Quantitative Analyses of Coupling in Hybrid Zones

Abstract

In hybrid zones, whether barrier loci experience selection mostly independently or as a unit depends on the ratio of selection to recombination as captured by the coupling coefficient. Theory predicts a sharper transition between an uncoupled and coupled system when more loci affect hybrid fitness. However, the extent of coupling in hybrid zones has rarely been quantified. Here, we use simulations to characterize the relationship between the coupling coefficient and variance in clines across genetic loci. We then reanalyze 25 hybrid zone data sets and find that cline variances and estimated coupling coefficients form a smooth continuum from high variance and weak coupling to low variance and strong coupling. Our results are consistent with low rates of hybridization and a strong genome-wide barrier to gene flow when the coupling coefficient is much greater than 1, but also suggest that this boundary might be approached gradually and at a near constant rate over time.

Figure 1.

Coupling and its consequences. (A) The coupling coefficient θ = s/r determines the extent to which barrier loci operate independently or as a unit, depicted here with a few example loci. As coupling (θ) increases the total selection (s*, represented by the size of the vertical arrows) experienced by barrier loci (red bars) and neutral loci (gray bars) increases because of increased linkage disequilibrium. With low coupling, geographic (B) and genomic (C) clines vary across the genome, whereas with high coupling geographic clines steepen and exhibit similar cline centers and widths and genomic clines converge to the genome-average admixture gradient (hybrid index). Geographic and genomic clines at barrier loci and neutral loci are shown in red and gray, respectively, in B and C. Dashed lines in C denote the genome-average admixture gradient.

Figure 2.

Relationship between geographic clines and coupling in simulated hybrid zones. Panel A shows estimates of mean cline slope (μ_β) and the standard deviation (SD) in slopes (σ_β) from the simulated hybrid zones. Points are colored by the known coupling coefficient. Panels B–D show the relationship between the coupling coefficient (θ) and μ_β (B), σ_β (C), and the coefficient of variation (CV) for cline slopes (D). Points denote results from individual simulations and are colored based on the migration rate (m) between neighboring demes. The best fit line from polynomial regression is shown along with the corresponding coefficient of determination (r²).

Figure 3.

Relationship between genomic clines and coupling in simulated hybrid zones. Panel A shows estimates of the standard deviation (SD) for genomic cline center (σ_c) and slope (σ_v) from the simulated hybrid zones. Points are colored by the known coupling coefficient. Panels B and C show the relationship between the coupling coefficient (θ) and σ_c (B) or σ_v (C). Points denote results from individual simulations and are colored based on the migration rate (m) between neighboring demes (see Fig. 2). The best fit line from polynomial regression is shown along with the corresponding coefficient of determination (r²).

Figure 4.

Summary of genomic clines empirical hybrid zones. The plots show estimated genomic clines (gray lines) for each of 25 hybrid zones. Each cline denotes the probability of ancestry for species 2 as a function of hybrid index (the total proportion of the genome inherited from species 2). Clines for 100 randomly chosen loci (or all loci if there were fewer than 100) are shown. The dashed one-to-one line denotes an ancestry probability equal to hybrid index. Our estimate of the coupling coefficient based on the variability among clines is reported.

Figure 5.

Cline variances and coupling in empirical hybrid zones. Panel A shows estimates of the standard deviation (SD) for genomic cline center (σ_c) and slope (σ_v) from the simulated (light gray points) and empirical (colored points) hybrid zones. Coupling coefficients for the 25 empirical hybrid zones were estimated from a linear regression model fit with the simulated hybrid zone data. Panel B plots the estimated coupling coefficients (θ) for the empirical hybrid zones sorted from smallest to largest. The horizontal line denotes θ = 1. Panels C and D show estimates of the SD for genomic cline center (σ_c) and slope (σ_v) for each of the 25 empirical hybrid zone data sets, here sorted from the largest (high variability among clines) to smallest (low variability among clines). These data suggest a continuum of estimated coupling coefficients (B) and cline parameters SDs (C,D).