Theoretical Models of the Influence of Genomic Architecture on The

Molecular Ecology (2014) 23, 4074–4088 doi: 10.1111/mec.12750

Theoretical models of the inﬂuence of genomic architecture on the dynamics of speciation

SAMUEL M. FLAXMAN,* AARON C. WACHOLDER,*† JEFFREY L. FEDER‡ and PATRIK NOSIL§ *Department of Ecology and Evolutionary Biology, University of Colorado, Boulder, CO 80309, USA, †Interdisciplinary Quantitative Biology, University of Colorado, Boulder, CO 80309, USA, ‡Department of Biological Sciences, University of Notre Dame, Notre Dame, IN 46556, USA, §Department of Animal and Plant Sciences, University of Shefﬁeld, Shefﬁeld S10 2TN, UK

Abstract A long-standing problem in evolutionary biology has been determining whether and how gradual, incremental changes at the gene level can account for rapid speciation and bursts of adaptive radiation. Using genome-scale computer simulations, we extend previous theory showing how gradual adaptive change can generate nonlinear population transitions, resulting in the rapid formation of new, reproductively isolated species. We show that these transitions occur via a mechanism rooted in a basic property of biological heredity: the organization of genes in genomes. Genomic organization of genes facilitates two processes: (i) the build-up of statistical associations among large numbers of genes and (ii) the action of divergent selection on persistent combinations of alleles. When a population has accumulated a critical amount of standing, divergently selected variation, the combination of these two processes allows many mutations of small effect to act synergistically and precipitously split one population into two discontinuous, reproductively isolated groups. Periods of allopatry, chromosomal linkage among loci, and large-effect alleles can facilitate this process under some conditions, but are not required for it. Our results complement and extend existing theory on alternative stable states during population divergence, distinct phases of speciation and the rapid emergence of multilocus barriers to gene ﬂow. The results are thus a step towards aligning population genomic theory with modern empirical studies.

Keywords: divergent selection, genomic architecture, linkage disequilibrium, population genetics, speciation phases Received 13 August 2013; revision received 17 March 2014; accepted 19 March 2014

be more common and evolve more quickly than previ- Introduction ously thought (e.g. Lawniczak et al. 2010; Michel et al. Evolutionary biologists have long sought the mecha- 2010; Hancock et al. 2011; Hohenlohe et al. 2012; Jones nisms by which gradual adaptive change, as epitomized et al. 2012; Nosil et al. 2012; Roesti et al. 2012). However, by Darwin (1859), can be reconciled with observations a predictive theory of the population genomics of diver- that population divergence can occur in sudden bursts, gence and speciation is far from solidified. Towards as for example, during adaptive radiations (Schluter that goal, we sought to extend multilocus speciation 2000; Gavrilets & Losos 2009). The past several decades theory (e.g. Felsenstein 1981; Barton 1983; Barton & Ben- have seen significant advances in this regard. Most gtsson 1986; Barton & de Cara 2009; Yeaman & Whit- recently, next-generation DNA sequencing and the lock 2011) in order to make predictions about the emerging field of population genomics have revealed genome-level dynamics that should be observed in a that widespread differentiation across the genome may variety of conditions involving selection, migration, recombination and drift. Specifically, our aim is to make Correspondence: Samuel M. Flaxman, Fax: 303 492 8699; steps towards enhancing theory so that statistical distri- – E-mail: samuel.fl[email protected] butions of measurable metrics such as FST, linkage

© 2014 John Wiley & Sons Ltd GENOMES AND THE ORIGIN OF SPECIES 4075 disequilibrium (LD), allele frequency differences loci and thus may contribute to the build-up of linkage between demes, effective migration rates, local disequilibrium (LD). LD is crucial for speciation (Felsen- adaptation – can eventually be predicted with greater stein 1981), and our simulations thus focused on two accuracy. Hence, here we investigate: How should gen- general properties of genomic architecture that enable ome-wide patterns of differentiation for divergently nonrandom combinations of alleles to persist over gen- selected loci change across the speciation continuum? erations and hence may facilitate the build-up of LD. Much previous theory has demonstrated how genetic First, the most general hereditary consequence of the discontinuities can evolve between populations and gen- organization of genes in genomes is that offspring are erate new species, even in the face of gene flow. Multilo- not formed gene-by-gene from a population ‘beanbag’ cus theory has shown how genetic barriers to gene flow of alleles (Mayr 1959), but rather, parents pass on sets of can become coupled to sharply differentiate populations genes to offspring. We show below with comparisons to with nonlinear transitions between species in space, and null models that this well-known property of genetics implicitly, in time (Barton 1983; Barton & Bengtsson 1986; can have profound consequences for the dynamics of Gavrilets 2004; Barton & de Cara 2009; Gavrilets & Vose population divergence. Second, subsets of genes within 2009; Bierne et al. 2011; Abbott et al. 2013). However, genomes are physically linked on chromosomes, creat- explicit predictions from previous theory are generally ing variation in recombination rates among different sets limited to restricted numbers of loci (a few dozen or less) of loci. The importance of chromosomal linkage relative and/or specific relationships between the strength of to other factors for speciation is debated (Feder & Nosil selection, the migration rate and the recombination rate. 2010; Feder et al. 2011; Via 2012; Yeaman 2013), because Furthermore, these theoretical works generally predict a while LD is crucial for speciation (Felsenstein 1981), single state of divergence for a given set of parameters, chromosomal linkage is not necessarily required for LD but as noted by Barton (2010), there is a need for theory to build. Our results help to clarify in which situations that makes predictions about when alternative stable chromosomal linkage strongly affects the pace of build- population genomic states – that is, a well-mixed popula- up of LD and thus has effects on speciation dynamics. tion vs. nearly reproductively isolated species – are possi- To generate reference points for quantitatively deter- ble. Barton (2010) found alternative stable states in a mining how these genomic features contribute to speci- Levene-type model, due to the build-up of LD, although ation, we created a null ‘beanbag’ model of no genomic the parameter range where such states were possible was architecture, in which the dynamics of alleles at all loci limited. We address these issues here. were independent of each other (see Methods and Fig. Although progress has been made, a broad range of S1, Supporting information). We also created two mod- general theoretical considerations for genomic diver- els in which genes were organized in genomes: (i) a gence remains unexplored. Indeed, next-generation ‘genome-only’ model lacking chromosomal linkage of genomic data collection is to some degree currently loci, implying independent assortment of all loci (i.e. a outpacing the development of theoretical frameworks model of unlinked loci, or as if each locus was situated for interpreting the results (Hudson 2008; Stapley et al. on its own unique chromosome) and (ii) a ‘linkage’ 2010). In essence, the ability to gather extensive gen- model in which loci were arrayed at map locations ome-level data sets presents a problem of scale (sensu along linear chromosomes (i.e. the usual biological con- Levin 1992): Mendelian inheritance at individual loci figuration of genes in the genome). We refer to the lat- has long formed the basis of population genetics, but ter two models as those with genomic architecture. The how do we turn knowledge about individual genes into ‘beanbag’ and ‘genome-only’ models are not meant to predictions about genome-level patterns? When do be representations of real biological systems, per se; gene-level predictions fail to scale up to the genome rather, they are in silico experiments representing null level, and what mechanisms could explain such discon- models that are required to partition the effects of fea- tinuities across scales? Instead of restricting analyses to tures of genomic architecture on divergence (akin to an groups of loci that show exceptional characteristics empirical experiment with a treatment involving the (‘statistical outliers’), can we make predictions about presence of a factor, and a control with the factor the genome-wide distributions across all loci? absent). Hence, comparing results between these three Here, we begin to address these issues using com- models allowed us to determine the roles of the two puter simulations that examine how genomic architec- general genomic features described above – sets of ture affects speciation dynamics. By ‘genomic genes organized together in genomes but on separate architecture’, we are referring to the features of how chromosomes and subsets of genes linked on chromo- genes are organized and arranged in genomes. This somes – in driving speciation dynamics. term is meant to include any features of the genome that We report how gradual adaptive change can generate may affect statistical relationships between some or all nonlinear population transitions, resulting in the rapid

© 2014 John Wiley & Sons Ltd 4076 S. M. FLAXMAN ET AL. formation of new, reproductively isolated species. We consequential events during the evolution of popula- show that these transitions occur via a mechanism facil- tions (e.g. mutations arising, by chance, in very tight itated by genomic structure and involving two interact- linkage with one another). The model versions used to ing processes: (i) the build-up of statistical associations generate results were programmed by the first author among large numbers of genes and (ii) the action of (SMF). For the purposes of validation, the second divergent selection on persistent combinations of alleles. author (ACW) independently programmed another ver- When a population has accumulated a critical amount sion of BU2S and was able to produce the same results of standing, divergently selected variation, the combina- as obtained by the first author (i.e. it is thus extremely tion of these two processes allows many mutations of unlikely that our results could be artefacts of ‘bugs’ in small effect to act synergistically and result in the ge- the program). nomes of populations diverging into discontinuous, Since the mechanics of BU2S are essentially the same congealed and reproductively isolated entities. Impor- as those presented by Flaxman et al. (2013), we here dis- tantly, these results give insights about the mechanisms cuss only a brief overview of key features relevant to that underlie shifts of populations between well-mixed our results and refer the reader to the original publica- and strongly diverged states. Furthermore, by making tion for a full description. Default modelling choices ‘apples-to-apples’ comparisons with null models, we and parameter values are described here; the next sub- are able to demonstrate circumstances under which section describes deviations from these used to address genomic features are or are not important for the specific issues. BU2S simulates the evolution of a finite dynamics of divergence. population in a spatially heterogeneous environment. Individuals migrate between demes with probability m per individual per generation. A divergently selected Materials and methods mutation arises in a randomly chosen individual in a We used simulations to explore the de novo build-up of randomly chosen deme once per generation, consistent population divergence in a two-deme environment with with empirically observed beneficial mutation rates divergent selection. Reproductive isolation occurred due (Halligan & Keightley 2009). The selection coefficient to divergent local adaptation (i.e. ‘extrinsic’ isolation) for the jth locus was denoted sj and was drawn from an and was quantified by the expected effective backward exponential distribution with mean s. The contribution migration rate (Vuilleumier et al. 2010), denoted me and to fitness of each locus was similar to schemes used by defined as the expected proportion of reproduction in a Felsenstein (1981) and Barton & de Cara (2009): the fit- + deme attributable to migrants. Populations consisted of ness contribution of a locus was 1 sj if homozygous + N discrete individuals each having their own, potentially for the favoured allele, 1 0.5sj if it was heterozygous, unique, explicit diploid genomes subject to soft selection and 1 if homozygous for the disfavoured allele. The fit- and regulation of total population size (constant N). Evo- ness of an individual i in deme k (Wik) was calculated lutionary dynamics resulted from the combination of multiplicatively as the product of contributions of all mutation, selection, migration, recombination and drift. QL individual loci: Wik ¼ wjkðgijÞ, where L is the total j¼1 number of divergently selected loci with two alleles Individual-based modelling segregating, and wjk is the fitness contribution of locus j We used an individual-based model built upon a previ- to the individual’s fitness given its genotype at that ously published computer program referred to as ‘BU2S’ locus (gij) and its current deme (k). (‘Build Up to Speciation’) by Flaxman et al. (2013). In a number of instances, we contrast results obtained Source code for the original version and our novel when s < m with results obtained when s > m. Our extensions of it are available at http://sourceforge.net/ focus on the s:m ratio is mainly for heuristic purposes projects/bu2s/files/. Our novel extensions were mainly in examining the dynamics of population divergence technical: we changed the handling of core data struc- (rather than the fates of individual alleles or polymor- tures so that potentially millions of loci could be simu- phisms per se), although for some cases we were also lated on practical computing timescales. This was able to estimate explicit relationships between s, m and accomplished by only keeping track of loci that had L that approximately define thresholds for the diver- two alleles segregating (rather than tracking all loci, gence of populations. including those with no variation). The biological Simulations were run for 1 200 000 generations – the importance of this change is that it allowed us to simu- maximum possible run time on the supercomputer for late the build-up of standing variation over long peri- the slowest running parameter combinations – or until ods of evolutionary time and to observe rare but an a priori threshold of low effective migration rate

= [me 1/(10 000N)] was reached indicating near mutation effect sizes: As the (default) exponential complete reproductive isolation, whichever came first. distribution of mutation effect sizes has an ‘extended Consequences of genomic architecture for divergence tail’ allowing mutations of large effect to occasionally = were determined by comparing results among the three arise, we also obtained results using constant sj s in different scenarios discussed above (see also Fig. S1, our full stochastic model (Fig. S10, Supporting informa- Supporting information). In the null ‘beanbag’ model, tion) and in the infinite-population model described when offspring were formed, the fitness-weighted fre- below. (vi) Periods of allopatry: Most simulations pro- quency of a given allele in a deme determined its proba- ceeded uninterrupted with continuous gene flow at rate bility of being represented in the offspring formed in the m per generation. We simulated secondary contact fol- deme. In the other two genomic models, an individual’s lowing allopatric divergence by having divergence for a relative fitness in its resident deme determined the prob- set number of generations with m = 0, followed by ability that it contributed a gamete to an offspring pro- migration each generation. We also constructed runs in duced in the deme. In the ‘genome-only’ model, gametes which migration was temporarily halted after a period were produced with the assumption of independent of divergence and then restored to determine whether assortment of all loci during meiosis. In the ‘linkage’ genomes could be induced to move from a mixed to a model, gametes were produced assuming that allelic congealed state during a period of allopatry using the combinations on a given chromosome could only be bro- available stores of standing variation. ken up by recombination events, which were independently identically distributed with a mean of 50 cM Metrics of divergence between consecutive events. The default genome in the

‘linkage’ model runs had total map length M=100 cM Reproductive isolation was quantified by me as and C = 4 chromosomes (i.e. each chromosome was explained above. Average, genome-wide linkage dis- 25 cM long); this small map size was chosen to reveal equilibrium (LD) was calculated as the average LD cor- cases in which linkage could have pronounced effects. relation coefficient (i.e. the ‘correlation of allelic states’: Freeman & Herron 2004), pffiffiffiffiffiffiffiffiffiffiffiffiD , for all possible pair- piqipjqj wise comparisons of polymorphic loci i and j. We calcu- Additional, alternative scenarios and parameter choices lated and analysed single-locus FST values using the HTHS FST ¼ Several additional scenarios and parameter choices standard formula HT , where HT was the total were considered in addition to the default conditions observed heterozygosity at a given locus at a given time described above. These additional simulations allowed step and HS was the expected heterozygosity based us to explore how general genome-wide congealing upon each deme’s observed heterozygosity (Hartl & may be during speciation, helping resolve the condi- Clark 2007). The time to speciation transition (a metric tions under which it is most likely to happen and the used in figures below) was operationally defined as the factors that facilitate its occurrence. In this regard, we time when residents reached a state of local adaptation. examined the consequences of (i) Large-effect loci: in In technical terms, this was measured as the last point simulations in which the goal was to consider the in time at which the magnitude of the average fitness of effects of alleles subject to very strong divergent selec- residents in demes was closer to the magnitude of the tion, we initialized simulations with such alleles already average fitness of migrants than it was to that of the present as standing variation. To accomplish this, the maximum possible fitness of a resident in the absence two alleles at each ‘large-effect locus’ were started with of gene flow (for further elaboration, see Fig. S2, Sup- a frequency of 0.5 so that they would not be lost by porting information). Alternative metrics of waiting drift. (ii) Variation in recombination maps: While most time (e.g. the time to reach an me threshold) gave simi- results from the linkage model used genomes of 100 cM lar results. We fit exponential or logistic statistical mod- total map length, additional simulations with larger ge- els for waiting times to the speciation transition using 4 nomes (ranging up to 10 cM) were conducted and the fit() function in MATLAB (version 7.14.0.739). these show how the effects of linkage diminish with increases in map size. (iii) Mutation rate: Effects of Statistical methods changing the number of mutations per generation are shown in Fig. S8 (Supporting information). (iv) Asym- We tested for nonrandom clustering of diverged loci metric selection: Unless otherwise noted, divergent within chromosomes using Ripley’s K function. We fol- selection was symmetric across the demes. Results with lowed Dixon’s (2013) approach for computing K as fol- – asymmetric selection having per-locus selection con- lows: suppose that the nth chromosome has Ln sistently stronger in one deme – are shown in Fig. S9 divergent loci and that the total length of the chromo- (Supporting information). (v) Statistical distribution of some (in map units) is l. Let x be any map distance

© 2014 John Wiley & Sons Ltd 4078 S. M. FLAXMAN ET AL. ranging from 0 to l. Then, from a genetic map of simplifying the problem to a haploid model. The divergent loci along the chromosome,PP Ripley’s K func- frequency trajectories of alleles at L loci, each with ð \ Þ I dij x selection effect s, were then simulated in an infinite- 6¼ tion is computed as KðxÞ¼ i j i ; where the sum- ðLn1Þk population, two-deme model with divergent selection, k = mations are over all Ln loci on the chromosome, Ln/ similar to the set-up of BU2S. Under our haploid model, l is the average ‘density’ of diverged loci on the chromo- two alleles existed in the population at every locus, some, and I is the indicator function which takes a value each favoured in one of the demes (where its relative + of 1 if the map distance, (dij) separating a pair of loci fitness contribution was 1 s) and selected against in (locus i and locus j) is less than x and 0 otherwise. The the other (fitness contribution = 1). For clarity, we note statistical null for these analyses was spatial randomness that there was no variance between selection coefficients of diverged loci. Null expectations for K(x), including within simulation runs; that is, for a given simulation, = means and 95% quantiles, were determined by ran- we set sj s for all j. This offered two advantages. First, domly distributing Ln diverged loci 10 000 times along a it provided results for a very different distribution of chromosome of length l and estimating K(x) as above. selection coefficients compared with the exponential We then computed K(x) for our actual results from the distribution used in most runs of the full BU2S model BU2S simulations. If K(x) values derived from the BU2S (i.e. no ‘extended tail’ of large-effect mutations). Second, simulations rose significantly above the null expectation it enabled massive simplification of the computation for (i.e. above the 97.5% quantile) for a given distance x, the infinite-population, haploid model because all the then they were considered to represent nonrandom different genotypes with k locally adapted alleles clustering of divergently selected loci along the chromo- (0 ≤ k ≤ L) had identical dynamics within a deme, and some. This analysis does not address the issue of non- therefore, only 2(L + 1) variables needed to be tracked. random distributions of diverged loci between different Simulations of this model were initialized in one of chromosomes (i.e. one chromosome having significantly two ways: either each deme started with all genotypes more diverged loci than others); the latter can be exam- containing L/2 favourable alleles (a completely mixed ined with multinomial tests (e.g. Flaxman et al. 2013), population) or each deme started with the frequency of but was not addressed here due to space limitations. genotypes in it containing L favourable alleles (maxi- mally diverged populations). Essentially, the two starting conditions could be considered to represent a An infinite-population analogue of the ‘genome-only’ primary contact scenario possessing extensive amounts model of standing variation not differentiated between popula- Ideally, combining findings from simulations with tions vs. a pair of populations immediately following analysis can help confirm in silico results and provide secondary contact that were alternatively fixed for dif- additional theoretical details into the evolutionary pro- ferent adaptive alleles. Each generation in the simula- cess. Unfortunately, for several reasons, deriving ana- tion began with migration between the demes, in which lytic results from a model with the complexity of BU2S a proportion m of each genotype class switched demes. was not feasible. However, we were able to derive After migration, selection occurred, in which the rela- numeric results about alternative stable population tive frequency of each genotype was multiplied by its genomic states from a deterministic, infinite-population- relative fitness within the deme. Last, the populations size model representing an analogue of the ‘genome- of each deme reproduced sexually. For any given mat- only’ model (i.e. genes in genomes, but no chromosomal ing pair type, independent assortment of all loci was linkage; constructing an infinite-population analogue of assumed, as in the ‘genome-only’ individual-based dip- the ‘linkage’ model was not tractable). Critically, this loid model described above. Simulations were contin- model allowed us to more fully investigate the parame- ued until the difference in allele frequency change ter space, including the number of segregating loci between generations became <10 9 in both demes, at under selection and a broad range of migration rates, which point the system was operationally defined to be when transitions to accelerated adaptive population at equilibrium. divergence and genome congealing were enabled. Barton and Shpak (2000) found that in certain condi- To model an infinite-sized population with genes in tions, the equilibria of ‘symmetric’ models like our infi- genomes requires tracking not just allele frequencies, nite-population model could be unstable to asymmetric but also the frequencies of genotypes. As others have fluctuations in allele frequencies. Hence, to address the noted (Kirkpatrick et al. 2002), this is often not feasible issue of stability, we made a new version of the infi- on practical timescales because of the exponential nite-population model where a single locus was growth of possible genotypes as increasing numbers of allowed to vary in frequency separately from the other loci are considered. Here, we accomplished this by first loci. We perturbed the frequency of either the single

© 2014 John Wiley & Sons Ltd GENOMES AND THE ORIGIN OF SPECIES 4079 locus or all other loci under a variety of conditions after where mixed starting populations remained mixed and the system reached a stable state (varying the number highly diverged starting populations remained diverged of loci, the migration rate, the per-locus strength of represented conditions under which rapid adaptive divergent selection, the size and direction of perturba- transitions could occur dependent upon the population tion and whether perturbations were in the same direc- structuring of standing variation in the genome. tion or opposite direction). In all cases, stability held. This is consistent with Barton and Shpak (2000), where Results they found that stability was violated only in some spe- cial cases. Divergence with strong selection relative to gene flow Under the deterministic, infinite-population, haploid model, two final equilibrium states were sometimes As would be expected based on previous theory (Hal- possible. In one state, the two populations were well dane 1948; Barton 1983; Barton & Bengtsson 1986; Gav- mixed, loci were polymorphic within demes, and little rilets 2004; Yeaman & Otto 2011; Yeaman & Whitlock divergence was observed between demes. In the other 2011), we found that populations readily diverged state, representing between-population genome congeal- when s ≥ m (Fig. 1A, D). In such cases, the probability ing, the two populations were highly diverged and loci that any given mutation would establish was consistent were near fixation for alternate adaptive alleles within with expectations from single-locus theory (Yeaman & demes. Comparing when populations (i) always became Otto 2011), and the incremental accumulation of muta- well mixed, (ii) always became highly diverged or (iii) tions with relatively large fitness effects scaled up to an assumed alternate equilibrium states depending upon incremental and generally linear rate of population- starting conditions allowed us to explore the combina- level divergence. Although genomic architecture some- tions of s, m and L values where one or multiple popu- times sped up divergence, it was not necessary for the lation genomic states could be equilibria. Thus, when evolution of strong reproductive isolation (Fig. 1; Figs populations became mixed under both starting condi- S3–S5, Supporting information). tions, this implied that a threshold number of selected loci had yet to establish to trigger genome-wide diver- Divergence with weak selection and strong gene flow: gence. In contrast, when populations became highly phase shifts in speciation diverged regardless of starting conditions, this implied the threshold conducive to genome congealing had Mutations with small adaptive effects should generally already been passed. Finally, the parameter space be more common than mutations with large adaptive

(A) (B) (C)

(D) (E) (F)

Fig. 1 Time courses of (A–C) effective migration rates (semi-log plots) and (D–F) numbers of loci with divergently selected alleles segregating for the three models of genomic architecture. Each column shows a different combination of m and s (parameters common to all panels: N = 20 000, C = 4 chromosomes, M = 100 cM; additional parameter combinations: Figs S3–S5, Supporting information). Each line is the median of 50 independent simulation runs.

© 2014 John Wiley & Sons Ltd 4080 S. M. FLAXMAN ET AL. effects (Orr 2003), and thus, populations may often have After this transition, mutation accumulation rates m > s. Such circumstances are challenging to investigate were substantially greater than prior to the transition analytically because rare, stochastic events increasingly (Fig. 1E, F). That is, rather than following expectations contribute to evolutionary dynamics as s values for single-locus probabilities (e.g. Yeaman & Otto 2011), decrease relative to m. In such cases, our simulations mutation accumulation and the reduction in effective found that incremental genetic change at individual loci migration reflected strong, multilocus barriers to gene did not scale to population-level divergence until after flow (Barton 1983; Barton & Bengtsson 1986; Barton & the population underwent a transition. Before the tran- de Cara 2009). Results from the genome-only model sition, the establishment of any individual mutation highlighted that this reduction in effective migration contributed very little to population-level divergence took place without any physical linkage of selected loci (Fig. 2), because each mutation’s effect on fitness was on chromosomes but did require the organization of weaker than the homogenizing force of migration (gene genes in genomes: no transition was observed for the flow). As expected from previous theory (Barton 1983; beanbag model, and prior to the transition, the beanbag Barton & Bengtsson 1986; Yeaman & Otto 2011; Yeaman model followed a similar trajectory as the two genomic & Whitlock 2011), each individual mutation had a very models (Fig. 1). low probability of establishment and genetic polymor- What caused these transitions and why did they phisms were often transient. occur only with genomic architecture? The answers to However, when many such mutations accumulated, a these questions involved the mechanism underlying the nonlinear transition occurred in which the rates of evolu- transitions: the combination of critical levels of genome- tion of genetic divergence and reproductive isolation dra- wide (i) divergent selection and (ii) linkage disequilib- matically increased (Fig. 2B, C). During the transition, rium (LD) triggered a positive feedback process of population differentiation emerged as a property of the population divergence (Fig. 2). For convenience, we entire genome rather than just of individual loci as the refer to this mechanism as ‘genome-wide congealing’ dynamics of many loci became coupled. In effect, the ge- (GWC). Specifically, GWC is a nonlinear transition of nomes of the two populations ‘congealed’ into two dis- populations from a relatively undifferentiated state to a tinct entities across the divergently selected loci, each strongly differentiated state. Whether or not genomes entity representing a new, reproductively isolated species. ‘congeal’ has a long history of study in evolutionary

Beanbag Genome only Linkage

(A) (B) (C)

(D) (E) (F)

Fig. 2 Evolutionary bifurcations of populations coincide with sudden increases in genome-wide linkage disequilibrium (LD) and reproductive isolation. Each column of panels displays results from one simulation run, for one of the three models of genomic architecture (parameters as in Fig. 1C: s = 0.01; m = 0.1). (A–C) Fitness values over time within one deme. Orange line: average resident fitness; purple line: average immigrant fitness; cyan line: maximum possible fitness given the alleles that are segregating; grey points: fitnesses of a random subsample (200 individuals per generation sampled) of individuals in the deme. Fitness values are relative to – the expected fitness of a randomly assembled genotype (yellow centreline). (D F) Effective migration rates, me (black lines, log10 scal- ing), and average, genome-wide LD (green lines, right hand axes) over time.

© 2014 John Wiley & Sons Ltd GENOMES AND THE ORIGIN OF SPECIES 4081 genetics (Turner 1967; Hartl 1977; Maynard Smith 1977). examine the consequences of relaxing these conditions, Our ‘genome-wide’ emphasis focuses on the fact that we consider, in turn, divergence involving a few the ‘congealing’ we are talking about is in the whole- strongly selected mutations and divergence during peri- genome, between-population sense (i.e. between- ods of allopatry (no gene flow). population LD) rather than relating to reduced If numerous mutations have large, adaptive effects, recombination and units of selection emphasized in then conditions are like those in Fig. 1A, D: speciation debates about ‘congealing’ decades ago. occurs without a dramatic transition in the rate that The two causal factors comprising GWC go hand-in- mutations establish, and populations diverge gene-by- hand: strong genome-wide divergent selection is only gene because each individual mutation can overcome manifested when there are nonrandom allelic combina- gene flow on its own. Thus, when there is no shortage tions (LD) involving many loci, but with migration, mul- of large-fitness-effect mutations, speciation with gene tilocus LD is effectively maintained only if genome-wide flow is not constrained by migration, and there is no selection is sufficiently strong. Nonlinear transitions great consequence of genome structure for the dynam- require genomic architecture because inheritance of sets ics of population divergence. of genes is necessary to orchestrate and preserve the two However, what if just a few large-effect mutations causal factors across generations. Transitions are rapid arise early in the process of population divergence and because once LD and divergent selection reach threshold most adaptive changes are small thereafter, as expected levels, they enhance each other in a positive feedback under Fisher’s (1930) geometric model? To address this process (i.e. each amplifies the other). However, prior to issue, we initialized simulations with varying numbers this tipping point, neither is strong enough to prevent of mutations having large selection coefficients, S (≥ m), extensive gene flow from disrupting the other. Thus, and the remainder having small coefficients (m > s). transitions are not initiated until fit combinations of Although the initial existence of the large-effect muta- alleles arise by chance, which at first is difficult, but tions shortened the time until populations diverged becomes increasingly likely as standing variation builds. (Fig. 3A), GWC remained important for driving speciation if the initial, large-effect mutations were not suffi- cient to push populations beyond the transition Relaxing conditions for GWC threshold point and cause genome-wide reproductive Genome-wide congealing is predicted to be most prom- isolation (Fig. 3A; Fig. S6, Supporting information). inent and easily detected under two conditions: (i) These results emphasized the potential for standing var- many loci contribute to divergence and (ii) high gene iation to enable GWC and facilitate ecological speciation flow impedes population-level divergence (s < m). To with gene flow.

(A) (B) (C)

Fig. 3 Effects of strongly adaptive mutations, allopatry and map size of the genome on waiting time to speciation transitions. (A) Mutations with large adaptive effects, S, can speed the approach to, or be sufﬁcient for, speciation (points above and on x-axis, respectively; results from ‘genome-only’ model; see also Fig. S6, Supporting information). (B) With initial allopatric divergence followed by secondary contact, populations usually either remain distinct (points on x-axis) or fuse and re-diverge tens of thousands of generations later (points above x-axis; see also Fig. S7, Supporting information). (C) As map length of the genome (recombination rate) increases, the behaviour of the ‘linkage’ model approaches the average of the ‘genome-only’ model. Solid lines are ﬁts from (A) exponential or (B, C) logistic statistical models. In all panels m = 0.1 (when migration occurs), s = 0.01, N = 5000. Numbers of independent simulation runs shown: (A) 238, (B) 412, (C) 324. Note that effects of linkage (vs. ‘genome only’) are less pronounced under other parameter combinations (Fig. 1B; Figs S3–S5, Supporting information).

With respect to allopatry, two predictions follow from GWC. First, when divergence in allopatry is followed by hybridization upon secondary contact, GWC predicts that populations should either quickly fuse or remain isolated; intermediate levels of mixing should be com- paratively rare because populations are predicted to either be undifferentiated or highly divergent, (Fig. 3B; Fig. S7, Supporting information). This prediction follows from the result that when conditions for GWC are satisfied, populations should usually be in one of two alternative population genomic states: (i) one relatively well-mixed population or (ii) two (mostly) reproductively isolated species (Figs 4–5). Hence, in a period immediately following secondary contact, populations will either have already evolved past the transition point for GWC in allopatry and remain as diverged species or Fig. 5 Parameter space of bistability in the deterministic, infi- not. In the latter case, gene flow will tend to cause popu- nite-population, haploid, genome-only model. For indicated lations to revert to a relatively undifferentiated state = values of L divergently selected loci (all having sj s), filled from which additional, divergently selected variation regions indicate the combinations of m and s for which alterna- must subsequently build to reach GWC (Fig. 3B; Fig. S7, tive stable population genomic states (well-mixed or diverged) Supporting information). In empirical cases when sec- were found in simulations starting from different initial condi- ondary contact leads to intermediate levels of mixing, tions (Methods). Outside these regions, the system always reached a single equilibrium regardless of the starting popula- we predict that genes of large effect may therefore often tion genomic conditions. underlie divergence, as they would be less prone to homogenization (Fig. 3A; Fig. S6, Supporting information; see also Discussion). Despite homogenization of should elevate LD and thereby help trigger GWC, which small-effect variants, the period of allopatry may still be would keep populations diverged even if migration important for incrementally increasing the chances for resumed. Simulations supported this prediction as well speciation in sympatry compared with primary contact (Figs 3B and 4B; Fig. S7C, Supporting information). scenarios by infusing populations with standing variation above baseline levels resulting from drift (albeit at Alternative stable population genomic states very similar frequencies between populations). A second prediction about allopatry is that when suffi- Using simulations of the deterministic, infinite-popula- cient standing variation for GWC exists, a period of geo- tion, haploid model from starting points of maximum graphical isolation, even without additional mutations, or minimum divergence (Methods), we were able to

(A) (B)

Fig. 4 Alternative stable population genomic states in a deterministic, inﬁnite-population, haploid model having genomic organization of genes (but no chromosomal linkage; L = 60 loci, m = 0.1). In both panels, the y-axis shows the local frequency of an allele in the deme in which it is favoured. Alleles at all loci have the same dynamics owing to the assumption of constant s (necessary for numerical tractability). (A) Within a range of s values, there are two equilibria and the equilibrium state of the system depends upon past conditions (i.e. the system exhibits bistability and hysteresis), as indicated by the two sets of points, one when initializing populations with maximum possible divergence (blue, leftward pointing triangles) and the other when initializing with minimum possible divergence (red, rightward pointing triangles; see Methods). (B) Example of how a brief period (10 generations; arrow) of reduced gene ﬂow can cause the system to switch to an alternative state. Dashed grey lines show the positions of the two predicted equilibria from panel A for the value of s (=0.038) that was used in this example.

© 2014 John Wiley & Sons Ltd GENOMES AND THE ORIGIN OF SPECIES 4083 characterize combinations of s, m and L associated with one species to two (Figs 1–3; Figs S3–S5, Supporting alternative stable population genomic states (Fig. 5). information). Supporting this idea, genomes with more The space of multiple stable equilibria can be visualized compact chromosomes had shorter waiting times to as being contained between two lines in a plot of s vs. speciation (Fig. 3C). When chromosomal linkage had m, with the position of the lines dependent on L. strong effects on divergence times, we found that Results from simulations were used to estimate (via divergent loci were clustered nonrandomly within chro- regression) the equations of these lines: mosomes (Fig. 6; Videos S1 and S2, Supporting information). These findings support the ‘divergence ¼ 0:257 0:578 þ 1:87 0:614 ðeqn 1aÞ supper L mL hitchhiking’ hypothesis (Via 2012) and are consistent with empirical observations of clustered genomic diver- 2:39L0:489 0:338 slower ¼ 0:0064m expð9:97L Þ; ðeqn 1bÞ gence (Nadeau et al. 2012; Andrew & Rieseberg 2013). We emphasize here that pronounced effects of linkage where supper and slower are the upper and lower bounds were only found in the most difficult conditions for of the multiple equilibria region, respectively. The range divergence (s m). We also predict that other struc- of possible combinations of s and m generating alterna- tural features that can reduce recombination, such as tive stable population genomic states – and hence, the genomic rearrangements (Noor et al. 2001; Rieseberg potential for rapid speciation via GWC – increased with 2001; Yeaman 2013), will promote GWC. the number of loci L experiencing divergent selection As our example time series animations show (Videos (Fig. 5). Notably, alternative stable states and GWC S1 and S2, Supporting information), nonrandom clusters were even possible when there was full sympatry of divergent loci that could be considered genomic (m = 0.5) and with maximum recombination between ‘islands’ of divergence (Turner et al. 2005) could be seen all loci (i.e. genome only = no linkage). to flicker in and out during the build-up to GWC in the linkage model. While most of these clusters were tem- porally transient, our time series animations also show Reductions in recombination arising from chromosomal an example of an ‘island’ that persisted over hundreds linkage of thousands of generations. To address how frequently Because GWC depends upon the evolution of LD, islands arose and how persistent they were over time, aspects of genome structure that reduce recombination we analysed whether the incidence of nonrandom and promote LD can potentially aid the transition from clusters of loci over time was correlated with the

(A) (B) (C) (D)

(E) (F) (G) (H)

– Fig. 6 Nonrandom clusters of divergent loci are seen when chromosomal linkage promotes speciation. (A D) A snapshot of FST values across the entire genome, ~40 000 generations prior to the speciation transition (for time series, see Videos S1 and S2, Supporting information). (E–H) Ripley’s K function (a statistical test for spatial clustering, see Methods) computed for the positions of loci on each chromosome from ‘linkage’ model results (red line) and under the statistical null hypothesis of no signiﬁcant clustering of loci (solid blue line: null expectation; dashed blue lines: 2.5% (lower) and 97.5% (upper) quantiles around the null). (G) On chromosome #3, more loci than expected by chance are separated by distances on the order of 0.1 cM. Parameters: s = 0.01, m = 0.1, N = 1000, C = 4, M = 100.

© 2014 John Wiley & Sons Ltd 4084 S. M. FLAXMAN ET AL. acceleration of divergence found in the linkage model dependent selection and typically consider only one or compared with the genome-only model (Fig. 7). For a few genes, or avoid dealing with genes altogether by parameter cases in which linkage accelerated diver- focusing on phenotypes. Second, like GWC, other pro- gence (e.g. Fig. 1C), we indeed found a higher incidence posed mechanisms (Templeton 1980; Kirkpatrick & Ra- of nonrandom clustering of loci than in cases when vigne 2002) have involved multiple loci and LD as linkage did not strongly affect divergence times (com- drivers of speciation. However, they require founding pare triangle symbol with other symbols in Fig. 7). events, population bottlenecks, epistasis, allopatry or assortative mating to act. Indeed, assortative mating and/or epistasis have been central to most previous Discussion theoretical solutions for speciation with gene flow. The mechanism we refer to as ‘GWC’ provides a frame- GWC does not require these factors, but instead work for understanding how gradual, incremental depends only on divergent selection and the organiza- change at the individual gene level can produce rapid, tion of genes in genomes. nonlinear shifts in population-level divergence. GWC The effects of the coupling between loci and thresh- shares elements with several prominent theories of spe- olds for the establishment of divergently selected muta- ciation, but it is distinguished from each in important tions have been demonstrated in previous works ways. First, adaptive dynamics models have usefully (Barton 1983; Bierne et al. 2011; Burger & Akerman revealed how incremental adaptive changes can some- 2011; Yeaman & Otto 2011; Yeaman & Whitlock 2011; times lead to sudden ‘evolutionary branching’ resulting Abbott et al. 2013). Hence, the main contributions of the in new species (Geritz et al. 1998; Ito & Dieckmann present work reside not in showing that the coupled 2012). However, these models rely on frequency- effects of many loci can drive and keep populations apart, but rather in (i) characterizing temporal dynamics of sharp population transitions (e.g. Fig. 2), (ii) characterizing regions of parameter space in which population divergence proceeds in a linear, gene-by-gene manner (e.g. Fig. 1A) vs. in a nonlinear manner involving GWC (e.g. Figs 1C, 4 and 5), (iii) enabling predictions of conditions under which chromosomal linkage of alleles is important for divergence dynamics (e.g. Figs 1, 3, 6 and 7), and (iv) showing how periods of allopatry interact with divergent selection and LD to shape transitions from populations to species (e.g. Figs 3 and 4; Fig. S7, Supporting information). Our contributions thus respond to the need for these types of predictions highlighted in recent reviews (Barton 2010; Abbott et al. 2013; Feder et al. 2013). Additionally, while recent works (Kirkpatrick & Barton 2006; Burger & Akerman 2011; Yeaman & Whitlock 2011) have considered how linkage enables establishment of weakly adaptive muta- Fig. 7 When linkage speeds up speciation, nonrandom clusters tions in the face of gene flow, our results on GWC with of loci are more common in the genome. The number of gener- ‘genome-only’ models show how such mutations can ations required for simulations to reach a low effective migra- gradually build up and drive divergence even without tion rate threshold [m < 1/(10 000N)] was compared between e linkage. Finally, our explicit use of null models (‘bean- 150 independent pairs of runs of the linkage and genome-only models with s = 0.01 and N = 20 000. Fifty pairs of runs were bag’ compared with ‘genome only’) created ‘apples-to- used to calculate means 1 SD for each point. Within a pair of apples’ comparisons that enabled us to quantitatively runs, the linkage and genome-only runs had the same exact partition effects of direct selection vs. hitchhiking and sequence of mutations arise. Positive values on the x-axis indi- to therefore draw conclusions about when general geno- cate that divergence took longer in the genome-only model. mic features are or are not important for the dynamics Values on the y-axis come from the linkage runs, which we of speciation. used to calculate a time-averaged measure of how often non- In an investigation of the evolution of strong barriers random clusters of loci could be statistically detected, as determined by values of Ripley’s K function that exceeded the to gene flow, Barton and de Cara (2009) extended Fel- upper limit of the 95% confidence interval around the null senstein’s (1981) seminal analysis of divergent selection hypothesis of random locations of loci for the given number of and the evolution of reproductive isolation. Similar to loci on a chromosome at a given time. our results, they found thresholds (e.g. in the strength of

© 2014 John Wiley & Sons Ltd GENOMES AND THE ORIGIN OF SPECIES 4085 selection), above which loci would become coupled, LD Our simulations also considered cases in which condi- would build through time and populations would split tions for GWC were relaxed (e.g. Figs 1 and 3), yielding into reproductively isolated species. Indeed, Barton and general predictions about how the number and effect de Cara (2009: p. 1831) noted that, ‘...selection may act sizes of genes underlying divergent adaptation could to strengthen linkage disequilibrium ... So, we expect a help shape speciation dynamics. When most mutations strong feedback, potentially leading to rapid speciation’. had large effects, speciation proceeded incrementally This statement was made in the context of incompatibili- (more uniformly in rate) through time because success- ties, which we did not study, and the methods used by ful mutations could overcome gene flow due solely to Barton & de Cara (2009) differed from our own in sev- the direct effects of selection acting on them; indirect eral other technical and biological ways, precluding effects arising from genomic architecture were not nec- quantitative comparisons of the thresholds for diver- essary for divergence. Hence, we predict that a few gence. For example, a number of the explicit results genes of large effect underlie divergence when genomic obtained by Barton and de Cara were found for small regions show highly heterogeneous divergence, and a numbers of loci and/or under the assumption that geno- range of intermediate forms between taxa is observed. type frequencies of selected loci were at equilibrium, For example, the number of hybrids reported between neither of which is necessarily true for our simulations. Heliconius butterfly taxa declines gradually with time Additionally, the recursions in Barton and de Cara since divergence, with no obvious abrupt shifts in this (2009), while theoretically reasonable, do not scale in a relationship at the boundary between populations and computationally efficient way to the larger numbers of species (Mallet et al. 2007). Consistent with our predic- loci we considered. However, there is a key qualitative tions, closely related Heliconius taxa often differ geneti- difference between their models’ results and our own. cally from one another only or primarily in a few Barton and de Cara’s models (and indeed, most previ- genomic regions (Nadeau et al. 2012). ous related theory) found a single threshold for popula- We stress, however, that we are not advocating an all- tion divergence: above this threshold, LD builds and or-none perspective of large-effect mutations vs. GWC. strong reproductive isolation evolves; below the thresh- Genomes also congeal due to large-effect loci alone, but old, LD does not build and populations are not strongly in such cases we predict that there will be no pro- isolated. By contrast, we found two thresholds consist- nounced acceleration in the rate of adaptive divergence ing of the upper and lower bounds on a region of or jumps in levels of differentiation displayed by parameter space in which alternative stable population selected genes. In addition, our results show how large- genomic states are possible [i.e. and the upper and effect mutations can help seed the speciation process by lower borders of the filled regions in Fig. 5, approxi- moving populations closer to the transition point where mated by equations (1a) and (1b)]. Above the upper GWC can occur due to the accumulation of additional, threshold, only the diverged state is stable, and below small-effect mutations (Fig. 3). In other words, large- the lower threshold, only a well-mixed population is sta- effect genes and GWC can complement each other by ble. However, between these two thresholds, either state contributing to differentiation during different phases of may occur. Why did this result emerge from our mod- divergence. In the early, ‘genic’ phase, divergence is els? As Fig. 5 shows, alternative states were only found mostly a characteristic of individual loci rather than a for moderate to large numbers of loci; previous solu- property of genomes, and most loci approximately fol- tions have been found for limited conditions generally low the behaviour one would expect from classical one- involving smaller numbers of loci, restrictions on migra- or two-locus population genetic models. Then, when tion rates and/or restrictions on recombination maps. GWC begins, populations transition to a ‘genomic’ How general is our model of GWC for speciation? As phase in which divergence becomes more apparent gen- noted above, GWC is predicted to be most prominent ome-wide and the degree of reproductive isolation when (i) many loci contribute to divergence and (ii) increases nonlinearly due the aggregate consequences of high gene flow impedes population-level divergence the synergy between many loci (seen in Fig. 2B, C, E, F). (s < m). These conditions are possible for most organ- Because speciation unfolds over time, generating data isms. Moreover, results from many next-generation for arrays of multiple (>2) closely related taxa at vary- DNA sequencing studies imply that the first condition ing points along the speciation continuum is required may frequently exist in nature (Pinho & Hey 2010; Sha- to enable empirical testing of GWC. Such data sets will fer & Wolf 2013). Regarding the second condition, the- likely soon be available thanks to ongoing empirical ory and empirical observations suggest that mutations studies in a number of systems, such as Rhagoletis flies with small adaptive effects should be more common (Michel et al. 2010; Powell et al. 2012, 2013), Lake Victo- than those with large adaptive effects (Orr 2003; Halli- ria cichlids (e.g. Keller et al. 2013), threespine stickle- gan & Keightley 2009; Rodrigue et al. 2010). back (e.g. Roesti et al. 2012), Helianthus sunflowers (e.g.

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Fig. S9 Increasingly asymmetric selection makes speciation more difﬁcult. S.M.F. and A.C.W. conceptualized ideas, developed Fig. S10 Times to speciation in the BU2S model when all selec- models, analysed data, and wrote the manuscript. J.L.F. tion coefﬁcients are a constant value (rather than being drawn and P.N. conceptualized ideas and wrote the manu- from an exponential distribution) and selection is symmetric in script. (A) the linkage model and (B) the genome-only model.

Video S1 Example broad time-lapse view of the process of the buildup of de novo genome-wide divergence, as shown through the temporal dynamics of F across the genome. Data accessibility ST Video S2 A slowed down time-lapse view of the critical period 1 Source code for simulations is available at (http:// of genome wide congealing from the same example as in sourceforge.net/projects/bu2s/ﬁles/). This will include Video S1. source ﬁles and plain text instructions for compiling and running the program.