The multi-allelic Moran process as a multi-zealot voter model: exact results and consequences for diversity thresholds

TL;DR

This paper extends the exact analysis of the Moran process to multiple alleles by mapping it to a multi-zealot voter model, revealing how diversity thresholds depend on population structure and network topology.

Contribution

It generalizes the two-allele Moran process to multiple alleles using a voter model analogy, deriving exact stationary distributions and critical mutation rates for arbitrary allele numbers.

Findings

Exact stationary distribution for multiple alleles in well-mixed populations.

Critical mutation rate depends on the number of alleles, given by 1/(m+2n-2).

Network heterogeneity influences genetic diversity in structured populations.

Abstract

The Moran process is a foundational model of genetic drift and mutation in finite populations. In its standard two-allele form with population size $n$, allele counts, and hence allele frequencies, change through stochastic replacement and mutation, yet converge to a stationary distribution. This distribution undergoes a qualitative transition at the \emph{critical mutation rate} $\mu_c=1/(2n)$: at $\mu=\mu_c$ it is exactly uniform, so that the probability of observing $k$ copies of allele~1 (and $n-k$ of allele~2) is $\pi(k)=1/(n+1)$ for $k=0,\dots,n$. For $\mu<\mu_c$ diversity is low: the stationary distribution places most of its mass near $k=0$ and $k=n$, and the population is therefore typically dominated by one allele. For $\mu>\mu_c$, on the other hand, diversity is high: the distribution concentrates around intermediate values, so that both alleles are commonly present at comparable frequencies. Recently, the two-allele Moran process was shown to be exactly equivalent to the voter model with two candidates and $\alpha_1$ and $\alpha_2$ committed voters (\emph{zealots}) in a population of $n+\alpha_1+\alpha_2$, where mutation is played by zealot influence. Here we extend this equivalence to multiple alleles and multiple candidates. Using the mapping, we derive the exact stationary distribution of allele counts for well-mixed populations with an arbitrary number $m$ of alleles, and obtain the critical mutation rate $\mu_c = 1/(m+2n-2)$, which depends explicitly on $m$. We then analyze the Moran process on randomly connected populations and show that both the stationary distribution and $\mu_c$ are invariant to network structure and coincide with the well-mixed results. Finally, simulations on general network topologies show that structural heterogeneity can substantially reshape the stationary allele distribution and, consequently, the level of genetic diversity.