant elimination probability of established oblivious mutants. b, When the mutation is beneficial, being a replacer does not affect the fixation probability when the population size $N$ is large. However the replacers again do have diminished elimination probability, as compared to oblivious mutants.
• Figure 5: Proportion of mutants in the mutation-selection process. In the mutation-selection process, any time an individual reproduces, the offspring mutates to the other type with fixed probability $u\in(0,1)$. The long-term average mutant frequency $\lambda$ depends on the mutant fitness $r$ ($x$-axis) and also on the mutation rate $u$ (colors). When the mutants are oblivious (dashed lines), they form the majority in the population when they have a fitness advantage $r>1$, otherwise they are in a minority. In contrast, replacers (solid lines) form majority even for certain values $r<1$, e.g., for $r\ge 0.46$ when $N=10$ (blue solid line vs dotted horizontal line in the left panel). In particular, we show that when $u\to 0$ and $N$ is large, replacers dominate so long as their relative fitness is above a threshold $r^\star=1/e\doteq 0.37$ (dotted line). For other $u$ they dominate for even smaller values of $r$. Here the population size is a,$N=10$, b,$N=50$.

Theorems & Definitions (39)

• Theorem 1: Neutral fixation probability
• Theorem 2: Neutral elimination probability
• Theorem 3: Fixation probability for large $N$
• Theorem 4: Elimination probability for large $N$
• Theorem 5: Cycle
• Theorem 1: Neutral fixation probability
• Theorem 2: Neutral elimination probability
• Theorem 3: Fixation probability for large $N$
• Theorem 4: Elimination probability for large $N$
• Theorem 5: Cycle