The University of Chicago
Local Adaptation by Alleles of Small Effect*
Author(s): Sam Yeaman,
Source: The American Naturalist, (-Not available-), p. S000
Published by: The University of Chicago Press for The American Society of Naturalists
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vol. 186, supplement
the american naturalist
october 2015
Symposium
Local Adaptation by Alleles of Small Effect*
Sam Yeaman†
Department of Forest and Conservation Sciences, 2424 Main Mall, University of British Columbia, Vancouver, British Columbia V6T 1Z4,
Canada; and Department of Botany, 6270 University Boulevard, University of British Columbia, Vancouver, British Columbia V6T 1Z4,
Canada
Online enhancements: appendix, source code.
abstract: Population genetic models predict that alleles with small
selection coefficients may be swamped by migration and will not contribute to local adaptation. But if most alleles contributing to standing
variation are of small effect, how does local adaptation proceed? Here
I review predictions of population and quantitative genetic models
and use individual-based simulations to illustrate how the architecture of local adaptation depends on the genetic redundancy of the
trait, the maintenance of standing genetic variation (VG), and the susceptibility of alleles to swamping. Even when population genetic models
predict swamping for individual alleles, considerable local adaptation
can evolve at the phenotypic level if there is sufficient VG. However,
in such cases the underlying architecture of divergence is transient:
FST is low across all loci, and no locus makes an important contribution
for very long. Because this kind of local adaptation is mainly due to
transient frequency changes and allelic covariances, these architectures
will be difficult—if not impossible—to detect using current approaches
to studying the genomic basis of adaptation. Even when alleles are large
and resistant to swamping, architectures can be highly transient if genetic redundancy and mutation rates are high. These results suggest that
drift can play a critical role in shaping the architecture of local adaptation, both through eroding VG and affecting the rate of turnover of
polymorphisms with redundant phenotypic effects.
Keywords: local adaptation, genomic signature, allelic covariance,
linkage disequilibrium, gene flow, migration.
Introduction
What is the genetic basis of adaptation? How many variants
contribute to a given adaptive phenotype? How big are their
effects? Where are they in the genome? These questions are
currently the subject of intense empirical scrutiny and theoretical discussion (Stinchcombe and Hoekstra 2008; Rockman 2012; Martin and Orgogozo 2013; Savolainen et al.
2013; Lee et al. 2014). With recent advances in sequencing
* This issue originated as the 2014 Vice Presidential Symposium presented at
the annual meetings of the American Society of Naturalists.
†
Present address: Department of Biological Sciences, University of Calgary,
Calgary, Alberta T2N 1N4, Canada; e-mail: [email protected].
Am. Nat. 2015. Vol. 186, pp. S000–S000. q 2015 by The University of Chicago.
0003-0147/2015/186S1-55813$15.00. All rights reserved.
DOI: 10.1086/682405
technology, it is now possible to explore associations between millions of single-nucleotide polymorphisms (SNPs)
and the phenotypes and environments involved with local
adaptation (e.g., Hohenlohe et al. 2010; Fournier-Level et al.
2011; Soria-Corrasco et al. 2014). Beyond our mechanistic interest in connecting genotype to phenotype, characterizing these variants allows us to answer questions about
how evolution shapes a species. In particular, the way that
a species inhabits a landscape can profoundly shape the
number, size, and genomic location of the alleles that contribute to variation in an adaptive trait (i.e., its genetic
architecture) as well as the copy number and genomic location of the loci that could potentially contribute to a
trait (i.e., its genomic architecture). As individuals disperse
among regions, they may encounter different environments or competitors, which can result in natural selection
acting at different strengths or in different directions in
trait space. Depending on the relationship between spatial
variation in selection and the rate of migration across a
landscape, heterogeneous environments can lead to local
adaptation (Felsenstein 1976; Hedrick et al. 1976; Hedrick
1986; Linhart and Grant 1996; Hereford 2009), affect the
maintenance of standing genetic variance (Slatkin 1978;
Lytghoe 1997; Tufto 2000; Yeaman and Jarvis 2006), and
result in speciation (Schluter 2000; Nosil 2012). By formulating clear predictions about how migration, selection,
and drift combine to shape genetic and genomic architecture and testing these predictions with empirical data, we
can learn much about how landscape shapes these fundamental aspects of evolution.
Population genetic models have provided the most tractable approach to generating testable predictions about the
evolution of genetic architecture under the balance between divergent selection and migration. Haldane (1930)
showed that a locally favored allele would be lost from a
single-island population when the rate of immigration
(m) of a maladapted genotype exceeds the strength of selection (s) against it. Subsequent studies have identified
similar critical thresholds for the maintenance of locally
adapted polymorphisms in models with a wide range of de-
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S000
The American Naturalist
mographics and patterns of environmental variation (reviewed in Felsenstein 1976; Bürger 2014). Conceptually,
the results of the single-locus, two-patch model can be understood as follows: the mean fitness of an allele depends on
the proportion of evolutionary time (over multiple generations) that it spends in each environment and the fitness
that it experiences there. At higher migration rates, a locally
adapted allele will spend a larger proportion of its evolutionary time in environments where it is disfavored, and
above a critical migration rate it may have a lower mean fitness than an intermediate, nonlocally adapted allele (or if
both alleles are equally specialized, one will be lost). At this
point, the maintenance of polymorphism will be deterministically disfavored, although loss of polymorphism is also
possible due to drift when the mean fitness of the locally
adapted alleles is only slightly higher than that of an intermediate allele (Yeaman and Otto 2011). This basic dynamic
can be extended to asymmetrical migration rates, strengths
of selection, and population densities and to incorporate
the amount of evolutionary time spent in different genetic
backgrounds in a multilocus context.
The generality of the critical migration threshold result
implies that large-effect alleles will be more likely to contribute to local adaptation, as they are more able to overcome the
homogenizing effect of migration, or “gene swamping” (Lenormand 2002). When migration rates are below the critical
threshold and polymorphism tends to be maintained, larger
alleles also have higher probabilities of establishment and
longer persistence times, further suggesting that architectures evolving under migration-selection balance should be
enriched for large-effect alleles (Yeaman and Otto 2011;
Yeaman and Whitlock 2011). Simulation studies show that
the effect-size distribution of alleles that underlie a locally
adapted trait tends to be shifted toward larger alleles when
there is persistent migration-selection balance, so that the
same amount of local adaptation can occur with fewer, larger
alleles (Griswold 2006; Yeaman and Whitlock 2011). Thus,
one way of assessing the importance of migration in shaping adaptive evolution is to study the genetic architecture
of traits in comparisons of allopatric versus parapatric pairs
of populations inhabiting contrasting environments (as per
Renaut et al. 2013).
If we observe a given rate of migration between contrasting environments, we might predict that stable maintenance of allelic polymorphism would be effectively prevented by swamping if all genetic variants are of small
effect relative to the migration rate. However, it is unclear
whether this would present a barrier to adaptation at the
phenotypic level, in part because alleles that are prone to
swamping may still persist in the population for short
periods of time and may contribute ephemerally to local adaptation. Hereafter, I will use the term “swampingprone alleles” to denote alleles with selection coefficients
that fall below the critical threshold predicted by the appropriate population genetic model, analogous to s ! m
in Haldane’s continent-island model, and “swampingresistant alleles” for the converse. It is important to note
that the actual value of the critical threshold can differ substantially according to model assumptions and must include the effect of linkage for polygenic traits. In addition,
the transition from swamping resistant to swamping prone
can involve a gradual transition in dynamics (e.g., gradual
changes in the persistence time of a polymorphism or mean
allelic divergence at equilibrium; Yeaman and Otto 2011).
The aim of this article is to explore the concept of swamping
as a barrier to local adaptation and clarify how population
and quantitative genetic models relate to this issue. I will begin by reviewing theory on the importance of linkage disequilibrium (LD), as this modifies the predictions for when
swamping should occur, and provide a brief review of quantitative and population genetic models. I will then return to
the main issue, using simulations to illustrate how genetic
architecture and local adaptation evolves when individual
alleles are small and the effect of LD is insufficient to prevent
swamping.
The Effect of LD
When extrapolating to the whole-genome scale, predictions about swamping need to be modified to account
for the effect of LD between locally adapted alleles (Slatkin
1975; Barton 1983; Bürger and Akerman 2011; Yeaman
and Whitlock 2011; Aeschbacher and Bürger 2014). LD
can alter the mean fitness of an allele through associations
with other locally adapted alleles that arise due to either
physical linkage when the alleles are on the same chromosome or statistical association when the alleles are on different chromosomes. In either case, the magnitude of fitness increase for a focal allele depends on the number of
other alleles under selection, their frequencies, and their
linkage relationships (Barton and Bengtsson 1986; Bürger
and Akerman 2011; Aeschbacher and Bürger 2014). This
locally increased fitness results in a decreased rate of gene
flow at the focal locus relative to the migration rate, which
has been termed the “effective migration rate” (me; Petry
1983; Bengtsson 1985; Bengtsson and Barton 1986; Feder
and Nosil 2010; Huisman and Tufto 2013) and has also
been discussed in terms of reduced rates of “betweenpopulation recombination” (Via 2012; Feder et al. 2013).
The effects of LD for physically linked and unlinked loci
have also been discussed in terms of “divergence hitchhiking” and “genomic hitchhiking,” respectively, but these
terms have been applied to both causal processes and resulting patterns in the literature, so it is not clear how they
should be applied to discuss swamping. If the LD-mediated
indirect effect of divergent selection is sufficiently large, it
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Adaptation by Swamping-Prone Alleles
may cause a new locally adapted mutation that otherwise
would be prone to swamping to instead be resistant to
swamping (Yeaman and Whitlock 2011), as has been elegantly shown using two-locus analytical models (Bürger and
Akerman 2011; Aeschbacher and Bürger 2014; Akerman
and Bürger 2014).
Two broad predictions emerge from the extensive body of
theoretical work on the effects of LD on adaptation (reviewed
in Smadja and Butlin 2011; Feder et al. 2012a). First, physical
linkage should most strongly affect the mean fitness of a focal
allele when the strength of divergent selection on a linked
background locus is large relative to the recombination rate
between them (s 1 r; Petry 1983; Barton and Bengtsson
1986; Bürger and Akerman 2011; Yeaman and Whitlock
2011; Aeschbacher and Bürger 2014; Akerman and Bürger
2014). This means that if the rate of recombination among
loci is much greater than the strength of selection, then physical linkage should not substantially modify the predictions
for swamping. Second, statistical associations among locally
adapted alleles that are physically unlinked will have substantial effects on mean fitness at a focal allele only when
the overall strength of selection against the immigrant phenotype (S) is very strong (Bengtsson 1985; Barton and
Bengtsson 1986; Feder et al. 2012b; Flaxman et al. 2013).
While statistical LD among selected loci will modify the
predictions about swamping, the effect is typically very
slight unless selection is very strong. For example, Aeschbacher and Bürger (2014, eq. [10]) analyzed a discrete-time
continent-island model with migration rate m and found
that the critical migration threshold for a weakly selected locus (with strength of selection p s1) linked to a strongly selected locus (with strength of selection p s2) is m* p s1 (s2 2
s1 1 r)=½(s1 2 r)(s1 2 s2 ) 1 r(1 2 s1 ), where r is the rate of
recombination between the loci. Setting r p 0.5 and assuming that s2 represents the sum of the fitness effects of all
other loci (s2 p S 2 s1 ), then m* p 0.0101 for s1 p 0.01
and s2 p 0.1, which is only slightly higher than m* p 0.01
for the single-locus model, illustrating the limited importance of statistical LD when S is small. Similarly, Feder et al.
(2012b) found small effects of statistical LD on establishment probability (which is related to the critical threshold;
Yeaman and Otto 2011) when the total strength of phenotypic selection (S) was small.
Taken together, the results of the above-described studies suggest that if LD causes alleles that are independently
prone to swamping to instead be resistant to swamping, this
effect will likely be restricted to clustered regions in the genome unless the relative fitness of migrants and their offspring is very low (i.e., S ≫0.1). As the scope for either type
of LD to affect swamping would be especially limited at the
outset of an adaptive walk (when fitness differences between
populations are low), swamping presents an intriguing and
potentially significant theoretical hurdle to local adaptation.
S000
Does Swamping Limit Local Adaptation
in Polygenic Traits?
Despite the predicted importance of swamping, it is not
immediately clear how to reconcile this prediction with
the behavior of quantitative genetic models of local adaptation. Quantitative genetic models come from a very different perspective than population genetic models, beginning from the assumption that standing genetic variance
is due to many alleles of vanishingly small effect. This allows the details of genetic architecture to be subsumed into
parameters representing the mean, variance, and higher
moments of the distribution of genetic values (Turelli and
Barton 1994; Falconer and Mackay 1996). Models of local
adaptation predict the trait divergence at equilibrium by
finding the point where the response to divergent selection
is balanced by the homogenizing effect of migration (e.g.,
Tufto 2000; Hendry et al. 2001; Lopez et al. 2008; Yeaman
and Guillaume 2009; Huisman and Tufto 2012). The behavior of these quantitative genetic models can be seen
most easily by examining the two-patch model of Hendry
et al. (2001):
Dp
DV V G
,
VG (1 2 m) 1 (q2 1 VP )m
(1)
where the equilibrium trait divergence (D) is a function of
the difference in the optimal trait values of the two populations (DV), the strength of Gaussian stabilizing selection
within each population (q2), migration rate (m), and genetic
(VG) and phenotypic (VP) variance in the quantitative trait
under selection. Because D scales with VG and VG is a parameter, divergence cannot be predicted a priori from these
models without knowing what determines VG, which can itself be affected by migration-selection balance. A modified
version of this model that also accounts for genetic skew
yields more accurate predictions of the equilibrium divergence observed in simulations (Debarre et al. 2015).
If we examine a simplified case where two populations
experience different environments (V p 21 vs. 11), m p
0.005, q2 p 25, and VP p VG , this model predicts equilibrium divergence to be D=Dv p 8VG =(8VG 1 1), showing
that adaptation is expected to increase with VG but giving
little insight into how VG is maintained. If similar parameters are used in the single-locus model of Yeaman and
Otto (2011), then stable allelic divergence would be predicted only for alleles with phenotypic effect sizes of a ≫
0.0215, which is about 1% of the difference in environmental optima (V2 2 V1 p 2; see eq. [A1] in the appendix
[available online] for details). At first glance, these models
might seem to contradict each other: population genetic
models predict that only large alleles will contribute to di-
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S000
The American Naturalist
vergence, while quantitative genetic models predict that divergence will evolve as long as VG is maintained. Because
standing variation is nearly ubiquitous in quantitative traits
(Houle 1992; Falconer and Mackay 1996) and is commonly
due to alleles of small effect (Mackay et al. 2009; Rockman
2012), the quantitative genetic model would seem to predict that some local adaptation should always evolve, even
with small-effect alleles. While there are numerous technical
problems that arise when contrasting models with such different assumptions (see the appendix for further discussion),
this simple contrast suggests that we have an incomplete understanding of how polygenic traits will evolve when alleles
are small and prone to swamping.
A different approach to studying local adaptation using
quantitative genetics has been taken by Latta (1998, 2003)
and Le Corre and Kremer (2003, 2011, 2012), who showed
that a quantitative genetic measure of adaptation (QST)
can be decomposed into terms representing the amount of
divergence at individual loci affecting the trait (FST) and
the variance-scaled within-population (covW0 ) and betweenpopulation (covB0 ) allelic covariance among these loci:
QST p
FST (1 1 covB0 )
,
FST (cov 2 covW0 ) 1 1 1 covW0
0
B
(2)
PP
P
where cov 0 p i j covij = i j i2 , which is the covariance
among allele effects at locus i and j, scaled by the variance
at locus i and evaluated either between populations (covB0 )
or within populations (covW0 ). Whereas LD is based on covariance in allele frequencies, the variances and covariances
in this formula also incorporate allele effect size (covij p
LDij 4ai aj , where a is the allele effect size; Latta 1998).
This is an instantaneous solution that accounts for the relative contributions of allelic variances and covariances, but it
does not tell us how such contributions are expected to
evolve. Latta (1998, 2003) and Le Corre and Kremer (2003,
2012) showed that as the number of loci increases and the
effect size decreases, the importance of FST decreases while
that of covB0 increases (holding QST constant). As a result,
substantial phenotypic differentiation can occur with only
small changes in allele frequency at many loci, which would
be difficult to distinguish from FST at neutral loci. This is a
natural consequence of dividing a given phenotypic fitness
effect among an increasing number of loci: as the per-locus
s decreases, allele frequency differences decrease, so FST is
smaller and more of the overall variance must be increasingly due to allelic covariances. However, it was not clear
from this work whether the contributions of individual loci
to divergence were stable over time nor whether the alleles
involved were resistant to swamping, even though the differences from neutral FST were slight.
Simulation Model
To explore the evolution of local adaptation when alleles
are prone to swamping, I used a modified version of Nemo
(Guillaume and Rougemont 2006) to simulate a scenario
with two patches connected by migration (m), similar to
that described in Yeaman and Whitlock (2011; source
code is available in a .tbz file, available online).1 Each generation, new individuals are drawn and survive with a
probability proportional to their fitness until the carrying
capacity of the local patch (N ) is reached. Individual fitness is determined by the function w p 1 2 f(jV 2 Zj=2)g ,
where Z is the individual phenotype, V is the locally optimal
phenotype, f is the strength of selection on the phenotype,
and g is the curvature of the fitness function. Unless otherwise indicated, the optima are set to V1 p 2 1 and V2 p 1
and g p 1, resulting in linear stabilizing selection, so that an
individual that is perfectly adapted to one patch experiences
a fitness cost of f in the other patch (a summary of the
variables and parameter settings is shown in table 1).
To facilitate comparison to single-locus population genetic models, the recombination rate between loci is set
to r p 0.5 (so that all loci recombine as if they were physically unlinked) and all loci are diallelic, with mutation
effects of either 1a or 2a on the phenotype. Individual
phenotypes are determined by the additive effects of all
mutations present in their genome, with the total number
of diploid loci denoted by ntot. In most cases, simulations
are run using a p 0.01 and f p 0.1, which results in a critical migration threshold of m* p 0.0021 when N p 1,000
(from the single-locus prediction of eq. [A1]). While this
prediction should approximate the true threshold for subsequent mutations of the same size, due to g p 1, complications arise with multiple loci (Spichtig and Kawecki
2004; see also the appendix), and LD and indirect selection
would cause slight deviations. As described above in “The
Effect of LD,” statistical LD is expected to have little effect
when S ! 0.1, and the effect of f here is analogous to S in
a two-locus model. As such, the single-locus prediction
for the critical migration threshold (eq. [A1]) should be approximately accurate for multilocus traits given the above
assumptions, suggesting that persistent allelic divergence
should be prevented by swamping for fa p 0.01; f p 0.1g
when m ≫ 0.0021.
Because of the assumption of stabilizing selection, the
selection coefficients (s) experienced by individual mutations will depend on the genetic background and architecture underlying the trait. In the above case where all Z p 0,
then s ∼ fa. When a phenotype is at a local optimum, how-
1. Code that appears in The American Naturalist is provided as a convenience to the readers. It has not necessarily been tested as part of the peer review.
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Adaptation by Swamping-Prone Alleles
Table 1: Description of variables and symbols along with parameter settings used in the simulations, where appropriate
Symbol
Description
N
Population size per patch
ntot
Total number of loci underlying
quantitative trait
Number of mutations necessary to
yield optimal phenotype
Recombination rate between loci
Mutation effect size
Per-locus mutation rate
Locally optimal phenotype
Strength of phenotypic selection
Curvature of fitness function
Migration rate
Effective migration rate
Critical migration threshold
Individual phenotype
Mean phenotypic divergence
between populations
Contribution of an individual locus
to divergence
Within-population genetic variance
Within-population phenotypic
variance
Strength of Gaussian stabilizing
selection
Total strength of selection (twolocus model)
Locus-specific strength of selection
(two-locus model)
Scaled allelic covariance between
populations
Scaled allelic covariance within
populations
nopt
r
a
m
V
f
g
m
me
m*
Z
D
d
VG
VP
q2
S
si
covB0
covW0
a
Parameter
settings
500, 1,000,
5,000, 10,000
50, 500, 5,000
S000
diallelic and equal-effect-size assumptions (with some provision for rounding error), but this quantity would have a
less simple definition for more realistic distributions of effect size. It is worth noting that the simulations used in
Feder et al. (2012a), Flaxman et al. (2013, 2014), and other
recent articles assume pure divergent directional selection,
while those of Le Corre and Kremer (2003, 2011, 2012) and
others assume spatially variable stabilizing selection.
FVF/(2a)
.5
.01a
1028 to 1023
21/11
.1a
1
.005
...
...
...
...
...
...
...
...
...
...
...
...
Unless otherwise noted.
ever, any further mutations with the same sign as the optimum become deleterious. This highlights a critical assumption that arises in modeling quantitative traits: when ntot
is small enough that mutations at all loci
Pare required to
yield the locally fittest phenotype (i.e., 2 i ai ≤ jVj), then
all loci will experience only directional selection within each
patch. By contrast, when the number of loci is larger than
the number of allelesP
required to yield a locally optimal phenotype (i.e., when 2 i ai 1 jVj), the fitness effect of an allele depends on its genetic background. This phenomenon
is known as genetic redundancy (Goldstein and Holsinger
1992; Nowak 1997)—there are more loci contributing to
the phenotype than are needed to achieve the optimum phenotype. The number of loci required to yield a locally optimal phenotype here is simply nopt p jVj=(2a), due to the
Results
The Effect of Genetic Redundancy
When there is no genetic redundancy (nopt p ntot ), the point
of transition between low and high multilocus trait divergence is consistent with the predicted critical threshold from
a single-locus model (eq. [A1]). When the migration rate
and total strength of selection on the phenotype (f) are
held constant and only a and ntot are allowed to vary (holding constant the sum of the phenotypic effect size across all
mutations, 2antot p jVj), there is high divergence in the
mean trait values of the populations (D) when {ntot p 1
and a p 0.5} and very little divergence when {ntot p 50
and a p 0.01} (fig. 1). Individual selection coefficients are
weaker with small a, and therefore the critical migration
thresholds are lower and, as predicted by equation (A1), little divergence is observed when a falls below the critical
threshold for m p 0.005 (fig. 1, dashed gray lines), even
though the strength of selection on the phenotype is high.
To test whether statistical LD could be contributing to divergence in these cases, these simulations were rerun with only a
single locus but the same allele effect sizes. The total phenotypic divergence in the multilocus cases (fig. 1, black line with
squares) was indistinguishable from the phenotypic divergence in the single-locus case, after multiplying the singlelocus D by the value of ntot in the corresponding multilocus
run (fig. 1, gray triangles). This shows that statistical LD is
not significantly altering the maintenance of polymorphism
and the potential for swamping.
By contrast, when traits are highly genetically redundant, considerable trait divergence can be maintained when
swamping is predicted for individual alleles, especially when
the trait is highly polygenic. If we hold constant {nopt p 50
and a p 0.01} but increase ntot , considerable divergence
can evolve when ntot ≫ nopt and per-locus mutation rates
(m) are high (fig. 2A), even though single-locus models predict that these alleles should be prone to swamping. Similar amounts of divergence after 50,000 generations are observed whether the populations were initialized with no
genetic variation and no divergence between populations
or initialized with individuals optimally adapted to their local environments (appendix, fig. A1A; figs. A1–A5 are available online).
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S000
The American Naturalist
10
1
10
2
10
3
10
4
n=2
1.0
* for
m = 0.005
n=5
0.5
n = 10
n = 20
0.0
0.01
0.1
Critical migration rate (m*)
n=1
1.5
n
=
n 10
= 0
n 75
=
n 50
n=4
= 0
30
Divergence in trait means (D)
2.0
0.5
Effect size ( )
Figure 1: Trait divergence increases when alleles are larger than the
critical threshold for m p 0.005, shown by gray dashed lines. For the
black line and symbols, the effect size (a) and number of loci (n p ntot )
were varied simultaneously, holding constant 2antot. For the gray triangles, simulations were run with a single locus, a was allowed to vary,
and the amount of divergence (D) was multiplied by the number of loci
used in the multilocus simulations run with the same value of a. In all
cases, N p 1,000, m p 0.005, and m p 1025 .
How is this stable adaptive phenotypic divergence maintained when swamping is predicted for individual loci?
One possible explanation is that the conditions for the
maintenance of polymorphism are relaxed when there is
enough standing genetic variation that some individuals’
phenotypes overshoot the local optimum, resulting in stabilizing selection, which occurs only when ntot ≫ nopt . If
stabilizing selection changes the conditions for the maintenance of polymorphism and this is critical to the establishment of divergence, then we should see little to no divergence until standing variation builds up and a substantial
proportion of individuals have phenotypes beyond the local optima (dividing the total phenotypic effect of selection
among a larger number of loci seems unlikely to facilitate
the maintenance of polymorphism in a way that does not
occur with divergent directional selection, but in the absence of appropriate analytical theory it should be considered). In parameter sets where substantial adaptive divergence evolved, there was usually enough standing genetic
variation at the end of the simulations that stabilizing selection was occurring (fig. 3A; proportion of individuals
with Z 1 jVj). However, by examining the time course of
evolution for one parameter set that results in particularly
high adaptive divergence and standing variation, it is clear
that only pure directional selection was acting during the
time when the initial adaptive divergence was evolving
(fig. 3B, prior to arrow). Thus, while the transition to sta-
bilizing selection likely alters the dynamics underlying the
maintenance of polymorphism, it is not a necessary prerequisite for the maintenance of local adaptation when
ntot ≫ nopt .
Another possible explanation is that statistical LD among
locally beneficial alleles is strong enough that they are no
longer prone to swamping once sufficient standing variation has accumulated. This type of behavior was observed
by Flaxman et al. (2014), where very strong LD (r close to
1) among numerous divergently selected loci caused a rapid
transition to high divergence and low effective migration
rates (reductions in me of ≫10#). However, in the simulations shown here in fig. 2A, the effective migration rate
remains high (never more than a 3% decrease), the LDs
are low (typically, r ! 0.05; fig. A2), and there are only
transient changes in allele frequency (see below; fig. 4),
which all suggest very different dynamics from those operating in the simulations of Flaxman et al. (2014). In addition,
as described above the maximum fitness cost to a migrant is
bounded by f p 0.1 under the divergent stabilizing regime
used here. In population genetic models with divergent directional selection of similar magnitude (S ∼ 0.1), statistical
LD has only a limited effect on both the maximum potential
reduction in effective migration (Barton and Bengtsson
1986) and the magnitude of changes in the critical migration
threshold (Aeschbacher and Bürger 2014). Although there
is no corresponding analytical prediction for swamping under divergent stabilizing selection, it seems unlikely that the
effect of LD would be greater than that under divergent directional selection.
An alternative explanation that seems most consistent
with the data is that different alleles are constantly establishing and being lost, so that individual alleles make only
fleeting contributions to phenotypic divergence. As long as
sufficient genetic variance exists within the population and
phenotypic selection is strong relative to migration, phenotypic divergence may stably persist despite constant flux in
the underlying architecture. For the additive diallelic loci
simulated here, the contribution of a locus to adaptive
phenotypic divergence (d ) can be calculated from the frequency of the 1a allele in patch 1 (p1) and patch 2 ( p2),
d p ½ap2 2a(12p2 ) 2 ½ap1 2 a(1 2 p1 ) p 2a(p2 2 p1 ), so
that
P the total divergence in mean phenotypes is then D p
2d. In the highly polygenic simulations, d varies considerably among loci, with many contributions to divergence occurring in the direction opposite the difference
in environmental optima (i.e., loci with d ! 0; fig. 4B).
Alleles making large contributions to divergence, shown
in the tails of the distribution in figure 4B, usually do so
for only a few hundred generations, and the architecture
of allelic divergence is constantly shifting (fig. 4C; see also
fig. A3 for a more detailed plot). High flux in architecture
occurs for all the simulations plotted in figure 2A: a bout of
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Adaptation by Swamping-Prone Alleles
Divergence in trait means (D)
A
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high divergence for a given allele, where the value of d is in
the top 20% of the distribution, lasts less than 300 generations on average (fig. A4).
By contrast, for simulations where f p 0.5 and a p 0.1,
when selection on individual alleles is strong relative to migration and swamping is not predicted by equation (A1),
there are very few loci with d ≪ 0 (fig. 4E; note the log10
scale), and architectures are temporally stable with large
positive contributions to divergence that typically last many
thousands of generations (fig. 4F). Interestingly, high flux in
architecture can also occur when alleles are predicted to be
individually resistant to swamping if ntot ≫ nopt and mutation rates are high (fig. 5), a finding also reported by Yeaman
and Whitlock (2011; their fig. A2). In this case, there is a dramatic difference in the stability of the genetic architecture
underlying divergence with just an order of magnitude of
change in the per-locus mutation rate (fig. 5A vs. 5B), which
does not occur for less redundant architectures (fig. 5C, 5D).
This likely occurs because there are many alleles with equivalent fitness effects when there is high genetic redundancy
and a high mutation rate, and these alleles can then replace
each other due to drift. Because of the importance of drift,
this effect may be most important in smaller or demographically unstable populations. However, it is unclear how migration and selection also contribute to this transience, so
further study is required.
The Importance of Standing Genetic Variation
0.0
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10
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10
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When all alleles are prone to swamping (as in fig. 4A–4C),
standing genetic variance (VG) increases with m and ntot
(fig. 2B), and there is a strong and nearly linear relationship between VG and adaptive trait divergence (fig. 2C).
Plotting D on this log10 scale reveals large relative differences in the amount of adaptation that occurs at low mutation rates (which are not visible in fig. 2A), even if such
adaptation is so limited on an absolute scale as to be undetectable in an experimental context. This relationship
between D and VG is consistent with the behavior of quantitative genetic models (eq. [1]), but is VG actually determining the maintenance of D at equilibrium? If so, then
understanding the maintenance of local adaptation when
alleles are prone to swamping is closely linked to understanding the maintenance of VG.
However, because VG and the higher moments of genetic variance can be inflated by migration among diverged
1
Within-population variance (VG)
Figure 2: Effect of mutation rate and genetic redundancy on divergence in trait means (A), within-population variance (B), and the relationship between them (C). Black lines and symbols have no re-
dundancy (ntot p nopt ), while lines with triangles and lines with
circles are moderately and highly redundant, respectively. B shows that
homogeneous environments (V1 p 0, V2 p 0; lines) maintained similar amounts of VG as heterogeneous environments (V1 p 21, V2 p
11; symbols). In all cases, N p 1,000 and m p 0.005.
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Proportion of extreme phenotypes
A
The American Naturalist
0.5
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Figure 3: Proportion of individuals with phenotypes that fall beyond the local optimum (Z 1 jVj), shown at the end of the simulations (A) and in the first 5,000 generations of the simulations run
with ntot p 5,000, N p 1,000, and m p 0.00001 (B). The arrow in B
indicates the last time step at which there were no individuals with
Z 1 jVj. Divergence in trait means is shown in B for reference; all other
parameters are as in figure 2A.
populations above baseline expectations for mutationselection balance (Slatkin 1978; Turelli and Barton 1994;
Lythgoe 1997; Yeaman and Guillaume 2009; Debarre et al.
2015), it is not immediately clear whether increased levels of VG in figure 2B are attributable to mutation alone
or are also affected by migration. Rerunning the simulations from figure 2A with a homogeneous environment
(V1 p V2 p 0) and high migration (m p 0.5) does not substantially affect levels of VG (fig. 2B), which shows that
maintenance of VG under these parameters is mainly driven
by mutation, not migration. By contrast, when alleles are
resistant to swamping (as in fig. 4D–4F), a much larger
amount of VG is maintained in heterogeneous environments relative to homogeneous environments (fig. A5).
Migration should inflate VG only when there is substantial
allele frequency divergence between populations, so these
observations are consistent with theory: swamping-prone
alleles exhibit little divergence, and migration therefore
has little effect on VG, while the converse is true for simulations with swamping-resistant alleles. Taken together,
these results show that adaptive divergence when alleles
are prone to swamping depends on VG, which itself depends
in part on the supply of mutations, which is greatly increased when traits are highly polygenic and redundant.
On the other hand, when alleles are resistant to swamping,
considerable D can be maintained with little VG or input
from mutation.
When traits are highly polygenic, Latta (1998, 2003)
and Le Corre and Kremer (2003, 2011, 2012) showed that
considerable trait divergence can evolve with limited allelic divergence (FST) at individual loci but high betweenpopulation covariance in allelic effects, represented by
covB0 . However, it was not clear from their work whether
these results would extend to traits made up of swampingprone alleles and whether the underlying genetic architectures were temporally stable. The results here show that
swamping-prone alleles are contributing to divergence primarily through allelic covariances: comparing the results
of simulations with ntot p 5,000 and considerable trait divergence to those with ntot p 50 and little trait divergence,
both result in similar levels of FST, proportions of polymorphic loci, and expected heterozygosity (fig. 6A, 6B).
On the other hand, simulations where high ntot results in
high D have much higher between-population allelic covariances (fig. 6C). The relative insensitivity of VG to migration when adaptation is due to swamping-prone alleles
(shown in fig. 2B) occurs because there is very limited allelic
differentiation between populations, and phenotypic divergence is mainly driven by positive between-population allelic covariances. As a result, migration does not substantially
impact allele frequencies or inflate genetic variance within
populations.
Because the amount of divergence when alleles are
prone to swamping depends on VG, predicting the potential for local adaptation in this context becomes a problem
of understanding the maintenance of genetic variation.
Genetic variance can be greatly impacted by drift, and
much less divergence is seen in simulations with smaller
population size (fig. A1B), as per Blanquart et al. (2012).
While theoretical models of mutation-selection balance
generally predict that VG should increase linearly with
ntotm (Latter 1960; Bulmer 1972a; Turelli 1984; Bürger
et al. 1989), both VG and trait divergence increase more
with ntot than with m (e.g., {ntot p 500 and m p 1024 } vs.
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500
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Swamping prone alleles
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Adaptation by Swamping-Prone Alleles
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Figure 4: Comparison between genetic architectures when alleles are prone to swamping (A–C; f p 0.1, a p 0.01) or resistant to swamping
(D–F; f p 0.5, a p 0.1). A and D show the divergence in phenotypic trait means over the simulation, with the gray rectangle indicating the last
2,000 generations over which the data for the remaining panels were collected. B and E show the distribution of contributions to divergence (d )
made by individual loci (note the log10 scale in E), with red and blue colorations indicating those loci with larger values of d (cutoffs set as 5a/2
and 5a/4); note that the Y-axis shows the number of loci within each bin of size d sampled over the entire 2,000-generation window shown
in C and F. C and F show the values of d from B and E plotted over time at each of the ntot p 500 loci in the simulations; white spaces indicate
values of jdj ! a=4. In both simulations, N p 1,000, m p 0.005; in A–C, m p 1024 ; in D–F, m p 1025 .
{ntot p 5,000 and m p 1025 }; fig. 2A). This occurs because
expected heterozygosity becomes saturated at high mutation rates (fig. 6B), which limits the potential for increases
in VG with further increases in mutation. These results suggest that genetic drift plays two main roles in the evolution
of local adaptation by swamping-prone alleles: eroding VG
and thereby limiting D, and affecting the rate of flux in the
alleles that contribute to divergence. It seems likely that
different demographics or selection regimes would change
the amount of VG maintained under the balance between
migration, selection, mutation, and drift, thereby affecting
the amount of local adaptation at equilibrium.
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µ = 10-4
µ = 10-5
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Locus
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Time (generations)
Time (generations)
Figure 5: Contributions to divergence (d) for traits with swamping-resistant alleles (f p 0.5, a p 0.1). A reproduces the results shown in
figure 4F; other panels show results with different values of ntot and m. All other parameters are as in figure 4F.
Discussion
There is a long history of using population genetic models
to study the maintenance of polymorphism under migrationselection balance (Felsenstein 1976; Bürger 2014). In the
context of these models, we have come to view local adaptation as being synonymous with large and persistent allelefrequency differences among populations. It is within this
paradigm that most current genome-scan studies orient
their search for the loci responsible for local adaptation,
aiming to detect large frequency differences or associations between genetic variants and environments or traits.
However, it has been increasingly acknowledged that local
adaptation does not necessarily involve large differences in
allele frequency. Important articles by Latta (1998, 2003)
and Le Corre and Kremer (2003, 2011, 2012) showed that
as the number of loci contributing to a trait increases, the
relative contribution of FST at individual loci tends to decrease, while among-population covariances in allele effect
size tend to contribute more (with similar results found
for clines; Barton 1999). These studies showed that mutations underlying polygenic traits would be difficult to empirically detect by comparison to FST at neutral markers
but did not explicitly evaluate whether the individual loci
were swamping resistant or whether FST tended to persist
for long. Given the low rates of mutation (m p 1025 ) and
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Adaptation by Swamping-Prone Alleles
A
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mean
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cov'B
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Per locus mutation rate (µ)
Figure 6: Effect of ntot and m on mean and 95th percentile values of
FST (A), expected heterozygosity (HS) and proportion of nonpolymorphic loci (B), and scaled genetic covariances within (covW0 ) and between
(cov0B ) populations (C). All other parameters are as in figure 2A.
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small number of loci (n ! 30) in the simulations of Kremer
and Le Corre (2011), it seems likely that the alleles they
simulated were resistant to swamping, as little divergence
was observed here under similar parameter combinations
with swamping-prone alleles (fig. 2A).
The results shown here suggest that local adaptation can
also occur when the individual alleles underlying a trait
are prone to swamping if there is sufficient standing genetic variation. Although we have no exact analytical models of swamping under multilocus stabilizing selection to
confirm this, three lines of evidence support the assertion that the alleles simulated here were swamping prone:
(1) they are small enough to be swamping prone under a
single-locus directional selection regime; (2) the effect of
statistical LD on swamping is negligible, justifying the
use of single-locus predictions (fig. 1); and (3) divergence
occurs prior to the onset of stabilizing selection (fig. 3), suggesting that stabilizing selection itself is not critical for
the emergence of local adaptation. This shows that adaptation under migration-selection balance cannot always be
predicted by deriving traditional population genetic models
for the maintenance of polymorphism and extrapolating
to the whole genome. Rather, adaptation in such cases is
a quantitative genetic phenomenon that can best be understood in light of the maintenance of variation across all
loci and the response to selection that occurs as individuals
with fortuitous combinations of alleles are favored. This
kind of adaptive response is most pronounced in highly
polygenic and genetically redundant traits with relatively
high rates of mutation, conditions not generally considered in the studies mentioned above. It seems possible that
a population genetic model of adaptation via swampingprone alleles could be made by representing how the pull
of weak selection modifies the expected persistence time
and allele frequency trajectories of a polymorphism (sojourn time density, as per Aeschbacher and Bürger 2014),
which could also be used to calculate per-locus statistics
such as FST, cov 0, and QST.
Regardless of where the “swamping line” is drawn and
how we model adaptation, the simulations here challenge
the conception of local adaptation as a clearly defined and
stereotyped genetic strategy common to many individuals
in the same population. Rather, local adaptation via small
alleles can arise from a heterogeneous and temporally fluctuating pool, from which a multitude of equally fit but genetically distinct combinations may be drawn (fig. 4). If
there is high genetic redundancy and sufficient standing
variation, there can be high turnover in the loci that contribute to divergence even when individual alleles are highly
swamping resistant (fig. 5). Taken together with previous
theory, this suggests that the pattern of divergence under
migration selection can be broadly categorized into three
motifs: (I) highly swamping resistant and strongly differen-
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tiated, (II) moderately swamping resistant and difficult to
distinguish from neutral alleles, and (III) swamping prone
and transient (table 2). While the importance of physical
and statistical LD was limited in the simulations here,
small-effect alleles may commonly evolve by motif I if they
are tightly linked to mutations of large effect or by motif II if
statistical LD is strong across the whole genome.
Where do real examples of local adaptation fall along the
continua illustrated in table 2? Over long periods of time, if
traits have high genetic redundancy (ntot ≫nopt ), alleles that
are larger or more tightly clustered will tend to outcompete
and replace smaller alleles (Yeaman and Whitlock 2011;
Yeaman 2013), and the repeated establishment of more
swamping-resistant alleles (motif I) would likely reduce
the proportion of divergence contributed by more weakly
selected or swamping-prone alleles (motifs II and III). At
the beginning of a bout of local adaptation, there is typically
much less scope for linkage to affect local adaptation via
increased establishment probability (Feder et al. 2012b;
Yeaman 2013), and allele effect size is likely the primary determinant of genetic architecture. In such cases, the balance
between migration and selection would determine the minimum effect size likely to make stable contributions (motifs I and II), while alleles smaller than this threshold would
contribute only transiently (motif III). The availability of
mutations of different size classes (relative to migration) is
therefore a critical determinant of the genetic architecture
that would evolve. All else being equal, at higher migration
rates more alleles would be swamping prone (motif III), but
any alleles large enough to resist swamping would be more
readily distinguished from neutrally evolving loci (motif I).
At low migration rates more alleles would be swamping resistant, but because neutral loci would also be more strongly
differentiated it might also be difficult to distinguish locally
adapted from neutral loci (motif II) for all but the most
strongly selected alleles (motif I). Alleles evolving via motif
I or II could also undergo transient fluctuations if mutation
rates and genetic redundancy are high, but this may be unlikely, as large-effect mutations are likely rare.
Although the simulations explored here are useful for
building intuition about how local adaptation evolves, it
is difficult to make predictions about whether we should expect to see local adaptation via swamping-prone alleles in
empirical contexts. Beyond the sensitivity of simulation
results to parameters about which we know little, the very
assumptions about evolutionary dynamics represented in
these simulations may not accurately represent how genetic
variation is maintained. Classical models of mutationselection balance (Latter 1960; Bulmer 1972a; Turelli 1984;
Bürger et al. 1989) seem unable to explain observed levels
of genetic variation (Barton and Turelli 1989; Johnson and
Barton 2005), and because the simulations used here follow
similar assumptions, their results should not be considered
quantitatively representative of what we should expect to observe in nature. Nonetheless, it is reasonable to expect that
traits governed by very few loci are unlikely to harbor high
levels of VG and would therefore be unlikely candidates for
adaptation via swamping-prone alleles. Conversely, since
most quantitative traits have substantial levels of VG (Houle
1992; Falconer and Mackay 1996), local adaptation via
swamping-prone alleles may be quite common in highly
polygenic traits. While genetic redundancy may not be critical for adaptation via swamping-prone alleles, traits with limited redundancy are necessarily less polygenic and therefore
less likely to harbor high levels of VG. Ultimately, empirical
data will be required to evaluate whether local adaptation
via swamping-prone alleles commonly occurs in nature. It
seems most likely to occur when selection acts on metatraits,
such as height or body size, that aggregate over the effects of
many biochemical pathways. Conversely, this type of adaptation may be less likely to occur in traits with a less complex
genetic basis, such as color pattern.
Genomic Signatures of Local Adaptation
We know that local adaptation is common at the phenotypic level (Hedrick et al. 1976; Hedrick 1986; Linhart
and Grant 1996; Hereford 2009) and we are rapidly accumulating evidence about adaptation at the genomic level
(reviewed in Mitchell-Olds et al. 2007; Stapley et al. 2010;
Feder et al. 2013; Savolainen et al. 2013), but we are still
far from a comprehensive understanding of where real
Table 2: Three motifs for patterns of divergence in alleles contributing to local adaptation
Motif
Susceptibility to
swamping
Divergence pattern
I: Strongly differentiated
Highly resistant
Moderate to high FST
II: Weakly differentiated
Moderately resistant
III: Transient
Prone
Low to moderate FST, difficult
to distinguish from neutral FST
Low FST
Stability of the genetic
basis of divergence
High, unless redundancy and mutation
rates are high
Moderate to high, unless redundancy
and mutation rates are high
Transient
Note: Transitions between these motifs will be gradual with changing migration or strength of selection. These characterizations are intended to facilitate
description of predictions in empirical contexts rather than suggest discrete behavior of the underlying dynamics.
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Adaptation by Swamping-Prone Alleles
populations fall within the spectrum of table 2. A nowfrequently-used approach to studying the architecture of
local adaptation is to scan the genome for SNPs or genomic regions that have strong statistical associations with a
locally adapted phenotype or environmental variable of interest or that have extreme values of FST beyond the patterns in allele frequency due to demography (Storz 2005;
Schoville et al. 2012). These approaches have identified
signatures consistent with local adaptation in a range of
species (e.g., Hohenlohe et al. 2010; Fournier-Level et al.
2011; Soria-Corrasco et al. 2014), but it is not always clear
how well they avoid false positives due to the confounding
effects of demography (Schoville et al. 2012; Vilas et al.
2012; de Mita et al. 2013; Lotterhos and Whitlock 2014).
Furthermore, it has been argued that many examples of
clusters of loci with extreme FST may in fact be false positives driven by reductions in heterozygosity within populations due to selective sweeps or background selection
rather than migration-selection balance (Cruikshank and
Hahn 2014). Beyond the problem of identifying false positives, it is not clear how well such approaches can identify
true positives because alleles under relatively weak selection, even when resistant to swamping, will not have large
differences in allele frequency compared with putatively
neutral markers (Latta 1998; Le Corre and Kremer 2003;
Storz 2005). With swamping-prone alleles, these problems
are likely to be so severe that it will be very difficult to identify causal loci without also generating a large proportion of
false positives. Further simulation studies that explicitly
compare patterns at selected versus neutral loci will be required to evaluate the feasibility of applying genome-scan
methods to detect a genomic signature of local adaptation
via swamping-prone alleles.
Even with perfect knowledge of which loci contribute to
a given quantitative trait, when traits are highly genetically
redundant and divergence is due to swamping-prone alleles, the contributions of individual alleles tend to fluctuate over a few hundred generations (fig. 4C). These fluctuations would likely occur more gradually at larger population
sizes and be exacerbated by rapid changes in population
size. Fluctuation in architecture is not confined to swampingprone alleles and can occur when alleles are resistant to
swamping, which may be relatively rapid if there is sufficient genetic redundancy and a high enough rate of mutation (fig. 5; Yeaman and Whitlock 2011). This flux could result in very different genetic architectures of divergence in
pairs of populations spanning similar environmental gradients in different parts of a species range if the pairs of populations are connected by sufficiently low migration rates
(i.e., a genetically redundant, postestablishment analogue
of the scenario studied by Ralph and Coop [2015]). Thus,
while range-wide population genomic studies have greater
power to detect signals of parallel adaptation along multiple
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transects, such parallelism would not even be expected for
adaptation via swamping-prone alleles or for traits with high
genetic redundancy (given sufficient isolation by distance between transects), reducing the relative power of range-wide
studies.
An interesting alternative to the locus-by-locus approaches of genome scans has been proposed by Berg and
Coop (2014). This approach uses the results of a genomewide association study (GWAS) to estimate genotypic
values within populations by taking the frequency-weighted
sum of the individual SNP effects on a trait of interest. It is
then possible to test whether variance among populations
in these estimated genotypic values exceeds background expectations, estimated using putatively neutral markers. This
method is therefore effectively a genomic analogue of QST
versus FST tests. The genotypic values used in this test are
analogous to D here, as they represent the sum of all locus
effects (d), and include the effects of both allelic differentiation and covariance. This test may be especially useful for detecting a signature of adaptation via swamping-prone alleles,
which tend to contribute to divergence mainly through covariances. However, because the small-effect alleles likely
to be prone to swamping are also difficult to detect using
GWAS and because of the difficulties of controlling for population structure, this approach might require very large
sample sizes and careful design to attain sufficient power
(Berg and Coop 2014).
At the phenotypic level, when local adaptation is highly
polygenic and QST is driven primarily by allelic covariances,
migration does not substantially inflate standing genetic
variance within populations (fig. 2B). Thus, finding higher
genetic variance within populations in regionally heterogeneous environments (Yeaman and Jarvis 2006) or in populations experiencing higher migration might suggest that
local adaptation is at least partly due to substantial changes
in allele frequency and swamping-resistant alleles. In addition, if sufficiently powered genome scans are unable to detect a strong signature of local adaptation in a given species
despite observing phenotypic divergence in a common garden, this is suggestive of the importance of swampingprone alleles. However, such negative evidence is hardly
very convincing, and direct positive associations of alleles
or covariances with divergence are much more appealing—
and hopefully attainable.
Future Directions and Technical Considerations
While it seems likely that drift will affect the rate of allele
turnover in the transient architectures observed here, the
effect of migration and selection on this rate of turnover
is unclear. Understanding the importance of these factors
may lead to qualitative predictions about the likelihood of
parallel evolution in architectures on the landscape scale.
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The American Naturalist
In addition, it would be interesting to see whether small
genomic rearrangements yielding clustering (as per Yeaman
2013) would evolve for loci with alleles that are typically
swamping prone. Given how temporal heterogeneity can
favor architectures with very different distributions (Kopp
and Hermisson 2009; Matuszewski et al. 2014), it would
also be interesting to explore the relative fitness of linked
versus unlinked arrangements and of large versus small alleles in the context of spatial plus temporal heterogeneity.
Together with previous studies, the results shown here illustrate the critical importance of the details of genetic architecture when using simulations. Specifying directional
versus stabilizing selection, effect sizes of alleles relative to
migration and rates of mutation will all have important effects on the outcome of simulations.
Conclusions
High migration rates will prevent small mutations from
making persistent contributions to local adaptation and will
tend to favor architectures with fewer, larger, and more
tightly clustered alleles. However, if swamping-resistant mutations are not available, it is still possible for local adaptation to evolve via small swamping-prone alleles. While this
lends further theoretical support to Rockman’s (2012) argument for the importance of small-effect alleles, the general
advantage of large alleles in the context of local adaptation
still holds. Although local adaptation can evolve by smalleffect alleles, bigger (or more tightly linked) is still better.
Genetic redundancy and the rate and size of mutations therefore have a critical effect on the kind of genetic architecture
that evolves under migration-selection balance.
Acknowledgments
I thank M. Whitlock for many stimulating discussions that
led up to this article. S. Aitken, F. Débarre, K. Hodgins, J.
Holliday, K. Lotterhos, L. Rieseberg, and the Kassen laboratory, as well as an insightful reviewer, also provided helpful
comments and discussion. This work is part of the AdapTree
Project funded by the Genome Canada Large-Scale Applied
Research Project, with cofunding from Genome British Columbia and the Forest Genetics Council of British Columbia (co–Project Leaders S. N. Aitken and A. Hamann). Simulations were run on Westgrid.
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