GENETIC HITCHHIKING AND THE DYNAMIC BUILDUP OF GENOMIC DIVERGENCE DURING SPECIATION WITH GENE FLOW

Authors


Abstract

A major issue in evolutionary biology is explaining patterns of differentiation observed in population genomic data, as divergence can be due to both direct selection on a locus and genetic hitchhiking. “Divergence hitchhiking” (DH) theory postulates that divergent selection on a locus reduces gene flow at physically linked sites, facilitating the formation of localized clusters of tightly linked, diverged loci. “Genome hitchhiking” (GH) theory emphasizes genome-wide effects of divergent selection. Past theoretical investigations of DH and GH focused on static snapshots of divergence. Here, we used simulations assessing a variety of strengths of selection, migration rates, population sizes, and mutation rates to investigate the relative importance of direct selection, GH, and DH in facilitating the dynamic buildup of genomic divergence as speciation proceeds through time. When divergently selected mutations were limiting, GH promoted divergence, but DH had little measurable effect. When populations were small and divergently selected mutations were common, DH enhanced the accumulation of weakly selected mutations, but this contributed little to reproductive isolation. In general, GH promoted reproductive isolation by reducing effective migration rates below that due to direct selection alone, and was important for genome-wide “congealing” or “coupling” of differentiation (FST) across loci as speciation progressed.

Recent technical advances in DNA sequencing have opened the door to studying genome-wide patterns of genetic differentiation during speciation (Ellegren 2008; Hohenlohe et al. 2010; Nosil and Feder 2012a; Nosil et al. 2012). Indeed, the data deluge produced by “next generation” sequencing has spotlighted the need for theory that predicts (1) genomic patterns of divergence at various stages of the process of speciation, and (2) the relative importance of various evolutionary processes in generating those patterns (e.g., Gompert et al. 2012).

Many of the evolutionary effects of selection, migration, and physical linkage are well understood for one or a few loci considered in isolation (Felsenstein 1976, 1981; Barton 2000; Kirkpatrick et al. 2002; Yeaman and Otto 2011). In contrast, the consequences of the combination of these factors in multidimensional phenotypic or genotypic cases are less well understood (Doebeli and Ispolatov 2010; Yeaman and Whitlock 2011), although some general theory has recently been developed (Barton and de Vladar 2009). Theory on genetic hitchhiking (Barton 2000; Charlesworth et al. 2003) is a useful starting place to begin addressing issues concerning how genome structure may affect progress toward speciation. We note that our use of the term “hitchhiking” is broader than Maynard Smith and Haigh's (1974) original definition describing the effects of selective sweeps within populations. Since then, the term has taken on a larger meaning encompassing “…the indirect effects of selection at one or more loci on the rest of the genome” (Barton 2000, p. 1553).

A number of recent publications have focused on how hitchhiking may facilitate speciation with gene flow (e.g., Via 2009, 2012; Feder and Nosil 2010; Buonaccorsi et al. 2011; Dopman 2011; Feder et al. 2012a,b; Flaxman et al. 2012; Nosil and Feder 2012a,b; Roesti et al. 2012). Two kinds of hitchhiking, termed “divergence hitchhiking” (henceforth, DH) and “genome hitchhiking” (GH), have been implicated in this context. DH theory postulates that divergent selection acting on a locus can reduce gene flow at physically linked loci, shielding those loci from the homogenizing effects of hybridization. This effect is in turn predicted to promote the maintenance and further accumulation of divergence (selected and neutral) at linked sites. GH theory focuses on the effects of divergent selection regardless of linkage. Specifically, GH is defined by genome-wide (rather than localized) reductions in average gene flow and effective recombination rates caused by divergent selection.

DH and GH are not mutually exclusive, but rather may operate simultaneously or sequentially during speciation with gene flow (Feder et al. 2012a). Thus, although GH reduces effective migration across the genome, there can still be substantial variability among gene regions in the degree of reduced migration, depending on the distribution of selected sites. Nonetheless, DH and GH can be distinguished in their predictions: strong effects of DH should create isolated clusters of tightly linked, diverged loci (islands), whereas GH should elevate divergence across the genome, pushing both “islands” and the “baseline” level of differentiation to higher levels (Feder et al. 2012a).

Here, we partition the relative contributions that DH and GH make—above those of selection acting directly on a locus (direct selection, hereafter DS)—to the establishment of new mutations involved in habitat-related fitness trade-offs (which generate ecologically based reproductive isolation). This is a critical issue for predicting the feasibility of speciation with gene flow. Future work could consider standing genetic variation, which is also likely to be important during speciation (e.g., Jones et al. 2012; Powell et al. 2012).

GENERAL STRATEGY AND RELATION TO PAST WORK

Our overarching goal is to resolve how different processes—DS, GH, and DH—affect genomic patterns of differentiation through time between two populations diverging with gene flow. Within this context, we were interested in determining whether and how aspects of genome structure, the distribution of selection coefficients, gross migration levels, and mutation rates influenced the potential for DH and GH to facilitate speciation.

In a previous study, Feder et al. (2012b) estimated the probabilities of establishment for new mutations for taxa that had accumulated a set number of already diverged loci in their genomes with selection coefficients S. Varying recombination distances of new mutations to these already diverged loci made it possible to parse out the relative contributions that DS, GH, and DH had for the establishment of a new mutation. Although informative, these previous results provided only a static view of the speciation process (snapshots at specific stages in which populations had achieved a set level of differentiation). Moreover, the simulations analyzed a maximum of only six already diverged loci.

Here, we investigate the dynamic nature of the divergence-with-gene-flow process by characterizing how potentially thousands of new mutations sequentially build up throughout the genome over time. This allowed us to study the evolution of increased reproductive isolation as speciation unfolded. Earlier work suggested that DH could not generally facilitate population divergence for regions >1 centi-Morgan (cM) from a strongly selected locus (Feder et al. 2012b; Flaxman et al. 2012). However, it is possible that slight accentuations of divergence due to DH generated early in the speciation process could sequentially build through time and become increasingly exaggerated to strongly affect the future course of population divergence. Previous static models could not have picked up such effects of DH. Here, we rectify this potential shortcoming by performing computer simulations actively following the evolutionary fates of new mutations as they sequentially enter the gene pools of continuously diverging populations. These new simulations therefore provide a novel, dynamic, and more realistic view of the effects of direct selection and hitchhiking on speciation than offered by previous studies. Our approach bears some similarities to previous individual-based models focused on genetic architecture (Griswold 2006; Barfield et al. 2011; Yeaman and Whitlock 2011), but is unique with respect to partitioning the effects of DS, GH, and DH. We also consider numbers of loci that are 1–2 orders of magnitude greater than previous simulations.

Specifically, we performed computer simulations under three different scenarios that represented populations diverging due to (1) the effects of DS only (DS scenario), (2) the effects of DS and GH together (GH scenario), and (3) the combined effects of DS, GH, and DH (“DH scenario”; see Fig. 1 and Methods). These scenarios allowed us to determine the consequences of GH and DH for progress toward speciation, where “progress” was measured by (1) the number of mutations differentially established between populations, (2) levels of genetic differentiation at each locus (FST), and (3) effective backward migration rates (sensu Vuilleumier et al. 2010). The latter represents a good proxy for reproductive isolation (Barton and Bengtsson 1986; Gavrilets 2004). Furthermore, to make results among simulation scenarios maximally comparable, we introduced identical sets of randomly generated sequences of mutations in each of three DS, GH, and DH scenario simulation runs (details below).

Figure 1.

The scenarios used to partition effects of (a) DS, (b) GH, and (c) DH. Rectangles indicate demes, large circles indicate individuals, dots indicate alleles, and straight lines indicate chromosomes. Aj represents an ancestral allele at locus j, and Bj represents a derived allele recently produced by mutation (lightning bolt). (a) To isolate effects of DS in the absence of hitchhiking, “individuals” were assembled for the purposes of migration, but then disintegrated into their component alleles for the purposes of reproduction. (b) Effects of GH in the absence of DH were established by assuming that alleles at each locus assorted independently. (c) In the DH scenario, physical linkage of loci occurred (but note this is the only difference from the GH scenario).

We generated results on how four main factors affect divergence over time. The first three were the average per locus strength of selection (S), the gross migration rate (m), and the rate of origin of divergently selected mutations. The fourth was “genome architecture,” a shorthand expression referring to the combination of the overall size and structure of the genetic map, the order and locations in which mutations arose, and the occurrence of physical linkage. By varying these factors we arrived at general predictions about the relative importance of DS, GH, and DH for the establishment of divergently selected mutations, for patterns of locus-specific and genome-wide differentiation, and for the evolution of reproductive isolation in the face of gene flow.

We report that hitchhiking could cause many more mutations to establish than DS alone. In cases when at least a few divergently selected mutations established, hitchhiking could lead to much higher values of FST and much lower values of effective migration rates than DS alone, indicating that hitchhiking can be of paramount importance in generating reproductive isolation. However, these effects of hitchhiking were mostly attributable to GH.

Methods

GENERAL OUTLINE OF THE MODEL

We used an individual-based model (termed BU2S for “Build Up to Speciation”) written by the first author. Random numbers for stochastic processes were generated using the Mersenne Twister (Saito and Matsumoto 2006). Source code (C programming language) is archived at http://sourceforge.net/projects/bu2s/files/. Data are archived at the Dryad Digital Repository (Flaxman et al. 2013).

Representation of individuals and their genomes

An individual in the model was defined by the combination of alleles it possessed at L diploid loci in its genome. These L loci were spread across a map of total length M cM having C autosomal chromosomes. For simplicity, we assumed that each chromosome had the same map length; that is, the length of the kth chromosome was Mk = M/C, and we ignored sex chromosomes, centromeres, inversions, or any other recombination modifiers. L/C loci were allocated to each chromosome, and the location of each locus was drawn from a uniform random distribution on [0, Mk]. This method creates distances between adjacent loci that follow an exponential distribution with mean Mk/(1 + L/C).

The ith individual's genotype at the jth locus was denoted gij. We made the simplifying assumption that at any given time, there could only be a maximum of two alleles segregating at a given locus (i.e., the only alleles possible at locus j were Aj and Bj), as commonly observed in SNP-based genome scans (e.g., Nosil et al. 2012). Generations were nonoverlapping, and the total population was fixed at N individuals. We studied genomes consisting of L = 1000 or 6000 loci, with new, divergently selected mutations introduced once every G = 1 or 200 generations in populations of N = 1000 or 20,000 individuals (Table 1).

Table 1. Parameters and variables used in simulation results presented here
Image

Populations and demes

We examined a two-deme, discrete space scenario, where within a deme the population was well mixed, and the average gross migration rate between demes was m (independent of genotype). We focused on a two-deme scenario because it is the logical starting point used in many models, it makes our work have maximum comparability with previous studies (Feder and Nosil 2010; Feder et al. 2012b), and many empirical systems—such as herbivorous insect populations feeding on alternative host plant species or adjacent lake and stream forms of fish—approximately fit this framework.

We assumed that there was divergent selection between the two demes, setting the stage for population differentiation and ecological speciation. Specifically, we assumed that all of the L loci could be targets of divergent selection, though with the distributions of selection coefficients we used, many of the loci were nearly neutral in the simulations (i.e., selection coefficients close to zero; see below and Fig. S1). Ancestral alleles, Aj, were favored in one deme (Deme 1), whereas derived alleles, Bj, were favored in the other (Deme 2).

Let wj1(gij) represent the contribution locus j makes to an individual's fitness in Deme 1. We defined wj1(AjAj) = 1+Sj, wj1(AjBj) = 1+(1 - Hj)Sj, and wj1(BjBj) = 1, where Hj is the dominance coefficient associated with allele Bj. In results shown here, we assumed codominance (Hj = 0.5) and that divergent selection was symmetrical across the demes, so that in Deme 2 we had wj2(AjAj) = 1, wj2(AjBj) = 1 + HjSj = 1 + 0.5Sj, and wj2(BjBj) = 1+Sj.

Initialization of simulation runs

We initialized simulations with individuals evenly distributed between demes (i.e., N/2 in each). In one set of simulations, Demes 1 and 2 were initially homogeneous such that all individuals in both demes were homozygous for ancestral alleles at all loci (i.e., gij = AjAj for all i, j), and divergence thus built up solely from new mutations. In a second set of simulations, we started with populations in which 1, 3, 5, 8, or 10 mutations of large effect (Sj = 0.5) were present initially at migration-selection balance (though only results with three mutations of large effect are shown due to space limitations). We explored the latter because it is possible that (1) a small number of large-effect mutations might act as important early seeds to further divergence, reflecting a key element of the verbal argument for DH (Via 2012), and (2) mutations of large effect are more likely to be favored and establish when a population is far away from an adaptive peak, as would be the case at the start of the simulations for Deme 2 (Fisher's geometric argument: Fisher 1930; Barton et al. 2007; Schluter et al. 2010; Rogers et al. 2012). Due to the assumptions above, in either initialization scenario, new mutations represented novel adaptations in Deme 2 (and maladaptations in Deme 1).

Sequence of iterated steps following initialization

  • 1.Demographic processes.

In each generation, each individual could migrate to the other deme with probability m. After migration, sexual reproduction occurred within each deme, after which the offspring replaced the parents, completing the life cycle and producing the population for the start of the next generation. Individuals were hermaphroditic and selfing was not allowed. Natural selection was simulated to occur together with reproduction: an individual's probability of contributing a gamete to a given offspring was proportional to the individual's relative fitness in its current deme (but see description of DS inheritance mechanics below). We assumed a multiplicative fitness scheme: given that it was in Deme l, the ith individual's fitness, Wil, was defined as math formula. The effective migration rate for Deme l in generation t, math formula, was calculated as the proportion of reproducers that had just migrated from the other deme (i.e., “effective backward migration rate,” sensu Vuilleumier et al. 2010). We note that this is an individual-level measure of effective migration, not a locus-specific measure. We calculated and analyzed FST values as locus-specific measures of divergence using the standard formula math formulawhere HT was the total observed heterozygosity at a given locus at a given time step and HS was the expected heterozygosity based upon each patch's observed heterozygosity (Hartl and Clark 2007).

We assumed that recombination events during meiosis were independently identically distributed along chromosomes. The locations and number of recombination events per chromosome were determined by drawing distances between consecutive recombination events from an exponential distribution with a mean distance of 50 cM. This procedure implies that a chromosome can have from zero to any number of recombination events in a given round of gamete formation and that there is no minimum distance between consecutive events.

  • 2.Mutation.

Once every G generations, a new mutation, Bj, that was favored in Deme 2 was introduced at a random locus j in a random individual in either deme. We used G = 200 generations to reflect the fact that beneficial mutations should be relatively rare; we used G = 1 to explore the effects of removing mutation limitation. For runs with G = 1, standing variation built up across the genome over time in a natural way due to the interaction of mutation, migration, selection, and drift (Yeaman and Otto 2011). That is, with G = 1, at any given time there were multiple loci that were variable whose alleles were not necessarily at equilibrium frequencies. When combined with our population sizes, our mutation rates span a wide and realistic range of values (reviewed by Sung et al. 2012). Based on the empirical expectation that most beneficial mutations will be of small effect, the selection coefficients, Sj, associated with new mutations, were drawn from an exponential distribution with mean S (Fig. S1).

During a given simulation run, a specific mutation was introduced only once at each of the L loci, with only one mutation introduced at a given time point. As expected—because a new mutation starts at a frequency of only 1/(2N)—many new mutations were lost quickly due to drift (Yeaman and Otto 2011). A new mutation could be lost due to selection if it arose in the “wrong” deme or linkage group. Although only a single mutation was introduced at each locus, the large number of loci considered means that many mutations occurred within any small span of the genome. For example, in runs with L = 6000 loci in a genome of M = 50 cM, the expected distance between adjacent loci is only ∼ 0.0083 cM. That implies that were around 240 (= 2 × 1/0.0083) loci within 1 cM of most loci. We further note that 0.0083 cM between adjacent loci is the average. As explained above, the distances between adjacent loci followed an exponential distribution with mean Mk/(1 + L/C). With that distribution, the majority of distances would have been even smaller than the average (the median of an exponential distribution is less than the mean; see Fig. S1), and there was no lower limit on how close together adjacent loci could be. Hence, just by chance, there were numerous very tight clusters of loci along each chromosome.

USING THE MODEL TO STUDY HITCHHIKING AND EFFECTS OF GENOME ARCHITECTURE

As noted above, we compared results among three scenarios (1) direct selection only (DS scenario), (2) DS + GH in the absence of DH (GH scenario), and (3) DS + GH + DH (DH scenario; Fig. 1). For the DS scenario, we constructed simulations in which the fate of each mutation was independent of divergence at any other loci in the genome. That is, an allele's probability of being selected to be inherited by a given offspring was proportional to the "allele's" weighted fitness in its current deme (i.e., the probability was derived from a standard population genetic model of divergent natural selection operating on one diallelic locus; Fig. 1a). The DS scenario was "not" intended to represent a real organismal system. Rather, it was conceived as an in silico system that allowed us to model and partition the effects of DS in the absence of hitchhiking (i.e., this is a statistically useful experiment generally not possible to perform with organisms other than perhaps those differing only by a single mutation).

In the GH scenario, all new mutations were constrained to be unlinked (Fig. 1b), essentially arising on separate chromosomes. Independent assortment of all loci (mutations) in the GH scenario was also not meant to be biologically realistic, but rather represented a computer experiment that enabled us to measure the combined effects of DS and GH without DH. Finally, to consider the combined effects of DS, GH, and DH, new mutations arose in a map of fixed total size, so that many loci would be on the same chromosome with distances < 50 cM; that is, in this latter case we considered explicit chromosomes with explicit maps and physical linkage (Fig. 1c). Thus, differences observed between the DH and GH scenarios could be attributed to the effects of physical linkage.

To make proper “apples-to-apples” comparisons among scenarios, we set up parallel runs of the three scenarios that had the exact same sequence of new mutations occurring (i.e., the same Sj values occurring in the same chronological order and generation times in the runs). Specifically, for a given set of parameters (i.e., a specific combination of C, M, L, G, S, and m), we created at least 10 distinct mutation sequences, and then carried out parallel DS, GH, and DH simulations with each of these sequences.

We quantified the cumulative effects of DS, GH, and DH using standard metrics of divergence and progress toward speciation: (1) the number of mutations that successfully established, (2) effective migration rates math formula, and (3) FSTj(t) (the value of FST for locus j in generation t). Many previous studies have been able to predict or measure these quantities after some period of divergence. Our work is novel in representing one of the few studies to examine not just an endpoint, but also the process of the buildup of divergence across hundreds of loci from a starting point of complete homogeneity across demes. That is, with our initial conditions (above), we started from a point where there was no divergence or linkage disequilibrium for all (or nearly all) loci, and the effective migration rate equaled the gross migration rate (i.e., math formula). However, by the end of some of the runs with larger S, we reached a point where the expected effective migration rate per generation was much less than one individual per generation (see points in figures below where math formula) and the demes became effectively reproductively isolated from one another (i.e., different biological species).

For the DH scenario runs, we analyzed the relationship between physical linkage and mutation establishment by comparing the observed distances between adjacent diverged loci to the null expectation of diverged loci being randomly distributed on a chromosome. We derived the null expectation using two methods. The first was a Monte Carlo simulation method. Let n represent the number of diverged loci on a chromosome at the end of a given DH simulation run. For chromosomes observed to have n ≥ 2, we chose n of the loci "in the same chromosomal map at random" and calculated the distances between adjacent loci. By repeating the process 1000 times for each chromosome, we obtained the expected null distribution of distances between adjacent diverged loci for the absence of DH effects. The second method was a simple analytic approximation that ignored the specific maps. For n ≥ 2 loci on a chromosome of length Mk, the expected distance between adjacent independently identically distributed loci would be math formula, and the distances should follow a truncated exponential distribution, with probability density function:

display math(1)

(Cox and Hinkley 2000), where math formula is the inverse of the expected distance between loci (i.e., math formula). To use equation (1) to approximate the observed data from multiple runs in which n varied from run to run, we used

display math(2)

where math formula was the weighted mean value of math formula over the runs, and the weighting was by the number of distances observed from each run. Equation (2) is meant to be an approximation, not an exact prediction. We used this approximation because it is the simplest possible (it has just two parameters, math formula and Mk) and because it could be easily integrated (with respect to x) to give an approximation for the null cumulative distribution function of distances.

Results

LIMITED MUTATIONS IN LARGE, INITIALLY HOMOGENEOUS POPULATIONS

We first present the results obtained from an initially homogeneous population of N = 20,000 individuals that is mutation limited (G = 200 generations between mutations). The genetic map in these results consists of C = 5 chromosomes, with each chromosome containing 200 loci randomly distributed along a length of Mk = 100 cM (total M = 500; “mutation limited” runs, hereafter). This corresponds to a mutation rate of 2.5 × 10−10 mutations per locus (site) per individual per generation. Figure 2 shows the accumulation of mutations over time in these runs. S and m both varied from 0.001 to 0.1, and the S/m ratio varied from 0.01 to 100. Each line in Fig. 2 represents the mean result from 10 or more independent simulation runs having different sequences of mutations. As expected, when most mutations have Sj values that were ≪ m and/or when S was small (Fig. 2a–f,i), very few mutations were able to establish (1% or less), with or without hitchhiking. So there was little progress toward speciation, as measured by the absence of any noticeable reduction in effective migration below the gross migration rate (Fig. 3a–f,i).

Figure 2.

The number of mutations that establish over time for different combinations of S and m in the “mutation limited” runs (parameters: C = 5, M = 500, L = 1000, N = 20,000, G = 200, T = 200,000). Each line shows the mean (of 10–40 different simulation runs) number of mutations that established over multiple independent simulation runs, each run having a different sequence of 1000 randomly generated mutations. All panels have three lines, one each for DS, GH, and DH scenarios (see Methods), though in some panels there appear to be fewer lines because of overlap. In all of these results, the distribution of Sj values followed the standard exponential with mean S (as given above the panel; see Methods).

Figure 3.

Changes in the effective migration rate over timemath formula(t), for the same runs and combinations of parameters as in Figure 2. Note that the y-axis is logarithmic, and in all cases, the effective migration rate starts (at time 0) at the given gross migration rate of m.

In contrast, when S was increased to 0.05 or 0.1 (Fig. 2g–l), many more mutations established. The effects of hitchhiking (GH and DH scenarios) in accelerating speciation were most pronounced when S = m = 0.1 (Fig. 2l): around 3.5% of the mutations established in the GH and DH runs, whereas about 2.6% of the mutations established in the DS runs. When m was lowered such that S = 0.1 = 10m, even more mutations established, but the differences with and without hitchhiking were not as great (Fig. 2k). Similar trends for effective migration rates were observed in these runs (Fig. 3g–l).

FREQUENT MUTATIONS IN SMALL POPULATIONS INITIALIZED WITH A FEW DIVERGED LOCI

In contrast with Figures 2 and 3, Figures 4 and 5 show results with very different parameters: more loci and mutations (6000 instead of 1000) introduced much more frequently (one mutation every generation) in smaller populations (1000 instead of 20,000 individuals). We also used a much smaller genetic map in these runs: C = 2 chromosomes, each with a length of Mk = 25 cM. Henceforth, we refer to this parameter set as the “many mutations” set of runs. We initialized populations in these runs with three divergently selected mutations of large effect (Sj = 0.5; explained above). Previous work (Feder and Nosil 2010; Feder et al. 2012b) suggested that these parameter changes should have magnified the potential strength of DH relative to the “mutation limited” runs (i.e., Figs. 2 and 3). In addition to having much tighter clusters of originating mutations, the “many mutations” runs had a much higher mutation rate: 1.67 × 10−7 mutations per locus per individual per generation. Indeed, the “many mutations” runs represented an extreme case: one in which the rate of new adaptive mutations is higher than it is likely to be in nearly any real population of just 1000 individuals.

Figure 4.

The number of mutations that establish over time for different combinations of S and m in the “many mutations” runs (parameters: C = 2, M = 50, L = 6000, N = 1000, G = 1, T = 6000). Each line shows the mean from 50 different simulation runs, and the interpretation of lines is the same as in Figure 2. In all of these results, the distribution of Sj values followed the standard exponential with mean S (as given above the panel; see Methods), but the runs were initialized with three mutations of large effect (Sj = 0.5 for each of the three) already established.

Figure 5.

Changes in the effective migration rate over timemath formula(t), for the same runs and combinations of parameters as in Figure 4. Interpretation of lines is the same as in Figure 3.

In spite of the large changes in parameters, several results were qualitatively similar to those in the “mutation limited” runs. It was still the case that few mutations (< 1%) established when selection was very weak (Fig. 4a–c) and that having many mutations of small effect contributed very little to reducing effective migration (Fig. 5a–f). Additionally, there were cases when hitchhiking (DH or GH) greatly accentuated mutation accumulation and reproductive isolation compared to DS (Figs. 4e,i,l and 5h,i,l); note that these accentuations occurred for the same or similar values of S and m as in the mutation limited case.

There were two main differences, however. First, with many mutations, individual mutations were more likely to establish at lower mean values of selection coefficients than under the mutation-limited scenario (compare Fig. 2d–f to Fig. 4d–f). Accordingly, effective migration declined more in the “many mutation” runs, owing simply to the greater number of divergently selected mutations that established (compare decreases in Fig. 3 to corresponding panels of Fig. 5). This was true even when S < m (Figs. 4l, 5l). Second, there were now cases in which DH enhanced the establishment of more mutations than GH. In particular, in Figure 4f, many more mutations established with DH than GH, whereas the number establishing with GH was only slightly greater than DS. However, this did not translate into noticeably greater reproductive isolation being produced by DH because the mutations were of such small effect (Fig. 5f). As was true in the mutation-limited case (Figs. 2 and 3), when significant decreases in effective migration occurred, the GH scenario was as effective as the DH scenario in facilitating mutation establishment (Fig. 4g–l) and reducing the effective migration rate (Fig. 5g–l).

THE DYNAMICS OF DIFFERENTIATION AT INDIVIDUAL LOCI AND OF GENOME-WIDE COUPLING OF LOCI

Insights into the dynamics of individual loci versus the genome as a whole can be gleaned from time series of the FST values. Twelve examples are shown in Figure 6. Six of the examples depicted (Fig. 6a–f) were from mutation-limited runs and the other six (Fig. 6g–l) from many mutation runs. In the former case (Fig. 6a–f), the first mutation to successfully establish reached an FST value reflecting selection-migration balance. In the DS cases (Fig. 6c,f), this was also true for all subsequent successful mutations. However, in the hitchhiking examples (Fig. 6a,b,d,e), there was a synergy between successful mutations: each successful mutation contributed to elevating the FST values of all other successful mutations in both the DH (Fig. 6a,d) and GH (Fig. 6b,e) scenarios. Two features of this synergy created by hitchhiking were noteworthy, especially in the case where S = m = 0.1 (Fig. 6a,b). First, the overall trajectory of FST values followed a sigmoid shape, making a somewhat abrupt evolutionary transition from lower to higher values. Second, as this transition occurred, the FST values for different loci went from being low and variable (reflecting gene flow and the different Sj values of the loci) to high and overlapping (reflecting that allele combinations were rarely broken up, regardless of the Sj values of individual loci).

Figure 6.

FSTj(t) in 12 different example runs of the model. The top row (a–c) gives examples from three of the runs whose results are included in Figures 2l and 3l; the second row (d–f) gives examples from Figures 2k and 3k; the third row (g–i) from Figures 4f and 5f; and the bottom row (j–l) from Figures 4i and 5i. (a)–(f) are mutation limited runs, whereas (g)–(l) are runs with many mutations; the x-axes differ accordingly. FSTj(t) values for each locus in the genome are plotted separately, though most stay at zero (a default value) because mutations at those loci are quickly lost. Successful establishment of a mutation is indicated by a line that rises nearly vertically from zero and then levels off.

Panels g–i of Figure 6 focus on examples from cases with many mutations and parameters that caused the biggest differences between DH and GH in mutation accumulation (i.e., three examples from runs shown in Fig. 4f). In all three panels, the uppermost lines depict loci initialized with mutations of large effect (Sj = 0.5; see Methods). These lines were higher in Fig. 6g,h (DH and GH cases) than in Fig. 6i (DS case) because of GH: the three large effect mutations worked in synergy to reduce gene flow. Subsequent mutations of small effect (mean S = 0.01, exponential distribution) also reached higher levels of FST with hitchhiking for the same reason. Additionally, these mutations reached higher FST values with DH than with GH alone. However, genome-wide divergence never congealed in either case because collectively these mutations did little to reduce effective migration rates (Fig. 5f).

In contrast, with slightly larger mean S ( = 0.05; Fig. 6j,k) and DH or GH, divergence was rapidly coupled across all loci in the genome. After several mutations established, FST values for all loci, whether mutations were large or small, quickly approached one in spite of the high migration rate (m = 0.1) because effective migration rates genome-wide plummeted (Fig. 5i). The presence or absence of linkage had no detectable effect on this result (i.e., GH and DH were equally effective: Figs. 4i, 5i).

DH AND THE CLUSTERING OF MUTATIONS IN “ISLANDS” OF DIVERGENCE

Using results from the many mutations runs, we explored whether successful mutations tended to accumulate in localized “islands” of divergence, or whether they were randomly distributed along the chromosomes. Statistical signatures of DH in the locations of established loci were most obvious for the smallest values of S and the S/m ratio (Fig. 7a–c,f). In these cases, there were more pairs of loci than expected that were separated by 0.01 to 1 cM (Fig. 7a–c,f). However, in any parameter sets that led to significant drops in effective migration rates (Fig. 5g–l), the agreement between our null distributions and the observed distances was very high (Fig. 7g–l).

Figure 7.

Tests for genomic clustering of divergence. Shown are cumulative distribution functions (CDFs) for the distances between adjacent diverged loci on chromosomes in the DH scenario with “many mutations.” Each panel plots observed results (red line with “x” symbols) from a combined 50 simulation runs for a given combination of S and m values (shown above panel). The green line is the expected CDF calculated from 1000 Monte Carlo simulated datasets assuming the mutations were randomly distributed within the same genetic maps as those that produced the observed data. The dashed black lines are 95% confidence limits from the latter simulated datasets. The thick dashed blue line is the analytical prediction made from equation (2). The x-axis stops at an upper limit set by chromosome length (25 cM). See text for more explanation of methods and interpretations.

Discussion

Our simulations made it possible to partition the contributions of DS, GH, and DH over time during population divergence with gene flow. We started with populations that were completely or nearly monomorphic and fixed for ancestral alleles at each locus. Thus, rather than viewing a snapshot of evolutionary time, we gained insights into the temporal dynamics of genomic divergence during speciation. The inclusion of these dynamics, combined with the large numbers of mutations and recombination distances used, extends previous theory in ways that have recently been highlighted as priorities for investigation (Yeaman and Whitlock 2011).

EFFECTS OF HITCHHIKING ON THE ACCUMULATION OF MUTATIONS AND REPRODUCTIVE ISOLATION

The effects of hitchhiking for accelerating population divergence were highly sensitive to S and the S/m ratio. The importance of the S/m ratio for gene flow and the establishment of mutations has been studied for decades (e.g., Crow and Kimura 1970; Bengtsson 1985; Barton 2000; Yeaman and Otto 2011). However, previous work has not highlighted how the S/m ratio affects the contributions of GH versus DH to generating reproductive isolation over time. Hence, with regard to the S/m ratio, we focus our discussion on how it impacted the magnitude of hitchhiking effects.

In any parameter set, very few mutations (< 1%) established unless S ≥ 0.05. For larger values of S with mS (Figs. 2g,j,k and 3g,j,k), direct selection was primarily responsible for determining how many mutations established; neither form of hitchhiking accentuated mutation accumulation or reproductive isolation compared to DS (panels g, j, and k of Figs. 2-5).

However, with S ≥ 0.05 and mS, the effects of hitchhiking were pronounced. For example, for S = m = 0.1, around 50–70% more mutations established with hitchhiking (GH and DH scenarios) compared to the DS scenario (Figs. 2l, 4l). The effective migration rate, math formula, was also dramatically affected by hitchhiking in such cases. When hitchhiking scenarios displayed steady declines in math formula, so did DS. However, even in the mutation-limited runs, the magnitude of the drop in math formula could be huge with hitchhiking compared to DS. For example, while the difference between DS and the hitchhiking scenarios in Figure 3k may appear small on the logarithmic scale, math formula in the DH and GH scenarios is still around five times lower than in the DS scenario. In Figure 3l, the difference is ∼100-fold. In the latter case, this meant that the number of effective migrants per deme per generation had dropped to about three with hitchhiking, whereas with only DS, there were over 300 effective migrants per deme per generation. Under the many mutation simulations, these differences were further accentuated (Fig. 5i,l).

GENERALLY MINOR EFFECTS OF DH

The only difference between the GH and DH scenarios is that the DH scenario allowed for the physical linkage of mutations (Fig. 1). Hence, in cases when results from GH and DH overlap, this indicates the absence of significant effects of linkage (and thus of DH). When divergence went beyond individual loci—that is, when effective migration rates dropped over time and FST values were uniformly elevated—the divergence was attributable to GH with no obvious accentuation from DH. Indeed, the major differences visible in panels g–l of Figures 2-5 are between DS and hitchhiking (the GH and DH results are largely overlapping). Furthermore, in some of the cases in which hitchhiking had the greatest effect relative to DS, we observed that if anything, DH hindered mutation establishment slightly compared to GH (Fig. 4h–l), consistent with the static time simulations of Feder et al. (2012b).

We note here that our “many mutations” parameter set modeled situations that, based upon previous results (Feder et al. 2012a; Via 2012), are highly favorable for DH to accelerate population divergence. These situations included (1) early mutations of large effect followed by many frequent mutations, (2) a “small map” with just two chromosomes and greatly reduced recombination distances between loci, and (3) small populations. Yet, for various combinations of m and S, results from these situations did not reveal large effects of DH in promoting reproductive isolation beyond that generated by DS or GH. We also note that our results on the magnitude of DH effects agree quite well with what has been described for DH in a single population: it typically operates over very small genomic scales, not big blocks (Charlesworth et al. 1997; Kim and Stephan 2002).

There are at least two reasons that DH may have played a relatively minor role. First, the benefits of linkage can be partially cancelled out by Hill–Robertson effects (Hill and Robertson 1966; Feder et al. 2012b; Via 2012). In early or intermediate stages of divergence, alleles are not diagnostically fixed between populations but rather are present at selection-migration equilibrium. Hence, there is a reasonable chance that a new mutation could occur in the “wrong” linkage phase on a chromosome with an allele beneficial for the alternate habitat. Tight linkage would thus increase the probability of the new mutation being lost. Similarly, with equal population sizes in the two demes (as we assumed), new mutations occur half the time in the population they are favored in and half the time in the population they are disfavored in. In the latter case, tight linkage can reduce the mutation's chances because recombination away from the wrong genetic background is less likely.

Second, DH rests mainly on particular mutation conditions: new mutations of minor effect (low Sj values) establishing due to their tight linkage to already diverged, strongly selected loci. This creates a bit of a “catch 22”: the effects of DH are postulated to be most pronounced for mutations of small effect (as was, indeed, observed: Fig. 7), yet small-effect mutations contribute relatively little to increasing reproductive isolation (Figs. 3a–f, 5a–f).

EFFECTS OF HITCHHIKING ON GENOMIC HETEROGENEITY DURING DIVERGENCE

With regard to the differentiation of individual loci as measured by FST, hitchhiking produced results that were dramatically different than DS. For the DS cases, FSTj(t) values for a given locus j reached migration-selection equilibrium, regardless of other loci (Fig. 6c,f,i,l). However, the dynamics of FSTj(t) with hitchhiking were quite different. Early in the process of divergence, FSTj(t) values were relatively low and heterogeneous across loci (just as they appeared for the duration of a DS run; Fig. 6a,b,d,e). However, after just a few mutations with moderate or large Sj values had accumulated, their effects on each other—that is, the effects of GH—became apparent: the establishment of each subsequent mutation led to a jump in FSTj(t) for all diverged loci. The result was a transient period where there was a range of elevated but still heterogeneous FSTj(t) values. Weakly selected mutations still displayed lower FSTj(t) values, whereas strongly selected mutations displayed very high FSTj(t), but each was elevated compared to what it would have been with DS only. The further establishment of new, divergently selected mutations led to further increases in FSTj(t), but also decreased the heterogeneity of FSTj(t) among loci as GH became increasingly enabled: all FSTj(t) values began to converge to 1. That is, with GH, we observed the congealing of the genome as individual loci converged to a high mean level of genomic divergence and reproductive isolation increasingly became a characteristic of the genome, rather than of individual loci, as embodied by some formulations of the biological species concept (Wu 2001; Feder et al. 2012a).

At least two additional aspects of the dynamics of FSTj(t) are noteworthy. First, early mutations of large effect and reductions in the gross migration rate greatly accelerated the evolutionary transition from low to generally high FSTj(t). That is, in runs with hitchhiking, very strong differentiation, even at weakly selected loci, could be observed very early in the process of divergence and before there had been much decline in math formula. Second, even after FSTj(t) had nearly reached its maximum across all diverged loci, new mutations continued to accumulate and very rapidly reached maximum FSTj(t) values. Although the later mutations had little effect on observed FSTj(t), they continued to cause log-linear declines in math formula (compare Fig. 6a,b to Fig. 3l and Fig. 6j,k to Fig. 5i). These declines in math formula and the dynamics of FSTj(t) (especially the sigmoid shapes seen in Fig. 6a,b) support a recently articulated “four phase” conceptual model of speciation with gene flow (Feder et al. 2012b). The congealing of FSTj(t) values across all the differentiated loci also supports the notion that an important transition that occurs during speciation is the “coupling” of diverging loci that initially have somewhat independent dynamics to a phase in which they largely have a shared dynamic (Barton 1983; Bierne et al. 2011; Abbott et al. 2013).

SUMMARY AND PROSPECTS FOR FUTURE WORK

We have shown that when hitchhiking facilitates speciation with gene flow, the effects predominantly involve GH. However, hitchhiking effects, even associated with GH, are sensitive to the distribution of S values relative to m. Once several mutations establish, GH manifests as a synergy between loci, whether the genes are physically linked or not. GH lowers the effective migration rate and thereby increases the probability that subsequent mutations—especially those with math formulaSj < m—can establish.

However, there are empirical examples in nature in which genes conferring reproductive isolation appear to be concentrated in localized areas of the genome (Feder et al. 2012a). Moreover, even when divergence is generally widespread across chromosomes, there is still often a disproportionate clustering of accentuated divergence into a few regions (Strasburg et al. 2009; Hohenlohe et al. 2010; Lawniczak et al. 2010; Heliconius Genome Consortium 2012; Jones et al. 2012; Nadeau et al. 2012; Roesti et al. 2012). Do such examples indicate an important contributing role for DH? In our results, there were indeed cases in which DH enhanced mutation accumulation (Fig. 4) and in which there was statistical evidence of clustering (Fig. 7), but the relative magnitudes of the mutations were so small that reproductive isolation was not greatly impacted.

Of course, with the addition of specific features of a system, a role for DH in accelerating reproductive isolation could be possible. For example, epistasis involving a two-allele system in which the fitness of segregating variants is dictated by an individual's genotype at other loci, would be expected to place a greater premium on linkage and enhance the role of DH in population divergence. An example could be a locus for habitat choice in which the fitness values at the choice gene are affected by the alleles at associated performance genes the individual possesses and the habitat it finds itself in. Certain types of phenotype-dependent assortative mating systems and systems of intragenomic conflict such as meiotic drive could similarly increase the significance of physical linkage (Burt and Trivers 2006; Crespi and Nosil 2013). These types of scenarios merit further theoretical investigation, particularly as the aforementioned empirical trends require explanation.

However, while hypothetical examples that might increase the role of DH can be constructed, the implication from our current results is clear: “islands” of divergence contributing to RI should not be automatically assumed to be products of DH alone. DH is one possible hypothesis that should be considered along with other hypotheses about genomic features associated with reduced recombination such as centromeres, meiotic drive, chromosomal rearrangements (including inversions), and/or genomically concentrated large-effect mutations replacing genomically dispersed small-effect mutations in cases of quantitative traits with an intermediate phenotypic optimum (Yeaman and Whitlock 2011). Our simulations did not include the possibility for large genomic rearrangements, nor did we make any assumptions about optimal trait values. These and several other extensions of theory represent very promising areas for future work. Specifically, additional extensions of the work presented here could include (1) additional structural features of the genome, (2) epistatic interactions, (3) elements of genomic conflict, and (4) nonrandom distributions of genes. We note, however, that the latter extensions are all special cases; our general model implies limited effects of DH. Thus, in empirical cases of nonrandom clustering, it is perhaps likely that one of these other factors is involved.

Finally, while our intention was to study de novo speciation under continuous gene flow, better integrating secondary contact models involving initial periods of allopatric divergence is also important. In such cases, DH might assume a greater role, for example, in impeding the erosion of differentiated gene regions containing many small effect mutations that accumulated in allopatry. Thus secondary contact scenarios could also provide a potential explanation for certain cases where reproductive isolation appears to be concentrated in localized areas of the genome.

Associate Editor: B. Fitzpatrick

ACKNOWLEDGMENTS

This work used the Janus supercomputer, which is supported by the National Science Foundation (award number CNS-0821794) and the University of Colorado Boulder. The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver, and the National Center for Atmospheric Research. SMF was supported by the University of Colorado Boulder and its Department of Ecology and Evolutionary Biology. JLF was supported by grants from NSF and the USDA. PN is funded by a European Research Council Starter Grant (NatHisGen). The authors declare no conflicts of interest.

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