Recombination Rate Modifiers under Stochastic Transmission

TL;DR

It is demonstrated that temporal variability in transmission constitutes an independent and qualitatively distinct force in the evolution of recombination rates.

Abstract

The Reduction Principle states that, near a stable equilibrium under fixed viability selection, a selectively neutral modifier allele that reduces recombination rate among selected loci is favored, whereas one that increases recombination rate is eliminated. This result assumes constant transmission parameters across generations, so that invasion is determined by the dominant eigenvalue of a single transmission-selection matrix. Here we analyze a minimal departure from this framework. In a diploid model, two loci experience symmetric multiplicative viability selection and a third, neutral locus modifies their recombination rate. All parameters are fixed except that recombination in modifier heterozygotes varies randomly across generations according to a stochastic process. When the recombination rate in modifier heterozygotes is constant, the Reduction Principle holds exactly: invasion occurs if the rare modifier allele reduces recombination relative to the resident rate. When recombination varies randomly across generations, invasion is governed by the top Lyapunov exponent of a product of random matrices. We show that temporal variation in recombination rate alone, in the absence of fluctuating viability selection, can reverse the direction of selection on the modifier locus predicted by the deterministic model. The mean recombination rate is insufficient to determine invasion of $M_2$; instead, outcomes depend on the full distribution of recombination rates and their ordered accumulation across generations. Parameters that affect only the magnitude of selection under constant transmission - including resident recombination, selection strength, and background linkage - can alter its sign under stochastic transmission. These results demonstrate that temporal variability in transmission constitutes an independent and qualitatively distinct force in the evolution of recombination rates.

Figures (2)

• Figure 1: Top Lyapunov exponent $\gamma$ governing invasion of a rare recombination modifier allele under temporally varying recombination in $M_1M_2$ heterozygotes. In each generation, $r_{12,t}$ is drawn i.i.d. from a scaled Beta distribution on $[0,\tfrac{1}{2}]$. The vertical axis shows the mean $\mathbb{E}[r_{12,t}]$ (restricted to $\mathbb{E}[r_{12,t}]\le r_{11}$), and the horizontal axis shows the variance $\mathrm{Var}(r_{12,t})$; blank cells indicate infeasible mean--variance combinations for this distribution. Colors denote the magnitude and sign of $\gamma$, with deeper orange indicating larger positive values (stronger invasion), deeper blue indicating more negative values (stronger elimination), and lighter shades indicating weaker effects. The black line marks $\gamma=0$. Parameters are $s=0.8$, $r=0.12$, and $r_{11}=0.05$, for which the resident polymorphism is locally stable with $\hat{D}\neq0$ and $\hat{\bar{w}}=0.47$.
• Figure 2: Distributional effects on stochastic invasion of a recombination modifier.Top row: Heatmaps of the Lyapunov exponent $\gamma$ as a function of $\mathbb{E}[r_{12,t}]$ and $\mathrm{Var}(r_{12,t})$ for three distributions supported on $[0,\tfrac{1}{2}]$ (scaled Beta, truncated Gamma, truncated LogNormal; truncated distributions are moment-matched in the truncated mean and variance). Colors indicate invasion – more orange ($\gamma>0$) – or extinction – more blue ($\gamma<0$); the black line marks $\gamma=0$. Blank cells correspond to infeasible mean--variance pairs. Other parameters are fixed at $s=0.8$, $r_{11}=0.05$, and $r=0.12$. Bottom row: Example truncated distributions with matched mean $\mathbb{E}[r_{12,t}]=0.040<r_{11}$ and (approximately) maximal feasible variance within each distribution. Shaded regions indicate probability mass. Vertical lines mark $\mathbb{E}[r_{12,t}]$ (black) and $r_{11}$ (red dashed). The top-left labels reports the tail probability $P(r_{12,t}>r_{11})$ estimated from i.i.d. samples.