Divergence with Gene Flow: Models and Data

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Divergence with Gene Flow:
Models and Data
Catarina Pinho1 and Jody Hey2
CIBIO, Centro de Investigação em Biodiversidade e Recursos Genéticos, Universidade do
Porto. Campus Agrário de Vairão, 4485-661 Vairão, Portugal; email: [email protected]
Department of Genetics, Rutgers University, Piscataway, New Jersey 08854;
email: [email protected]
Annu. Rev. Ecol. Evol. Syst. 2010. 41:215–30
Key Words
First published online as a Review in Advance on
August 19, 2010
disruptive selection, divergence population genetics, isolation with
migration, linkage disequilibrium, speciation, species, sympatry
The Annual Review of Ecology, Evolution, and
Systematics is online at ecolsys.annualreviews.org
This article’s doi:
c 2010 by Annual Reviews.
Copyright All rights reserved
Since Darwin first proposed that new species could arise without geographic
separation, biologists have debated whether or not divergence occurs in the
presence of gene exchange. Today we understand that new species can diverge while exchanging genes, depending on the strength of disruptive natural selection and the factors that affect the linkage relationships of genes
under disruptive selection. This mode of diversification—divergence with
gene flow—includes sympatric speciation, in which gene exchange occurs
since onset of divergence, and secondary contact following a period of geographic isolation, as well as all sorts of situations in which gene flow happens
intermittently. In recent years, statistical tools have been developed that can
reveal the action of gene flow during divergence. Isolation-with-migration
(IM) models include parameters for population size, time of population separation, and gene exchange, and they have been used extensively to estimate
levels of gene exchange. A survey of studies that have used these models
shows that a plurality find little evidence of gene flow; however, many report
nonzero gene exchange.
7 October 2010
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Gene flow: the
movement of alleles
between partially
separated populations
caused when
individuals who have
one parent from each
population reproduce
and pass genes from
one population into
the other
association of alleles
among different genes,
such that some alleles
tend to be found
linked to particular
alleles at other genes
As clear and persuasive as Darwin was on natural selection and adaptation in On the Origin of
Species . . . , he was uncharacteristically difficult to understand when, in Chapter IV, he explained
how one species gives rise to two. Over several pages, Darwin walked the reader through his only
figure and described how natural selection could create multiple species from a single species that
inhabits a continuous large area, without the geographic separation of populations. Darwin called
his model the principle of divergence, but exactly how it is supposed to work has sometimes eluded
readers (Gould 2002, Tammone 1995).
Darwin’s emphasis on his principle of divergence (what today we would identify as a model
of sympatric speciation) invoked a strong critique by Moritz Wagner (1873), who argued that
migration and reproduction across the range of the species (what we would today identify as gene
flow) would impede the divergence process. Darwin acknowledged this point in the sixth edition
of this book but maintained that his model was appropriate for most cases of speciation.
It is striking that the Darwin/Wagner debate anticipated, by roughly 100 years, the energetic
debate over sympatric speciation that emerged in the late twentieth century. Neither had an
inkling of genes or even the rudiments of an appropriate inheritance model, and yet Wagner put
his finger precisely on that factor that can prevent divergence. If Darwin’s model was to be correct,
and if his principle of divergence truly described a mechanism for the origin of species, then it
must be possible (and common) for two species to arise from one species in the presence of gene
In this review of gene exchange and divergence, we begin by developing a general genetic perspective on how populations become genetically differentiated in the presence of gene flow. The
literature on this process is quite extensive [see recent reviews by Bolnick & Fitzpatrick (2007),
Butlin (2005), Coyne & Orr (2004), Gavrilets (2003), and Via (2001)]; however, by focusing narrowly on the factors shaping linkage disequilibrium, we hope to present a simplified and accessible
summary. We then turn to a review of recent methodological advances and applications to see
what has been learned in recent years about the frequency and magnitude of gene exchange during
We conclude that speciation has occurred if two closely related populations are genetically incompatible in a way that prevents hybrids from being formed or that causes low hybrid fitness.
But just how much divergence is required to create a genetic barrier? At minimum, we can say that
a reproductive barrier requires that populations be different at two genes (Dobzhansky 1937). It
is difficult, though not impossible (Orr 1991), to explain speciation and low hybrid fitness with a
model in which two populations are fixed for different alleles at just a single gene. This is because
a difference between populations at one gene, for alleles that cause low fitness in a heterozygote, is
unlikely to come to pass simply because any new allele that is harmful in heterozygous condition
is quickly lost soon after it arises.
Bateson (1909), Dobzhansky (1937) and Muller (1940) independently suggested that more than
one gene is required for a new species and that the genetic basis of reproductive isolation is likely to
involve at least two interacting loci (Orr 1996). The Bateson-Dobzhansky-Muller (BDM) model
begins with recently separated populations that are both completely homozygous for allele A at
one locus and for allele B at another locus (all individuals have genotype AABB). Then at the first
locus, one population has a mutation to allele a that replaces all other gene copies at this locus,
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Figure 1
A genetic perspective on divergence. Two loci, A and B, each have two alleles favored in different
populations or portions of the range. For simplicity, both genes are represented as occurring on a single
chromosome. Chromosomal genotypes are shown in the population or geographic region where they are
common. (a) Two allopatric populations, each fixed for different alleles at the two loci. (b) Divergence under
disruptive selection that leads to geographic differentiation at both genes. The range of the species includes
areas at the extremes of the selective gradients, in which only a single chromosome is common (on the left
and right), as well as a central region where all four possible chromosomes may occur. The numbers of each
chromosome type in the central region depend on the strength of disruptive selection, levels of assortative
mating, and rates of recombination in individuals that are heterozygous at both genes. (c) If heterozygotes
are not rare and recombination breaks down linkage disequilibrium between the A and B genes, then
polymorphisms at other genes not under selection may be widespread throughout the range of the species.
(d ) If recombination is low or is suppressed in heterozygotes (indicated by a black bar representing a
chromosomal rearrangement on one chromosome type), then other genes will also come into linkage
disequilibrium with the selected genes.
whereas in the second population at the second locus, a new allele b replaces all other copies of
the gene. Figure 1a shows the two chromosome types that would be found in their respective
populations (assuming, for simplicity, that both loci are on the same chromosome). If a hybrid is
produced, it will be heterozygous at both loci and will have genotype AaBb. But alleles a and b
have never been together on the same individual before, hence their interaction has never been
subject to the scrutiny of selection. It is possible that they interact negatively, causing the hybrids
to have reduced or possibly zero fitness, in which case speciation would be complete. If two genes
are not sufficient and hybrid fitness is only partly reduced, we can then imagine adding more loci
to the model with new alleles becoming fixed in just one population and further contributing to
low hybrid fitness.
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7 October 2010
Divergence Without Geographic Separation
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Disruptive selection
selection): when
selection favors
individuals with traits
at the opposite ends of
a distribution and
removes individuals
with intermediate
The BDM model invokes two mechanisms for preventing the spread of alleles that have become
fixed in one population into the other population. The first is geographic separation, and the
second, which kicks in when hybrids are produced, is epistatic incompatibility between alleles that
have become fixed in different populations.
The flip side of the BDM model is that if hybrids are produced and are not completely sterile,
then it may happen that an allele that is fixed in one population does spread through the other
population. When this occurs, then both the actual amount of divergence and the potential for
epistatic incompatibilities are reduced.
These ideas can be used when we turn to the case of two populations that are not isolated from
each other, but that overlap or abut each other and that exchange genes with each other. Although
the BDM model does not apply directly to this case, we learn from the BDM model that if the
two populations are to diverge in the face of gene exchange, then (a) there must be forces acting
against the spread of genes across both populations, and (b) there must be forces that lead to a
stronger barrier to gene exchange and that allow divergence to increase.
Natural selection can certainly create some divergence, at least at the level of individual genes,
within an interbreeding population. For example, Figure 1b depicts a single population in which
a new allele a is favored and replaces the A allele, but only over a portion of the population’s
range (that is, the A allele is favored over the remainder of the range). In this kind of model,
the disruptive selection itself is the direct cause of the limited spread of alleles, and there are no
physical barriers to gene movement, nor is there an epistatic cause of low hybrid fitness as in the
BDM model. Rather, low fitness individuals occur when specific genotypes occur in the wrong
part of the range of the population.
An investigator coming upon the situation in Figure 1b would perceive a single species that
has a strong reciprocal shift in A and a allele frequencies across the range. Other polymorphic loci
that are not under disruptive selection cannot be expected to show the same regional limitations.
For example, Figure 1c shows the entire range of the species being polymorphic for two alleles,
B and b, at a second locus that, in this case, is not under disruptive selection.
However, the situation in Figure 1b can be the seeds for continuing divergence that spreads
across the genome to affect more than just the genes that are initially under disruptive selection.
To see how this works, suppose that the B locus is actually closely linked to the A locus and that
the b allele arises on a chromosome bearing the A allele. Then, for two reasons, we may not expect
to find b alleles on the same chromosomes as a alleles. First, the opportunity for recombination,
which could move a b allele onto a chromosome carrying an a allele, is low because the geographic
distribution of the A and a alleles, due to disruptive selection, means that there are fewer Aa
heterozygotes than would be expected under random mating. Second, even when Aa heterozygotes
occur, the tight linkage between the two loci means that recombination events are rare. In short,
the disruptive selection on the A and a alleles can cause divergence at linked loci, and it can cause
a shortage of some gene combinations (that is, ab and AB chromosomes) relative to that expected
under random mating and free recombination. This shortage is called linkage disequilibrium, and
it is a key prerequisite for speciation in most models of divergence in the absence of a geographic
barrier (Bolnick & Fitzpatrick 2007, Felsenstein 1981, Gavrilets 2003, Rice & Hostert 1993).
The central point here is that disruptive selection on a locus essentially creates a place in the
genome where additional divergence can accumulate. The length of this region depends on the
recombination rate around the selected locus, but more importantly the selection regime itself can
directly reduce the number of heterozygotes that are required for recombination to breakdown
linkage disequilibrium.
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7 October 2010
In the absence of geographic separation, disruptive selection must be an ultimate cause of
divergence. However, there are other key factors that come into play because of their effect on
the level of linkage disequilibrium around the gene or genes that are the targets of disruptive
selection. For example, initial linkage disequilibrium can occur when two geographically separated populations, which have diverged slightly due to selection or drift acting differently upon
each population, come back into contact. Upon coming into contact and exchanging genes, the
divergence process may possibly continue if disruptive selection continues to act sufficiently to
maintain linkage disequilibrium.
Another important factor that can facilitate low recombination is assortative mating associated
with the traits that are the subject of disruptive selection. In such circumstances, individuals tend
to mate with others that share their genotype because of indirect effects of the alleles that are under
disruptive selection. This could be simply due to the geographic structuring that occurs because
of disruptive selection, so that most individuals come into contact only with others of the same
genotype (Kirkpatrick & Ravigné 2002), or it could be because of a pleiotropic effect in which the
alleles under disruptive selection also affect other traits related to mate choice. Probably the most
discussed example of this, which applies to many phytophagus insects, is when genetic variation
that affects food choice also leads to assortative mating because organisms tend to mate at feeding
sites (Bush 1994).
Yet another major factor can be a chromosomal rearrangement that is itself in linkage disequilibrium with the locus under disruptive selection. This is represented in Figure 1d as a black
bar that occurs on chromosomes with the a allele and that spans the A and B loci. Such a suppressor of recombination can have a major impact on the divergence process. In the first place,
the low recombination and the regime of disruptive selection causes and sustains strong linkage
disequilibrium around the A locus. Second, this disequilibrium comes to affect other loci that
lie within the span of the rearrangement. Third, additional alleles at other loci that interact with
the polymorphisms at the A locus can arise and contribute to an even stronger regime of disruptive selection. To understand this, suppose that allele b, which is linked to allele A, is favored over
exactly the same range as allele A. This could be because this allele is suited to the same particular
regime of disruptive selection (e.g., as a modifier of the effects of the A allele), or it could be that
allele b reduces the rate at which individuals with allele A chromosomes reproduce with individuals
bearing chromosomes with the a allele. If it is the latter, then we have the beginnings of a process
of reinforcement of a reproductive barrier and an increase in assortative mating.
The stronger the reduction in recombination or the more genes affected, then the more opportunity there is for the recruitment of variation at additional loci that respond to the regime of
disruptive selection; thus, the faster will the two populations diverge and the sooner will speciation
be complete. The association can also act to limit the effects of selective sweeps, which would contribute to the similarity of the two populations by hindering the spread of alleles occuring in the
region of reduced recombination that would be favored in both populations (Navarro & Barton
2003). In effect, the reduced recombination that is represented in Figure 1d plays a role that is
analogous to that of the population separation in Figure 1a. They both reduce the segregation of
recombination chromosomes and the spread of alleles between the populations.
We can appreciate gene flow and diversifying natural selection as two forces in opposition, with
each force tending to limit the other. On the one hand, we have disruptive selection and whatever
factors contribute to linkage disequilibrium among loci that are differentiated by the selective
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7 October 2010
regime. But on the other hand, gene exchange and recombination reduce the linkage disequilibrium that can, in turn, prevent the recruitment of additional loci to the divergence process.
Will gene exchange prevent divergence? Not necessarily, because strong selection may prevent
some alleles from passing between populations whereas other unlinked genes that are not affected
by selection can pass between populations. In this way, there may be genes that effectively do not
experience gene exchange and behave as islands of differentiation in an otherwise uniform genome
(Wu 2001).
Evidence of a Role for Reduced Recombination in Divergence with Gene Flow
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Models that invoke a role for reduced recombination in divergence with gene flow predict that
the genetic targets of disruptive selection or genes that cause low hybrid fitness are more likely to
be found near chromosomal regions with reduced recombination (Navarro & Barton 2003, Noor
et al. 2001b, Rieseberg 2001, Trickett & Butlin 1994). In the case of the divergence of Drosophila
pseudoobscura and D. persimilis, the genes associated with partial reproductive barriers are indeed
linked to chromosomal inversions that differentiate the species (Noor et al. 2001a). The idea is that
among populations that have diverged to some extent in allopatry, and that then come together
and exchange genes, those that differ from chromosomal inversion differences are less likely to
merge back into a single species. Consistent with this is the observation in the genus Drosophila
that young pairs of sympatric species are more likely to differ by chromosomal inversions than are
species pairs that are allopatric (Noor et al. 2001b).
If divergence with gene flow is facilitated by reduced recombination, then we also expect to
see evidence of more gene flow in parts of the genome where recombination is high. These types
of patterns have been found to be associated with chromosomal breakpoints and inversions in
the apple maggot Rhagoletis pomonella (Feder 2003), in D. pseudoobscura and D. persimilis (Machado
et al. 2002), and in the sunflowers Helianthus annuus and H. petiolaris (Yatabe et al. 2007). Low gene
flow relative to other parts of the genome has also been found in low recombination areas near
chromosomal centromeres between different forms of the mosquito Anopheles gambiae (Slotman
et al. 2006, Stump et al. 2005) and between subspecies of the european rabbit Oryctolagus cuniculus
(Carneiro et al. 2009).
Gene Flow Mapping: Identifying Targets of Disruptive Selection
by Studying Gene Flow
The intertwined and oppositional relationship between gene flow and natural selection means that
we can study one of these mechanisms to learn more about the other. For an investigator trying
to reconstruct how divergence occurred for a particular pair of sister species the first question
is often whether or not gene flow occurred during the process. If it is found that gene flow has
occurred, then the investigator is placed in the paradoxical but useful position of studying the
forces driving divergence by identifying loci that have not diverged. Like an artist working with
negative images, the investigator may not be able to identify or directly study those loci most
targeted by selection, but by studying the gene exchange at other loci, the action of disruptive
selection can be revealed. Evaluating which loci have experienced gene exchange and which have
not can help to identify genomic locations that are contributing more than the average to divergence, as well as to reveal aspects of the genetic architecture that seem to be important for
speciation (Geraldes et al. 2006, Machado et al. 2002, Rieseberg et al. 1999, Turner et al. 2005,
Wu 2001).
7 October 2010
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Historically, before recent advances in population genetic methods, an inference that gene exchange was occurring during the divergence process usually depended on the biogeographic circumstances of the populations concerned. If sister species or members of a species complex are
sympatric, and if they are entirely restricted to a small geographic area, then a conclusion of gene
flow during species formation may seem parsimonious. The most famous such case is that of the
cichlid fishes of the great African lakes (Tanganyika, Victoria, and Malawi), each of which harbors
an endemic group of hundreds of closely related species (Kocher 2004, Salzburger & Meyer 2004).
If genetic data are available, such as allele or haplotype frequencies at some genes for each of
two closely related species, then population genetic methods can be brought to bear on gene flow
questions. However, estimates and conclusions regarding gene flow during divergence remain
difficult to come by. The challenge is to understand what kinds of patterns of genetic variation are
left in the genes of diverging species under different kinds of histories. For example, one of the main
lessons to emerge from studying the population genetics of divergence is that a finding of shared
variation between two species is not necessarily a signal of gene exchange. When a population
splits into two, genetic variation continues to be shared by the daughter populations for a period
of time thereafter (possibly a long period of time if populations are large and genetic drift is
slow), even in the absence of gene exchange. As divergence proceeds, loci that were polymorphic
in the ancestral population experience fixation of alleles in the descendant populations, and this
sorting of gene lineages is part of the way the populations become different. The question of gene
exchange often comes down to the challenging task of distinguishing among the two main causes
of similarity between the genes from different populations: (a) incomplete lineage sorting since
divergence began and (b) actual gene exchange.
Gene Flow Can Cause a High Variance in Divergence among Loci
Gene exchange during divergence can add significantly to the normal variance that occurs among
genes in levels of shared polymorphism between diverging populations. Genes that are linked to
those under disruptive selection may not experience any gene exchange, whereas other unlinked
genes may experience substantial gene flow. Taken together, the variance among genes in apparent
divergence between related species may be too much for a simple divergence model that does not
include gene flow. In some cases a pattern of extreme variation among genes in levels of divergence
may be so striking that a conclusion of gene exchange seems warranted even without the benefit
of statistics. Such seems to be the case for the M and S forms of Anopheles gambiae. A genome-wide
study showed that most of the genome has been homogenized through gene flow between the two
forms, except for three small “islands” of differentiation, which are thought to contain genes that
are important in keeping these forms distinct in sympatry (Turner & Hahn 2007, Turner et al.
This general approach, which focuses on variation in levels of divergence among genes, can
be statistical if we have an understanding of how much different loci should vary in a null model
that does not have any gene flow. Wakeley & Hey (1997) developed the theory for the numbers
of shared and fixed differences between two populations diverging under a simple isolation model
with no gene flow (Figure 2a). By fitting a data set to this model, and then by comparing the
actual numbers of shared and fixed differences to data simulated without gene flow under the fitted
model, a statistical test of gene flow can be conducted (Wang et al. 1997). Some of the limitations
of this approach include the fact that it lacks a way to estimate gene flow levels and that it cannot
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7 October 2010
Descendant populations
(from which data are collected)
4N2 u
M1 u –1
4N2 u
M2 u –1
time (Tu)
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Figure 2
(a) The isolation model with three population size parameters and a splitting time parameter. Effective
population sizes (N), migration rates per gene copy per generation (M), and time since population splitting
(T ) are all scaled by the mutation rate (u) of the genes being studied. Effective population sizes are assumed
to be constant. The ancestral population is assumed to have been present indefinitely back into the past.
(b) The isolation-with-migration (IM) model includes two parameters for gene flow between the sampled
populations. Gene flow rates are assumed to be constant over the time period since population splitting.
Migration arrows represent the movement of genes as time moves forward; alternative representations of the
IM model show migration arrows in the coalescent direction backward in time.
be applied to data that departs from the infinite sites mutation model under which the theory of
fixed and shared differences was developed.
Quantifying Gene Flow Using Isolation-with-Migration Models
Figure 2b shows the isolation-with-migration (IM) model studied by Nielsen & Wakeley (2001).
It is the same as the isolation model in Figure 2a; however, in addition it allows that the two
daughter populations have been exchanging genes at a constant rate since splitting. This model
includes six parameters, three for population size, two for gene exchange, and one splitting time
term. To estimate the parameters of the IM model, Nielsen & Wakeley (2001) devised a way to
approximate the likelihood:
P(X | G)P(G | ) d G,
P(X | ) =
IM: isolation-withmigration model or
where X represents the data (a genetic data set with samples from each of two related populations),
denotes the set of six parameters that are to be evaluated, and G represents a gene tree or
genealogy. The calculation of P(X |G), the probability of the data set given a particular genealogy,
requires an appropriate mutation model for the genetic data, whereas the estimation of P(G|),
the probability of the genealogy given the parameters, is based on the coalescent process under
the IM model. However, because the true genealogy is not known, it is necessary to consider all
possible genealogies in order to fully connect the data X and the model parameters ; hence,
there is the integration in Equation 1 over ψ, the set of possible genealogies (Felsenstein 1988,
Griffiths 1989).
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7 October 2010
The challenge of Equation 1 is that, because genealogies are complex, it is generally not
possible to treat the integration over genealogies in a literal sense for purposes of calculation. The
spectrum of genealogical possibilities (in terms of topology, branch lengths, and gene flow events)
is so vast that a complete evaluation of Equation 1 is virtually impossible. Nielsen & Wakeley’s
(2001) idea was to adapt the Bayesian approach of Wilson & Balding (1998), in which both G
and (that is, genealogies and model parameters) are included in a Markov chain Monte Carlo
(MCMC) simulation, to the IM model. This simulation yields an approximation that converges
on a full likelihood-based solution. In addition, because it is likelihood-based, the method brings
with it the benefits and flexibility of likelihood-based statistics, including parameter estimates and
likelihood-ratio tests, and it uses all of the information in the data that bears on the model (Fisher
1922, 1925).
One of the main results from an analysis under this method is a curve for each migration
parameter with an estimated posterior probability for every value of migration within the range
specified by the user. The location of the peak of this curve can be taken as an estimate of the
parameter. The difference between the probabilities at that peak and at a migration rate of zero
can be used for a likelihood-ratio test of the null hypothesis of zero gene flow (Hey 2010, Nielsen
& Wakeley 2001).
There have been several extensions and alterations of the original method of Nielsen & Wakeley (2001), including changes that permit the use of data sets with multiple loci (Hey & Nielsen
2004), relax the assumption of constant population sizes (Hey 2005), reduce the state space of
MCMC simulation to just genealogies (Hey & Nielsen 2007), and allow the study of more than
two related populations (Hey 2010). The methods have been implemented as various publicly
available computer programs including MDIV (Nielsen & Wakeley 2001), IM (Hey 2005, Hey
& Nielsen 2004), IMa (Hey & Nielsen 2007), and IMa2 (Hey 2010). All of these methods share
the assumption that the history of sampled populations can be reasonably represented by an IM
model. Additionally, they share the following assumptions:
Markov chain Monte
Carlo (MCMC):
a method of computer
simulation to generate
random values from a
complex probability
IM: a computer
program that performs
an IM analysis
IMa: a computer
program that performs
an IM analysis
1. The sampled genes are under selective neutrality. This does not necessarily mean strict
neutrality (e.g., a model in which deleterious mutations occur but are removed by natural
selection can still be approximated by the model); however, genes that have been the target
of recent selective sweeps will be in violation.
2. Mutation has followed the assumptions of the particular mutation model in use. The programs can accommodate the infinite sites mutation model (Kimura 1969), the HKY finite
sites mutation model (Hasegawa et al. 1985), and the stepwise mutation model for use with
microsatellites (Ohta & Kimura 1973).
3. No recombination within sampled genes.
4. Free recombination between sampled genes.
Most of the practical difficulties that arise when using these methods fall into three areas:
1. Not having sufficient data. Rarely has history been such that data from a single locus can
provide resolution on all six parameters, and even multilocus data sets may not generate
parameter estimates with small confidence intervals.
2. Having data that does not meet the assumptions of the model. Failures of this sort can be
of a great many types.
3. Slow mixing of the Markov chain simulation. Large data sets and data from some kinds
of histories usually require very long run times, and figuring out how long to run a program can be difficult. Investigators typically need to do repeated independent runs to assess
convergence of their MCMC simulations.
www.annualreviews.org • Divergence with Gene Flow
7 October 2010
Recently, other methods, which avoid some of these difficulties, have been developed for the
IM model. Although they do not use all of the information in the data, because of their reliance
on summary statistics, the MCMC method of Becquet & Przeworski (2007) and the Approximate
Bayesian Computation (ABC) method of Lopes et al. (2009) do not require the assumption of no
recombination within loci.
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Supplemental Material
We conducted a meta-analysis of published research articles that used the IM and IMa programs
to estimate gene flow among recently diverged taxa. We identified ∼250 papers that used either of
these programs and that were in the Web of Science database as of August 4, 2009. Because we were
interested in obtaining information on gene flow during divergence, we did not further consider
articles about closely related populations of the same species. All studies addressing questions of
divergence (either between species, subspecies, or lower level differentiation) were included as
long as there was also some evidence reported in the paper of differentiation (e.g., phenotypic
or ecological). We excluded several studies in which the overall inference design appeared to be
circular because groups were defined based on the same genetic markers that were used to fit the
IM model. There were also several cases where multiple studies (often by different researchers)
addressed the divergence among the same group of taxa. In these cases, we selected one study
that seemed most complete in terms of the amount of data or thoroughness of IM analysis.
Finally, we kept in our survey only articles that reported quantitative parameter estimates that
were relevant for subsequent analysis. The final list used for the meta-analysis includes 49 studies
(see Supplemental Table 1; follow the Supplemental Material link from the Annual Reviews
home page at http://www.annualreviews.org).
Many of the studies reported on multiple analyses that differed, for example, in the number
of loci included, or the data set that was used, or on the parameters estimated. In these cases, we
chose the set of parameter estimates that most fit the particular question we addressed (see below).
Many studies dealt with more than two taxa and, hence, presented multiple pairwise analyses. To
avoid the nonindependence of data points from studies with k > 2 taxa, we randomly selected
k − 1 of the pairwise comparisons. In cases where the researchers performed separate analyses for
different kinds of loci, we randomly selected one of the data sets for further analysis.
For each of the articles in the final set, we collected all the available information regarding
parameter estimates obtained using the programs mentioned above, as well as noting other relevant
methodological aspects (e.g., whether or not tests of IM assumptions were performed, whether or
not there was a test of nonzero gene flow, etc.). In many cases, only a subset of estimated parameters
were reported. We tried to maximize the available information by performing conversions using
an appropriate mutation rate (provided by the researchers, calculated based on information in
the paper, or estimated from other conversions). Because we were interested in evaluating overall
patterns of divergence with gene flow, we computed the mean of the locus-specific migration
rates in articles in which the researchers did not report an overall migration rate. The final dataset
(Supplemental Table 1) thus includes a mixture of estimates taken directly from the articles as well
as some values based on calculations from those estimates and other values reported in the studies.
The Distribution of Estimated Population Migration Rates among Studies
To understand the impact of gene flow on divergence, we would like to know the effective number
of gene migrations received by a population per generation. This value is also known as the
7 October 2010
Number of observations
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– 0 .1– 0 .2– 0 .3– 0 .4 – 0 .5 – 0 .6 – 0 7– 0 .8 – 0 .9 –1 1– 2– 3–
4 – 5– 6 – 7– 8– 9 –1 –10
2NM range
Figure 3
Counts of estimated mean 2NM values among studies. From those studies listed in Supplemental Table 1
that reported 2NM values, or from which these values could be calculated, the mean of the values for each
direction is given. Note the changing X-axis scale for larger values.
population migration rate or 2NM (Wright 1931). One simple way to estimate this is take the
product of an estimated population size parameter (4Nu) and the estimated migration parameter
for genes moving into that population (M/u) and divide by two. Figure 3 shows a histogram for
the average 2NM value (that is, the mean of the values for each direction between two populations)
from the studies that met the criteria described above. Each of the estimated values has a variance,
and any individual estimated 2NM that is not zero may or may not be statistically different from
zero. However, the histogram in Figure 3 may still be a useful estimate of the distribution of
2NM values in nature. At least two features merit notice: the sharp falloff of counts for values
much above zero and the substantial fraction and long tail of gene flow estimates that are above
In a neutral model without selection, values of 2NM greater than or equal to one can prevent
populations from accumulating much divergence (Wright 1931). For studies of divergence, where
disruptive selection may be acting, a finding of even a small amount of gene flow can be quite
interesting. However, because IM analyses fit a neutral model, without selection, they do not
reveal how much selection is acting against gene flow. Any particular value of 2NM may reflect
the actual rate of hybrid formation and reproduction, if hybrids and their progeny have high
fitness, or it may reflect only the movement of genes that manage to pass between populations
despite low fitness in hybrids and their progeny. For this reason, the estimated values of 2NM
must be treated as measures of net gene flow, because they reflect the passage of genes after some
unknown amount of selection against gene flow has acted.
Gene Flow Versus Population Splitting Time
If we take as given that complete reproductive isolation comes eventually to all sister species, then
we expect gene flow levels to decline for increasing times since populations began to separate.
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Time since common ancestor (2Ne units)
Figure 4
Mean 2NM against splitting time by taxonomic group. Five major taxonomic groups were represented by
five or more studies. Points and linear regression (LR) trend lines are shown (a small number of points fell
outside the plotted area). Splitting time is in units of 2Ne generations and is calculated as the estimated value
of the splitting parameter, divided by one half of the mean of the estimated population size parameters for
the sampled populations.
To see if this pattern is found among the surveyed studies, we calculated the mean value of 2NM
for each pair of sampled populations and plotted these values against estimates of the population
splitting time in units of 2Ne generations (that is, the timescale on which genetic drift leads to
divergence). The overall trend is for 2NM values to decline with larger splitting times; however,
there is a wide scatter to the points, and the slope of the linear regression is not significantly
different from zero (results not shown). Figure 4 shows points and trend lines determined by
linear regression for those large taxonomic groups that are represented by more than three points.
All of the slopes are negative (though the regression lines for insects and birds are very flat), so
there is a suggestion of the expected relationship. However, the main impression gained from this
analysis is that the process of gene flow decline over time occurs with a wide variance.
Can Other Factors Cause the Appearance of Gene Flow
in an Isolation-with-Migration Analysis?
Without the addition of more parameters, there are obviously many kinds of demographic history
that may be overlooked by the six-parameter IM model in Figure 2b. Structure within populations,
changes in the sizes of populations, and changes in the amount of gene flow over time are just some
of the demographic complexities that investigators might wish to study and which are not directly
accessible in a six-parameter IM model. One way to address this concern, particularly if a lot of data
are available, is to use more complicated models (e.g., Fagundes et al. 2007, Hey 2005). Another
general approach to this issue is to simulate data under models that violate the assumptions of the
basic IM model and evaluate the performance of the programs on the simulated data. Two recent
studies have examined by simulation the performance of the IM and IMa programs (Becquet &
7 October 2010
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Size of ancestral population: Na/[(N1+N2)/2]
Figure 5
Mean 2NM plotted against the relative size of the ancestral population. The size of the ancestral population
is calculated by dividing the estimated value by the mean of the estimated sizes of the descendant populations.
Przeworski 2009, Strasburg & Rieseberg 2010) in the face of violations of many of the assumptions.
Not surprisingly, the ancestral population size parameter, which is the parameter that pertains to
the oldest times in the model and, thus, one to which relatively little information in the data apply,
was often the one most likely to be affected by departures from model assumptions.
Becquet & Przeworski (2009) identified a scenario in which a structured ancestral population
becomes panmictic immediately prior to splitting into two descendant populations, which can
create falsely positive signals of gene exchange after the splitting event. This kind of history is
similar to one that has been discussed in which the ancestral population has exchanged genes with
other nonsampled populations and ends up as a result having an estimated size that is considerably
larger than the sampled populations (Won et al. 2005). The basic IM model assumes that the
ancestral population, prior to splitting, had persisted in panmixia and isolation from all other
populations forever. Failures of this assumption, particularly ones that add variation to the depth
of the genealogies in that ancestral population, can be expected to impact estimates of parameters
that apply to times more recent than the splitting event. One way to accommodate such histories
is to include more than two sampled populations and, thus, more than one population splitting
event (Hey 2010), although such models also require much more data.
We can partly assess whether or not the studies that show signs of nonzero gene flow do so
because of added ancestral variation. If an ancestor had added variation (e.g., due to population
structure or to gene exchange with another population), then its estimated size might be substantially larger than those of descendant populations (Won et al. 2005). If this same phenomenon
also contributed to nonzero gene flow estimates, we might observe a positive association between
these variables. In Figure 5, we plot the mean value of 2NM against the size of the ancestral
population as a fraction of the size of descendant populations. There is a suggestion of a positive
relationship, but this is largely driven by a single point with high values for both variables, and the
regression does not approach statistical significance.
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Divergence can and does lead to new species in the presence of gene flow. However, this process
is expected to arise by a complex interaction between disruptive selection, and many questions
remain. At least two main areas of questions and research can be foreseen—one that is functional
and a second that is historical and demographic. First, what are the details of the interaction between disruptive selection and linkage? Answering this question requires identification of the gene
targets of selection and the recombinational landscape around them in the genome. Second, what
has been the detailed history of gene exchange during the divergence process? In particular, we
would like to know how often disruptive selection itself can be the initial trigger for disequilibrium
or if most cases of divergence with gene flow began with divergence arising between separated
populations. We would also like to know how gene flow rates have varied over time as well as
along the chromosomes in relation to the targets of selection.
1. The question of whether or not two species can diverge from one another while they share
a geographic area and while they exchange genes has interested evolutionary biologists
since Darwin proposed such a model as his “principle of divergence.”
2. Models of divergence in sympatry are more complex than models of divergence in allopatry, but both classes of models generally require that selection have affected two or
more genes, and both classes of models require features that create linkage disequilibrium
between loci affected by selection.
3. Under sympatry, linkage disequilibrium can be caused directly by disruptive selection,
particularly if multilocus heterozygotes have reduced fitness. It can also be facilitated by
tight linkage, assortative mating, and chromosomal rearrangements that reduce crossing
over in multilocus heterozygotes.
4. Statistical methods for studying exchange under an isolation-with-migration (IM) model
have been widely used in recent years. This model includes population-size parameters
for an ancestral and two descendant populations, as well as a splitting time term and two
parameters for gene exchange.
5. A review of papers that have used the IM and IMa computer programs shows that (a) a
plurality of studies find low or zero gene flow, while (b) many studies do find evidence
of nonzero gene flow (sometimes at quite high levels) between diverging populations or
The authors are not aware of any affiliations, memberships, funding, or financial holdings that
might be perceived as affecting the objectivity of this review.
This work was partially supported by a research grant from the Fundação para a Ciência e a Tecnologia [FCT, Portugal: PTDC/BIA-BDE/66210(2006)] and by a postdoctoral fellowship from
the FCT [SFRH/BPD/28869(2006)] to C.P. This work was also supported by a National Institutes
7 October 2010
of Health grant (R01GM078204) and a National Science Foundation grant (DEB-0949561) to
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