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

TL;DR

We show that a neutral multi-allele Moran process with mutation maps exactly to a multi-candidate voter model with zealots on fully mixed populations, yielding a Dirichlet–multinomial stationary distribution with $α_i = rac{2(n-1)u_i}{1-\Lambda}$ and a critical mutation rate $u_c = rac{1}{2n-2+m}$. This framework extends to randomly connected populations via an ER mean-field reduction, where zealot counts become $α_i^{ ext{ER}}=rac{2d u_i}{1-\Lambda}$ and effective counts satisfy $α_i^{ ext{eff}}=(n-1)/d \, α_i^{ ext{ER}}$, preserving the Dirichlet–multinomial form at leading order. The stationary distribution is thus robust to network structure under mean-field assumptions, though degree fluctuations and spatial locality modulate diversity (with second-order corrections providing closer matches on heterogeneous graphs). Simulations across diverse topologies reveal that degree heterogeneity tends to suppress diversity while local interactions on rings promote it, yet the Moran–voter correspondence remains a guiding principle for understanding multisite neutrality across architectures.

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} $μ_c=1/(2n)$: at $μ=μ_c$ it is exactly uniform, so that the probability of observing $k$ copies of allele~1 (and $n-k$ of allele~2) is $π(k)=1/(n+1)$ for $k=0,\dots,n$. For $μ<μ_c$ 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 $μ>μ_c$, 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 $α_1$ and $α_2$ committed voters (\emph{zealots}) in a population of $n+α_1+α_2$, 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 $μ_c = 1/(m+2n-2)$, which depends explicitly on $m$. We then analyze the Moran process on randomly connected populations and show that both the stationary distribution and $μ_c$ 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.

Figures (7)

• Figure 1: Simplex depicting the set of states $(k_1,k_2,k_3)$ with $k_1+k_2+k_3=n$. Each point in the simplex corresponds to frequencies $(v_1,v_2,v_3) = (k_1/n,k_2/n,k_3/n)$. The corners represent all individuals of type 1 (upper corner), or type 2 (left corner) or type 3 (right corner). An internal point has frequencies corresponding to the length of the line drawn from the point to the sides as indicated.
• Figure 2: Heat-maps comparing the exact Dirichlet--multinomial distribution with the distribution obtained from $10^7$ samples generated from the exact distribution for 3 alleles and $n=201$. The critical mutation probability (used in the corresponding Moran model) is $u_c=1/(2n+1)$ (middle row), corresponding to $\alpha_c=1$. Top row: $u = 1/(4n-1) < u_c$, corresponding to $\alpha=0.5$; Bottom row: $u = 1/(n+2) > u_c$, corresponding to $\alpha=2.0$.
• Figure 3: Simulations on Erdős--Rényi graphs with connection probability $p=0.3$. The critical mutation probability is $u_c=1/(2n+1)$ (middle row), corresponding to $\alpha_c=1$. Top row: $u = 1/(4n-1) < u_c$ and $\alpha=0.5$; Bottom row: $u = 1/(n+2) > u_c$ and $\alpha=2.0$. Columns show the mean-field prediction, and the simulated stationary distributions for the Moran and voter models. Color bar is the same as in Fig. \ref{['fig:exact']}.
• Figure 4: Simulations on Barabasi--Albert scale-free graphs with parameters $m_0=m=3$. The critical mutation probability is $u_c=1/(2n+1)$ (middle row), corresponding to $\alpha_c=1$. Top row: $u = 1/(4n-1) < u_c$ and $\alpha=0.5$; Bottom row: $u = 1/(n+2) > u_c$ and $\alpha=2.0$. Columns show the mean-field prediction, the corrected mean-field approximation (Appendix \ref{['app:second-order-degree']}), and simulations for the Moran and voter models, respectively. For the voter model the values of $\alpha$ were rescaled by $d/(n-1)$. Color bar is the same as in Fig. \ref{['fig:exact']}.
• Figure 5: Simulations for 3 alleles on a simple ring graph, where each node is connected to the 2 nearest neighbors. The critical mutation probability is $u_c=1/(2n+1)$ (middle row) corresponding to $\alpha_c=1$. Top row: $u = 1/(4n-1) < u_c$ and $\alpha=0.5$; Bottom row: $u = 1/(n+2) > u_c$ and $\alpha=2.0$. Columns show the mean-field prediction, and the simulated stationary distributions for the Moran and voter models. Color bar is the same as in Fig. \ref{['fig:exact']}.