Abstract
When isolated populations adapt to novel environments, they may do so using the same or different alleles. Use of the same alleles—parallel genetic evolution—is increasingly likely if populations have shared ancestral standing genetic variation. Here, we investigate the conditions under which populations undergo parallel genetic evolution from standing variation and the associated implications for speciation via environment-specific hybrid fitness. Using computer simulations, we find that populations adapting to identical environments tend to fix the same alleles from standing variation unless there is a deficient or excess amount of variation in the founding population. We also find that the degree of parallel genetic adaptation decreases faster-than-linearly as selection deviates from fully parallel toward divergent. This rapid decrease in parallelism occurs because seemingly small environmental differences among populations correspond to steep reductions in the fraction of alleles that are mutually beneficial. Lastly, we show that adaptation from standing variation reduces the phenotypic segregation variance of hybrids under both parallel and divergent selection. Under parallel selection, parental populations fix the same alleles which do not segregate in hybrids. In addition, adaptation from standing variation proceeds via a greater number of alleles with individually smaller effects than when adaptation is from new mutation—this further reduces segregation variance even under divergent selection. This reduced segregation variance improves mean hybrid fitness when parents adapt to similar environments but reduces mean hybrid fitness under divergent adaptation. Therefore adaptation from standing variation forestalls speciation via parallel selection and promotes speciation via divergent selection.
Impact summary Among all of the questions that evolutionary biologists have posed, some of the most lingering concern the predictability of evolution. For example, if two groups of organisms independently evolve similar traits in similar habitats, is it likely that this evolution proceeded using the same genes? If so, what can we learn about the probability that these groups have evolved reproductive barriers causing them to become distinct species? We investigate these two questions from a theoretical perspective in our article and focus on the viability of hybrids as measured in the environments to which their parents are adapted. In particular, we investigate the role of standing genetic variation—the genetic variability that exists within a population at any given time—in causing populations to evolve using the same genes. We have three main conclusions. First, we find that the greatest parallel evolution results when there is a match between the quantity of standing genetic variation and the selective demands of the environment, and significant redundancy in the standing variation readily leads to non-parallelism. Our second conclusion is that parallel evolution from standing genetic variation is likely only when different populations adapt to very similar environments; small environmental differences largely preclude parallelism at the genetic level. Finally, whereas standing genetic variation decreases the likelihood of speciation when two populations adapt to similar environments, it increases the likelihood of speciation when populations adapt to different environments. In sum, our article indicates that adaptation from standing variation can have an important role for speciation that depends on whether populations adapt to similar or different environments.
Introduction
The process of adaptation involves positive natural selection on beneficial alleles that improve the fit of a population to its environment. Adaptation can proceed via selection on new mutations or on pre-existing standing genetic variation (Orr and Betancourt 2001). If populations experience divergent natural selection, which favours distinct phenotypes in different environments, then different alleles are expected to fix regardless of their origin. Parallel natural selection, by contrast, favours the same phenotypes and alleles in different populations. Under parallel selection, distinct populations might experience and fix alternative adaptive alleles by chance (Mani and Clarke 1990), but the probability of genetic parallelism is expected to be greater if populations share a common pool of ancestral standing variation (Schluter and Conte 2009; Conte et al. 2012). Here, we investigate how the availability of standing genetic variation for adaptation affects parallel genetic evolution—defined as reuse of the same alleles during independent bouts of adaptation—and the associated implications for speciation via reduced hybrid fitness.
Adaptation facilitates progress towards speciation when adapting populations evolve reproductive isolating barriers as a by-product. One reason these reproductive isolating barriers can arise is because genetic differences between populations reduce the fitness of their hybrids, thereby reducing gene flow upon secondary contact. Under ecological speciation driven by divergent selection (Schluter 1996), F1 hybrids might be intermediate in phenotype and unfit in both parental environments (e.g., Hatfield and Schluter 1999). Under an alternative process, populations diverge genetically by chance in the course of adapting to similar environments despite parallel natural selection (‘mutation-order’ speciation; Schluter 2009). In this case, distinct populations fix different adaptive alleles that—when combined—render hybrids less fit than either parent because they have novel phenotypes poorly suited to the parental environment. A central goal of research on speciation is to characterize the conditions that can facilitate and impede progress toward ecological and mutation-order speciation.
Adaptation from standing variation is common and is responsible for some of the most spectacular adaptive radiations found in nature (e.g., African cichlids; Brawand et al. 2015). Nevertheless, the implications of standing variation for the process of speciation have not been explored. We hypothesized that—besides speeding up the process—adaptation from standing variation would reduce the evolution of reproductive isolation under parallel selection because adapting populations would fix the same alleles and therefore fewer incompatibilities. We also hypothesized that standing variation would have less of an effect on genetic parallelism and the evolution of isolating barriers under divergent selection.
In this article, we address the following question: Under what conditions do populations undergo parallel genetic adaptation from standing variation and what are the associated implications for the evolution of reproductive isolation between them? We note that ‘divergent’ and ‘parallel’ natural selection—and their corollaries of ecological and mutation-order speciation—are useful concepts but are the endpoints of a continuum of possible environmental differences (Langerhans and Riesch 2013; Martin and Lenormand 2015; Stuart et al. 2017; Bolnick et al. 2018). Few theoretical studies have investigated patterns of parallel genetic evolution and speciation across environments of varying similarity, and our focus is largely on determining how patterns change across the continuum of parallel to divergent. We use individual-based simulations to investigate our research question and include analytical derivations to explain simulation results where relevant. Our results suggest that standing variation has important implications for speciation that vary across environments.
Methods
We used computer simulations to investigate parallel genetic adaptation and speciation from ancestral standing variation. Our simulations were carried out using the conceptual framework of Fisher’s (1930) geometric model of adaptation (reviewed by Tenaillon [2014]). Fisher’s model makes explicit and testable predictions that have received support in laboratory and field studies (MacLean et al. 2010; Rogers et al. 2012; Stearns and Fenster 2016). While originally developed to make inferences about the genetics of adaptation, the model has been adapted to study the evolutionary genetics of speciation (Barton 2001; Chevin et al. 2014; Fraïsse et al. 2016; Simon et al. 2017). We considered the case of a single ancestral population that founds two identical populations in novel environments, which then adapt without gene flow between them (i.e., allopatry; see Fig. 1A). After adaptation, populations are brought into secondary contact. This general colonization history is modelled around the process of adaptation as it can occur in nature, for example in insects with respect to their host-plants (‘micro-allopatry’, e.g., Drès and Mallet 2002), fish populations in post-glacial lakes (Bell and Foster 1994) and in birds or plants isolated within glacial refugia (e.g., Weir and Schluter 2004; Pettengill and Moeller 2012).
Genotype to phenotype
The phenotype of a haploid individual is represented by an m-dimensional vector, z = [z1, z2, …, zm], with m being the number of independent ‘traits’ (Orr 2000). Each trait value, zi, is determined by the summed effects of all underlying loci, which are initially fixed for alleles with an effect of 0 on all m traits. We focus on m = 2 in our numerical analyses, and present analytical derivations in higher dimensions in the relevant contexts to investigate how trait dimensionality (i.e., ‘organism complexity’; Orr 2000) affects our conclusions.
Life-cycle
We assume non-overlapping generations. Viability selection occurs at the beginning of each generation, during the haploid phase. The probability that an individual with phenotype z survives is a Gaussian function of its Euclidean distance from the optimum, ||z – o||, and the strength of selection, s: P = exp(−s||z – o||2/2) (equation 1; Fraïsse et al. [2016]). We note that some empirical studies support the use of Gaussian fitness functions (e.g., Martin and Lenormand [2006]). The simulation is ended if no individuals survive. If the number of survivors exceeds the carrying capacity, K, then survivors are randomly culled to a population size of K.
Haploids that survive viability selection then randomly conjugate (each with any individual but itself; i.e., self-incompatible) and undergo meiosis with free recombination between all loci. Each mating pair is a diploid individual that generates 2B haploid offspring, which each acquire a new mutation with probability μ. An infinite number of loci is assumed, such that each mutation arises at a new locus (‘infinite-sites’ sensu Kimura [1969]). For simplicity we assume that the effect of a mutation on each of the m traits is independently drawn from a normal distribution with a mean of 0 and SD of σ. Thus all mutations are unique (‘continuum-of-alleles’ sensuKimura [1965]). The effects of mutational covariance and modularity (restricted pleiotropy sensuChevin et al. [2010]) are left out but could be readily explored with the provided simulation code (Data accessibility). See Table 1 for descriptions of all parameters and values used in simulations.
Generating ancestral standing genetic variation
We initiate focal populations with standing genetic variation from a common ancestor. To generate ancestral standing variation, we conducted burn-in simulations of a large population (Kanc = 10,000) under weak stabilizing selection (sanc = 0.01) at the origin (oanc = [0, 0, … 0]) for 100,000 generations. All other parameters (e.g., mutation rate) were identical to adapting populations (Table 1). Ancestral populations reached mutation-selection balance and had, on average, a mean of 192 segregating alleles (Fig. S1A). Each segregating allele was present in, on average, 7.3 % of individuals (Fig. S1B).
Adaptation to a new environment
Following the burn-in we generated two initially-identical founder (i.e., parental) populations. The founder population size was equal to the carrying capacity (here 1000 individuals). Founder populations were fixed for alleles that were fixed in the ancestor, and the amount of standing variation was determined by sampling n segregating alleles from the ancestor. Populations adapted from only new mutation when n = 0 and otherwise contained some ancestral standing variation. Each of the n segregating alleles were selected for inclusion in the founder populations with a probability proportional to their frequency in the ancestor, and the expected frequency of an allele in a founding population was equal to its frequency in the ancestor. Specifically, each founding individual inherited a particular allele with a probability equal to its frequency in the ancestor. For each parameter combination, we began each simulation from a unique realization of the ancestor (i.e., distinct burn-in). This randomly generated population was duplicated and used to found both adapting populations.
After initialization, founder populations adapted to their respective specified phenotypic optima without any gene flow (Fig. 1B). Adaptation proceeded via the reassortment of ancestral standing variation (if n > 0) and via new mutation (probability of mutation, μ = 0.001; all simulations) simultaneously. Individuals gain new mutations with identical probability and properties as in the ancestor. There are two key features of the new phenotypic optima. The first is the initial distance to the optima, calculated as the Euclidean distance between the optima and the origin, d. More distant optima require a greater amount of phenotypic change and initially exert stronger selection. The second key feature is the angle of divergence, θ, which captures the difference in direction from the origin to the optima of the two populations (see Fig. 1B). Angle is used to describe the continuum of parallel to non-parallel evolution and is explicitly invoked in most metrics that quantify phenotypic parallelism (see Bolnick et al. [2018]). Optima differing by a small angle impose relatively parallel natural selection on founder populations, whereas optima whose direction differ by large angles impose divergent selection.
We ended simulations after 2000 generations, at which time all populations had reached their phenotypic optima (Fig. S2A) and mutation-selection balance (Fig. S2B). An unavoidable and important effect of standing variation is that it quickens adaptation because populations do not have to ‘wait’ for beneficial mutations to arise (Barrett and Schluter 2008). In our model and others like it (e.g., Barton [2001] & Chevin et al. [2014]), reproductive isolation evolves rapidly during the initial stages of adaptation and then decelerates as populations approach their optima. Accordingly, any reduction in hybrid fitness that evolves does so more rapidly when evolution occurs from standing variation. We allowed sufficient time in our simulations for all populations to reach their optima even when adaptation is from only new mutation. Populations continue to diverge indefinitely after reaching their optima, but this divergence is weak and constant across all simulations in our study. Therefore, our conclusions refer to equilibrium conditions rather than transient states and are unaffected by standing variation’s influence on the speed of adaptation.
Formation of hybrids and quantification of segregation variance & fitness
After the adaptation phase of simulations had ended, derived populations ‘met’ in secondary contact and produced hybrids. We randomly chose a haploid individual from each of the two populations to form a diploid (F1) that produce recombinant haploid hybrids through meiosis with free recombination among loci. If an allele was shared by both parents—which is possible only if it originated from the ancestral standing variation because new mutations are unique and thus cannot arise de novo in both populations—then the allele was present in the hybrid. Alleles present in only one parent—all alleles arising from new mutation during adaptation and alternative alleles fixed from the standing variation—were inherited by the hybrid offspring with probability 0.5 (i.e., fair Mendelian segregation). This process was repeated to create 100 hybrids between the pair of parental populations.
We calculated two quantities of interest in the hybrids: segregation variance and fitness. Segregation variance is the phenotypic variation of hybrids (Wright 1968; Slatkin and Lande 1994) and is calculated here as the mean phenotypic variance across all m traits. Higher segregation variance results when parents are differentiated by a greater number of alternative alleles (holding effect size constant) or alleles of individually-larger effect (holding number of alleles constant) (Slatkin and Lande 1994; Chevin et al. 2014). Segregation variance captures the phenotypic consequences of hybridization and is therefore more biologically meaningful than simply quantifying the number of different alleles that distinguish populations or their average effect size. Phenotypic variance in parental populations before hybridization is near zero and is not affected by the quantity of standing variation nor the initial distance to the optimum (Fig. S2C).
An individual hybrid’s fitness in a given parental environment was calculated from its phenotype in the same manner as the fitness of their parents (Fig. 1C). We quantified the fitness of each hybrid in both parental environments and recorded its final fitness as the larger of the two values. That is, we calculated the survival probability (equation 1) of each hybrid with respect to both optima, and an individual hybrid’s survival probability was the higher of the two values. This can be imagined as, for example, giving the hybrid a choice of alternative host-plants (Drès and Mallet 2002). Our fitness metric reflects that which is traditionally recognized as ‘extrinsic’ isolation, and explicitly considers epistasis for fitness between alleles at different loci. Accordingly, two alleles that reduce fitness when combined in a hybrid can be considered Bateson-Dobzhansky-Muller incompatibility loci (Bateson 1909; Dobzhansky 1937; Muller 1942; Chevin et al. 2014; Fraïsse et al. 2016) with environment-specific effects on fitness (see Arnegard et al. [2014], Schumer et al. [2014], and Ono et al. [2017] for discussion of environment-specific incompatibilities).
Results and discussion
Effects of standing variation on the segregation variance in hybrids following parallel and divergent selection
We first investigated how the quantity of standing variation in the founding populations affects the segregation variance in hybrids (hereafter simply ‘segregation variance’). To investigate these dynamics, we ran simulations in which populations underwent either parallel (identical optima, θ = 0°) or divergent (opposite optima, θ = 180°) evolution, and repeated this with different amounts of ancestral standing variation (via n). A representative set of simulations is plotted in Figure 2A.
Several conclusions emerge from these simulations. First, segregation variance is consistently high when populations undergo divergent (θ = 180°) adaptation. This is because alleles that are beneficial in one environment are deleterious in the other. Thus populations fix alternative adaptive alleles regardless of whether they originate in the standing variation or arise from new mutation. This is true for any dimensionality, m. Second, under parallel evolution (θ = 0°), standing variation generally reduces segregation variance. This is largely because the same alleles are fixed in both populations and therefore do not segregate in hybrids. Third, segregation variance under parallel selection is minimized when populations are founded with intermediate quantities of standing variation. When there is little or no standing variation, populations undergo non-parallel genetic evolution due to their reliance on alternative new mutations. When there is substantial standing variation, there are many redundant alleles in the standing variation and populations again fix alternative beneficial variants. The quantity of standing variation that minimizes segregation variance under parallel selection (arrow in Fig. 2A) is greater for more distant optima (Fig. 2B) because a greater number of mutations are required for adaptation.
It has been widely suggested in the literature that parallel genetic evolution is most likely when populations are founded from a common ancestor with the same pool of standing variation (Conte et al. 2012). Our results largely confirm this intuition because the simulations indicate that the populations adapted using the same alleles under parallel selection. For maximal parallel evolution, however, there must be a match between the supplied standing variation and the demands of the environment. Significant redundancy in the standing variation can lead to non-parallel genetic evolution from standing variation even when optima are identical.
Parallel genetic evolution across environments
To investigate the relationship between the angle of divergence (i.e., θ) and segregation variance, we conducted simulations across a range of angles between 0° and 180°. In this section, simulations had either no standing variation (i.e., only de novo mutation) or the amount of standing variation that minimized segregation variance under parallel selection (θ = 0°; arrow in Fig. 2A). We plot a representative set of simulations in Figure 3A.
Several key results emerge from these simulations. First, segregation variance is not affected by the angle of divergence when there is no standing variation (n = 0 in Fig. 2; light green line and points in Fig. 3A). This occurs because all new mutations are unique and arise at different loci in our simulations. Second, when there is standing variation, segregation variance increases faster-than-linearly with the angle of divergence (compare dark green and black lines in Fig. 3A). This result is partially caused by the specific geometric properties of angles: a given change in θ causes a larger change in the Euclidean distance between optima, δ, when θ is smaller (Fig. S3A; see also Fig. S3B for same data as in Fig. 3A plotted with δ on x-axis) than when θ is large.
The geometric properties of adaptation also contribute to the rapid decrease in genetic parallelism as the angle of divergence increases. During adaptation, a given mutation is only expected to fix in two allopatric populations if it is beneficial in both. For a given population, beneficial mutations bring populations closer to the centre of a hypersphere (circle when m = 2) centered at the optimum with the adapting population a point on the surface (Fisher 1930; see cartoon inset of Fig. 3B). The radius of the hypersphere is equivalent to the initial distance to the optimum, d. Considering two populations, a given mutation is mutually beneficial if it its phenotypic effect brings a population’s phenotype into in the intersecting volume of the two hyperspheres (pink region in Fig. 3B inset). The expected degree of non-parallelism is therefore captured by the fraction of mutations that are beneficial in one population but deleterious in the other (e.g., for ‘red’ population in Fig 3B inset: area of the red region divided by the area of the pink and red regions combined). That is, the alleles favoured by selection in only one of two populations are those with phenotypic effects leading into the non-overlapping regions of the hyperspheres. Analytically (see Appendix), we show that this ‘fraction of non-overlap’ is zero when θ= 0°, one when θ= 180°, and increases faster-than-linearly with θ (Fig. 3B; see also Fig. S3A for the relationship between θ, δ, and the fraction of non-overlap). This fraction of non-overlap increases even more rapidly with θ in higher dimensions (compare solid line to dashed in Fig. 3B). Because the fraction of non-overlap is independent of the distance to an optimum (see Appendix), this pattern is expected to remain constant over the course of an adaptive ‘walk’ (sensu Orr [1998]) and regardless of whether populations have the same or different initial phenotype.
In addition, the average probability of fixation for mutually beneficial mutations decreases with the angle of divergence. This is intuitive: when θ = 0°, the region of overlap contains mutations that are highly beneficial in both populations (e.g., their shared optima). As θ increases, the mutations remaining in the region of overlap have more deleterious pleiotropic consequences and are thus less mutually beneficial.
We note that the above arguments reflect the probability of parallel genetic evolution generally and not from standing variation specifically. Therefore, populations adapting to similar environments have a higher probability of fixing similar mutations whether the mutations arise de novo or were segregating in the ancestor. The arguments above are thus component pieces of a comprehensive theory of the probability of genetic parallelism across environments during adaptation without gene flow. Integrating the processes discussed above with estimates of both the distribution of fitness effects of new mutations (Eyre-Walker and Keightley 2007) and the fitness effects of mutations across environments (Martin and Lenormand 2015) will be valuable.
The simulations depicted in Fig. 3 also illustrate an unanticipated general effect of standing variation: a reduction of segregation variance for all angles of divergence. This is most apparent by examining the right side of Fig. 3A. When selection is divergent, θ = 180°, there are no beneficial alleles shared between adapting populations, yet segregation variance is lower than when adaptation occurs only by new mutations. The reason standing variation reduces segregation variance in our simulations is that past stabilizing selection in the ancestor changes the effect sizes of alleles during adaptation by removing large effect mutations (see Fig. 1C). Supporting this hypothesis, we conducted a separate set of simulations initiated with standing variation but without past selection and found that segregation variance under divergent selection was the same as when adaptation was from new mutation only (Fig. S4). In the absence of large-effect mutations, adaptation from standing genetic variation proceeds via a greater number of alleles of relatively small effect compared to adaptation from new mutation. This, in turn, reduces segregation variance because individuals recover a more intermediate phenotype as a greater number of smaller effect alleles contribute to character divergence (Castle 1921; Barton et al. 2017). In addition to past selection removing large-effect alleles, small-effect alleles are more likely to be used in adaptation when present in the standing variation because they are less likely to be lost due to drift (Orr and Betancourt 2001; Hermisson and Pennings 2005). Large-effect alleles are more costly when dimensionality is greater (Orr 2000) and therefore small-effect alleles in the standing variation should further reduce segregation variance in higher dimensions.
Effect of standing variation on environment-specific hybrid fitness
In this section, we use data from the simulations in the previous section (see Fig. 3A) to evaluate the effect of standing variation on hybrid fitness. Specifically, we focus on differences in mean hybrid fitness between simulations that were initiated with and without standing variation. For an individual simulation, mean relative hybrid fitness was calculated as: [mean probability of an individual hybrid surviving] / [probability of an individual parent surviving]. The results of these simulations with respect to mean fitness are plotted in Figures 4 A & B.
The simulations reveal that standing genetic variation improves mean hybrid fitness when parental populations adapt to similar optima, but reduces hybrid fitness when parents undergo divergent adaptation. This result is caused by environment-specific effects of segregation variance on hybrid fitness (Fig. 4C). When the hybrid phenotype distribution is centred at the phenotypic optimum, as it is under parallel selection (i.e., θ = 0°), segregation variance is deleterious and the use of shared standing variation causes hybrids to have uniform phenotypes that resemble both parents. By contrast, when selection is divergent hybrids fall in a fitness valley between parental optima and some segregation variance is beneficial for mean hybrid fitness. This is because phenotypic variation in hybrids causes some individuals on the tails of the phenotype distribution to approach an optimum (see Fig. 1C). (Maximum hybrid fitness is not affected by standing variation under parallel selection and is reduced by standing variation under divergent selection [Fig. S5]).
These results indicate that, relative to when adaptation is from only new mutation, the effect of standing variation is to forestall the accumulation of reproductive isolation in response to parallel natural selection and facilitate the evolution of reproductive isolation in response to divergent natural selection. Standing variation necessarily speeds up adaptation (Fig. S2; see also Fig. S6 for plots of segregation variance over time) and so by necessity the evolution of isolating barriers occurs more quickly. Our conclusions about the effects of standing variation on hybrid fitness are reported at mutation-selection balance when populations are locally-adapted. Therefore our results may not hold for cases where populations are still experiencing strong selection in their new environment. Our conclusions about mutation-order speciation do not change in higher dimensions. Under divergent selection, however, the amount of segregation variance that maximizes mean hybrid fitness declines as dimensionality increases because the number of ways hybrids can ‘go wrong’ (Orr 2000) increases (see online Mathematica notebooks for approximations in higher dimensions for mean hybrid fitness under parallel and divergent selection). Maximum hybrid fitness, by contrast, always increases with higher segregation variance under divergent selection.
Additional factors influencing the probability of parallel genetic evolution
Several processes that we did not model could affect the probability of parallel genetic evolution and the evolution of hybrid fitness. First, we assumed there was no recurrent mutation at the same locus (e.g., Chan et al. 2010). Recurrent mutation would not qualitatively change our results but would increase genetic parallelism even without standing variation. Second, we assumed no gene flow between populations. Gene flow is expected to reduce the likelihood that populations fix alternative alleles under parallel natural selection (Nosil and Flaxman 2011; Anderson and Harmon 2014) and also impede local adaptation in response to divergent selection (Nosil 2012). Third, we assume that the ancestral standing variation is random with respect to the new environment. In some cases, however, alleles that are beneficial in a particular derived habitat may seed into—and be maintained within—the ancestral population via a balance between regular introgression and selection (Schluter and Conte 2009). Such variants could be maintained at a higher frequency in the ancestral populations than those maintained by mutation selection balance. When founders bring such variation to a new environment that is similar to that of the source populations, the effects of such ‘transported’ alleles are no longer random with respect to the direction of selection but instead are highly likely to be beneficial and evolve in parallel (see Nelson and Cresko 2018).
Fourth, we assumed that distinct populations were founded from the same ancestor and hence had access to an identical pool of standing variation. If the ancestral population is structured, such that new populations are founded with different subsets of ancestral standing variation, we expect reduced genetic parallelism under parallel selection compared to the case where founder populations are identical (Conte et al. 2015). Differences in allele frequencies between ancestral populations also reduces parallel genetic evolution because for a given effect size mutations at highest frequencies are most likely to be used for adaptation (MacPherson and Nuismer 2017). Founder events are therefore expected to lead to greater genetic divergence between populations undergoing parallel natural selection.
The interpretation of our simulation results has several additional caveats. In particular, we considered only haploid selection and strict additivity of allelic effects on phenotypes— future work should investigate the effects of ploidy on parallel genetic evolution and on speciation from standing variation. We also assumed that the sole fitness optima available to hybrids are those that the two parents are adapted to. Hybrids often perform best when there is an additional adaptive peak not being exploited by the parental forms (Rieseberg et al. 1999). Last, we only considered environment-specific hybrid fitness rather than ‘intrinsic’ isolating barriers. Intrinsic isolating barriers are a function of genetic divergence among populations (Orr 1995). Since segregation variance in our study directly reflects genetic divergence between populations, our results of segregation variance across environments (Fig. 3) could have implications for the evolution of intrinsic barriers.
Implications for studies of parallel evolution
Our results predict that subtle differences in the environments to which populations are adapting can substantially preclude parallel genetic evolution. This is because seemingly small differences in habitat similarity (via θ) greatly reduce the quantity of alleles that are expected to be beneficial in both populations. Even for cases termed ‘parallel’, there can be a wide range of phenotypic variation among replicates (for quantitative summary in fishes see Oke et al. 2017). If researchers wrongly assume that environments or phenotypes are similar or even identical, they may be misled into attributing genetic non-parallelism to various biological mechanisms when a better explanation is that the allele is not mutually beneficial. This rapid loss of shared beneficial alleles is especially pronounced for cases with high dimensionality. Dimensionality is difficult to quantify but estimates are typically between 1 and 10 for bacteria (Tenaillon 2014). In complex organisms (e.g., vertebrates) dimensionality is expected to be higher and therefore genetic parallelism should be less common.
Conclusion
Our analyses confirm the intuition that allopatric populations adapting from a common pool of ancestral standing genetic variation tend to undergo parallel genetic evolution in response to parallel natural selection. We show that the probability of parallel genetic evolution rapidly decreases as selection tends from parallel toward divergent and that this reduction in genetic parallelism occurs most quickly in complex organisms. Parallel genetic adaptation from standing variation reduces the segregation variance of hybrids under parallel selection. We also find that adaptation from standing variation leads to reduced segregation variance regardless of the angle of divergence because adaptation proceeds via more alleles of individually smaller effect. This reduced segregation variance further improves mean hybrid fitness under parallel natural selection but reduces mean hybrid fitness under divergent selection. In sum, adaptation from standing genetic variation can affect the evolution of reproductive isolating barriers under both parallel and divergent natural selection and this is mediated by its effects on the segregation variance of hybrids.
Author contributions
K.A.T. and D.S. developed the original ideas upon which the paper is based. K.A.T. wrote the first draft of the manuscript with input from M.M.O. and D.S., and all authors contributed to subsequent revisions. M.M.O. wrote the simulations and analytical derivations with input from K.A.T. K.A.T. performed, optimized, and analyzed the simulations, with input from M.M.O.
Data accessibility
Python (version 3.6.1) scripts and resulting data, R (version 3.4.1) scripts to process and plot the simulated data, and a Mathematica notebook (version 9; and PDF copy) to derive analytical results will be archived on Dryad. For now, these are hosted on GitHub (https://github.com/Ken-A-Thompson/SVS).
Acknowledgements
Discussions with S. Otto motivated and refined the approach used in this study. Feedback from S. Arnold, M. Chapuisat, L. Chavarie, R. Holzman, A. MacPherson, S. Otto, L. Rieseberg, J. Rolland, M. Urquhart-Cronish and R. Yamaguchi improved the manuscript. K.A.T. was funded by The University of British Columbia, the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Izaak Walton Killam Memorial Fund for Advanced Studies. M.M.O. was funded by The University of British Columbia. D.S. was funded by the Canada Foundation for Innovation, Genome BC, and NSERC.
Appendix 1
Here we outline an explanation for why segregation variance increases faster-than-linearly with the angle of divergence, θ (Fig. 3A). In our simulations, non-zero segregation variance after hybridization can only arise from non-parallel genetic evolution in the parental populations. Therefore, this explanation is equivalent to explaining why the extent of non-parallel genetic adaptation increases faster-than-linearly with the angle of divergence.
Our explanation focuses on the extent of phenotypic space wherein mutations are mutually beneficial, that is, improve the fitness both adapting populations in their respective environments. At the time of founding both adapting populations are expected to have the same mean phenotype, which is the mean ancestral phenotype. Mutations that move this ancestral mean phenotype into the region that has higher fitness in both parental environments are thus mutually beneficial (at least when phenotypes are sufficiently clustered around the mean). The region of phenotypic space that has higher fitness than the mean in one environment is a hypersphere (of dimension m), centered on the optimum with a radius equal to the distance between the mean phenotype and the optimum, d. A second hypersphere describes the phenotypic space that has higher fitness than the mean phenotype in the other parental environment. The region that is mutually beneficial is then the intersection of two hyperspheres, which is the union of two hyperspherical caps.
Fortunately, the volume of a hyperspherical cap is known for any dimension, m (Li 2011). It depends on the dimensionality, the radii of the two hyperspheres, and the distance between their centers. In our case the radii are equivalent and equal to d and the distance between the two centres is = d [2(1 – Cosθ)]1/2. Thus, the amount of phenotypic space that is mutually beneficial can be written as a function of m, d, and θ. The amount of phenotypic space that is beneficial in a given environment is simply the volume of one of the hyperspheres. Dividing the volume of the mutually-beneficial space by the volume of the space beneficial in a given environment gives the fraction of beneficial mutations which are mutually beneficial Ix[(1 + m)/2, 1/2] (A1) where Ix[a, b] is the regularized incomplete beta function (Equation 6.6.2 in Abramowitz and Stegun [1972]) and here x = Cos(θ/2)2. Eq. A1 depends on only m and θ, i.e., it is independent of the distance from the ancestor to the new optima, d. We refer to Eq. A1 as the fraction of overlap, and one minus Eq. A1 as the fraction of non-overlap.
The fraction of non-overlap (one minus Eq. A1) exhibits a faster-than-linear increase with θ for all values of m> 0, and the increase is faster for greater values of m (Fig. 3B). Thus, if standing genetic variation was uniformly distributed throughout the beneficial hyperspheres, the percent of segregating beneficial mutations that were beneficial, and thus expected to fix, in only one adapting population would increase faster-than-linearly with the angle of divergence.
The above analysis considers only the very onset of adaptation, when the two adapting populations have the same mean phenotypes, such that the fraction of phenotypic space that is beneficial in one population that is also beneficial in the other population (call this X) is equivalent to the fraction of beneficial mutations (if uniformly distributed across the hyperspheres) that are mutually beneficial (call this Y). As adaptation proceeds the mean phenotypes of the adapting populations depart from one another and X therefore no longer equals Y. This is because mutations are vectors that move a phenotype in a particular direction, and thus a mutually beneficial point in phenotypic space is only a mutually beneficial mutation if both populations have the same mean phenotype. To see this, imagine two populations whose mean phenotypes begin at the origin but move towards (1,0) and (0,1) respectively. As these populations adapt and depart from the origin along the x and y axes, there remains for some time a fraction of mutually beneficial phenotypic space (X) around the line x=y between the two optima. But a mutation that moves the (1,0) population into this space is a vector pointing primarily right, while a mutation that moves the (0,1) population into this same space is a vector pointing primarily up. Another way to see that X does not equal Y is to consider what happens as these two populations approach their optima. The fraction of mutually beneficial space then approaches zero, yet very sufficiently small mutations that move phenotypes up and to the right are mutually beneficial.
To account for the inequality between phenotypic space (X) and mutational vectors (Y) during adaptation we must shift the mean phenotypes so that they are at the same point in phenotypic space, and move their optima by an identical translation. We then have X=Y. An easy way to imagine this is to keep the mean phenotypes in place at the mean ancestral phenotype (the origin) and consider adaptation as the movement of the optima closer to the mean phenotypes (i.e., the difference between the optimal and current phenotype). From this perspective, as adaptation proceeds the radii of the hyperspheres shrinks (roughly equivalently in the two populations) but there remains some region of overlap. In fact, because the fraction of overlap (Eq. A1) does not depend on the radii of the hyperspheres, the fraction of non-overlap is expected to remain constant throughout adaptation as long as θ is constant.
In reality and in our simulations, standing genetic variation is not uniformly distributed, the probability of fixation varies across the region of overlap, and adaptation uses up some of the standing variation so that the distribution of standing variation changes with time. Taking the first two complications into account would require weighted averages across the space contained in the hyperspherical caps, which is beyond the scope of our study. The third complication is yet more involved, and would require an analysis of how standing genetic variation is used as adaptation proceeds (i.e., how the distribution of segregating effects and allele frequencies shift as alleles fix). Such a calculation is also beyond the scope of this article. Despite these complications, it seems that the simple analysis above qualitatively captures the essence of why segregation variance increases faster-than-linearly with the angle of divergence.