Science Paper Summary

3902  |  wileyonlinelibrary.com/journal/mec Molecular Ecology. 2019;28:3902–3914.© 2019 John Wiley & Sons Ltd

1   |   I N T R O D U C T I O N

Maynard‐Smith and Haig (1974) recognized the influence of selection on linked neutral sites, proposing that strong positive selection could reduce genetic diversity at nearby sites. This process is now referred to as a ‘selective sweep’. Much later, Charlesworth, Morgan, and Charlesworth (1993) proposed that deleterious mutations could also affect genetic diversity at nearby sites, because some haplotypes would be removed from the population as selection acts against linked deleterious alleles. They named this process background se‐ lection (BGS). Both selective sweeps and background selection af‐ fect genetic diversity; they both reduce the effective population size and distort the site frequency spectrum (SFS) of linked loci. Empirical evidence of a positive correlation between genetic diversity and

recombination rate has been reported in several species (Cutter & Payseur, 2013), including Drosophila melanogaster (Begun & Aquadro, 1992; Elyashiv et al., 2016), humans (Spencer et al., 2006), collared flycatchers, hooded crows and Darwin's finches (Dutoit et al., 2017; see also Vijay et al., 2017).

BGS is also expected to affect FST (Charlesworth, Nordborg, & Charlesworth, 1997; Cruickshank & Hahn, 2014; Cutter & Payseur, 2013; Hoban et al., 2016). The negative relationship between effec‐ tive population size Ne and FST is captured in Wright's classical in‐ finite island result; FST =

1

1+4Ne(m+�) (Wright, 1943), where m is the

migration rate and µ is the mutation rate. One might therefore ex‐ pect that loci under stronger BGS would show higher FST.

Many authors have also argued that, because BGS reduces the within‐population diversity, it should lead to high FST (Cruickshank

Received: 20 May 2018  |  Revised: 19 June 2019  |  Accepted: 3 July 2019 DOI: 10.1111/mec.15197

O R I G I N A L A R T I C L E

Background selection and FST: Consequences for detecting local adaptation

Remi Matthey‐Doret  | Michael C. Whitlock

Department of Zoology and Biodiversity Research Centre, University of British Columbia, Vancouver, BC, Canada

Correspondence Remi Matthey‐Doret, Department of Zoology and Biodiversity Research Centre, University of British Columbia, Vancouver, BC V6T 1Z4, Canada. Email: [email protected]

Funding information The work was funded by NSERC Discovery Grant RGPIN‐2016‐03779 to MCW and by the Swiss National Science Foundation via the fellowship Doc.Mobility P1SKP3_168393 to RMD.

Abstract Background selection is a process whereby recurrent deleterious mutations cause a decrease in the effective population size and genetic diversity at linked loci. Several authors have suggested that variation in the intensity of background selection could cause variation in FST across the genome, which could confound signals of local ad‐ aptation in genome scans. We performed realistic simulations of DNA sequences, using recombination maps from humans and sticklebacks, to investigate how varia‐ tion in the intensity of background selection affects FST and other statistics of popu‐ lation differentiation in sexual, outcrossing species. We show that, in populations connected by gene flow, Weir and Cockerham's (1984; Evolution, 38, 1358) estimator of FST is largely insensitive to locus‐to‐locus variation in the intensity of background selection. Unlike FST, however, dXY is negatively correlated with background selec‐ tion. Moreover, background selection does not greatly affect the false‐positive rate in FST outlier studies in populations connected by gene flow. Overall, our study in‐ dicates that background selection will not greatly interfere with finding the variants responsible for local adaptation.

K E Y W O R D S

adaptation, evolutionary theory, local adaptation, population genetics—theoretical, population structure

     |  3903MATTHEY‐DORET AnD WHITLOCK

& Hahn, 2014; Cutter & Payseur, 2013; Hoban et al., 2016). Expressed in terms of heterozygosities, FST =

HT−HS

HT =1−

HS

HT , where

HT is the expected heterozygosity in the entire population and HS is the average expected heterozygosity within subpopulations (HS and HT are also sometimes called πS and πT, e.g. Charlesworth, 1998). All else being equal, a decrease in HS would indeed lead to an increase in FST. However, all else is not equal; HT is also affected by BGS (Charlesworth et al., 1997). Therefore, in order to under‐ stand the effects of BGS on FST, we must understand the relative impact of BGS on both HS and HT. (This paragraph has expressed FST by one definition that mirrors Nei's GST, but Weir and Cockerham's (1984) θ estimates a similar quantity, FST =

�D

�S+�D ,

where �D = d−1

d

(

�T −�S )

and d is the number of populations sam‐ pled. For the majority of this paper, we consider this more broadly used measure from Weir and Cockerham.) By either measure of FST, if HS and HT are changed in the same proportion by BGS, there would be no effect on FST.

Charlesworth et al. (1997) as well as Zeng and Corcoran (2015) have performed more sophisticated analysis measuring the effect of BGS on FST through numerical simulations. Charlesworth et al. (1997) reported that BGS reduces the within‐population hetero‐ zygosity HS slightly more than it reduces the total heterozygosity HT, causing a net increase in FST. The effect on FST reported is quite substantial, but, importantly, their simulations were not meant to be realistic. The authors highlighted their goal in the methods:

The simulations were intended to show the quali‐ tative effects of the various forces studied […], so we did not choose biologically plausible values […]. Rather, we used values that would produce clear‐cut effects.

For example, talking about their choice for the deleterious muta‐ tion rate of 8 × 10−4 per site:

This unrealistically high value was used in order for background selection to produce large effects [...]

Much of the literature on the effect of BGS on FST is based on the results in Charlesworth et al. (1997), even though they only intended to provide a proof of concept (see also Zeng & Charlesworth, 2011 and Zeng & Corcoran, 2015). They did not at‐ tempt to estimate how strong of an effect BGS has on FST in real genomes.

The intensity of BGS varies throughout the genome as a conse‐ quence of variation in recombination rate, selection pressures and mutation rates. Therefore, if BGS significantly affects FST, we should expect that baseline FST to vary throughout the genome. It is im‐ portant to distinguish two separate questions when discussing the effect of BGS on FST: (a) How does BGS affect the average genome‐ wide FST? and (b) How does locus‐to‐locus variation in the intensity of BGS affect locus‐to‐locus variation in FST? The second question is

of particular interest to those trying to identify loci under positive selection (local selection or selective sweep). Locus‐to‐locus varia‐ tion in FST due to BGS potentially could be confounded with the FST peaks created by positive selection. In this paper, we focus on this second question.

The identification of loci involved in local adaptation is often performed via FST outlier tests (Hoban et al., 2016; Lotterhos & Whitlock, 2014). Other tests exist to identify highly divergent loci such as cross‐population extended haplotype homozygosity (XP‐ EHH; Sabeti et al., 2007), comparative haplotype identity (Lange & Pool, 2016) and cross‐population composite likelihood ratio (XP‐ CLR; Chen, Patterson, & Reich, 2010). FST outlier tests, such as fdist2 (Beaumont & Nichols, 1996), bayescan (Foll & Gaggiotti, 2008) or flk (Bonhomme et al., 2010), look for genomic regions showing par‐ ticularly high FST values to find candidates for local adaptation. If BGS can affect FST unevenly across the genome, then regions with a high intensity of BGS could potentially have high FST values that could be confounded with the pattern caused by local selection (Charlesworth et al., 1997; Cruickshank & Hahn, 2014). BGS could therefore inflate the false‐positive rate when trying to detect loci under local selection.

The potential confounding effect of BGS on signals of local ad‐ aptation has led to an intense effort trying to find solutions to this problem (Aeschbacher, Selby, Willis, & Coop, 2017; Bank, Ewing, Ferrer‐Admettla, Foll, & Jensen, 2014; Huber, Degiorgio, Hellmann, & Nielsen, 2016). Many authors have understood from Cruickshank and Hahn (2014) that dXY should be used instead FST in outlier tests (e.g. McGee, Neches, & Seehausen, 2015; Yeaman, 2015; Whitlock & Lotterhos, 2015; Brousseau et al., 2016; Picq, Mcmillan, & Puebla, 2016; Payseur & Rieseberg, 2016; Hoban et al., 2016; Vijay et al., 2017; see also Nachman & Payseur, 2012). FST is a measure of pop‐ ulation divergence relative to the total genetic diversity, while dXY is an absolute measure of population divergence defined as the prob‐ ability of nonidentity by descent of two alleles drawn in the two different populations averaged over all loci (Nei, 1987; Nei, 1987 originally called it DXY but, here, we follow Cruickshank & Hahn's, 2014 terminology by calling it dXY ). The argument is that because FST is a measure of divergence relative to the genetic diversity and dXY an absolute measure of divergence and because BGS reduces genetic diversity (Cruickshank & Hahn, 2014; Hoban et al., 2016), then BGS must affect FST but not dXY, a claim that we will investigate in this paper.

Whether BGS can affect genome‐wide FST under some con‐ ditions is not in doubt (Charlesworth et al., 1997), but whether locus‐to‐locus variation in the intensity of BGS present in natural populations substantially affects variation in FST throughout the genome is very much unknown. Empirically speaking, it has been very difficult to measure how much of the genome‐wide varia‐ tion in genetic diversity is caused by BGS, as opposed to selective sweeps or variation in mutation rates (Cutter & Payseur, 2013; see also attempts in humans by Cai, Macpherson, Sella, & Petrov, 2009, McVicker, Gordon, Davis, & Green, 2009 and Elyashiv et al., 2016). We are therefore in need of realistic simulations that

3904  |     MATTHEY‐DORET AnD WHITLOCK

can give us more insight into how BGS affects genetic diversity among populations and how it affects the statistics of population divergence.

In this article, we investigate the effect of BGS in structured pop‐ ulations with realistic numerical simulations. Our two main goals are (a) to quantify the impact of locus‐to‐locus variation in the intensity of BGS on FST (Weir & Cockerham, 1984) and dXY (Nei, 1987) and (b) to determine whether BGS inflates the false‐positive rate of FST outlier tests.

2   |   M E T H O D S

Our goal is to perform biologically plausible simulations of the local genomic effects of background selection in biological settings with ongoing gene flow among populations. BGS is expected to vary with strength of selection (itself affected by gene density), mutation rate and recombination rate across the genome. We used data from real genomes to simulate realistic covariation in recombination rates and gene densities. We chose to base our simulated genomes on two eu‐ karyote recombination maps derived from sticklebacks and humans, because these two species have attracted a lot of attention in stud‐ ies of local. The recombination rate variation in humans is extremely fine scale, but it presents the potential issue that it is estimated from linkage disequilibrium data. As selection causes linkage disequilib‐ rium to increase, estimates of recombination rate at regions under strong selection may be under‐estimated, which might bias the simu‐ lated variance in the intensity of BGS. Although the recombination map for stickleback is much less fine‐scaled, the estimates are less likely to be biased as they are computed from pedigrees.

Our simulations are forward in time and were performed using the simulation platform simbit version 3.69. We simulated nonover‐ lapping generations with hermaphroditic, diploid individuals and random mating within patches. Selection occurred before disper‐ sal. The code and user manual are available at https ://github.com/ RemiM atthe yDore t/SimBit. The rational for using a new simulation platform is because all existing simulation platforms today were too slow for our needs (See Appendix S1). To double check our results, we also ran some simulations with slim (Haller & Messer, 2017) and nemo (Guillaume & Rougemont, 2006) (see Appendix S1), confirming that we obtain consistent distributions of genetic diversity and of FST among simulations. For simulations with slim and nemo, independent simulations were parallelized with gnu parallel (Tange, 2011).

2.1 | Genetics

For each simulation, we randomly sampled a sequence of about 10 cM from either the stickleback (Gasterosteus aculeatus) genome or the human genome (see Treatments below) and used this genomic location to determine the recombination map and exon locations for a simulation replicate. For the stickleback genome, we used the gene map and recombination map from Roesti, Moser, and Berner (2013). Ensembl‐retrieved gene annotations were obtained from

Marius Roesti. For the human genome, we used the recombination map from The International HapMap Consortium (2007) and the gene positions from NCBI and positions of regulatory sequences on Ensembl (Zerbino et al., 2018). We excluded sex chromosomes to avoid complications with haploid parts of the genomes. As estimates of mutation rate variation throughout the genome are very limited, we assumed that the haploid mutation rate varies from site to site following an exponential distribution with mean of 2.5 × 10−8 per generation (mean estimate from Nachman & Crowell, 2000).

More specifically, we first randomly sampled a sequence of 105 nucleotides, which we will refer to as the focal region. All of the sta‐ tistics (defined under the section Statistics below) are calculated only on this focal region of each simulation. Nucleotides that occur in locations determined to be exons in the sampled genomic map are subject to selection (see Selection), while all other nucleotides are assumed to be neutral. The focal region itself contained on average ~0.44 genes for the human genome and ~3.15 genes for the stick‐ leback genome.

We simulated a 5 cM region on each side of the focal region (re‐ sulting in a window of 10 cM plus the map distance covered by the specific focal region of 105 sites) in order to capture the local effects of background selection. In these 10 cM flanking regions, we only tracked exons (or in the case of the human genetic map, exons and regulatory sequences). In the nearest 1 cM on each side of the focal region, as well as within the focal region, we individually simulated each nucleotide as a bi‐allelic locus. On the remaining outer 4 cM, to improve the speed and RAM usage of the process, we tracked the number of mutations in blocks of up to 100 nucleotides. For these blocks, we tracked only the number of mutations but not their loca‐ tion within the block. Ignoring recombination within a block likely had little effect on the results because the average recombination distance between the first and last site of a block is of the order of 10−6 cM. The expected number of segregating sites within a block is

4N� ∑2N−1

i=1

�

1

i

�

, which for a mutation rate per block of 10−6 and a

population size of N = 10,000 is ~0.42. The probabilities of having more than one mutation and more than two mutations (based on a Poisson approximation) are therefore only approximately 6.7% and 0.9%, respectively. Overall, the level of approximation used is very reasonable.

2.2 | Selection

As we are interested in the effect of BGS, we modelled the effects of purifying selection against novel deleterious mutations. Each nucleotide in the exons (and regulatory sequences for the human genetic map) is subject to purifying selection with a selection coef‐ ficient against mutant alleles determined by a gamma distribution described below. For focal regions that include exons, statistics are computed over a sequence that is at least partially under direct pu‐ rifying selection.

To create variance in selection pressures throughout the ge‐ nome, each exon (and regulatory sequence for the human genetic

     |  3905MATTHEY‐DORET AnD WHITLOCK

map) has its own gamma distribution of heterozygous selection co‐ efficients s. The mean and variance of these gamma distributions are drawn from a bivariate uniform distribution with correlation coefficient of 0.5 (so that when the mean is high, so is the vari‐ ance) bounded between 10−8 and 0.2 for both the mean and the variance. These bounds were inspired by the methodology used in Gilbert et al. (2017). The gamma distributions are bounded to one. Figure S1 shows the overall distribution of selection coefficient s, with 2% of mutations being lethal and an average deleterious selection coefficient for the nonlethal mutations of 0.074. In the treatment Low selection pressure (see Treatments below), the upper bounds for the mean and variance of the gamma distributions were set to 0.1 instead of 0.2. To improve the performance of our simulations, we assumed multiplicative fitness interactions among alleles, where the fitness of heterozygotes at locus i is 1 − si and the fitness of the double mutant is (1 − si)

2. Any mutation changes the state of the locus to the other possible allele.

As a consequence of our parameter choices, our genome‐wide deleterious mutation rate was about 1.6 in sticklebacks and about 3 in humans. 9.8% of the stickleback genome and 2.7% of the human genome were under purifying selection. For comparison, the ge‐ nome‐wide deleterious mutation rate is estimated at 2.2 in humans (Keightley, 2012) and 0.44 in rodents (Keightley & Gaffney, 2003). To our knowledge, there is currently no such estimation for stick‐ lebacks. Note, however, that the above estimates cannot reliably detect mutations that are quasi‐neutral (s « 1/(2N)). By our distri‐ bution of selection coefficients, 49% of all deleterious mutations have a heterozygote selection coefficient lower than 1/(2Ne) when Ne = 1,000 (42% when Ne = 10,000). The fraction of selection coeffi‐ cients that are of intermediate effect (between 1/(2Ne) and 10/(2Ne)) is 10% when Ne = 1,000 (7% when Ne = 10,000).

It is worth noting that, in rodents, about half of the deleteri‐ ous mutations occur in noncoding sequences (Keightley & Gaffney, 2003). Our simulations using human genetic map had all exons and all regulatory sequences under purifying selection. With our simu‐ lations based on the stickleback genome, however, only exons were under purifying selection. It is therefore possible that we would have over‐estimated the deleterious mutation rate in gene‐rich regions and under‐estimated the deleterious mutation rate in other regions, especially in stickleback. This would artificially increase the locus‐ to‐locus variation in the intensity of BGS in our simulations, which is conservative to our conclusions.

2.3 | Demography

In all simulations (except for those matching the Charlesworth et al. (1997) results), we started with a burn‐in phase with a single popula‐ tion of N diploid individuals, lasting 5 × 2N generations. The popula‐ tion was then split into two populations of N individuals each with a migration rate between them equal to m. After the burn‐in phase, each simulation was run for 5 × 2N more generations for a total of 10 × 2N generations.

2.4 | Treatments

We explored the presence and absence of deleterious muta‐ tions over two patch sizes, three migration rates, two genomes and three selection scenarios. We considered a default design and explored variations from this design. The Default design had a population size per patch of N = 1,000, a migration rate of m = 0.005 and used the stickleback genome for its recombination map and gene positions. As deviations from this default design, we explored modification of every variable, one variable at a time. The Large N treatment has N = 10,000. The Human Genetic Map treatment uses the human genome for gene positions, regulatory sequences and recombination map. The treatments No Migration and High Migration have migration rates of m = 0 and m = 0.05, respectively. The Constant µ treatment assumes that all sites have a mutation rate of 2.5 × 10−8 per generation. The Low selection pressure treatment simulates lower selection coefficients (see sec‐ tion Selection above).

To test the robustness of our results and because it may be rel‐ evant for inversions, we also performed simulations where the re‐ combination rate for the entire 10 cM region was set to zero. As a check against previous work, we qualitatively replicated the results of Charlesworth et al. (1997) by performing simulations with similar assumptions as they used. We named this treatment CNC97. In our CNC97 simulations, N = 2,000, m = 0.001 and 1,000 loci were all equally spaced at 0.1 cM apart from each other with constant selec‐ tion pressure with heterozygotes having fitness of 0.98 and mutant homozygote fitness of 0.9 and constant mutation rate μ = 0.0004. Note that Charlesworth et al. (1997) used partially recessive alleles which we mimic for this case, but the simulations for the other cases considered in this paper use multiplicative interactions between al‐ leles. We performed further checks against previous works that are presented in Appendix S1. A full list of all treatments can be found in Table 1.

In all treatments (except Large N), we performed 4,000 simula‐ tions: 2,000 simulations with BGS and 2,000 simulations without se‐ lection (where all mutations were neutral). For Large N, simulations took more memory and more CPU time. For the Large N case, we performed 1,000 simulations with background selection and 1,000 simulations without selection.

We set the generation 0 at the time of the split. The state of each population was recorded at the end of the burn‐in period (gen‐ eration −1) and at generations 0.001 × 2N, 0.05 × 2N, 0.158 × 2N, 1.581 × 2N and 5 × 2N after the split. For N = 1,000, the sampled generations are therefore −1, 2, 100, 316, 3,162 and 10,000.

2.5 | Predicted intensity of background selection

In order to investigate the locus‐to‐locus correlation between the predicted intensity of BGS and various statistics, we computed B, a statistic that approximates the expected ratio of the coalescent time with background selection over the coalescent time without

3906  |     MATTHEY‐DORET AnD WHITLOCK

background selection (

B= TBGS

Tneutral

)

. B quantifies how strong BGS is ex‐

pected to be for a given simulation (Nordborg, Charlesworth, & Charlesworth, 1996). Furthermore, B has been used to predict the effect of BGS on FST (Zeng & Corcoran, 2015). A B value of 0.8 means that BGS has caused a drop of neutral genetic diversity of 20% com‐ pared to a theoretical absence of BGS. Lower B values indicate stronger BGS.

Both Hudson and Kaplan (1995) and Nordborg et al. (1996) have derived the following theoretical expectation for B.

where ri is the recombination rate between the focal site and the ith site under selection, and si is the heterozygous selection coefficient at that site, and ui is the (haploid) mutation rate at the ith site. By this formula, B is bounded between 0 and 1, where 1 means no BGS at all and low values of B mean strong BGS. We computed B for all sites in the focal region and report the average B for the region. While B is cal‐ culated just for the focal region, it is based on the effects of all selected sites in the simulated region of just over 10 cM.

For the stickleback genome, B values ranged from 0.03 to 0.99 with a mean of 0.84 (Figure S2). For the human genome, B values ranged from 0.20 to 1.0 with a mean at 0.91. In the No Recombination treatment, B values range from 10−10 to 0.71 with a mean of 0.07.

Excluding the treatments No Recombination and CNC97, we ob‐ served that simulations with BGS have a genetic diversity (whether HT or HS; HT data not shown) 6%–25% lower than simulations without BGS. Messer and Petrov (2013) simulated a panmictic

population, looking at a sequence of similar length inspired from a gene‐rich region of the human genome, and reported a similar de‐ crease in genetic diversity. Under the No Recombination treatment, this average reduction of genetic diversity due to BGS is 53%. In empirical studies, linked selection is estimated to reduce genetic diversity by up to 6% according to Cai et al. (2009) or 19%–26% according to McVicker et al. (2009) in humans. In D. melanogaster, where gene density is higher, the reduction in genetic diversity due to linked selection is estimated at 36% when using Kim and Stephan (2000)'s methodology and is estimated at 71% reduction using a composite likelihood approach (Elyashiv et al., 2016). In mice, BGS alone cannot explain fully the reduction in genetic di‐ versity at low recombination sites, and selective sweeps due to positive selection are responsible for the majority of the reduction in diversity due to linked selection (Booker & Keightley, 2018). It is worth noting that, because we were interested in the locus‐to‐ locus variation of various statistics in response to varying intensity of BGS, we did not simulate a whole genome worth of BGS, and hence, the overall reduction in genetic diversity that we observe should not be understood as a genome‐wide effect of BGS.

2.6 | FST outlier tests

In order to know the effect of BGS on outlier tests of local adapta‐ tion, we used a variant of fdist2 (Beaumont & Nichols, 1996). We chose fdist2 because it is a simple and fast method for which the assumptions of the test match well to most of the demographic sce‐ narios simulated here (although it is a poor match to the No migra‐ tion scenario, which has not reached equilibrium in our simulation conditions). Because the program fdist2 is not available through

B=exp

(

− ∑

i

uisi

(si +ri) 2

)

Treatment N m Genome Other BGS

Default 1,000 0.005 Stickleback – Yes

No

High migration 1,000 0.05 Stickleback – Yes

No

Large N 10,000 0.005 Stickleback – Yes

No

Human genetic map

1,000 0.005 Human – Yes

No

Low selection pressure

1,000 0.005 Stickleback Low selection pressure

Yes

No

Constant µ 1,000 0.005 Stickleback Constant mutation rate

Yes

No

No migration 1,000 0 Stickleback – Yes

No

No recombination 1,000 0.005 Stickleback Absence of crossover

Yes

No

CNC97 2,000 0.001 NA See Methods section

Yes

No

T A B L E 1   Summary of treatments. For all treatments but CNC97, the average mutation rate was set to 2.5 × 10−8 per site, per generation and the mean heterozygous selection coefficient as described in the text

     |  3907MATTHEY‐DORET AnD WHITLOCK

the command line, we rewrote the fdist2 algorithm in r and c++. Source code can be found at https ://github.com/RemiM atthe yDore t/Fdist2.

Our fdist2 procedure is as follows: first, we estimated the migra‐ tion rate from the average FST of the specific set of simulations consid‐ ered (m= 1−FST

4 (

d

d−1

)2

NFST

; Charlesworth, 1998) and then running 50,000

simulations each lasting for 50 times the half‐life to reach equilibrium FST given the estimated migration rate (Whitlock, 1992). For each SNP, we then selected the subset of fdist2 simulations for which allelic di‐ versity was <0.02 away from the allelic diversity of the SNP of interest. The p‐value is computed as the fraction of fdist2 simulations within this subset having a higher FST than the one we observed. The false‐ positive rate is then defined as the fraction of neutral SNPs for which the p‐value is lower than a given α value, using α = 0.05. We confirmed that the results were similar for other α values.

For the outlier tests, to avoid issues of pseudo‐replication, we considered only a single SNP (randomly sampled from the focal re‐ gion) per simulation whose minor allele frequency is >0.05. Then, we randomly assembled SNPs from a given treatment into groups of 500 SNPs to create the data file for fdist2. We have 4,000 simula‐ tions (2,000 with BGS and 2,000 without BGS) per treatment (Large N is an exception with only 2,000 simulations total), which allowed eight independent false‐positive rate estimates per treatment (four estimates with BGS and four without BGS). In each treatment, we tested for different false‐positive rate with and without BGS with both a Welch's t test and a Wilcoxon test.

2.7 | Statistics

FST and dXY are both measures of population divergence. In the litera‐ ture, there are several definitions of FST, and we also found potential misunderstanding about how dXY is computed.

There are two main estimators of FST in the literature: GST (Nei, 1973) and θ (Weir & Cockerham, 1984). In this article, we focus on θ as an estimator of FST (Weir & Cockerham, 1984). There are also two methods of averaging FST over several loci. The first method is to sim‐ ply take an arithmetic mean over all loci. The second method consists at calculating the sum of the numerator of θ over all loci and dividing it by the sum of the denominator of θ over all loci. Weir and Cockerham (1984) showed that this second averaging approach has lower bias than the simple arithmetic mean. We will refer to the first method as the ‘average of ratios’ and to the second method as ‘ratio of the aver‐ ages’ (Reynolds, Weir, & Cockerham, 1983; Weir & Cockerham, 1984). In this article, we use FST as calculated by ‘ratio of the averages’, as advised by Weir and Cockerham (1984). To illustrate the effects of BGS on the biased estimator of FST, we also computed FST as a simple arithmetic mean (‘average of the ratio’), and we will designate this sta‐ tistic with a subscript FST (average of ratios).

dXY is a measure of genetic divergence between two populations X and Y. Nei (1987) defined dXY as,

where L is the total number of sites, Al is the number of alleles at the lth site, and xl,k and yl,k are the frequency of the kth allele at the lth locus in the population X and Y, respectively.

Some population genetics software packages (e.g. egglib; De Mita & Siol, 2012) average dXY over polymorphic sites only, instead of averaging over all sites, as in Nei's (1987) original definition of dXY. This measure averaged over polymorphic sites only will be called dXY‐SNP; otherwise, we use the original definition of dXY by Nei (1987).

We report the average FST, dXY and within‐population genetic

diversity HS = ∑L

l=1

�

1− ∑Al

k=1 x2 l,k

�

∕L. Our main results lie in the com‐

parison between simulations with BGS and simulations without BGS within each treatment. Because theoretical expectations exist for the strength of BGS on genetic diversity within popula‐ tions, we also investigated the relationships between this theoret‐ ical expectation, B and FST, FST (average of ratios), dXY, dXY‐SNP, and HS in five independent tests for each treatment and at each generation. For this, we used Pearson correlation test, Spearman rank correla‐ tion tests, ordinary least squares regressions, robust regressions (using M‐estimators; Huber, 1964) and permutation tests. The re‐ sults were systematically consistent. Permutations tests of Pearson's correlation coefficients were performed with 50,000 iterations. Because all tests were congruent, we only report the Pearson correlation coefficient and the p‐values from permutation tests.

The data presented below are available on Dryad (https ://doi. org/10.5061/dryad.44vr58d).

3   |   R E S U LT S

The distributions of FST values from simulations with BGS are extremely similar to the distribution of FST values of simulations where all mutations were neutral. This remains true even in the most extreme treatment with no recombination. This general

dXY =

∑L

l=1

�

1− ∑Al

k=1 xl,kyl,k

�

L

F I G U R E 1   Violin plots showing the distribution of FST values for the Default treatment (labelled ‘With Recombination’) and for the No Recombination treatment. Simulations with BGS are shown in red, and simulations without BGS are in blue. The means and standard errors are displayed with dots and error bars (although the error bars are barely visible because the standard errors are so small)

3908  |     MATTHEY‐DORET AnD WHITLOCK

result is exemplified in Figure 1 by comparing the Default treat‐ ment to the No Recombination treatment. As we have a large number of simulations, the means of the distributions of FST are significantly different between simulations with BGS and simulations without BGS for both the Default (Wilcoxon tests: W = 47,875,000; p = .00002) and the No Recombination (Wilcoxon tests: W = 47,804,000; p = .002) treatments, but the increase in mean FST due to BGS is only 4.3% for the Default treatment and 2.6% in the No Recombination treatment.

Figure 2 shows the means and standard errors for FST, dXY and HS for all treatments. For most treatments, FST is very little changed by BGS, even when HS and dXY are strongly affected.

Figure 3 shows the correlation between the predicted B and the statistics FST, dXY and HS for Default at the last generation. FST is not correlated with B (p = .24, r = −.02). The strongest correlations with B are observed for the statistics dXY (p = 4 × 10

−5, r = .06) and HS (p = 4 × 10

−5, r = .06). In fact, the two statistics dXY and HS are very highly correlated (p < 2.2 × 10−16, r = .99). This high correlation

F I G U R E 2   Comparisons of mean FST (left column), dXY (central column) and HS (right column) between simulations with (black) and without (grey) BGS. Error bars are 95% CI derived from permutations. Significance of difference between mean FST with and without BGS based on permutation tests is indicated with stars (***p < .001; **p < .01; *p < .05)

F I G U R E 3   Correlation between B and FST, HS and dXY for the last generation (5 × 2N generations after the split) of the Default treatment. Each grey dot is a single simulation where there is BGS. The large black dot is the mean of the simulations with no BGS. The p‐values are computed from a permutation test, and r is the Pearson's correlation coefficient. p‐values and r are computed on both simulations with and without BGS. Results are congruent when computing the correlation coefficients and p‐value on the subset of simulations that have BGS

Low BGS

High BGS

r : .02 P : .24

0

0.05

0.1

0.15

0.0 0.5 1.0 B

F S

T

Low BGS

High BGS

r : .06 P : 4 x 10 5

0.0000

0.0005

0.0010

0.0015

0.0 0.5 1.0 B

d X

Y

Low BGS

High BGS

r : .06 P : 4 x 10 5

0.0000

0.0005

0.0010

0.0015

0.0 0.5 1.0 B

H S

     |  3909MATTHEY‐DORET AnD WHITLOCK

explains the resemblance between the central and right graphs of Figure 3. All correlations between the statistics HS, FST, FST (aver‐ age of ratios), dXY, and dXY‐SNP and B are summarized in Tables S1–S5, respectively.

When looking at correlations between B and the statistics of population divergence, the No Migration treatment is an exception to the other treatments. For the No Migration treatment, FST is not sig‐ nificantly correlated with B at early generations but become slightly correlated as divergence rises to an FST of 0.6 and higher. dXY shows an opposite pattern. dXY is very significantly correlated with B at early generations and seemingly independent of B at the last generation. Note that for FST all correlation coefficients are always very small. The largest r2 observed for FST is r

2 = 0.01 (found for FST No Migration). As expected, in the CNC97 simulations, there is a strong differ‐

ence between simulations with BGS and simulations without BGS for all three statistics (FST, dXY and HS) at all generations (Welch's t tests; all p < 2.2 × 10−16; Figure 2).

FST averaged over loci as advised by Weir and Cockerham (1984) was generally less sensitive to BGS than FST calculated as an average of ratios (compare Tables S2 and S3). This effect is again partially visible in the correlations with B. Figure S2 illustrates the sensitivity of FST (average of ratios) in the worst case, the No Recombination treat‐ ment. This sensitivity is driven largely by rare alleles and goes away when minor alleles below a frequency of 0.05 are excluded.

The observed false‐positive rate (FPR) for the FST outlier test is relatively close to the α values except for No Migration (with and without BGS) and CNC97 (with BGS). Excluding the intentionally unrealistic treatment CNC97, we do not see more significant dif‐ ferences between the FPR with and without BGS than we would expect by chance (Figure 4). There are other statistics of inter‐ est that one can consider to investigate whether BGS causes FST outliers. Among all treatments (excluding CNC97), the fraction of SNPs that are associated with a FST that is >10 times the average FST in its particular treatment is 0.075% with BGS and 0.085% without BGS. These numbers go up to 1.7% and 1.8% for SNPs that have a FST five times greater than the average FST, for treat‐ ments with BGS and without BGS, respectively. We have also computed, in each treatment, the ratio of the largest FST to the average FST. Among treatments without BGS, the largest FST was on average 12.2 times the average FST, while among treatments with BGS, the largest FST was on average 12.1 times the average FST. With an α threshold of 0.001, we observe that 0.080% and 0.083% of the SNPs turn out false positives among treatments without BGS and among treatment with BGS, respectively. For the conditions considered in this paper, BGS does not signifi‐ cantly increase the rate of FST outliers.

4   |   D I S C U S S I O N

In agreement with previous works (e.g. Charlesworth, 2012; Elyashiv et al., 2016; Messer & Petrov, 2013; Nordborg et al., 1996; Vijay et al., 2017; Zeng & Charlesworth, 2011), we show

that background selection reduces genetic diversity, both within and among populations. The magnitude of this effect is very simi‐ lar to previous realistic simulations (Messer & Petrov, 2013). The effect of BGS on FST is however rather small and does not seem

F I G U R E 4   Comparison of false‐positive rate (FPR) returned by fdist2 between simulations with BGS (black) and without BGS (grey) for all treatments by generation. The significance level is 0.05 and is represented by the horizontal dashed line. Significance based on a Welch's t test is indicated with stars (***p < .001; **p < .01; *p < .05). With Wilcoxon tests, none of the treatments displayed here comes out as significant

3910  |     MATTHEY‐DORET AnD WHITLOCK

to impact the overall distribution of FST in the sexual outcrossing conditions that we study here (Figure 1). The relative robustness of FST to variation in the intensity of BGS is contrary to what has been found in less realistic simulations (Charlesworth et al., 1997; Zeng & Corcoran, 2015).

FST was also generally not significantly correlated with B. The only exception is for the No Migration treatment, where, after many generations, as the average FST becomes very high (FST > 0.5), we observe a slight, yet significant, negative correlation between the expected effects of BGS, B and FST (intense BGS lead to high values of FST). This highlights that FST is not completely insensitive to BGS, but FST is largely robust to BGS in cases of reasonable levels of gene flow among populations. The observed correlation coefficients are always very small with not a single r2 value >1%. It is important to highlight that B has not been defined in order to estimate the effect of BGS onto FST but only for the effect of BGS on HS in a panmictic population, although Zeng and Corcoran (2015) predict that B can be used to determine the effects of BGS on FST.

In this study, we investigated the locus‐to‐locus variation in the intensity of BGS and how it affects FST. Future research is needed to attempt a theoretical estimate of the genome‐wide effect of BGS on FST (but see Charlesworth et al., 1997; Zeng & Corcoran, 2015). Our work has been restricted to the stickleback and human recombination maps. While these two genomes are reasonable representatives of many eukaryotic genomes, they are not good representatives of more compact genomes such as bacterial genomes or yeasts. Our simulations used randomly mat‐ ing diploid populations. Nonrandom mating, selfing and asexual reproduction could also affect our general conclusion and poten‐ tially strongly increase the effects of BGS on FST (Charlesworth et al., 1997). Also, we did not explore the effect of haploid selec‐ tion as we only considered autosomes (see Charlesworth, 2012). We have explored two population sizes, but we could not explore population sizes of the order of a million individuals (like D. melan‐ ogaster) and still realistically simulate such long stretches of DNA. It is not impossible that a much greater population size or a more complex demography could result in BGS having a greater effect on FST than what we observed here (Torres, Szpiech, & Hernandez, 2018). In the No Recombination treatment, we have explored cases of complete suppression of recombination over stretches of DNA of, on average, 0.8% of the stickleback genome and our results were still consistent. However, we have not explored the effect of suppression of recombination over greater regions, such as a whole chromosome that does not recombine. We have also not explored such suppression of recombination in perfectly isolated populations as we mainly focused on interconnected popula‐ tions. It is not impossible that in such cases we might observe a greater impact of BGS on FST such as those observed in Zeng and Corcoran (2015) and reproduced in Appendix S1.

Some have argued that, because BGS reduces the within‐popula‐ tion diversity, it should lead to high FST (Cruickshank & Hahn, 2014; Cutter & Payseur, 2013; Hoban et al., 2016). All else being equal, this statement is correct. However, BGS reduces HT almost as much as HS

(Figure S4). It is therefore insufficient to consider only one compo‐ nent, and we must consider the ratio of these two quantities cap‐

tured by the definition of FST, FST = (�T−�S) �T+

� S

d−1

. This ratio, as we have

shown, appears to be relatively robust to BGS. While genome‐wide BGS might eventually be strong enough to affect FST values, it ap‐ pears that locus‐to‐locus variation in the intensity of BGS is not strong enough to have much impact on FST in outcrossing sexual or‐ ganisms, at least as long as populations are not too highly diverged.

We also investigated the consequences of BGS on the widely used but imperfect estimator, FST (average of ratios), for which FST mea‐ sures for each locus are averaged to create a genomic average. It is well known that FST (average of ratios) is a biased way to average FST over several loci (Weir & Cockerham, 1984); however, its usage is rela‐ tively common today. In our simulations, FST (average of ratios) is more af‐ fected by BGS than FST. Interestingly, FST (average of ratios) is most often higher with weaker BGS. The directionality of this correlation may seem unintuitive at first. To understand this discrepancy, remember that BGS affects the site frequency spectrum; we observed that BGS leads to an excess of loci with low HT (results not shown, but see Charlesworth, Charlesworth, & Morgan, 1995; see also contrary ex‐ pectation in Stephan, 2010). Loci associated with very low HT also have low FST (Figure S3), a well‐known result described by Beaumont and Nichols (1996). As BGS creates an excess of loci with low HT and loci with low HT tend to have low FST, BGS can actually reduce FST (average of ratios). After filtering out SNPs with a minor allele frequency lower than 5%, most of the correlation between FST (average of ratios) and B is eliminated (Figure S2).

The absolute measure of divergence dXY is more sensitive to BGS than FST (Figure 2; Tables S2 and S4). Regions of stronger BGS are associated with low dXY. This is in agreement with correlation tests between B and dXY. The effect, although significant, is of relatively small size. BGS should affect dXY by its effect on the expected het‐ erozygosity, and this effect should be greater early in divergence and with migration, compared to divergence occurring because of new mutations in isolated populations. This is consistent with the results of our simulations. This result is in agreement with Vijay et al. (2017) who reported a strong correlation between HS and dXY when FST is low (FST ≈ 0.02), but this correlation breaks down when studying more distantly related populations (FST ≈ 0.3).

As BGS also leads to a reduction of the number of polymor‐ phic sites, BGS has an even stronger effect on dXY‐SNP than on dXY (Figure S2). (The measure that we call dXY‐SNP is dXY improperly calcu‐ lated based only on polymorphic sites, as is done in some software packages.)

Zeng and Corcoran (2015) approximated the effect of back‐ ground selection on FST accounting for the effect that BGS has on the effective population size (FST =

1

4NBm+1 ). With our simulations,

we report only little to no association between this expectation and the actual FST (see Figure S5). We think that BGS is a complex process and reducing its effect on FST by its theoretical expecta‐ tion of the statistic B is misleading. Below, we discuss some of the reasons.

     |  3911MATTHEY‐DORET AnD WHITLOCK

Because most deleterious mutations are recessive (García‐ Dorado et al., 2000; Peters, Halligan, Whitlock, & Keightley, 2003; Shaw & Chang, 2006), the offspring of migrants, who enjoy an in‐ creased heterozygosity compared to local individuals, will be at a se‐ lective advantage. The presence of deleterious mutations therefore leads to an increase in the effective migration rate (Ingvarsson & Whitlock, 2000), which leads to a decrease in FST. In our simulations however, mutational allelic effects are close to additive, and hence, we should not expect to see much effect of BGS on the effective mi‐ gration rate. Additionally, patches that have a lower average fitness will receive migrants that are comparably fitter and hence will enjoy a higher effective migration rate than other patches. This should also lead to decrease FST.

There is one additional factor that may weaken the connection between the FST predicted based on Ne = NB used in the Zeng and Corcoran approximation. This approximation implicitly assumes that the effects of new deleterious alleles on the effective population size are immediate. However, in a structured population connected by migration (and when migration is strong relative to the strength of selection), alleles will typically migrate before selection acts to elim‐ inate them. As a result, BGS affects the global genetic variation and the local genetic variation in a similar way. In other words, if alleles migrate before the effects of selection on linked loci is fully realized, then BGS will affect coalescent times of alleles between populations in a parallel way to how it affects coalescent times of alleles in the same population, resulting in a weakened effect on FST compared to that predicted by the Zeng and Corcoran approximation. We an‐ ticipate that much stronger selection coupled with weaker migra‐ tion would show a larger effect on FST, as is in fact the case in the parameters chosen by Zeng and Corcoran in their simulations (see Appendix S1).

Another reason why one might not observe a correlation be‐ tween intensity of BGS and FST throughout the genome is because mutation rate is auto‐correlated throughout the genome; neutral sequences closely linked to sequences that frequently receive dele‐ terious mutation are also likely to experience frequent neutral muta‐ tions. As a high mutation rate leads to low FST values (FST ≅

1

1+4Ne(m+�) ,

Wright, 1943), autocorrelation in mutation rate may also act as to reduce the effect of BGS on FST. This effect is however likely to be negligible as long as m » µ.

Recently, evidence of a correlation between recombination rate and FST has been interpreted as likely being caused by deleterious mutations rather than positive selection, whether the divergence between populations is very high (e.g. Cruickshank & Hahn, 2014), moderately high (Vijay et al., 2017) or moderately low (Torres et al., 2018). Here we showed the BGS is unlikely to explain all of these correlations between FST and recombination rate, especially in cases with relatively low divergence among populations. As positive se‐ lection has been shown to also have an important effect on ge‐ netic diversity (Eyre‐Walker & Keightley, 2009; Hernandez, Kelley, Elyashiv, Melton, & Auton, 2011; Macpherson, Sella, Davis, & Petrov, 2007; Sattath, Elyashiv, Kolodny, Rinott, & Sella, 2011; Wildman, Uddin, Liu, Grossman, & Goodman, 2003), it would be important to

investigate whether positive selection (selective sweeps and local adaptation) could be an important driver of the correlations between FST and recombination rate. More research would be needed to in‐ vestigate whether this is true.

McVicker et al. (2009) attempted an estimation of B values in the human genome (see also Elyashiv et al., 2016). They did so using equations from Nordborg et al. (1996). As there is little knowledge about the strength of selection throughout the genome, to our un‐ derstanding, this estimation of B values should be highly influenced by the effects of beneficial mutations as well as deleterious muta‐ tions. Torres et al. (2018) reused this data set and found a slight asso‐ ciation between B and FST among human lineages. It is plausible that this correlation between B and FST could be driven by positive selec‐ tion rather than by deleterious mutations; this possibility should be further explored.

Our fdist2 analysis shows that the false‐positive rate does not differ in simulations with BGS or without BGS (Figure 4). The only exceptions concern the unrealistic CNC97 treatment. The average FST at the last generation of the No Migration treatment is >0.8. With such high FST, the FST outlier method, fdist2, does not seem to perform well and both the simulations without BGS and with BGS lead to very high false‐positive rates (0.472 without BGS and 0.467 with BGS; Figure 4). Beaumont and Nichols (1996) showed that the fdist2 procedure is problematic for nonequilibrium pop‐ ulations, especially when FST is >0.5 (see section nonequilibrium populations from their results), especially when the number of subpopulations sampled is low. It therefore sounds plausible that fdist2 would poorly apply to our No Migration treatment. While other issues may intervene in FST outlier methods (Lotterhos & Whitlock, 2014), our results show that BGS should not represent any significant issue for outcrossing sexual species with moder‐ ately low mean FST.

We have shown that BGS affects HS and dXY but has only a very minor effect on FST among sexual outcrossing populations connected by gene flow. Many authors (Cutter & Payseur, 2013; Hoban et al., 2016) have raised concerns that BGS can strongly reduce our ability to detect the genomic signature of local adap‐ tation. Our analysis shows that BGS is not a strong confounding factor to FST outlier tests.

A C K N O W L E D G E M E N T S

Many thanks to Loren H. Rieseberg, Sarah P. Otto and Amy L. Angert for their help in discussing the design of the project and for feedback. Thanks to Sarah P. Otto, Darren E. Irwin and Bret A. Payseur as well as two anonymous reviewers for helpful comments on the manuscript. We also thank Yaniv Brandvain and Tom Booker for their feedback and Marius Roesti for help with the Ensembl‐ retrieved gene annotations. We also want to thank Nick Barton for his help in interpreting why our results differ from Zeng and Corcoran (2015) theoretical expectation. We also acknowledge ComputeCanada that provided the hardware resources for running our simulations.

3912  |     MATTHEY‐DORET AnD WHITLOCK

D ATA AVA I L A B I L I T Y S TAT E M E N T

R. Matthey‐Doret, M. C. Whitlock, Data from: Background selec‐ tion and FST: consequences for detecting local adaptation, 2019, Molecular Ecology, https ://doi.org/10.5061/dryad.44vr58d

A U T H O R C O N T R I B U T I O N S

R.M.‐D. wrote the simbit software, performed the simulations and the statistical analyses. Both R.M.‐D. and M.C.W. designed the pro‐ ject. M.C.W. formulated the original idea and supervised the project all the way through. R.M.‐D. wrote the MS and M.C.W. made sub‐ stantial revisions on the MS.

O R C I D

Remi Matthey‐Doret https://orcid.org/0000‐0001‐5614‐5629

Michael C. Whitlock https://orcid.org/0000‐0002‐0782‐1843

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S U P P O R T I N G I N F O R M AT I O N

Additional supporting information may be found online in the Supporting Information section at the end of the article.

How to cite this article: Matthey‐Doret R, Whitlock MC. Background selection and FST: Consequences for detecting local adaptation. Mol Ecol. 2019;28:3902–3914. https ://doi. org/10.1111/mec.15197