Catch Me if You Can: Adaptation from Standing Genetic ... · Catch Me if You Can: Adaptation from Standing Genetic Variation to a Moving Phenotypic Optimum Sebastian Matuszewski,*,1

HIGHLIGHTED ARTICLEGENETICS | INVESTIGATION

Catch Me if You Can: Adaptation from StandingGenetic Variation to a Moving Phenotypic Optimum

Sebastian Matuszewski,*,1 Joachim Hermisson,*,† and Michael Kopp‡

*Mathematics and BioSciences Group, Faculty of Mathematics, University of Vienna, A-1090 Vienna, Austria, †Max F. PerutzLaboratories, A-1030 Vienna, Austria, and ‡Aix Marseille Université, Centre National de la Recherche Scientifique, Centrale

Marseille, I2M, Unité Mixte de Recherche 7373, 13453 Marseille, France

ABSTRACT Adaptation lies at the heart of Darwinian evolution. Accordingly, numerous studies have tried to provide a formalframework for the description of the adaptive process. Of these, two complementary modeling approaches have emerged: While so-called adaptive-walk models consider adaptation from the successive fixation of de novo mutations only, quantitative genetic modelsassume that adaptation proceeds exclusively from preexisting standing genetic variation. The latter approach, however, has focused onshort-term evolution of population means and variances rather than on the statistical properties of adaptive substitutions. Our aim is tocombine these two approaches by describing the ecological and genetic factors that determine the genetic basis of adaptation fromstanding genetic variation in terms of the effect-size distribution of individual alleles. Specifically, we consider the evolution ofa quantitative trait to a gradually changing environment. By means of analytical approximations, we derive the distribution of adaptivesubstitutions from standing genetic variation, that is, the distribution of the phenotypic effects of those alleles from the standingvariation that become fixed during adaptation. Our results are checked against individual-based simulations. We find that, comparedto adaptation from de novo mutations, (i) adaptation from standing variation proceeds by the fixation of more alleles of small effectand (ii) populations that adapt from standing genetic variation can traverse larger distances in phenotype space and, thus, havea higher potential for adaptation if the rate of environmental change is fast rather than slow.

KEYWORDS adaptation; standing genetic variation; natural selection; models/simulations; population genetics

ONE of the biggest surprises that has emerged from evo-lutionary research in the past few decades is that, in

contrast to what has been claimed by the neutral theory(Kimura 1983), adaptive evolution at the molecular levelis widespread. In fact, some empirical studies concludedthat up to 45% of all amino acid changes between Drosoph-ila simulans and D. yakuba are adaptive (Smith and Eyre-Walker 2002; Orr 2005b). Along the same line, Wichmanet al. (1999) evolved the single-stranded DNA bacterio-phage FX174 to high temperature and a novel host andfound that 80 2 90% of the observed nucleotide substitu-tions had an adaptive effect. These and other results have

led to an increased interest in providing a formal frameworkfor the adaptive process that goes beyond traditional popu-lation- and quantitative-genetic approaches and considersthe statistical properties of suites of substitutions in termsof “individual mutations that have individual effects” (Orr2005a, p. 121). In general, selection following a change inthe environmental conditions may act either on de novomutations or on alleles already present in the population,also known as standing genetic variation. Consequently,from the numerous studies that have attempted to addressthis subject, two complementary modeling approaches haveemerged.

So-called adaptive-walk models (Gillespie 1984; Kauffmanand Levin 1987; Orr 2002, 2005b) typically assume that se-lection is strong compared to mutation, such that the popu-lation can be considered monomorphic all the time and allobserved evolutionary change is the result of de novo muta-tions. These models have produced several robust predictions(Orr 1998, 2000; Martin and Lenormand 2006a,b), which aresupported by growing empirical evidence (Cooper et al. 2007;

Copyright © 2015 by the Genetics Society of Americadoi: 10.1534/genetics.115.178574Manuscript received February 24, 2015; accepted for publication May 26, 2015;published Early Online June 1, 2015.Available freely online through the author-supported open access option.Supporting information is available online at www.genetics.org/lookup/suppl/doi:10.1534/genetics.115.178574/-/DC1.1Corresponding author: EPFL SV IBI-SV UPJENSEN, AAB 0 43 (Bâtiment AAB), Station15, CH-1015 Lausanne, Switzerland. E-mail: [email protected]

Genetics, Vol. 200, 1255–1274 August 2015 1255

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Rockman 2012; Hietpas et al. 2013; but see Bell 2009), andhave provided a statistical framework for the fundamentalevent during adaptation, that is, the substitution of a residentallele by a beneficial mutation. Specifically, the majority ofmodels (e.g., Gillespie 1984; Orr 1998; Martin and Lenormand2006a) consider the effect-size distribution of adaptive sub-stitutions following a sudden change in the environment.Recently, Kopp and Hermisson (2009b) and Matuszewskiet al. (2014) extended this framework to gradual environ-mental change.

In contrast, most quantitative-genetic models consideran essentially inexhaustible pool of preexisting standinggenetic variants as the sole source for adaptation (Lande1976). Evolving traits are assumed to have a polygenic basis,where many loci contribute small individual effects, suchthat the distribution of trait values approximately followsa Gaussian distribution (Bulmer 1980; Barton and Turelli1991; Kirkpatrick et al. 2002). Since the origins of quantita-tive genetics lie in the design of plant and animal breedingschemes (Wricke and Weber 1986; Tobin et al. 2006;Hallauer et al. 2010), the traditional focus of these modelswas on predicting short-term changes in the populationmean phenotype (often assuming constant genetic variancesand covariances) and not on the fate and effect of individualalleles. The same is true for the relatively small number ofmodels that have studied the contribution of new mutationsin the response to artificial selection (e.g., Hill and Rasbash1986a) and the shape and stability of the G matrix [i.e., theadditive variance–covariance matrix of genotypes (Joneset al. 2004, 2012a)].

It is only in the past decade that population geneticistshave thoroughly addressed adaptation from standing ge-netic variation at the level of individual substitutions (Orrand Betancourt 2001; Hermisson and Pennings 2005;Chevin and Hospital 2008). Hermisson and Pennings(2005) calculated the probability of adaptation from stand-ing genetic variation following a sudden change in the se-lection regime. They found that, for small-effect alleles, thefixation probability is considerably increased relative to thatfrom new mutations. Furthermore, Chevin and Hospital(2008) showed that the selective dynamics at a focal locusare substantially affected by genetic background variation.These results were experimentally confirmed by Lang et al.(2011), who followed beneficial mutations in hundreds ofevolving yeast populations and showed that the selectiveadvantage of a mutation plays only a limited role in deter-mining its ultimate fate. Instead, fixation or loss is largelydetermined by variation in the genetic background—whichneed not to be preexisting, but could quickly be generatedby a large number of new mutations. Still, predictions be-yond these single-locus results have been verbal at best,stating that “compared with new mutations, adaptationfrom standing genetic variation is likely to lead to fasterevolution [and] the fixation of more alleles of small effect[. . .]” (Barrett and Schluter 2008, p. 38). Thus, despite re-cent progress, one of the central questions still remains

unanswered: From the multitude of standing genetic var-iants segregating in a population, which are the ones thatultimately become fixed and contribute to adaptation, andhow does their distribution differ from that of (fixed) denovo mutations?

The aim of this article is to contribute to overcomingwhat has been referred to as “the most obvious limitation”(Orr 2005b, p. 6) of adaptive-walk models and to study theecological and genetic factors that determine the geneticbasis of adaptation from standing genetic variation. Specif-ically, we consider the evolution of a quantitative trait ina gradually changing environment. We develop an analyti-cal framework that accurately describes the distribution ofadaptive substitutions from standing genetic variation anddiscuss its dependence on the effective population size, thestrength of selection, and the rate of environmental change.In line with Barrett and Schluter (2008), we find that, com-pared to adaptation from de novo mutations, adaptationfrom standing genetic variation proceeds, on average, bythe fixation of more alleles of small effect. Furthermore,when standing genetic variation is the sole source for adap-tation, faster environmental change can enable the popula-tion to remain better adapted and to traverse larger distancesin phenotype space.

Model and Methods

Phenotype, selection, and mutation

We consider the evolution of a diploid population of N indi-viduals with discrete and nonoverlapping generations char-acterized by a single phenotypic trait z, which is underGaussian stabilizing selection with regard to a time-dependentoptimum zoptðtÞ;

wðz; tÞ ¼ exp

"2

�z2zoptðtÞ

�22s2

s

#; (1)

where s2s describes the width of the fitness landscape (see

Table 1 for a summary of our notation and Figure 1 fora graphical abstract of the model). Throughout this articlewe choose the linearly moving optimum,

zopt ðtÞ ¼ vt; (2)

where v is the rate of environmental change.Mutations enter the population at rate Q=2 (with

Q ¼ 4Nu; where u is the per-haplotype mutation rate),and we assume that their phenotypic effect size a followsa Gaussian distribution with mean 0 and variance s2

m (whichwe refer to as the distribution of new mutations); that is,

pðaÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi2ps2

m

p exp

2

a2

2s2m

!: (3)

All mutations that are segregating in the population at thetime the environment starts changing are considered as

1256 S. Matuszewski, J. Hermisson, and M. Kopp

standing genetic variants, whereas all mutations introducedafter that point are considered as de novo mutations.

Throughout this article we equate genotypic with pheno-typic values and, thus, neglect any environmental variance.Note that this model is, so far, identical to the moving-optimum model proposed by Kopp and Hermisson (2009b)(see also Bürger 2000).

Genetic assumptions and simulation model

To study the distribution of adaptive substitutions fromstanding genetic variation, we conducted individual-basedsimulations (IBS) (available from the corresponding authorupon request; see Bürger 2000; Kopp and Hermisson 2009b)that explicitly model the simultaneous evolution at multipleloci, while making additional assumptions about the geneticarchitecture of the selected trait, the life cycle of individuals,and the regulation of population size. This serves as our mainmodel.

Genome: Individuals are characterized by a linear (contin-uous) genome of diploid loci, which determine the pheno-type z additively (i.e., there is no phenotypic epistasis; note,however, that there is epistasis for fitness). Mutations occurat constant rate u per haplotype. In contrast to the majorityof individual-based models (e.g., Jones et al. 2004; Kopp andHermisson 2009b; Matuszewski et al. 2014), we do not fixthe number of loci a priori, but instead assume that eachmutation creates a unique polymorphic locus, whose posi-tion is drawn randomly from a uniform distribution over theentire genome (where genome length is determined by therecombination parameter r described below). Consequently,each locus contains only a wild-type allele with phenotypiceffect 0 and a mutant allele with phenotypic effect a, whichis drawn from Equation 3. Thus, we effectively design a bial-lelic infinite-sites model with a continuum of alleles (seealso Figure 1B).

To monitor adaptive substitutions, we introduce a pop-ulation-consensus genome G that keeps track of all poly-morphic loci, that is, of all mutant alleles that are segregatingin the population. If a mutant allele becomes fixed in thepopulation, it is declared the new wild-type allele and itsphenotypic effect is reset to 0. The phenotypic effects of allfixed mutations are taken into account by a variable zfix; whichcan be interpreted as a phenotypic baseline effect. Thus, thephenotype z of an individual i is given by

zi ¼ zfix þX

h2f1;2g

Xl2G

1ði; l; hÞal; (4a)

where

1ði; l;hÞ¼(1 if individual i carries mutant allele a at locus l on haplotype h0 otherwise:

(4b)

Life cycle: Each generation, the following steps areperformed:

1. Viability selection: Individuals are removed with probabil-ity 12wðzÞ (see Equation 1).

2. Population regulation: If, after selection, the populationsize N exceeds the carrying capacity K, N2K randomlychosen individuals are removed.

3. Reproduction: The surviving individuals are randomlyassigned to mating pairs, and each mating pair producesexactly 2B ¼ 4 offspring. Note that under this scheme,the effective population size Ne equals 4=3 times the cen-sus size (Bürger 2000, p. 274). To account for this differ-ence, Q in the analytical approximations needs to becalculated on the basis of this effective size; i.e.,Q ¼ 4Neu: The offspring genotypes are derived fromthe parent genotypes by taking into account segregation,recombination, and mutation.

Table 1 A summary of notation and definitions

Notation Definition

a Phenotypic effect of mutationpðaÞ (Gaussian) distribution of new mutationsz PhenotypezB Mean genetic background phenotypen Rate of environmental changewðz; zoptðtÞÞ (Gaussian) fitness functions2s Width of Gaussian fitness function

s2m Variance of new mutations

s2g (Background) genetic variance

sða; tÞ Time-dependent selection coefficient for allele with phenotypic effect ax Frequency of mutant alleleNe Effective population sizeu Per-locus mutation rateQ Population-wide mutation rate (per trait)Pfix Fixation probabilityrðx;aÞ Distribution of mutant allele frequency at a single locus with phenotypic effect aPSGVðaÞ Probability to adapt from standing genetic variationpSGVðaÞ Distribution of adaptive substitutions from standing genetic variationdeq Equilibrium lag

Adaptation from Standing Variation 1257

Recombination: For each reproducing individual, the num-ber of crossing-over events during gamete formation (i.e.,the number of recombination breakpoints) is drawn froma Poisson distribution with (genome-wide recombination)parameter r (i.e., the total genome length is r � 100 cM;see Supporting Information, File S1). The genomic positionof each recombination breakpoint is then drawn from a uni-form distribution over the entire genome, and the offspringhaplotype is created by alternating between the maternaland the paternal haplotype, depending on the recombina-tion breakpoints. Free recombination (where all loci are as-sumed to be unlinked) corresponds to r/N: In this case,for each locus a Bernoulli-distributed random number isdrawn to determine whether the offspring haplotype willreceive the maternal or the paternal allele at that locus.

Simulation initialization and termination: Starting froma population of K wild-type individuals with phenotypez ¼ 0 (i.e., the population was perfectly adapted at t ¼ 0),we allowed for the establishment of genetic variation, s2

g; byletting the population evolve for 10; 000 generations understabilizing selection with a constant optimum. Increasing thenumber of generations had no effect on the average s2

g forthe population sizes considered in this study. Following thisequilibration time, the optimum started moving under on-going mutational input, and the simulation was stoppedonce all alleles from the standing genetic variation wereeither fixed or lost (i.e., when s2

sgv ¼ 0). Simulations werereplicated until a total number of 5000 adaptive substitu-tions from standing genetic variation were recorded.

The individual-based simulation program is written inC++ and makes use of the GNU Scientific Library (Galassiet al. 2009). Mathematica (Wolfram Research, Champaign,IL) was used for the numerical evaluation of integrals and tocreate plots and graphics, making use of the LevelSchemepackage (Caprio 2005).

Analytical approximations: Evolution of a focal locus inthe presence of genetic background variation

To obtain an analytically tractable model, we need toapproximate the multilocus dynamics. Clearly, simple in-terpolation of single-locus theory will fail, because whenalleles at different loci influencing the same trait segregateas standing genetic variation, the selective dynamics of any

individual allele are critically affected by the collectiveevolutionary response at other loci. In particular, any allelethat brings the mean phenotype closer to the optimumsimultaneously decreases the selective advantage of othersuch alleles (epistasis for fitness). Thus, if simultaneousevolution at many loci allows the population to closelyfollow the optimum, large-effect alleles at any given locusare likely to remain deleterious (as their carriers wouldovershoot the optimum). To account for these effects, weadopt a quantitative-genetics approach originally developedby Lande (1983) and introduce a genetic background whosemean phenotype zB evolves according to Lande’s equation

DzB ¼ s2gb; (5a)

where DzB is the change between generations, s2g is the

(additive) genetic variance, and

b ¼ @logðwÞ@z

(5b)

denotes the selection gradient, which measures the changein log mean fitness per unit change of the mean phenotype(Lande 1976). Furthermore, under the simplifying assump-tion that s2

g remains constant over time, the mean back-ground phenotype evolves according to

zBðtÞ � vt2vg

�12 ð12gÞt

�; (6a)

with

g ¼ s2g

s2s þ s2

g(6b)

(Bürger and Lynch 1995).Given the dynamics of the genetic background, we then

choose a single focal locus and derive the time-dependentselection coefficient sða; tÞ for a mutant allele with pheno-typic effect a (for details see below; see also Figure 1C). Wethen use theory for adaptation from standing genetic varia-tion (Hermisson and Pennings 2005) and for fixation undertime-inhomogeneous selection (Uecker and Hermisson2011) (summarized in Appendix B) to estimate the proba-bility that this allele contributes to adaptation. If there is no

Figure 1 Illustration of our main model assumptions.(A) Phenotypic evolution in the moving-optimummodel: The black curve and colored circles give thedistribution of phenotypes for three different snap-shots in time for a population that adapts to a time-dependent phenotypic optimum (green circles) underGaussian stabilizing selection (see equation). Note thatthe population mean phenotype typically lags behindthe optimum. (B) The biallelic infinite-sites model: Eachmutation creates a new allele at a unique site. (C) Thefocal-locus genetic-background model used in our an-alytical approximation (Lande 1983).


http://www.genetics.org/lookup/suppl/doi:10.1534/genetics.115.178574/-/DC1/genetics.115.178574-1.pdf


linkage (i.e., there is free recombination between all loci),each locus can be viewed as the focal locus (with a specificphenotypic effect a), allowing us to obtain an estimate forthe overall distribution of adaptive substitutions from stand-ing genetic variation. Thus, in these approximations, ourmultilocus model is effectively treated within a single-locusframework. Note that a similar focal-locus approach has re-cently been used to analyze the effect of genetic backgroundvariation on the trajectory of an allele sweeping to fixation(Chevin and Hospital 2008) and to study the probability ofadaptation to novel environments (Gomulkiewicz et al.2010), with both studies stressing the fact that genetic back-ground variation cannot be neglected and critically affectsthe adaptive outcome.

Results

In the following, we calculate, first, the probability thata focal allele from the standing genetic variation becomesfixed when the population adapts to a moving phenotypicoptimum. Based on this intermediate result, we then derivethe effect-size distribution of such alleles and discuss theoverall potential for adaptation from standing genetic varia-tion and its importance relative to de novo mutations.

Note that, for the first result, we assume the focal alleleto arise from a wild-type allele by recurrent mutation withthe per-locus (population-wide) mutation rate u. For all sub-sequent results, we let u/0; in accordance with the infinite-sites model described above.

The probability for adaptation from standinggenetic variation

The probability that a focal mutant allele from the standinggenetic variation contributes to adaptation depends on thedynamics of its selection coefficient in the presence ofgenetic background variation. For an allele with pheno-typic effect a and a genetic background with mean zB and(constant) variance s2

g; the selection coefficient can be cal-culated as

sða; tÞ ¼ wðaþ zBðtÞ; tÞwðzBðtÞ; tÞ 2 1

� 2a2

2�s2s þ s2

g�þ a

s2s þ s2

gðvt2 zBðtÞÞ:

(7)

Note that the genetic background variance has the effect ofbroadening the fitness landscape experienced by the focalallele (the term s2

s þ s2g). Plugging Equation 6a into Equa-

tion 7 then yields the time-dependent selection coefficient,

sða; tÞ � 2a2

2�s2s þ s2

g�þ av

g�s2s þ s2

g� �12 ð12gÞt

�: (8)

Assuming that the population is perfectly adapted at t ¼ 0(zB ¼ 0), the initial (deleterious) selection coefficient isgiven by

sða; 0Þ¼2a2

2�s2s þ s2

g�:

Unlike in the model without genetic background variation(Kopp and Hermisson 2009b), sða; tÞ does not increase lin-early, but instead depends on the evolution of the pheno-typic lag d between the optimum and the mean backgroundphenotype. In particular, the population will reach a dynamicequilibrium with DzB ¼ v; where it follows the optimumwith a constant lag

deq ¼ vg

(9)

(Bürger and Lynch 1995). Consequently, the selection co-efficient for a approaches

limt/N

sða; tÞ¼2a2

2�s2s þ s2

g

�þ av

g�s2s þ s2

g

�: (10)

Note that the right-hand side can be written assða; 0Þ þ abeq; where beq is the equilibrium selection gradi-ent (Kopp and Matuszewski 2014). In this case, the largestobtainable selection coefficient is for a ¼ deq and evalu-ates to

smax ¼ limt/N

s�deq; t

� ¼ v2

2g2�s2s þ s2

g

�: (11)

The range of allelic effects a that can reach a positive selec-tion coefficient is bounded by amin ¼ 0 and amax ¼ 2deq:Note that in previous adaptive-walk models (e.g., Koppand Hermisson 2009b; Matuszewski et al. 2014) there wasno strict amax; since the population followed the optimum bystochastic jumps, whereas in the present model, the geneticbackground evolves deterministically and establishes a con-stant equilibrium lag.

Assuming the mutant allele was initially deleterious andarises recurrently at rate u=2; its allele frequency spectrumprior to the onset of environmental change, rðx;aÞ; is givenby Equation B5. The fixation probability for a mutation start-ing with n initial copies, Pfix;sgða; nÞ; is then given by Equa-tion B7a, where Equation B7b is given by

2uðaÞ ¼1þRN0 ð1þ sða; tÞÞ3 exp

"2

2

a2

2�s2s þ s2

g

�!

þ �

12 ð12gÞt� 1

loghð12gÞt

iþ1

!3

av

g�s2s þ s2

g

�!t

#dt

(12)

(using Equation 8). In the general case, this expressionneeds to be evaluated numerically. Only in the limit s2

g/0 isthere an explicit solution


Pfix;s2g¼0ða; nÞ

¼ 12

12

"1þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip

2�av�s2s�rexp

sða; 0Þ22�av�s2s�!

3 erfc

sða; 0Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�av�s2s�q!#21!n

; (13)

where erfcðxÞ ¼ ð2= ffiffiffiffip

p Þ RNx exp½2t2�dt denotes the Gauss-ian complementary error function.

Finally, combining Equations B5, B7, and 12, the proba-bility that an allele with allelic effect a is present in thestanding variation and contributes to adaptation can thenbe calculated as

PSGVðaÞ ¼

12CðaÞ R 10 xu21exph24Nejsða; 0Þjx

i

3

�12 1

uðaÞ

2Nx

dx if 0,a,amax

0 otherwise;

8>>>>><>>>>>:

(14)

where CðaÞ ¼ ðg½u; 4Nejsða; 0Þj�=ð4Nejsða; 0ÞjÞuÞ21:

The analytical approximation (Equation 14) representsan important intermediate result toward our goal of derivingthe distribution of adaptive substitutions from standinggenetic variation. Yet, unlike the main model, it assumesthat the focal locus is subject to recurrent mutation (seeEquation B5). To test its accuracy, we therefore used analternative simulation approach based on the Wright–Fishermodel (see Appendix A for details). As shown in Figure 2 andFigure S3_1 (File S3), Equation B5 performs generally verywell. The only exception occurs when the background varia-tion is high (large s2

g) and stabilizing selection is weak (i.e., ifs2s is large). In this case, Equation 14 underestimates PSGVðaÞ

for small a � 0:5sm: The reason is that, under a constantoptimum (i.e., before the environmental change), the geneticbackground compensates for the deleterious effect of a (i.e.,zB , 0, in violation of our assumption that zBð0Þ ¼ 0), effec-tively reducing the selection strength against the deleteriousmutant allele. Consequently, a is, on average, present athigher initial frequencies than predicted by Equation B5.

Note that, if a is small compared to the genetic back-ground variation (i.e., in the limit of a=sg/0), selectionis weak (s2

s large), and environmental change is slow (i.e.,v � 1025), PSGVðaÞ will approach the probability of fixationfrom standing genetic variation for a neutral allele [i.e.,a ¼ 0; see Figure S3_1, bottom right (File S3)]. This prob-ability can be calculated as

PSGV; neutral ¼Z 1

0xrðxÞdx ¼ Hu2 1

~g þ cðuÞ; (15)

where rðxÞ is given by Equation B3, Hn denotes the nthharmonic number, ~g � 0:577 is Euler’s gamma, and cð�Þ is

the polygamma function [see dashed lines in Figure 2 andFigure S3_1 (File S3)].

Figure 2 and Figure S3_1 (File S3) show some generaltrends: First, the probability for a mutant allele to become fixedincreases with the rate of environmental change, v (irrespectiveof its effect size a, the per-locus mutation rate u and the widthof the fitness landscape s2

s ), since only the positive term inEquation 8 depends (linearly) on v. Second, PSGVðaÞ is propor-tional to u as long as u is small [compare u ¼ 0:004 andu ¼ 0:04 in Figure S3_1 (File S3)], simply because the proba-bility that a is present in the population is linear in u. Thus,Figure 2 is representative for the limit u/0; which is usedbelow. Indeed, only if the per-locus mutation rate is fairly large(u. 0:1), does the shape of the distribution of allele frequen-cies become important, and the increase in PSGVðaÞ with u

becomes less than linear (Figure S3_1 (File S3)). Third,changes in the width of the fitness landscape, s2

s ; have a dualeffect: While increasing s2

s promotes the initial frequency of thefocal allele in the standing genetic variation (because stabilizingselection is weaker), the selection coefficient increases moreslowly after the onset of environmental change (such that theallele is less likely to be picked up by selection; see Equation 7).Our results, however, show that the former effect always out-weighs the latter (as PSGVðaÞ increases with s2

s ). Finally, if thegenetic background variation s2

g is below a threshold value(e.g., s2

g , 0:005; the exact threshold should depend on u ands2s ), it only marginally affects the fixation probability of the focal

allele a. Once s2g surpasses this value, however, it critically

affects PSGVðaÞ (in accordance with the results by Chevin andHospital 2008). In particular, as s2

g increases PSGVðaÞ decreases,because most large-effect alleles remain deleterious even if en-vironmental change is fast. Thus, enlarged background variationacts as if reducing the rate of environmental change v.

The distribution of adaptive substitutions from standinggenetic variation

We now derive the distribution of adaptive substitutionsfrom standing genetic variation over all mutant effects a. Inthe previous section, we derived the probability of adapta-tion at a focal locus (with a given effect a) by treating thegenetic background variance s2

g as an independent modelparameter. In the full model (as in the individual-based sim-ulations), this variance emerges from a balance of mutation,selection, and drift at all background loci. As such, it isa function of the basic model parameters for these forces.Since we use an infinite-sites model, there is no recurrentmutation and each allele originates from a single mutation.In this situation, the background variance s2

g is accuratelypredicted by the stochastic house-of-cards (SHC) approxi-mation (results not shown; Bürger and Lynch 1995)

s2g ¼ Qs2

m

1þ Nes2m�s2s; (16)

where mutation is parametrized by the total (per trait)mutation rate Q and the variance of de novo mutations s2

m;








the width of the fitness landscape is given by s2s ; and the

effective population size Ne is a measure for genetic drift.To derive the probability that an allele with a given

phenotypic effect a contributes to adaptation, we first needto calculate the probability that such an allele segregates inthe population at time 0. Following Hermisson and Pennings(2005), the probability P0 that the allele is not presentcan be approximated by integrating over the distributionof allele frequencies rðx;aÞ (Equation B5) from 0 to 1=2Ne;

yielding

P0ðaÞ��

2Ne

4Nejsða; 0Þj þ 1

2u

¼ exp2u log

2Ne

4Nejsða; 0Þj þ 1

�� (17)

(equation 7 and appendix of Hermisson and Pennings2005). The fixation probability can then be calculated byconditioning on segregation of the allele in the limit u/0(due to the infinite-sites assumption). Using Equation 14,this probability reads

PsegðaÞ

¼ limu/0

PSGVðaÞ12 P0ðaÞ

� limu/0

12CðaÞ R 10 xu21exp½24Nejsða; 0Þjx��12 1

uðaÞ�2Nx

dx

12 exp½2ulog½2Ne=ð4Nejsða; 0Þj þ 1Þ�� ;

(18)

where CðaÞ ¼ ðg½u; 4Nejsða; 0Þj�=ð4Nejsða; 0ÞjÞuÞ21 (see alsoEquation B5) and uðaÞ is given by Equation 12. The limit inEquation 18 can be approximated numerically by setting u toa very small, but positive value.

Multiplying by the rate of mutations with effect a [i.e.,Q pðaÞ], the distribution of adaptive substitutions fromstanding genetic variation is given by

pSGVðaÞ � QpðaÞPsegðaÞR amax0 QpðaÞPsegðaÞda

¼ C1ðaÞpðaÞPsegðaÞ;(19)

where C1ðaÞ is a normalization constant [black line in Figure3, Figure 4, Figure 5, and Figure S3_2 and Figure S3_3 (FileS3)]. Note that Equation 19 still depends on Q through itseffect on the background variance s2

g [which affects PsegðaÞ].In particular, in the SHC approximation (Equation 16), s2

gscales linearly with Q: Furthermore, Equation 19 should bevalid for any distribution of mutational effects pðaÞ:

In the limit where the equilibrium lag is reached fast (i.e.,when g is large; Equation 6b), the moving-optimum modelreduces to a model with constant selection for any focalallele (i.e., as in Hermisson and Pennings 2005). UsingEquations B6 and 17, the fixation probability for a segregat-ing allele can be calculated as

Pseg;SGV;deqðaÞ

� limu/0

12 exp½2ulog½1þ 4Nesða;NÞ=ð4Nejsða; 0Þj þ 1Þ��12 P0ðaÞ :

(20)

Plugging Equation 20 into Equation 19, the distribution ofadaptive substitutions from standing genetic variation canbe approximated by

pSGV;deqðaÞ � C2ðaÞpðaÞPseg;SGV;deqðaÞ; (21)

where C2ðaÞ is a normalization constant [red line in Figure3, Figure 4, and Figure S3_2 and Figure S3_3 (File S3)].

Figure 2 The probability for a mutant allele to adapt from standing genetic variation as a function of the rate of environmental change v. Solid linescorrespond to the analytical prediction (Equation 14), the gray dashed line shows the probability for a neutral allele (a ¼ 0; Equation 15), and symbolsgive results from Wright–Fisher simulations (see Appendix A). The phenotypic effect size a of the mutant allele ranges from 0:5sm (top line; black) to3sm (bottom line; purple) with increments of 0:5sm: The numbers in each parameter box (per locus mutation rate u, width of fitness landscape s2

s )correspond to different values of the genetic background variation s2

g with s2g ¼ 0 (no background variation; top left), s2

g ¼ 0:005 (top right), s2g ¼ 0:01

(bottom left), and s2g ¼ 0:05 (bottom right). Other parameters: Ne ¼ 25;000; u ¼ 0:004; s2

m ¼ 0:05:





Similarly, the fixation probability of de novo mutationsunder the equilibrium lag deq can be derived [using Equation10 and Equation B2 with an initial frequency of 1=ð2NÞ] as

Pfix;DNM;deqðaÞ ¼ 12 exp

"2a�2deq 2a

�s2s þ s2

g

#!; (22)

yielding the distribution of adaptive substitutions

pDNM;deqðaÞ � pðaÞC3ðaÞPfix;DNM;deqðaÞ; (23)

where C3ðaÞ is a normalization constant [gray-shaded areain Figure 3, Figure 4, and Figure S3_2 and Figure S3_3 (FileS3)].

In contrast, if the environment changes very slowly, wecan calculate the limit distribution of adaptive substitutions

from standing genetic variation by approximating thefixation probability by that of a neutral allele, which isequal to its initial allele frequency x (even though the fre-quency spectrum, i.e., the distribution of x, is not that ofa neutral allele). In this case,

Pseg;v/0ðaÞ � limu/0

ℱ ðaÞ12 P0ðaÞ (24a)

with

ℱ ðaÞ ¼Z 1

0rðx;aÞ xdx ¼ 1Fð0; uþ 1; 4Nejsða; 0ÞjÞ1

1Fð0; u; 4Nejsða; 0ÞjÞ1;

(24b)

where rðx;aÞ is given by Equation B4 and the right-handside is a ratio of hypergeometric functions.

Figure 3 The distribution of adaptive substitutions from standing genetic variation. Histograms show results from individual-based simulations. Theblack line corresponds to the analytical prediction (Equation 19), with the genetic background variation s2

g determined by the SHC approximation(Equation 16). The red line gives the analytical prediction for the limiting case where the equilibrium lag deq is reached fast (Equation 21). The blue line isbased on the analytical prediction (Equation 25), which assumes a neutral fixation probability, but has been shifted so that it is centered around theempirical mean. The gray-shaded area gives the analytical prediction for substitutions from de novo mutations under the assumption that thephenotypic lag deq has reached its equilibrium (Equation 23). The asterisks indicate where Nesmax $10: Fixed parameter: s2

m ¼ 0:05:




Using Equation 24a, the distribution of substitutions fromstanding genetic variation reads

pSGV;v/0ðaÞ � C4ðaÞpðaÞPseg;v/0ðaÞ; (25)

where C4ðaÞ again denotes a normalization constant (blueline in Figure 3; Figure 4; and Figure S3_2, Figure S3_3 andFigure S3_4 (File S3)).

The accuracy of the approximation: When compared toindividual-based simulations, our analytical approximation forthe distribution of adaptive substitutions from standinggenetic variation (Equation 19) performs, in general, verywell as long as selection is strong; that is, the rate ofenvironmental change v is high and/or the width of the fitnesslandscape s2

s is not too large [Figure 3 and Figure S3_2 (FileS3)]. Under weak selection, however, Equation 19 fails tocapture the fixation of alleles with neutral or negative effects(“backward fixations”; a# 0). The reason is that Equation B7considers only the fixation of alleles whose selection coeffi-cient sða; tÞ becomes positive in the long term. But if the rateof environmental change is slow (or s2

s is very large), mostalleles get fixed or lost simply by chance, that is, genetic drift.In particular, if genetic drift is the main driver of phenotypicevolution [i.e., Nejsða; tÞj, 1], the distribution of adaptivesubstitutions is almost symmetric around 0 [see Figure S3_4(File S3)]. This distribution is described very well by Equation25, which assumes that the fixation probability of an allele isproportional to its initial frequency in the standing variation.In addition, even for cases where environmental change imposesmodest directional selection, Equation 25 still captures theshape of the distribution of adaptive substitutions reasonablywell, when centered around the empirical mean (blue line inFigure 3, Figure 4, and Figure S3_2 and Figure S3_3 (File S3)).

With a moving phenotypic optimum, the selection co-efficient (Equation 8) is initially very small. Accordingly,there is always a phase during the adaptive process wheregenetic drift dominates, that is, where Nejsða; tÞj,1 for allmutant alleles. The length of this phase (i.e., the time ittakes until selection becomes the main force of evolution)depends on the interplay of multiple parameters, notablyv;s2

s ; Ne; and Q: A good heuristic to determine whether evo-lution will ultimately become dominated by selection is tocalculate Nesmax (Equation 11), which gives the maximal pop-ulation-scaled selection coefficient. Since the selection coeffi-cient of most mutations will be smaller than this value, onecan consider as a rule of thumb that selection is the maindriver of evolution as long as Nesmax $ 10: In this case, Equa-tion 19 matches the individual-based simulations very well[see asterisks in Figure 3, Figure 4, and Figure S3_2 andFigure S3_3 (File S3)], and this holds true even when themutation rate Q is an order of magnitude larger than therange used in the majority of our individual-based simulations[Figure S3_5 (File S3)]. In summary, the accuracy of our ap-proximation crucially depends on the efficacy of selection.

While our analytical approximations assume that thegenetic background variation remains constant, s2

g in ourIBS is an emergent property of the model parameters (no-tably Ne;s

2s ;Q) and, thus, is expected to fluctuate (e.g., due

to genetic drift). Furthermore, with a moving phenotypicoptimum, s2

g is expected to increase—the larger v is, thestronger the increase in s2

g—because of initially rare large-effect alleles increasing in frequency (Bürger 1999). In gen-eral, however, our approximations seem to be robust tothese effects, suggesting that these temporal changes in s2

gdo not affect the adaptive process strongly.

The effects of linkage on the distribution of adaptivesubstitutions from standing genetic variation are discussed

Figure 4 The distribution of adaptive substitutions from standing genetic variation for various rates of environmental change. For further details seeFigure 3. Fixed parameters: Q ¼ 2:5; N ¼ 2500; s2

m ¼ 0:05:









in File S1. The main result is that only tight linkage hasa noticeable effect, namely to reduce the efficacy of selectionand increase the proportion of backward fixations (movingthe distribution closer to the prediction from Equation 25).

Biological interpretation: As shown in Figure 3 and Figure4 [see also Figure S3_2, Figure S3_3 and Figure S3_5 (FileS3)], adaptive substitutions from standing genetic variationhave, on average, smaller phenotypic effects than those fromde novo mutations. There are two reasons for this result.First, in the standing genetic variation, small-effect allelesare more frequent than large-effect alleles and might al-ready segregate at appreciable frequency (increasing theirfixation probability). Second, substitutions from standingvariation occur in the initial phase of the adaptive process,where the phenotypic lag is small, whereas our approxima-tion for de novo mutations (Equation 23) assumes that thephenotypic lag has reached its maximal (equilibrium) value(which need not be large, depending on the amount of ge-netic background variation). The relative importance ofthese two effects can be seen in Figure 3 and Figure 4[see also Figure S3_2 and Figure S3_3 (File S3)]: Compar-ing the gray-shaded area (Equation 23; de novo mutationsunder the equilibrium lag) with the red line (Equation 21;standing genetic variation under the equilibrium lag) showsthe effect of larger starting frequencies of small-effect muta-tions from the standing genetic variation. The difference ofthe black (Equation 19; standing genetic variation) and red(Equation 21; standing genetic variation under the equilib-rium lag) lines shows the effects of the initially smaller lag(i.e., the effect of the dynamical selection coefficient). Notethat the first effect is always important (even if Q and s2

s arelarge and v is small, where the red line and the gray-shadedarea almost coincide—although this is only because the

approximation is bad). The second effect, however, becomesparticularly important if g ¼ s2

g=ðs2g þ s2

s Þ is small (i.e., ifthe time to reach the equilibrium lag is large), such thatselection coefficients are dynamic and small-effect allelesare selected earlier than large-effect alleles, explaining therelative lack of large-effect alleles in the distribution ofadaptive substitutions.

Generally, the distribution of adaptive substitutions isunimodal and resembles a log-normal distribution [Figure 3,Figure 4, and Figure S3_5 (File S3)]. Only if selection is veryweak (i.e., when s2

s is large and/or v is small), does it containa significant proportion of backward fixations [with negativea; Figure 4 and Figure S3_3 (File S3); see The accuracy ofthe approximation]. As the rate of environmental change vincreases, the mean phenotypic effect of substitutions increases(Figure 5, top row), too, but the mode may actually decrease(Figure 4); that is, the distribution becomes more asymmetricand skewed, resembling the “almost exponential” distributionof substitutions from de novo mutations in the sudden-changescenario (Orr 1998). A likely explanation is that small-effectalleles, which are common in the standing variation, are understronger selection and have an increased fixation probability ifv is large (see Figure 2).

Interestingly, if the environment changes very fast, thesimulated distribution of adaptive substitutions from stand-ing genetic variation almost exactly matches the onepredicted by Equation 23 for de novo mutations [Figure 6;see also Figure 3, Figure 4, and Figure S3_2 and Figure S3_3(File S3)]. However, this seems to be an artifact rather thana relevant biological phenomenon. The reason is that theenvironment changes so fast that the population quickly diesout. Thus, the resulting distribution of adaptive substitutionsis that for a dying population and might not necessarily re-flect the adaptive process. In an experimental setup, though,

Figure 5 The mean size of adap-tive substitutions from standinggenetic variation, measured inunits of mutational standard devi-ations (sm) as a function of therate of environmental change n

(A, C, and E) and for various n

as a function of the population-wide mutation rate Q (B), thewidth of the fitness landscapes2s (D), and the population size

N (F). Lines show the analyticalprediction (the mean of thedistribution, Equation 19), and sym-bols give results from individual-based simulations. Error bars forstandard errors are containedwithin the symbols. For n ¼ 0:1;no simulation results are shown,as these constitute a degeneratecase (for details see The accuracyof the approximation). Fixedparameter: s2

m ¼ 0:05:









where populations evolve until they go extinct, the distribu-tion of adaptive substitutions from standing genetic varia-tion might truly be indistinguishable from that from de novomutations.

In the following, we discuss the influence of the othermodel parameters (Q; s2

s ; and N) on the distribution ofadaptive substitutions from standing genetic variation and,in particular, its mean a (Figure 5).

Increasing the rate of mutational supply Q generallydecreases a via its effect on the equilibrium lag deq ¼ v=g:More precisely, increasing Q increases the background geneticvariance s2

g (Equation 16) and hence g, which in turndecreases deq and reduces the likelihood for the fixation oflarge-effect alleles. This effect is strongest if deq is small rela-tive to the mutational standard deviation sm; but weakens ifdeq is large. Note that, as long as s2

s � s2g; deq is approxi-

mately proportional to v=Q: Consequently, in Figure 5, Aand B, a decreases with Q for small but not large values ofthe rate of environmental change v, but, when increasing Q tovery large values, a decreases even for comparably large v[Figure S3_5 (File S3)]. In summary, the effect of the rateof mutational supply on the mean size and the distributionof adaptive substitutions from standing genetic variationdepends strongly on the rate of environmental change.

The width of the fitness landscape s2s affects different

aspects of the adaptive process, but its net effect is an in-crease of the mean effect size of fixed alleles as s2

s increases(i.e., as stabilizing selection gets weaker), especially if therate of environmental change is intermediate (Figure 5D).The reason is that weak stabilizing selection increases thefrequency of large-effect alleles in the standing variation. Inaddition, weak selection also increases the phenotypic lag(Equation 9; see also Kopp and Matuszewski 2014), againfavoring large-effect alleles. Note that the latter point holdstrue even though weak selection increases the backgroundvariance s2

g: Finally, the effect of s2s is strongest for interme-

diate v, because for small v, large-effect alleles are neverfavored, whereas for large v, all alleles with positive effecthave a high fixation probability.

Similar arguments hold for Ne (when the rate of muta-tional supply, Q; is held constant). First, increasing Ne willalways increase the efficacy of selection, resulting in lowerinitial frequencies of mutant alleles (Equation B4) and de-creased s2

g (Equation 16). If the environment changesslowly, a increases with Ne; because the equilibrium lagincreases (caused by the decrease in s2

g). In contrast, ifthe rate of environmental change is fast, a slightly decreaseswith Ne due to the lower starting frequency of large-effectalleles and because small-effect alleles are selected moreefficiently (i.e., they are less prone to get lost by geneticdrift; Figure 5F).

The potential for adaptation from standing geneticvariation and the rate of environmental change

So far, we have focused on the distribution of adaptivesubstitutions for individual fixation events. We now address

what can be said about the total progress that can be madefrom standing genetic variation following a moving pheno-typic optimum. The overall potential for adaptation fromstanding genetic variation depends on the mean number ofalleles segregating as standing genetic variation, which canbe accurately approximated as

jG j ¼ 1þQlog2s2

ss2m

�(26)

(results not shown; Foley 1992). The mean number ofalleles that become fixed can then be calculated as

jG jfix ¼ jG jZ amax

0pðaÞPsegðaÞda; (27)

where the integral equals the normalization constant inEquation 19 (i.e., the proportion of fixed alleles). Finally,using Equation 27, the average distance traveled in pheno-type space before standing variation is exhausted is given by

z* ¼ 2jG jfix a ¼ 2jG jZ amax

0apðaÞPsegðaÞda; (28)

where a is the mean phenotypic effect size of adaptive sub-stitutions from standing genetic variation, and the factor 2in Equation 28 comes from the fact that we are consideringdiploids (and a denotes the phenotypic effect per haplo-type). Note that, once the shift of the optimum considerablyexceeds z*; the population will inevitably go extinct withoutthe input of new mutations.

Figure 7 [see also Figure S3_6, Figure S3_7, Figure S3_8,Figure S3_9, Figure S3_10 and Figure S3_11 (File S3)] illus-trates these predictions and compares them to results fromindividual-based simulations (where, unlike in the rest ofthis article, new mutational input was turned off after theonset of the environmental change).

Both the mean number of fixations jG jfix and the meanphenotypic distance traveled z * increase with the rate of en-vironmental change, reflecting the fact that more and larger-effect alleles become fixed if the environment changes fast.

Figure 6 The distribution of adaptive substitutions from standing geneticvariation in the case of fast environmental change. For further details seeFigure 3. Fixed parameters: s2

s ¼ 100; Q ¼ 10; N ¼ 2500; n ¼ 0:1;s2m ¼ 0:05:




Only for very large v do jG jfix and z * decrease sharply,because the population goes extinct before fixations can becompleted. Note that in these cases the rate of environmen-tal change exceeds the “critical rate of environmentalchange” (Bürger and Lynch 1995, p. 151) [shaded dashedline in Figure 7 and Figure S3_9, Figure S3_10 and FigureS3_11 (File S3)], which for our choice for the number ofoffspring B ¼ 2 equals

vcrit ¼ s2g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2log

h2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2s��

s2g þ s2

s�q i

s2g þ s2

s

vuut: (29)

At small values of v, jG jfix matches the “neutral” prediction(gray dashed line in Figure S3_6, Figure S3_7 and FigureS3_8 (File S3)). Note that these fixations have almost noeffect on z*; because their average effect is zero. At inter-mediate v, Equation 28 slightly underestimates z * for param-eter values leading to large background variance s2

g (i.e.,high Q and s2

s ). The likely reason is that the analyticalapproximation assumes s2

g to be constant, while it obviouslydecreases in the simulations (since there are no de novomutations). All these results are qualitatively consistentacross different values of s2

s and Q [Figure 7 and FigureS3_9 and Figure S3_11 (File S3)].

The relative importance of standing genetic variationand de novo mutations over the course of adaptation

Until now we have compared adaptation from standinggenetic variation to that from de novo mutations in terms oftheir distribution of fixed phenotypic effects. We now turn toinvestigating their relative importance over the course ofadaptation. For this purpose, we recorded (in individual-based simulations) the contributions of both sources of var-iation to the phenotypic mean and variance. An averagetime series for both measures is shown in Figure 8. Asexpected, the initial response to selection is almost entirelybased on standing variation, but the contribution of de novomutations increases over time. As a quantitative measure forthis transition, we define tDNM;50ðzÞ as the point in timewhere the cumulative contribution of de novo mutationshas reached 50%: Indeed, we find that, beyond this time,adaptation almost exclusively proceeds by the fixation ofde novo mutations (Figure 8A). As expected, tDNM;50ðzÞdecreases with v [Figure 9 and Figure S3_12 (File S3), firstrow], while the total phenotypic response z increases (Fig-ure 9 and Figure S3_12 (File S3), second row). The reason isthat faster environmental change induces stronger directionalselection and increases the phenotypic lag, such that standingvariation is depleted more quickly and de novo mutations con-tribute earlier. Note that, as in Figure 7, the total phenotypicresponse at time tDNM;50ðzÞ decreases once v exceeds the max-imal sustainable rate of environmental change, for the samereasons as discussed above. Furthermore, tDNM;50ðzÞ increaseswith both Q and s2

s (due to the increased standing variation;see Equation 16). Interestingly, the relative contribution of

original standing genetic variation to the total genetic varianceat time tDNM;50ðzÞ remains largely constant (at �20%) overa large range of v and does not show any dependence on Q ors2s [Figure 9 and Figure S3_12 (File S3), third row]. Devia-

tions occur only if v is either very small or very large. Inparticular, if v is small, standing variation is almost completelydepleted before new mutations play a significant role. Con-versely, if v is very large, standing genetic variation still formsthe majority of the total genetic variance. As mentioned above,this is most likely because the population goes extinct beforefixations can be completed, that is, before the entire (stand-ing) adaptive potential is exhausted. All these results remainqualitatively unchanged if, instead of tDNM;50ðzÞ; we definetDNM;50ðs2

gÞ as the point in time where 50% of the currentgenetic variance goes back to de novo mutations [FigureS3_13 and Figure S3_14 (File S3)].

Discussion

When studying the genetic basis of adaptation to changingenvironments, most theoretical work has focused on adap-tation from new mutations (e.g., Gillespie 1984; Orr 1998,2000; Collins et al. 2007; Kopp and Hermisson 2007, 2009a,b; Matuszewski et al. 2014). Consequently, very little isknown about the details of adaptation from standing geneticvariation (but see Orr and Betancourt 2001; Hermisson andPennings 2005), that is, which of the alleles segregating ina population will become fixed and contribute to the evolu-tionary response. Here, we have used analytical approxima-tions and stochastic simulations to study the effects ofstanding genetic variation on the genetic basis of adaptationin gradually changing environments. Supporting a verbalhypothesis by Barrett and Schluter (2008), we show that,when comparing adaptation from standing genetic variationto that from de novo mutations, the former proceeds, onaverage, by the fixation of more alleles of small effect. Inboth cases, however, the genetic basis of adaptation cruciallydepends on the efficacy of selection, which in turn is de-termined by the population size, the strength of (stabilizing)selection, and the rate of environmental change. Whenstanding genetic variation is the sole source for adaptation,we find that fast environmental change enables the popula-tion to traverse larger distances in phenotype space thanunder slow environmental change, in contrast to studies thatconsider adaptation from new mutations only (Perron et al.2008; Bell and Gonzalez 2011; Bell 2013; Lindsey et al.2013). We now discuss these results in greater detail.

The genetic basis of adaptation in themoving-optimum model

Introduced as a model for sustained environmental change,such as global warming (Lynch et al. 1991; Lynch and Lande1993), the moving-optimum model describes the evolutionof a quantitative trait under stabilizing selection towarda time-dependent optimum (Bürger 2000). A large numberof studies have analyzed both the basic model and several









modifications, for example, models with a periodic or fluc-tuating optimum or models for multiple traits (Slatkin andLande 1976; Charlesworth 1993; Bürger and Lynch 1995;Lande and Shannon 1996; Kopp and Hermisson 2007,2009a,b; Gomulkiewicz and Houle 2009; Zhang 2012;Chevin 2013; Matuszewski et al. 2014). Following tradi-tional quantitative-genetic approaches, the majority of thesestudies assumed that the distribution of genotypes (and phe-notypes) is Gaussian with constant (time-invariant) geneticvariance, and they have mostly focused on the evolution ofthe population mean phenotype and on the conditions forpopulation persistence (Bürger and Lynch 1995; Lande andShannon 1996; Gomulkiewicz and Houle 2009). None ofthese models, however, allows one to address the fate ofindividual alleles (i.e., whether they become fixed or not).In a recent series of articles on the moving-optimum model,Kopp and Hermisson (2007, 2009a,b) studied the geneticbasis of adaptation from new mutations and derived thedistribution of adaptive substitutions (i.e., the distributionof the phenotypic effects of those mutations that arise andbecome fixed in a population); this approach has recentlybeen generalized to multiple phenotypic traits by Matuszewskiet al. (2014). The shape of this distribution resemblesa gamma distribution with an intermediate mode. Thus,most substitutions are of intermediate effect with onlya few large-effect alleles contributing to adaptation. Thereason is that small-effect alleles—despite appearing morefrequently than large-effect alleles—have only small effectson fitness (and are, hence, often lost due to genetic drift),while large-effect alleles might be removed because they“overshoot” the optimum (Kopp and Hermisson 2009b). A de-tailed comparison and discussion of the distribution of adap-tive substitutions from de novo mutations with (Equation 23)

and without (Kopp and Hermisson 2009b) genetic backgroundvariation are given in File S2.

Here, we have studied the genetic basis of adapta-tion from standing genetic variation. We find that thedistribution of substitutions from standing genetic variationdepends on the distribution of standing genetic variants(i.e., distribution of alleles segregating in the populationprior to the environmental change) and the intensity of se-lection. The former is shaped primarily by the distribution ofnew mutations and the strength of stabilizing selection,which removes large-effect alleles. Depending on the speedof change v, we find two regimes that are characterized byseparate distributions of standing substitutions. If the envi-ronment changes sufficiently fast, the distribution of adap-tive substitutions resembles a lognormal distribution witha strong contribution of small-effect alleles [Equation 19;Figure 3; and Figure S3_2 (File S3)]. The reason is that,in the standing genetic variation, small-effect alleles aremore frequent than large-effect alleles and might alreadysegregate at appreciable frequency (so that they are not lostby genetic drift). With a moving optimum, they furthermoreare the first to become positively selected, hence reducingthe time they are under purifying selection. Finally, epistaticinteractions between cosegregating alleles (or between a fo-cal allele and the genetic background) also favor alleles ofsmall effect. Consequently, when adapting from standinggenetic variation, most substitutions are of small phenotypiceffect.

The second regime occurs if the rate of environmentalchange v is very small. In this case, allele-frequency dynam-ics are dominated by genetic drift, and the distribution ofadaptive substitutions reflects the approximately Gaussiandistribution of standing genetic variants [Equation 25; Figure

Figure 7 The average distance traversed inphenotype space, z*; as a function of the rateof environmental change n, when standing ge-netic variation is the sole source for adaptation.Symbols show results from individual-basedsimulations (averaged over 100 replicate runs).The solid line gives the analytical prediction(Equation 28), with s2

g taken from Equation16. The shaded dashed line gives the criticalrate of environmental change (Equation 29).Error bars for standard errors are containedwithin the symbols. Fixed parameters: N ¼ 2500;s2m ¼ 0:05:




S3_4 (File S3)]. It should be noted, however, that fixationsunder this regime take a very long time, similar to that ofpurely neutral substitutions (i.e., on the timescale of 4Ne).

Finally, we have studied the relative importance ofstanding genetic variation and de novo mutations over thecourse of adaptation. As shown in Figure 4 and Figure 5,the initial response to selection is almost entirely basedon standing variation, with de novo mutations becominggradually more important. The timescale of this transitionstrongly depends on the rate of environmental change, butfor slow or moderately fast change, it typically occurs over atleast hundreds of generations [Figure 9 and Figure S3_12,Figure S3_13 and Figure S3_14 (File S3)]. This observationis in contrast to results by Hill and Rasbash (1986b), whofound that under strong artificial (i.e., truncation) selectionin small populations (N ¼ 20), new mutations might con-tribute up to one-third of the total response after as little as20 generations. Our results show that the situation is verydifferent for large populations under natural selection ingradually changing environments. The likely reason for thisdifference is that truncation selection induces strong direc-tional selection (corresponding to large v) and only extremephenotypes reproduce. Thus, truncation selection is muchmore efficient in maintaining large-effect de novo mutations,while eroding genetic variation more quickly (because itintroduces a large skew in the offspring distribution). How-ever, the similarities and differences in the genetic basis ofresponses to artificial vs. natural selection are an interestingtopic—in particular, for the interpretation of the largeamount of genetic data available from breeding programs(Stern and Orgogozo 2009)—that should be addressed infuture studies.

Throughout this study, we have focused on adaptation toa moving optimum, that is, a scenario of gradual environ-mental change. An obvious question is how our resultswould change under the alternative scenario of a one-timesudden shift in the optimum (as assumed in numerousstudies, e.g., Orr 1998; Hermisson and Pennings 2005;Chevin and Hospital 2008). While beyond the scope of this

article, our approach should, in principle, still be applicable.In particular, each focal allele still experiences a gradualchange in its selection coefficient, due to the evolution ofthe genetic background. Unlike in the moving-optimummodel, however, the selection coefficient decreases, as themean phenotype gradually approaches the new optimum.Hence, a suitably modified version of Equation 12 wouldgive the probability that a focal allele establishes in the pop-ulation (i.e., escapes stochastic loss), but in the absence ofcontinued environmental change, establishment does notguarantee fixation. In other words, alleles need to “racefor fixation” before other competing alleles get fixed andthey become deleterious (Kopp and Hermisson 2007,2009a). The dynamics of a mutation along its trajectoryshould therefore be even more complex than in the mov-ing-optimum model and show an even stronger dependenceon the genetic background (Chevin and Hospital 2008).

Extinction and the rate of environmental change

Recently, several experimental studies have explored howthe rate of environmental change affects the persistence ofpopulations that rely on new mutations for adapting toa gradually changing environment (Perron et al. 2008; Belland Gonzalez 2011; Lindsey et al. 2013). In line with theo-retical predictions (Bell 2013), all studies found that“evolutionary rescue” is contingent on a small rate of envi-ronmental change. In particular, Lindsey et al. (2013,p. 463) evolved replicate populations of Escherichia coli un-der different rates of increase in antibiotic concentration andfound that certain genotypes were evolutionarily inaccessi-ble under rapid environmental change, suggesting that “rap-idly deteriorating environments not only limit mutationalopportunities by lowering population size, but [. . .] alsoeliminate sets of mutations as evolutionary options”. Thisis in stark contrast to our prediction that faster environmen-tal change can enable the population to remain better adap-ted and to traverse larger distances in phenotype spacewhen standing genetic variation is the sole source for adap-tation [Figure 7 and Figure S3_9, Figure S3_10 and FigureS3_11 (File S3)]; in line with recent experimental observa-tions; H. Teotonio, private communication). The differencebetween these results arises from the availability of the“adaptive material.” While de novo mutations first need toappear and survive stochastic loss before becoming fixed,standing genetic variants are available right away and mightalready be segregating at appreciable frequency. Thus, inboth cases, the rate of environmental change plays a critical,although antagonistic, role in determining the evolutionaryoptions. While fast environmental change eliminates sets ofnew mutations, it simultaneously helps to preserve standinggenetic variation until it can be picked up by selection. Un-der slow change, in contrast, most large-effect alleles fromthe standing variation, by the time they are needed, arealready eliminated by drift or stabilizing selection.

Our results also mean that, if the optimum stops movingat a given value zopt;max; populations will achieve a higher

Figure 8 The contributions of standing genetic variation (light gray) andde novo mutations (dark gray) to the cumulative phenotypic response toselection z (A) and the current genetic variance (B) over time. Plots showaverage trajectories over 1000 replicate simulations. The red circle marksthe point in time where 50% of the total phenotypic responses were dueto de novo mutations. The inset in A shows a more detailed plot of thedynamics of z up to this point. Fixed parameters: s2

s ¼ 50; Q ¼ 5;N ¼ 2500; n ¼ 0:001; s2

m ¼ 0:05:





degree of adaptation (higher z *) if the final optimum isreached fast rather than slowly (see also Uecker et al.2014), at least if standing genetic variation is the sole sourcefor adaptation. While this assumption is an obvious simpli-fication, it may often be approximately true in natural pop-ulations. The same holds true in experimental populations,where selection is usually strong and the duration of theexperiment short, such that de novo mutations can fre-quently be neglected (see Figure 9).

Testing the predictions

The predictions made by our model can in principle betested empirically, even though suitable data might besparse and experiments challenging. There is, of course,ample evidence for adaptation from standing genetic vari-ation (e.g., Teotónio et al. 2009; Jerome et al. 2011; Joneset al. 2012b; Sheng et al. 2015). For example, Domingueset al. (2012) showed that camouflaging pigmentation ofoldfield mice (Peromyscus polionotus) that have colonizedFlorida’s Gulf Coast has evolved quite rapidly from a pre-existing mutation in the Mc1r gene, Limborg et al. (2014)investigated selection in two allochronic but sympatric line-ages of pink salmon (Oncorhynchus gorbuscha) and iden-tified 24 divergent loci that had arisen from differentpools of standing genetic variation, and Turchin et al.(2012) showed that height-associated alleles in humans

display a clear signal for widespread selection on standinggenetic variation.

However, testing the predictions of our model requires, inaddition, detailed knowledge of the genotype–phenotyperelation. Currently, there are only a small (yet increasing)number of systems for which both a set of functionally val-idated beneficial mutations and their selection coefficientsunder different environmental conditions are available(Jensen 2014). Thus, estimating the distribution of standingsubstitutions will be challenging, because of the often un-known phenotypic and fitness effects of beneficial mutationsand the large number of replicate experiments needed toobtain a reliable empirical distribution. Furthermore, froman experimental point of view it is often difficult to discrim-inate between phenotypic (or fitness) effects of individualmutations and phenotypic changes induced by phenotypicplasticity and environmental variance. However, even ifthese problems were solved, small-effect alleles might notbe detectable due to statistical limitations (Otto and Jones2000), and in certain limiting cases where the populationquickly goes extinct (i.e., when the environment changesvery fast), the distribution of adaptive substitutions fromstanding genetic variation might be indistinguishable fromthat from de novo substitutions (Figure 6).

Recent developments in laboratory systems (Morran et al.2009; Parts et al. 2011), however, have created opportunities

Figure 9 (Top row) The point in time tDNM;50ðzÞwhere 50% of the phenotypic responses to mov-ing-optimum selection have been contributed byde novo mutations as a function of the rate ofenvironmental change for various values of Q (left)and s2

s (right). Insets show the results for large n ona log scale. (Middle row) The mean total phenotypicresponse at this time. (Bottom row) The relativecontribution of original standing genetic variationto the total genetic variance at time tDNM;50ðzÞ:Data are means (and standard deviations) from1000 replicate simulation runs. Fixed parameters(if not stated otherwise): s2

s ¼ 50; Q ¼ 5; N ¼ 2500;s2m ¼ 0:05:


for experimental evolution studies in which population size,the selective regime, and the duration of selection can bemanipulated and adaptation from de novo mutations andstanding genetic variation can be recorded (Burke 2012;Schlotterer et al. 2015; see also Teotonio et al. 2012). Apply-ing these techniques in experiments in the vein of Lindseyet al. (2013), but starting from a polymorphic population,should make it possible to test the relation between the rateof environmental change and population persistence and toassess the probability of adaptation from standing genetic var-iation. First experiments along these lines are currently beingcarried out in populations of Caenorhabditis elegans, with theaim of determining the limits of adaptation to different ratesof increase in sodium chloride concentration (H. Teotonio,private communication; see also Theologidis et al. 2014). Fur-thermore, Pennings (2012) recently applied the Hermissonand Pennings (2005) framework to show that standing ge-netic variation plays an important role in the evolution of drugresistance in human immunodeficiency virus, affecting up to39% of patients (depending on treatment) and explainingwhy resistance mutations in patients who interrupt treatmentare likely to become established within the first year. A similarapproach should also be applicable to scenarios of gradualenvironmental change (e.g., evolution of resistance mutationsunder gradually increasing antibiotic concentrations).

Conclusion

As global climate change continues to force populations torespond to the altered environmental conditions, studyingadaptation to changing environments—both empirically andtheoretically—has become one of the main topics in evolu-tionary biology. Despite increased efforts, however, very lit-tle is known about the genetic basis of adaptation fromstanding genetic variation. Our analysis of the moving-optimum model shows that this process has, indeed, a verydifferent genetic basis than that of adaptation from de novomutations. In particular, adaptation proceeds via the fixationmany small-effect alleles (and just a few large ones). In ac-cordance with previous studies, the adaptive process criticallydepends on the tempo of environmental change. Specifically,when populations adapt from standing genetic variation only,the potential for adaptation increases as the environmentchanges faster.

Acknowledgments

We thank S. Aeschbacher, R. Bürger, L. M. Chevin,H. Teotonio, C. Vogl, and two anonymous reviewers forconstructive comments on this manuscript. This study wassupported by the Austrian Science Fund (grant P 22581-B17to M.K. and grant P22188 to Reinhard Bürger), the AustrianAgency for International Cooperation in Education and Re-search (grant FR06/2014 to J.H.), Campus France (grantPHC AMADEUS 31642SJ to M.K.), and a Writing-Up Fellow-ship from the Konrad Lorenz Institute for Evolution andCognition Research (to S.M.).

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Communicating editor: J. Wakeley


Appendix

Appendix A: Wright–Fisher Simulations: A Focal Locus with Recurrent Mutations

In Appendix A, we describe the Wright–Fisher (WF) simulations used (solely) for checking our analytical approximationsregarding the probability for adaptation from standing genetic variation (Equation 14). To simulate evolution at a focallocus, we followed Hermisson and Pennings (2005) and implemented a multinomial WF sampling approach (available fromthe corresponding author upon request). These simulations serve as an additional analysis tool that has been adjusted to theapproximation method and allows the adaptive process to be simulated fast and efficiently. In addition, they go beyond theindividual-based model in one aspect, as they do not make the infinite-sites assumption but allow for recurrent mutation atthe focal locus (but still with at most two alleles).

Genome

At the focal locus, mutations with a fixed allelic effect a appear recurrently at rate u and convert ancestral alleles into derivedmutant alleles. Accordingly, despite a genetic background with normally distributed genotypic values, there are at most twotypes of (focal) alleles in the population, where each type “feels” only the mean background zB; which evolves according toLande’s equation (Equation 5, see above). The genetic background variation s2

g is assumed to be constant and serves as a freeparameter that is independent of u, Ne; and s2

s : Note that the evolutionary response at the focal locus is influenced by that ofthe genetic background, and vice versa, meaning that the two are interdependent.

Procedure

We follow the evolution of 2Ne alleles at the focal locus. Each generation is generated by multinomial sampling, where theprobability of choosing an allele of a given type (ancestral or derived) is weighted by its respective (marginal) fitness.Furthermore, the mean phenotype of the genetic background zB evolves deterministically according to Equation 5 withconstant s2

g: To let the population reach mutation–selection–drift equilibrium, each simulation is started 4Ne generationsbefore the environment starts changing. Initially, the population consists of only ancestral alleles “0”; the derived allele “1” iscreated by mutation. If the derived allele reaches fixation before the environmental change (by drift), it is itself declared“ancestral”; i.e., the population is set back to the initial state. After 4Ne generations, the optimum starts moving, such that theselection coefficient of the derived allele, which is initially deleterious [i.e., sða; tÞ#0], increases and may eventually becomebeneficial [i.e., sða; tÞ. 0], depending on the response at the genetic background. Simulations continue until the derivedallele is either fixed or lost. Fixation probabilities are estimated from 100; 000 simulation runs.

Appendix B: Theoretical Background

In Appendix B, we briefly recapitulate results from previous studies that form the basis for our analytical derivations.

The probability of adaptation from standing genetic variation for a single biallelic locus after a suddenenvironmental change

Hermisson and Pennings (2005) studied the situation where the selection scheme at a single biallelic locus changesfollowing a sudden environmental change. In particular, they derived the probability for a mutant allele to reach fixationthat was neutral or deleterious prior to the change but has become beneficial in the new environment. In the continuum limitfor allele frequencies this probability is given by

PSGV ¼Z 1

0rðxÞPxdx; (B1)

where rðxÞ is the density function for the allele frequency x of the mutant allele in mutation–selection–drift balance and Px

denotes its fixation probability.For a mutant allele present at frequency x in a population with effective size Ne and with selective advantage sb in the new

environment, the fixation probability is given by

PxðsbÞ �12 exp½24Nesbx�12 exp½24Nesb�

(B2)

(Kimura 1957). There are two points to make here. First, mutational effects in the Hermisson and Pennings (2005) modelare directly proportional to fitness, whereas mutations in our model affect a phenotype under selection. Second, in ourframework, sb denotes the (beneficial) selection coefficient for heterozygotes.


Approximations for rðxÞ can be derived from standard diffusion theory (Ewens 2004; for details see Hermisson andPennings 2005). If the mutant allele was neutral prior to the change in the selection scheme,

rðxÞ ¼ CxQ2112 x12u

x2 1: (B3)

Here, u is the per-locus mutation rate, C ¼ ð~g þ cðuÞÞ21 denotes a normalization constant where ~g � 0:577 is Euler’sgamma, and cð�Þ is the polygamma function. Similarly, if the mutant allele was deleterious before the environmental change(with negative selection coefficient sd), the allele-frequency distribution is given by

rðxÞ ¼ C

�12 exp

�ð12 xÞ4Nejsdj �

xu21

x21; (B4)

where C ¼ ð1F1ð0; u; 4NejsdjÞÞ21 denotes a normalization constant and 1F1ða; b; cÞ is the hypergeometric function. If the allelewas sufficiently deleterious (4Nejsdj$ 10), Equation B4 can further be approximated as

rðxÞ ¼ Cxu21exp�24Nejsdjx

; (B5)

where C ¼ ðg½u; 4Nejsdj�=ð4NejsdjÞuÞ21 again denotes a normalization constant with g½a; b� ¼ R b0 ta21exp½2t�dt denoting thelower incomplete gamma function.

Finally, the probability that a population successfully adapts from standing genetic variation can be derived as

PSGV ¼ 12

1þ 4Nesb

4Nejsdj þ 1

!2u

¼ 12 exp

"2ulog

4Nesb

4Nejsdj þ 1

�#:

(B6)

Fixation probabilities under time-inhomogeneous selection

In gradually changing environments, the selection coefficient of a given (mutant) allele is not fixed but changes over time (i.e., as the position of the optimum changes). Uecker and Hermisson (2011) recently developed a mathematical frameworkbased on branching-process theory to describe the fixation process of a beneficial allele under temporal variation in pop-ulation size and selection pressures. They showed that the probability of fixation of a mutation starting with n initial copies isgiven by

PfixðnÞ ¼ 12 ð121uÞn; (B7a)

where

2u ¼ 1þZ N

0

�Nð0ÞNeðtÞ

exp2

Z t

0sðtÞdt

�dt: (B7b)

Assuming that the population size remains constant and that the selection coefficient increases linearly in time,sðtÞ ¼ sd þ svt; Equation B7a becomes

PfixðnÞ ¼ 12

12

"1þ 1

2

ffiffiffiffiffiffiffip

2sv

rexp

s2d2sv

!erfc

sdffiffiffiffiffiffiffi2sv

p!#21!n

; (B8)

where erfcð�Þ denotes the complementary Gaussian error function.


GENETICSSupporting Information

www.genetics.org/lookup/suppl/doi:10.1534/genetics.115.178574/-/DC1

Catch Me if You Can: Adaptation from StandingGenetic Variation to a Moving Phenotypic Optimum

Sebastian Matuszewski, Joachim Hermisson, and Michael Kopp

Copyright © 2015 by the Genetics Society of AmericaDOI: 10.1534/genetics.115.178574

Catch me if you can: Adaptation from standing genetic variation to amoving phenotypic optimum

Supporting Information

Sebastian Matuszewski, Joachim Hermisson, Michael Kopp

1

SUPPORTING INFORMATION 1: LIMITED RECOMBINATION

S. Matuszewski et al. 2 SI

Wendy

Typewritten Text

File S1

Figure S1_1 – The distribution of adaptive substitutions from standing genetic variation for free recombination (dark bins) compared to that for limited recombination (light bins). The black line correspondsto the analytical prediction (eq. 19). σ2

g is given by equation (19). Fixed parameters: σ2s = 50, N = 2500, v = 0.001, σ2

m = 0.05.

r = 0 r = 0.01 r = 0.1 r = 1

Θ=

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Figure S1_2 – The distribution of adaptive substitutions from standing genetic variation for free recombination (dark bins) compared to that for limited recombination (light bins). The black line correspondsto the analytical prediction (eq. 19). σ2

g is given by equation (16). Fixed parameters: Θ = 5, N = 2500, v = 0.001, σ2m = 0.05.

r = 0 r = 0.01 r = 0.1 r = 1

σ2 s

=50

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Figure S1_3 – The distribution of adaptive substitutions from standing genetic variation for complete linkage (no recombination). Theblack and the grey line corresponds to the analytical prediction (eq. 25) that are centred around the mean of theindividual-based simulation. For the grey line Ne has been adjusted by a factor 0.375 to match the distribution from theindividual-based simulations. Other parameters: v = 0.001, r = 0, N = 2500, σ2

m = 0.05.

The individual-based simulation results presented in the main text were obtained under the assumption of freerecombination. In this Supplementary Information, we relax this assumption and study the effects of linkage (i.e.,limited recombination).

We first clarify the meaning of the recombination parameter r, which determines the mean number of crossoverevents per meiosis. By definition, the simulated genome corresponds to a single chromosome of length DG(r) =r · 100 centimorgan (cM), and the mean distance between two randomly chosen sites is 1

3DG(r). The mean distancebetween two adjacent polymorphic loci is DG,adjacent(r) = 1

|G|+1DG(r), where |G| is the mean number of polymorphicloci, which depends on Θ and σ2

s (eq. 26).

The corresponding recombination rate r between two polymorphic loci is given by the inverse of Haldane’s mappingfunction (Speed 2005), that is,

r(r) = 12 (1− exp [−2DG(r)]) , (S1)

see Table S1_1. Please note the difference between r and r.


Table S1_1 – The classical population genetic recombination rate r (eq. S1) between two adjacent loci for different values of σ2s , Θ and r.

Other parameters: σ2m = 0.05.

r

0 0.01 0.1 1

( Θ,σ

2 s) (5/50) 0 0.019 0.16 0.49

(10/50) 0 0.01 0.089 0.43

(5/100) 0 0.017 0.15 0.49

The effect of limited recombination on the distribution of adaptive substitutions from standing genetic variation isillustrated in Figures S1_1 and S1_2. For r = 1 (corresponding to a genome length of 100cM and an averagerecombination rate r of close to 0.5, see table S1_1), the distribution is essentially identical to that for linkageequilibrium. As r decreases, the distribution progressively shifts to the left, becomes more symmetric and includesmore and more alleles with negative phenotypic effect. For r = 0 (corresponding to complete linkage or asexualreproduction), it resembles the distribution for “drift-driven” evolution (i.e., when selection is not efficient; Fig. S3_4).The reason is that fixation involves entire haplotypes carrying multiple mutations, whose (positive and negative) effectslargely cancel. From a different perspective, limited recombination leads to Hill-Robertson interference betweenco-segregating alleles (Hill and Robertson 1966), which in many respects corresponds to a decrease in effectivepopulation size Ne (Comeron et al. 2008), which in turn reduces the efficacy of selection. Note, however, that unlikein the case of a slowly changing environment (Fig. S3_4) reducing Ne also affects the equilibrium allele-frequencydistribution ρ(x, α) (by reducing the strength of selection against large-effect alleles). In line with previous simulationresults (Comeron et al. 2008), we find that equation (25) provides a very good fit, when Ne is set to 37.5% of itsoriginal value.


SUPPORTING INFORMATION 2: THE DISTRIBUTION OF ADAPTIVE SUBSTITUTIONS FROMDE-NOVO MUTATIONS WITH AND WITHOUT GENETIC BACKGROUND VARIATION

There are two ways in which the distribution of adaptive substitutions from standing genetic variation can be comparedto that from de-novo mutations. The first comparison considers a population without genetic background variation.This is the situation studied by Kopp and Hermisson (2009a), where an essentially monomorphic population performsan adaptive walk following a moving optimum. The second situation is the one described by equation (25), where newmutations interact with a genetic background of constant variance (this background is presumably itself constantlyreplenished by new mutations). Analytical predictions for all three distributions are compared in Figures. S2_1,S2_2 and S2_3. It can be seen that the adaptive-walk prediction (eq. 14 in Kopp and Hermisson 2009b; redline) is always shifted towards larger α compared to the distribution of adaptive substitutions from standing geneticvariation (eq. 19, black line). The predicted distribution from de-novo mutations in the presence of genetic backgroundvariation (eq. 23, grey curve) shifts from the latter to the former as v increases. The reason is that, for small v, thefixation of both standing variants and new mutations in the presence of background variation is strongly constrainedby the equilibrium lag (eq. 9). For large v, in contrast, the lag is large and adaptation is primarily limited bythe available alleles, independent of their source and initial frequency (mutation-limited regime sensu Kopp andHermisson 2009b). Note, however, that in both limiting cases, equation (19) is a poor predictor for the simulatedsubstitutions from standing variation (Fig. 6, S3_4). Nevertheless, it remains true that adaptive substitutions fromnew mutations are generally smaller than those from new mutations, with or without genetic background variation.


Wendy

Typewritten Text

File S2

Figure S2_1 – Comparison of the analytical predictions for the distribution of adaptive substitutions from standing genetic variation and de-novo mutations. The black line corresponds toeq. (19), with the genetic background variation σ2

g determined by the SHC approximation (eq. 16). The grey curve gives the analytical prediction for substitutions from de-novomutations under the assumption that the phenotypic lag δeq has reached an equilibrium (eq. 23). The red line gives the analytical prediction for the first substitution from de-novomutations under the adaptive-walk assumption that there is no genetic background variation (Kopp and Hermisson 2009b, eq. 14). Note that, for some parameter combinations, thesimulated distribution from standing variation deviates from eq. (19). In particular, for small v, it approaches the “neutral” prediction eq. (25, see Fig. 4, and for large v, it mayapproach the distribution from new mutations, eq. (23), see Fig. 6). Fixed parameters: Θ = 2.5, N = 2500, σ2

m = 0.05.

v=0.0001 v=0.000316 v=0.001 v=0.00316 v=0.01 v=0.0316

σ2 s=

10

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Figure S2_2 – Comparison of the analytical predictions for the distribution of adaptive substitutions from standing genetic variation and de-novo mutations. For further details see Fig. S2_1.Fixed parameters: Θ = 5, N = 2500, σ2

m = 0.05.

v=0.0001 v=0.000316 v=0.001 v=0.00316 v=0.01 v=0.0316

σ2 s=

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Figure S2_3 – Comparison of the analytical predictions for the distribution of adaptive substitutions from standing genetic variation and de-novo mutations. For further details see Fig. S2_1.Fixed parameters: Θ = 10, N = 2500, σ2

m = 0.05.

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SUPPORTING INFORMATION 3: SUPPORTING FIGURES


Wendy

Typewritten Text

File S3

Figure S3_1 – The probability for a mutant allele to adapt from standing genetic variation as a function of the rate of environmental change v. Solid lines correspond to the analytical prediction(eq. 14), the grey dashed line shows the probability for a neutral allele (α = 0; eq. 15), and symbols give results from Wright-Fisher simulations. The phenotypic effect size α ofthe mutant allele ranges from 0.5σm (top line; black) to 3σm (bottom line; purple) with increments of 0.5σm. The figures in each parameter box (per locus mutation rate θ, widthof fitness landscape σ2

s ) correspond to different values of the genetic background variation σ2g with σ2

g = 0 (no background variation; top left), σ2g = 0.005 (top right), σ2

g = 0.01(bottom left) and σ2

g = 0.05 (bottom right). Other parameters: Ne = 25000, σ2m = 0.05.

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Figure S3_2 – The distribution of adaptive substitutions from standing genetic variation. Histograms show results from individual-based simulations. The black line corresponds to the analytical prediction (eq. 19),with the genetic background variation σ2

g determined by the SHC approximation (eq. 16). The red line gives the analytical prediction for the limiting case where the equilibrium lag δeq is reached fast(eq. 21). The blue line is based on the analytical prediction (eq. 25)—which assumes a neutral fixation probability—but has been shifted so that it is centered around the empirical mean. The grey curvegives the analytical prediction for substitutions from de-novo mutations under the assumption that the phenotypic lag δeq has reached its equilibrium (eq. 23). The asterisks indicate where Nesmax ≥ 10.Fixed parameter: σ2

m = 0.05.

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Figure S3_3 – The distribution of adaptive substitutions from standing genetic variation for various rates of environmental change. For further details see Fig. S3_2. Fixed parameters: Θ = 2.5,N = 2500, σ2

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Figure S3_4 – The distribution of adaptive substitutions from standing genetic variation in the case of slow environmental change (v = 10−5). Histograms show results from individual-basedsimulations. The blue line gives the analytical prediction (eq. 25), with σ2

g given by equation (16), which assumes a neutral fixation probability. Fixed parameters: σ2m = 0.05.

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Figure S3_5 – The distribution of adaptive substitutions from standing genetic variation for various mutation rates Θ. The distributionshifts towards smaller values as Θ increases reflecting the fact that there is more genetic background variation σ2

g , whichallows the population to follow the optimum more closely, such that most large-effect alleles will remain deleterious. Forfurther details see Fig. S3_2. Fixed parameters: N = 2500, σ2

s = 10, v = 0.01, σ2m = 0.05.


Figure S3_6 – The average number of fixed adaptive substitutions from standing genetic variation, |G|fix, as a function of the rate ofenvironmental change v, when standing genetic variation is the sole source for adaptation. Symbols show results fromindividual-based simulations (averaged over 100 replicate runs). The black line gives the analytical prediction (eq. 27) andthe grey line corresponds to the average number of neutral fixations (|G|fix,v→0 = |G|

∫∞−∞

p(α)Πseg,v→0(α)dα.). In both

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Figure S3_9 – The average distance traversed in phenotype space, z∗, as a function of the rate of environmental change v, when standinggenetic variation is the sole source for adaptation. Symbols show results from individual-based simulations (averaged over100 replicate runs). The black line gives the analytical prediction (eq. 28), with σ2

g taken from equation (16). Thegrey-dashed line gives the critical rate of environmental change (eq. 29). Error bars for standard errors are containedwithin the symbols. Fixed parameters: N = 1000, σ2

m = 0.05.

Θ=2.5 Θ=5 Θ=10

σ2 s=

10

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Figure S3_10 – The average distance traversed in phenotype space, z∗, as a function of the rate of environmental change v, whenstanding genetic variation is the sole source for adaptation. Symbols show results from individual-based simulations(averaged over 100 replicate runs). The black line gives the analytical prediction (eq. 28), with σ2

g taken fromequation (16). The grey-dashed line gives the critical rate of environmental change (eq. 29). Error bars for standarderrors are contained within the symbols. Fixed parameters: N = 2500, σ2

m = 0.05.

Θ=2.5 Θ=5 Θ=10

σ2 s=

10

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Figure S3_11 – The average distance traversed in phenotype space, z∗, as a function of the rate of environmental change v, whenstanding genetic variation is the sole source for adaptation. Symbols show results from individual-based simulations(averaged over 100 replicate runs). The black line gives the analytical prediction (eq. 28), with σ2

g taken fromequation (16). The grey-dashed line gives the critical rate of environmental change (eq. 29). Error bars for standarderrors are contained within the symbols. Fixed parameters: N = 5000, σ2

m = 0.05.

Θ=2.5 Θ=5 Θ=10

σ2 s=

10

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10

v

z*

σ2 s=

100

ææ

æ

æ

æ

ææ

æ

æ

10-5 10-4 0.001 0.01 0.10.0

0.5

1.0

1.5

2.0

2.5

v

z*

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æ

æ

æ

æ

æ

æ

æ

10-5 10-4 0.001 0.01 0.10

1

2

3

4

5

v

z*

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æ

æ

æ

æ

æ

æ

æ

10-5 10-4 0.001 0.01 0.10

2

4

6

8

10

v

z*


10-5 10-4 0.001 0.01 0.10

2000

4000

6000

8000

10000

v

t DN

M,5

0

0.001 0.01 0.1

100

103

v

t DN

M,5

0

10-5 10-4 0.001 0.01 0.10

2000

4000

6000

8000

10000

v

t DN

M,5

0

0.001 0.01 0.1

100

103

v

t DN

M,5

010-5 10-4 0.001 0.01 0.10

2

4

6

8

10

12

v

z

10-5 10-4 0.001 0.01 0.10

1

2

3

4

5

6

7

v

z

10-5 10-4 0.001 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

v

ΣSG

V2

�Σg2

Q = 2.5Q = 5Q = 10

10-5 10-4 0.001 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

v

ΣSG

V2

�Σg2

Σs2

= 10

Σs2

= 50

Σs2

= 100

Figure S3_12 – First row: the point in time tDNM,50 (z̄) where 50% of the phenotypic response to moving-optimum selection have beencontributed by de-novo mutations as a function of the rate of environmental change for various values of Θ (left) and σ2

s(right). Insets show the results for large v on a log-scale. Second row: The mean total phenotypic response at this time.Third row: The relative contribution of original standing genetic variation to the total genetic variance at timetDNM,50 (z̄). Data are means and standard errors from 1000 replicate simulation runs. Fixed parameters (if not statedotherwise): σ2

s = 50, Θ = 5, N = 1000, σ2m = 0.05.


10-5 10-4 0.001 0.01 0.10

1000

2000

3000

4000

v

t DN

M,5

0

0.001 0.01 0.1

100

103

vt D

NM

,50

10-5 10-4 0.001 0.01 0.10

1000

2000

3000

4000

v

t DN

M,5

0

0.001 0.01 0.1

100

103

v

t DN

M,5

010-5 10-4 0.001 0.01 0.1

0.0

1.0

2.0

3.0

4.0

5.0

v

z SG

V

10-5 10-4 0.001 0.01 0.10.0

0.5

1.0

1.5

2.0

2.5

v

z SG

V

10-5 10-4 0.001 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

v

È�zSG

VÈ�

HÈ�z S

GV

È+È�z

DN

MÈL

Q = 2.5

Q = 5

Q = 10

10-5 10-4 0.001 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

v

È�zSG

VÈ�

HÈ�z S

GV

È+È�z

DN

MÈL

Σs2

= 10

Σs2

= 50

Σs2

= 100

Figure S3_13 – First row: the point in time tDNM,50(σ2g

)where 50% of the genetic variance is composed of de-novo mutations as a

function of the rate of environmental change for various values of Θ (left) and σ2s (right). Insets show the results for large

v on a log-scale. Second row: The mean total phenotypic response from standing genetic variation at this time. Thirdrow: The relative contribution of original standing genetic variation to the total genetic variance at time tDNM,50

(σ2g

).

Data are means and standard errors from 1000 replicate simulation runs. Fixed parameters (if not stated otherwise):σ2s = 50, Θ = 5, N = 1000, σ2

m = 0.05.


10-5 10-4 0.001 0.01 0.10

1000

2000

3000

4000

5000

v

t DN

M,5

0

0.001 0.01 0.1

100

103

vt D

NM

,50

10-5 10-4 0.001 0.01 0.10

1000

2000

3000

4000

5000

v

t DN

M,5

0

0.001 0.01 0.1

100

103

v

t DN

M,5

010-5 10-4 0.001 0.01 0.1

0.0

1.0

2.0

3.0

4.0

5.0

v

z SG

V

10-5 10-4 0.001 0.01 0.10.0

0.5

1.0

1.5

2.0

2.5

v

z SG

V

10-5 10-4 0.001 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

v

È�zSG

VÈ�

HÈ�z S

GV

È+È�z

DN

MÈL

Q = 2.5

Q = 5

Q = 10

10-5 10-4 0.001 0.01 0.10.0

0.2

0.4

0.6

0.8

1.0

v

È�zSG

VÈ�

HÈ�z S

GV

È+È�z

DN

MÈL

Σs2

= 10

Σs2

= 50

Σs2

= 100

Figure S3_14 – First row: the point in time tDNM,50(σ2g

)where 50% of the genetic variance is composed of de-novo mutations as a

function of the rate of environmental change for various values of Θ (left) and σ2s (right). Insets show the results for large

v on a log-scale. Second row: The mean total phenotypic response from standing genetic variation at this time. Thirdrow: The relative contribution of original standing genetic variation to the total genetic variance at time tDNM,50

(σ2g

).

Data are means and standard errors from 1000 replicate simulation runs. Fixed parameters (if not stated otherwise):σ2s = 50, Θ = 5, N = 2500, σ2

m = 0.05.


LITERATURE CITED

Comeron, J. M., A. Williford, and R. M. Kliman, 2008 The Hill-Robertson effect: evolutionary consequences ofweak selection and linkage in finite populations. Heredity 100: 19–31.

Hill, W. G., and A. Robertson, 1966 The effect of linkage on limits to artificial selection. Genet. Res. 8: 269–294.Kopp, M., and J. Hermisson, 2009a The genetic basis of phenotypic adaptation I: Fixation of beneficial mutationsin the moving optimum model. Genetics 182: 233–249.

Kopp, M., and J. Hermisson, 2009b The genetic basis of phenotypic adaptation II: The distribution of adaptivesubstitutions in the moving optimum model. Genetics 183: 1453–1476.

Speed, T. P., 2005 Genetic Map Functions. John Wiley & Sons, Ltd, 2nd edition.


Documents

Catch Me if You Can: Adaptation from Standing Genetic ... · Catch Me if You Can: Adaptation from Standing Genetic Variation to a Moving Phenotypic Optimum Sebastian Matuszewski,*,1