The geometry and genetics of hybridization

Abstract

When divergent populations form hybrids, hybrid fitness can vary with genome composition, current environmental conditions, and the divergence history of the populations. We develop analytical predictions for hybrid fitness, which incorporate all three factors. The predictions are based on Fisher's geometric model, and apply to a wide range of population genetic parameter regimes and divergence conditions, including allopatry and parapatry, local adaptation, and drift. Results show that hybrid fitness can be decomposed into intrinsic effects of admixture and heterozygosity, and extrinsic effects of the (local) adaptedness of the parental lines. Effect sizes are determined by a handful of geometric distances, which have a simple biological interpretation. These distances also reflect the mode and amount of divergence, such that there is convergence toward a characteristic pattern of intrinsic isolation. We next connect our results to the quantitative genetics of line crosses in variable or patchy environments. This means that the geometrical distances can be estimated from cross data, and provides a simple interpretation of the "composite effects." Finally, we develop extensions to the model, involving selectively induced disequilibria, and variable phenotypic dominance. The geometry of fitness landscapes provides a unifying framework for understanding speciation, and wider patterns of hybrid fitness.

Keywords: Fisher's geometric model; hybrid fitness; line crosses; quantitative genetics; speciation.

Figure 1

Under Fisher's geometric model, each genotype is associated with the values of $n$ quantitative traits (illustrated with $n=2$), and its fitness depends on the distance of this phenotype from an optimum. The position of the optimum can change over time and space. Shown are two parental lines, P1 and P2, which differ by $d=7$ substitutions, each represented by a vector, denoted ${\mathbf{m}}_j$ for $j=1..d$. These vectors represent the effects of the P2 allele, whether this is derived or ancestral. But they are ordered so that the chain passes through the most recent common ancestor (MRCA) of P1 and P2. Also shown are cartoons of the parental genomes, with the derived alleles as colored boxes. Hybrid genomes contain a mix of the parental alleles. In the hybrid shown, 1/7 of the divergent sites contains an allele from each line, so that $p_{12}=1/7$; and two further P2 alleles are present as homozygotes (one ancestral and one derived), yielding a hybrid index of $h=2.5/7$. The illustration shows that fixed differences can be physically linked. Such linkage reduces the variance in the hybrid indexes within a given cross, but plays no other role in our analyses.

Figure 2

Predictions for hybrid fitness depend on a small number of geometric quantities. The distances are defined in some multi‐dimensional trait space, but are estimable, in principle. (A) With additive phenotypes, and a single environmental optimum, predictions depend on just three distances: the distances of the two parental phenotypes from the optimum ($r_{1\mathrm{O}}^2$ and $r_{2\mathrm{O}}^2$), and the distance between the parental phenotypes ($r_{12}^2$). Results can also be written in terms of $\rho$, which measures the extent to which the parental populations are maladapted to the current environment in similar ways. Also shown is the midparental phenotype, P. This is the expected phenotype of balanced hybrids, and so hybrid advantage is maximized when P coincides with the optimum. (B) With two environments, A and B, characterized by different optima, one measure of local adaptation is ${\rho }^{\ast }$: the cosine similarity between the vectors linking the optima, and the parental phenotypes. When the two parental phenotypes are close to the two optima, results depend on $r_{12}^2$ alone. (C) With variable phenotypic dominance, results depend on the phenotype of the global heterozygote, G, which is equivalent to the initial F1 cross under strictly biparental inheritance and expression, and which may differ from the midparental phenotype, P. In the example shown, G is closer to the P2 phenotype than the P1 phenotype, this implies directional dominance, with P2 alleles being dominant on average.

Figure 3

(A)–(D) the distance $r_{12}^2$ can vary systematically with the mode of divergence between the parents. The variation depends on the chain of $d$ substitutions that differentiate the parental lines, and compares their trajectory to a random walk with the same number of steps, and distribution of effect sizes. (A) When substitutions form a more‐or‐less direct path between the parental phenotypes, the observed phenotypic difference is greater than would be predicted under a comparable random walk; this implies that $r_{12}^2>4$ with a maximum at $\max (r_{12}^2)=4d$. (B) When the true path of divergence really did resemble a random walk, then $r_{12}^2\approx 4$ is expected. This might happen if stabilizing selection on the phenotype was ineffective, or if the optimum value wandered erratically. Systematically smaller values of $r_{12}^2$ are predicted under two conditions. Either (C) genomic divergence continued, despite effective stabilizing selection on the phenotype, leading to “system drift.” Or (D) populations successfully tracked environmental optima, but without leading to a straight path of substitutions. (E)–(G) $r_{12}^2$ also plays a key role in determining patterns of hybrid fitness, especially with locally adapted parents. Results are shown for the standard crosses, in three different environments, where the optimal phenotype coincides with: (E) the P1 phenotype, (F) the P2 phenotype, and (G) the midparental phenotype. Color shows variation in $r_{12}^2$, including the inflection points at $r_{12}^2=4/3$ (equal fitnesses for the F1 and fitter backcross), and $r_{12}^2=4/7$ (equal fitnesses for the less fit parent and the less fit backcross). All results use equation (17), with $\mathrm{Var}(h)=0$, and the appropriate values of $\overline{h}$ and ${\overline{p}}_{12}$ for each cross; but results are shown on an arbitrary scale, such that fitter hybrids are higher on the plots.

Figure 4

The outcomes of hybridization over time and space. (A) A cartoon of the divergence process that was simulated, with two populations adapting in allopatry to abruptly shifting optima, and then continuing to accumulate divergence via system drift. (B)–(E) change in the composite effects with increasing divergence (Table 1), as measured with respect to (B) both parental environments, or (C)–(E) other single environments. (F)–(O) results for simulated hybrids, plotting $r_{\mathrm{HO}}^2$ on a reversed axis, such that fitter genotypes are higher. Points with error bars show the mean and 95% quantiles for 10,000 recombinant hybrids, generated for the reciprocal backcrosses, and the $\mathrm{F}2$. The dark central point (labeled Fn) shows the mean of 10,000 homozygous hybrids, derived from automictic selfing among F1 gametes. Red and blue lines show analytical predictions. These use equation (17), with the measured values of $r_{1\mathrm{O}}^2$ and $r_{2\mathrm{O}}^2$, and the assumption that $\rho =-1$ (for the “intermediate” environment) or $\rho =0$ (all other cases). Hybrids were scored in the (F)–(J) early stages of divergence ($d=100$; red lines), and (K)–(O) at later stages ($d=2000$; blue lines). Simulation procedure is described in Supporting Information Appendix 2, and used the following parameters: $N=1000$, $N\overline{s}=-10$, $2NU=1$, $n=2$, $k=2$, free recombination, and “bottom‐up” mutations.

Figure 5

Two extensions to the model, incorporating selection on early‐generation hybrids, and variable phenotypic dominance. Plots show the distance from the optimum, $r_{\mathrm{HO}}^2$, on a reverse axis, so that higher points are fitter. After the initial F1 cross, we simulated either random union of gametes among the hybrids (F2–F5), or repeated backcrossing to parental line P1 (BC1–BC4). For the later crosses, we chose parents either wholly at random (black points and lines), or with a probability proportional to their fitness (asterisks and red points and lines). In each case, results for 10,000 simulated hybrids (mean and 95% quantiles), are compared to analytical predictions. Blue lines show predictions that ignore the effects of selection on earlier hybrids (eq. 24 with $\beta =0$); red lines show predictions assuming that only optimally‐fit parents reproduce (eq. 24 with $\beta =1/2$). (A) results with quadratic selection (eq. 3 with $k=2$) such that there is limited variation in parental fitness. (B) results with truncation‐like selection (eq. 3 with $k=6$), and high variance in parental fitness. (C) and (D) equivalent results, when populations were simulated with variable phenotypic dominance (such that heterozygous effect of each new mutation was the homozygous effect, multiplied by a uniformly distributed random variable). The clearest consequence is that the F1 are suboptimal, even when the parental lines are optimal. Here, predictions use equations (27) and (28) with the observed $r_{\mathrm{GO}}^2$, and $v=1/12$, from the variance of a uniform distribution. All predictions assumed optimally‐fit parents ($r_{.\mathrm{O}}^2=0$). Simulations are described in Supporting Information Appendix 2, and used the following parameters: $N=1000$, $N\overline{s}=-0.1$, $2NU=1$, $n=2$, $\overline{c}=0.5$, “bottom‐up” mutations, stationary optima matching the ancestral state; hybrids were formed as soon as one of the diverging populations had fixed 1000 substitutions.