Valentin Hivert

Raphael Leblois

Eric J Petit

Mathieu Gautier

Renaud Vitalis
email: renaud.vitalis@inra.fr

Raphaël Leblois

Measuring Genetic Differentiation from Pool-seq Data

Keywords: F ST, genetic differentiation, pool sequencing, population genomics

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instead of individual genotypes raises statistical challenges in deriving correct estimates of genetic differentiation. In this article, we provide a method-of-moments estimator of F ST for Pool-seq data, based on an analysis-of-variance framework. We show, by means of simulations, that this new estimator is unbiased, and outperforms previously proposed estimators. We evaluate the robustness of our estimator to model misspecification, such as sequencing errors and uneven contributions of individual DNAs to the pools. Finally, by reanalyzing published Pool-seq data of different ecotypes of the prickly sculpin Cottus asper, we show how the use of an unbiased F ST estimator may question the interpretation of population structure inferred from previous analyses.

INTRODUCTION

It has long been recognized that the subdivision of species into subpopulations, social groups and families fosters genetic differentiation [START_REF] Wahlund | Zusammens etzung von populationen und korrelationserscheinungen vom standpunkt der vererbungslehre aus betrachtet[END_REF]Wright 1931). Characterizing genetic differentiation as a means to infer unknown population structure is therefore fundamental to population genetics, and finds applications in multiple domains, including conservation biology, invasion biology, association mapping and forensics, among many others.

In the late 1940s and early 1950s, [START_REF] Malécot | Les Mathématiques de l'Hérédité[END_REF] and Wright (1951) introduced F -statistics to partition genetic variation within and between groups of individuals [START_REF] Holsinger | Genetics in geographically structured populations: defining, estimating and interpreting F ST[END_REF][START_REF] Bhatia | Estimating and interpreting F ST : the impact of rare variants[END_REF]. Since then, the estimation of F -statistics has become standard practice (see, e.g., [START_REF] Weir | Genetic Data Analysis II[END_REF]Weir and Hill 2002;[START_REF] Weir | Estimating F -statistics: A historical view[END_REF], and the most commonly used estimators of F ST have been developed in an analysis-of-variance framework [START_REF] Cockerham | Variance of gene frequencies[END_REF][START_REF] Cockerham | Analyses of gene frequencies[END_REF][START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF], which can be recast in terms of probabilities of identity of pairs of homologous genes [START_REF] Cockerham | Analyses of gene frequencies[END_REF]Rousset 2007;[START_REF] Weir | An unified characterization of population structure and relatedness[END_REF].

Assuming that molecular markers are neutral, estimates of F ST are typically used to quantify genetic structure in natural populations, which is then interpreted as the result of demographic history [START_REF] Holsinger | Genetics in geographically structured populations: defining, estimating and interpreting F ST[END_REF]: large F ST values are expected for small populations among which dispersal is limited (Wright 1951), or between populations that have long diverged in isolation from each other [START_REF] Reynolds | Estimation of the coancestry coefficient: basis for a short-term genetic distance[END_REF]); when dispersal is spatially restricted, a positive relationship between F ST and the geographical distance for pairs of populations generally holds (Slatkin 1993;Rousset 1997). It has also been proposed to characterize the heterogeneity of F ST Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 estimates across markers for identifying loci that are targeted by selection [START_REF] Cavalli-Sforza | Population structure and human evolution[END_REF][START_REF] Lewontin | Distribution of gene frequency as a test of the theory of the selective neutrality of polymorphism[END_REF][START_REF] Beaumont | Evaluating loci for use in the genetic analysis of population structure[END_REF][START_REF] Vitalis | Interpretation of variation across marker loci as evidence of selection[END_REF]Akey et al. 2002;[START_REF] Beaumont | Adaptation and speciation: what can F ST tell us?[END_REF][START_REF] Weir | Measures of human population structure show heterogeneity among genomic regions[END_REF]Lotterhos andWhitlock 2014, 2015;[START_REF] Lotterhos | The relative power of genome scans to detect local adaptation depends on sampling design and statistical method[END_REF].

Next-generation sequencing (NGS) technologies provide unprecedented amounts of polymorphism data in both model and non-model species [START_REF] Ellegren | Genome sequencing and population genomics in nonmodel organisms[END_REF]. Although the sequencing strategy initially involved individually tagged samples in humans (The International HapMap Consortium 2005), whole-genome sequencing of pools of individuals (Pool-seq) is being increasingly used for population genomic studies (Schlötterer et al. 2014). Because it consists in sequencing libraries of pooled DNA samples and does not require individual tagging of sequences, Pool-seq provides genome-wide polymorphism data at considerably lower cost than sequencing of individuals (Schlötterer et al. 2014). However, non-equimolar amounts of DNA from all individuals in a pool and stochastic variation in the amplification efficiency of individual DNAs have raised concerns with respect to the accuracy of the so-obtained allele frequency estimates, particularly at low sequencing depth and with small pool sizes [START_REF] Cutler | To pool, or not to pool?[END_REF][START_REF] Ellegren | Genome sequencing and population genomics in nonmodel organisms[END_REF]Anderson et al. 2014). Nonetheless, it has been shown that, at equal sequencing effort, Pool-seq provides similar, if not more accurate, allele frequency estimates than individual-based analyses [START_REF] Futschik | The next generation of molecular markers from massively parallel sequencing of pooled DNA samples[END_REF][START_REF] Gautier | Estimation of population allele frequencies from next-generation sequencing data: pool-versus individual-based genotyping[END_REF]). The problem is different for diversity and differentiation parameters, which depend on second moments of allele frequencies or, equivalently, on pairwise measures of genetic identity: with Pool-seq data, it is indeed impossible to distinguish pairs of reads that are identical because they were Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 sequenced from a single gene, from pairs of reads that are identical because they were sequenced from two distinct genes that are identical in state (IIS) [START_REF] Ferretti | Population genomics from pool sequencing[END_REF].

Appropriate estimators of diversity and differentiation parameters must therefore be sought, to account for both the sampling of individual genes from the pool and the sampling of reads from these genes. There has been several attempts to define estimators for the parameter F ST for Pool-seq data (Kofler et al. 2011;[START_REF] Ferretti | Population genomics from pool sequencing[END_REF], from ratios of heterozygosities (or from probabilities of genetic identity between pairs of reads) within and between pools. In the following, we will argue that these estimators are biased (i.e., they do not converge towards the expected value of the parameter), and that some of them have undesired statistical properties (i.e., the bias depends upon sample size and coverage). Here, following [START_REF] Cockerham | Variance of gene frequencies[END_REF], [START_REF] Cockerham | Analyses of gene frequencies[END_REF], [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF], [START_REF] Weir | Genetic Data Analysis II[END_REF], Weir and Hill (2002) and Rousset (2007), we define a method-of-moments estimator of the parameter F ST using an analysis-of-variance framework. We then evaluate the accuracy and the precision of this estimator, based on the analysis of simulated datasets, and compare it to estimates defined in the software package PoPoolation2 (Kofler et al. 2011), and[START_REF] Ferretti | Population genomics from pool sequencing[END_REF]. Furthermore, we test the robustness of our estimators to model misspecifications (including unequal contributions of individuals in pools, and sequencing errors). Finally, we reanalyze the prickly sculpin (Cottus asper ) Pool-seq data (published by [START_REF] Dennenmoser | Adaptive genomic divergence under high gene flow between freshwater and brackish-water ecotypes of prickly sculpin (Cottus asper ) revealed by Pool-Seq[END_REF], and show how the use of biased F ST estimators in previous analyses may challenge the interpretation of population structure.

Note that throughout this article, we use the term "gene" to designate a Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 segregating genetic unit (in the sense of the "Mendelian gene" from [START_REF] Orgogozo | The "mendelian gene" and the "molecular gene": two relevant concepts of genetic units[END_REF]. We further use the term "read" in a narrow sense, as a sequenced copy of a gene. For the sake of simplicity, we will use the term "Ind-seq" to refer to analyses based on individual data, for which we further assume that individual genotypes are called without error.

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 MODEL F -statistics may be described as intra-class correlations for the probability of identity in state (IIS) of pairs of genes [START_REF] Cockerham | Analyses of gene frequencies[END_REF]Rousset 1996Rousset , 2007)), and F ST is best defined as:

F ST ≡ Q 1 -Q 2 1 -Q 2 (1)
where Q 1 is the IIS probability for genes sampled within subpopulations, and Q 2 is the IIS probability for genes sampled between subpopulations. In the following, we develop an estimator of F ST for Pool-seq data, by decomposing the total variance of read frequencies in an analysis-of-variance framework.

A complete derivation of the model is provided in the Supplemental File S1.

For the sake of clarity, the notation used throughout this article is given in Table 1. We first derive our model for a single locus, and eventually provide a multilocus estimator of F ST . Consider a sample of n d subpopulations, each of which is made of n i genes (i = 1, . . . , n d ) sequenced in pools (hence n i is the haploid sample size of the ith pool). We define c ij as the number of reads sequenced from gene j (j = 1, . . . , n i ) in subpopulation i at the locus considered. Note that c ij is a latent variable, that cannot be directly observed from the data. Let X ijr:k be an indicator variable for read r (r = 1, . . . , c ij ) from gene j in subpopulation i, such that X ijr:k = 1 if the rth read from the jth gene in the ith deme is of type k, and X ijr:k = 0 otherwise. In the following, we use standard dot notations for sample averages, i.e.: X ij•:k ≡ r X ijr:k /c ij , X i••:k ≡ j r X ijr:k / j c ij and X •••:k ≡ i j r X ijr:k / i j c ij . The analysis of variance is based on the computation of sums of squares, as fol-

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 lows:

n d i n i j c ij r (X ijr:k -X •••:k ) 2 = n d i n i j c ij r (X ijr:k -X ij•:k ) 2 + n d i n i j c ij r (X ij•:k -X i••:k ) 2 + n d i n i j c ij r (X i••:k -X •••:k ) 2 ≡ SSR :k + SSI :k + SSP :k (2)
As is shown in the Supplemental File S1, the expected sums of squares depend on the expectation of the allele frequency π k over all replicate populations sharing the same evolutionary history, as well as on the IIS probability Q 1:k that two genes in the same pool are both of type k, and the IIS probability Q 2:k that two genes from different pools are both of type k. Taking expectations (see the detailed computations in the Supplemental File S1), one has:

E(SSR :k ) = 0 (3)
for reads within individual genes, since we assume that there is no sequencing error, i.e. all the reads sequenced from a single gene are identical and X ijr:k = X ij•:k for all r. For reads between genes within pools, we get:

E(SSI :k ) = (C 1 -D 2 ) (π k -Q 1:k ) (4)
where C 1 ≡ i j c ij = i C 1i is the total number of reads in the full sample (total coverage), C 1i is the coverage of the ith pool and D 2 ≡ i (C 1i + n i -1) /n i .

D 2 arises from the assumption that the distribution of the read counts c ij is multinomial (i.e., that all genes contribute equally to the pool of reads;

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 see Equation A15 in Supplemental File S1). For reads between genes from different pools, we have:

E(SSP :k ) = C 1 - C 2 C 1 (Q 1:k -Q 2:k ) + (D 2 -D 2 ) (π k -Q 1:k ) (5)
where

C 2 ≡ i C 2 1i and D 2 ≡ [ i C 1i (C 1i + n i -1) /n i ] /C 1 (see Equa-
tion A16 in Supplemental File S1). Rearranging Equations 4-5, and summing over alleles, we get:

Q 1 -Q 2 = (C 1 -D 2 ) E(SSP ) -(D 2 -D 2 ) E(SSI) (C 1 -D 2 ) (C 1 -C 2 /C 1 ) (6)
and:

1 -Q 2 = (C 1 -D 2 ) E(SSP ) + (n c -1) (D 2 -D 2 ) E(SSI) (C 1 -D 2 ) (C 1 -C 2 /C 1 ) (7)
where n c ≡ (C 1 -C 2 /C 1 ) / (D 2 -D 2 ). Let M SI ≡ SSI/ (C 1 -D 2 ) and M SP ≡ SSP/ (D 2 -D 2 ). Then, using the definition of F ST from Equation 1, we have:

F ST ≡ Q 1 -Q 2 1 -Q 2 = E(M SP ) -E(M SI) E(M SP ) + (n c -1) E(M SI) (8)
which yields the method-of-moments estimator:

F pool ST = M SP -M SI M SP + (n c -1) M SI (9)
where

M SI = 1 C 1 -D 2 k n d i C 1i πi:k (1 -πi:k ) ( 10
)
Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 and:

M SP = 1 D 2 -D 2 k n d i C 1i (π i:k -πk ) 2 (11)
(see Equations A25 and A26 in Supplemental File S1). In Equations 10 and 11, πi:k ≡ X i••:k is the average frequency of reads of type k within the ith pool, and πk ≡ X •••:k is the average frequency of reads of type k in the full sample. Note that from the definition of X •••:k , πk ≡ i j r X ijr:k / i j c ij = i C 1i πi:k / i C 1i is the weighted average of the sample frequencies with weights equal to the pool coverage. This is equivalent to the weighted analysis-of-variance in [START_REF] Cockerham | Analyses of gene frequencies[END_REF] (see also [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF][START_REF] Weir | Genetic Data Analysis II[END_REF]Weir and Hill 2002;Rousset 2007;[START_REF] Weir | An unified characterization of population structure and relatedness[END_REF]. Finally, the full expression of F pool ST in terms of sample frequencies reads:

F pool ST = k (C 1 -D 2 ) n d i C 1i (π i:k -πk ) 2 -(D 2 -D 2 ) n d i C 1i πi:k (1 -πi:k ) k (C 1 -D 2 ) n d i C 1i (π i:k -πk ) 2 + (n c -1) (D 2 -D 2 ) n d i C 1i πi:k (1 -πi:k ) ( 12
)
If we take the limit case where each gene is sequenced exactly once, we recover the Ind-seq model: assuming c ij = 1 for all (i, j), then

C 1 = n d i n i , C 2 = n d i n 2 i , D 2 = n d and D 2 = 1. Therefore, n c = (C 1 -C 2 /C 1 ) / (n d -1),
and Equation 9 reduces exactly to the estimator of F ST for haploids: see Weir (1996), p. 182, andRousset (2007), p. 977.

As in [START_REF] Reynolds | Estimation of the coancestry coefficient: basis for a short-term genetic distance[END_REF], [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF], [START_REF] Weir | Genetic Data Analysis II[END_REF] and Rousset (2007), a multilocus estimate is derived as the sum of locus specific numerators over the sum of locus-specific denominators:

FST = l M SP l -M SI l l M SP l + (n c -1) M SI l (13) Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

where M SI and M SP are subscripted with l to denote the lth locus. For Ind-seq data, [START_REF] Bhatia | Estimating and interpreting F ST : the impact of rare variants[END_REF] refer to this multilocus estimate as a "ratio of averages" by opposition to an "average of ratios", which would consist in averaging single-locus F ST over loci. This approach is justified in the Appendix of [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF] and in [START_REF] Bhatia | Estimating and interpreting F ST : the impact of rare variants[END_REF], who analyzed both estimates by means of coalescent simulations. Note that Equation 13assumes that the pool size is equal across loci. Also note that the construction of the estimator in Equation 13 is different from [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF]. These authors defined their multilocus estimator as a ratio of sums of components of variance (a, b and c in their notation) over loci, which give the same weight to all loci, whatever the number of sampled genes at each locus. Equation 13 follows Genepop's rationale (Rousset 2008) instead, which gives instead more weight to loci that are more intensively covered.

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

MATERIALS AND METHODS

Simulation study

Generating individual genotypes: we first generated individual genotypes using ms [START_REF] Hudson | Generating samples under a Wright-Fisher neutral model of genetic variation[END_REF], assuming an island model of population structure (Wright 1931). For each simulated scenario, we considered 8 demes, each made of N = 5, 000 haploid individuals. The migration rate (m) was fixed to achieve the desired value of F ST (0.05 or 0.2), using Equation 6in Rousset (1996) leading, e.g., to M ≡ 2N m = 16.569 for F ST = 0.05 and M = 3.489 for F ST = 0.20. The mutation rate was set at µ = 10 -6 , giving θ ≡ 2N µ = 0.01.

We considered either fixed, or variable sample sizes across demes. In the latter case, the haploid sample size n was drawn independently for each deme from a Gaussian distribution with mean 100 and standard deviation 30; this number was rounded up to the nearest integer, with min. 20 and max. 300 haploids per deme. We generated a very large number of sequences for each scenario, and sampled independent single nucleotide polymorphisms (SNPs) from sequences with a single segregating site. Each scenario was replicated 50 times (500 times for Figures 3 andS2).

Pool sequencing: for each ms simulated dataset, we generated Pool-seq data by drawing reads from a binomial distribution [START_REF] Gautier | Estimation of population allele frequencies from next-generation sequencing data: pool-versus individual-based genotyping[END_REF]. More precisely, we assume that for each SNP, the number r i:k of reads of allelic type k in pool i follows:

r i:k ∼ Bin y i:k n i , δ i (14)
Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 where y i:k is the number of genes of type k in the ith pool, n i is the total number of genes in pool i (haploid pool size), and δ i is the simulated total coverage for pool i. In the following, we either consider a fixed coverage, with δ i = ∆ for all pools and loci, or a varying coverage across pools and loci, with δ i ∼ Pois(∆).

Sequencing error: we simulated sequencing errors occurring at rate µ e = 0.001, which is typical of Illumina sequencers [START_REF] Glenn | Field guide to next-generation DNA sequencers[END_REF]Ross et al. 2013).

We assumed that each sequencing error modifies the allelic type of a read to one of three other possible states with equal probability (there are therefore four allelic types in total, corresponding to four nucleotides). Note that only biallelic markers are retained in the final datasets. Also note that, since we initiated this procedure with polymorphic markers only, we neglect sequencing errors that would create spurious SNPs from monomorphic sites. However, such SNPs should be rare in real datasets, since markers with a low minimum read count (MRC) are generally filtered out.

Experimental error: non-equimolar amounts of DNA from all individuals in a pool and stochastic variation in the amplification efficiency of individual DNAs are sources of experimental errors in pool sequencing. To simulate experimental errors, we used the model derived by [START_REF] Gautier | Estimation of population allele frequencies from next-generation sequencing data: pool-versus individual-based genotyping[END_REF]. In this model, it is assumed that the contribution η ij = c ij /C 1i of each gene j to the total coverage of the ith pool (C 1i ) follows a Dirichlet distribution:

{η ij } 1≤j≤n i ∼ Dir ρ n i ( 15
)
Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

where the parameter ρ controls the dispersion of gene contributions around the value η ij = 1/n i , expected if all genes contributed equally to the pool of reads. For convenience, we define the experimental error as the coefficient of variation of η ij , i.e. ≡ V(η ij )/E(η ij ) = (n i -1)/(ρ + 1) (see [START_REF] Gautier | Estimation of population allele frequencies from next-generation sequencing data: pool-versus individual-based genotyping[END_REF]. When tends toward 0 (or equivalently when ρ tends to infinity), all individuals contribute equally to the pool, and there is no experimental error. We tested the robustness of our estimates to values of comprised between 0.05 and 0.5. The case = 0.5 could correspond, for example, to a situation where (for n i = 10) 5 individuals contribute 2.8× more reads than the other 5 individuals.

Other estimators

For the sake of clarity, a summary of the notation of the F ST estimators used throughout this article is given in Table 2. PP2 d : this estimator of F ST is implemented by default in the software package PoPoolation2 (Kofler et al. 2011). It is based on a definition of the parameter F ST as the overall reduction in average heterozygosity relative to the total combined population (see, e.g., [START_REF] Nei | Estimation of fixation indices and gene diversities[END_REF]:

PP2 d ≡ ĤT -ĤS ĤT ( 16
)
where ĤS is the average heterozygosity within subpopulations, and ĤT is the average heterozygosity in the total population (obtained by pooling together all subpopulation to form a single virtual unit). In PoPoolation2, ĤS is

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 the unweighted average of within-subpopulation heterozygosities:

ĤS = 1 n d n d i n i n i -1 C 1i C 1i -1 1 - k π2 i:k (17)
(using the notation from Table 1). Note that in PoPoolation2, PP2 d is restricted to the case of two subpopulations only (n d = 2). The two ratios in the right-hand side of Equation 17 are presumably borrowed from [START_REF] Nei | Estimation of average heterozygosity and genetic distance from a small number of individuals[END_REF] to provide an unbiased estimate, although we found no formal justification for the expression in Equation 17for Pool-seq data. The total heterozygosity is computed as (using the notation from Table 1):

ĤT = min i (n i ) min i (n i ) -1 min i (C 1i ) min i (C 1i ) -1 1 - k π2 k (18)
PP2 a ≡ Qr 1 -Qr 2 1 -Qr 2 (19)
where Qr 1 and Qr 2 are the frequencies of identical pairs of reads within and between pools, respectively, computed by simple counting of IIS pairs. These are estimates of Q r 1 , the IIS probability for two reads in the same pool (whether they are sequenced from the same gene or not) and Q r 2 , the IIS probability for two reads in different pools. Note that the IIS probabiliy Q r 1 is different from Q 1 in Equation 1, which, from our definition, represents the IIS probability between distinct genes in the same pool. This approach therefore confounds pairs of reads within pools that are identical because

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 they were sequenced from a single gene, from pairs of reads that are identical because they were sequenced from distinct, yet IIS genes.

FRP 13 : this estimator of F ST was developed by [START_REF] Ferretti | Population genomics from pool sequencing[END_REF] (see their Equations 3 and 10-13). [START_REF] Ferretti | Population genomics from pool sequencing[END_REF] use the same definition of F ST as in Equation 16above, although they estimate heterozygosities within and between pools as "average pairwise nucleotide diversities", which, from their definitions, are formally equivalent to IIS probabilities. In particular, they estimate the average heterozygosity within pools as (using the notation from Table 1):

ĤS = 1 n d n d i n i n i -1 1 -Qr 1i (20)
and the total heterozygosity among the n d populations as:

ĤT = 1 n 2 d n d i n i n i -1 1 -Qr 1i + n d i =i 1 -Qr 2ii (21)
Analyses of Ind-seq data:

For the comparison of Ind-seq and Pool-seq datasets, we computed F ST on subsamples of 5,000 loci. These subsamples were defined so that only those loci that were polymorphic in all coverage conditions were retained, and the same loci were used for the analysis of the corresponding Ind-seq data. For the latter, we used either the [START_REF] Nei | Estimation of fixation indices and gene diversities[END_REF] estimator based on a ratio of heterozygosity (see Equation 16above), hereafter denoted by NC 83 , or the analysis-of-variance estimator developed by [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF], hereafter denoted by WC 84 .

All the estimators were computed using custom functions in the R soft-

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 ware environment for statistical computing, version 3.3.1 (R Core Team 2017). All these functions were carefully checked against available software packages, to ensure that they provided strictly identical estimates.

Application example: Cottus asper [START_REF] Dennenmoser | Adaptive genomic divergence under high gene flow between freshwater and brackish-water ecotypes of prickly sculpin (Cottus asper ) revealed by Pool-Seq[END_REF] investigated the genomic basis of adaption to osmotic conditions in the prickly sculpin (Cottus asper ), an abundant euryhaline fish in northwestern North America. To do so, they sequenced the whole-genome of pools of individuals from two estuarine populations (CR, Capilano River Estuary; FE, Fraser River Estuary) and two freshwater populations (PI, Pitt Lake and HZ, Hatzic Lake) in southern British Columbia (Canada). We downloaded the four corresponding BAM files from the Dryad Digital Repository (doi: 10.5061/dryad.2qg01) and combined them into a single mpileup file using SAMtools version 0.1.19 [START_REF] Li | The Sequence Alignment/Map format and SAMtools[END_REF]) with default options, except the maximum depth per BAM that was set to 5,000 reads.

The resulting file was further processed using a custom awk script, to call SNPs and compute read counts, after discarding bases with a Base Alignment Quality (BAQ) score lower than 25. A position was then considered as a SNP if: (i ) only two different nucleotides with a read count > 1 were observed (nucleotides with ≤ 1 read being considered as a sequencing error);

(ii) the coverage was comprised between 10 and 300 in each of the four alignment files; (iii ) the minor allele frequency, as computed from read counts, was ≥ 0.01 in the four populations. The final data set consisted of 608,879 SNPs.

Our aim here was to compare the population structure inferred from pairwise estimates of F ST , using the estimator F pool ST on the one hand (Equa-Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 tion 12), and PP2 d on the other hand. Then, to conclude on which of the two estimators performs better, we compared the population structure inferred from F pool ST and PP2 d to that inferred from the Bayesian hierarchical model implemented in the software package BayPass [START_REF] Gautier | Genome-wide scan for adaptive divergence and association with population-specific covariates[END_REF]. Bay-Pass allows the robust estimation of the scaled covariance matrix of allele frequencies across populations for Pool-seq data, which is known to be informative about population history [START_REF] Pickrell | Inference of population splits and mixtures from genome-wide allele frequency data[END_REF]. The elements of the estimated matrix can be interpreted as pairwise and population-specific estimates of differentiation [START_REF] Coop | Using environmental correlations to identify loci underlying local adaptation[END_REF], and therefore provide a comprehensive description of population structure that makes full use of the available data.

Data availability

The authors state that all data necessary for confirming the conclusions presented in this article are fully represented within the article, figures, and tables. Supplemental Tables S1-S3 and Figures S1-S4 are available at FigShare, along with a complete derivation of the model in the Supplemental File S1 at FigShare.

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

RESULTS

Comparing Ind-seq and Pool-seq estimates of F ST Single-locus estimates F pool ST are highly correlated with the classical estimates WC 84 [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF]) computed on the individual data that were used to generate the pools in our simulations (see Figure 1). The variance of F pool ST across independent replicates decreases as the coverage increases. The correlation between F pool ST and WC 84 is stronger for multilocus estimates (see Figure S1A).

Comparing Pool-seq estimators of F ST

We found that our estimator F pool ST has extremely low bias (< 0.5% over all scenarios tested: see Tables 3 andS1-S3). In other words, the average estimates across multiple loci and replicates closely equal the expected value of the F ST parameter, as given by Equation 6in Rousset (1996), which is based on the computation of IIS probabilities in an island model of population structure. In all the situations examined, the bias does neither depend on the sample size (i.e., the size of each pool) nor on the coverage (see Figure 2).

Only the variance of the estimator across independent replicates decreases as the sample size increases and/or as the coverage increases. At high coverage, the mean and root mean squared error (RMSE) of F pool ST over independent replicates are virtually indistinguishable from that of the WC 84 estimator (see Table S1).

Figure 3 shows the RMSE of F ST estimates for a wide range of pool sizes and coverages. The RMSE decreases as the pool size and/or the coverage increases. The F ST estimates are more precise and accurate when differen-

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 tiation is low. Figure 3 provides some clues to evaluate the pool size and the coverage that is necessary to achieve the same RMSE than for Ind-seq data. Consider, for example, the case of samples of n = 20 haploids. For F ST ≤ 0.05 (in the conditions of our simulations), the RMSE of F ST estimates based on Pool-seq data tends to the RMSE of F ST estimates based on Ind-seq data either by sequencing pools of ca. 200 haploids at 20X, or by sequencing pools of 20 haploids at ca. 200X. However, the same precision and accuracy are achieved by sequencing ca. 50 haploids at ca. 50X.

Conversely, we found that PP2 d (the default estimator of F ST implemented in the software package PoPoolation2) is biased when compared to the expected value of the parameter. We observed that the bias depends on both the sample size, and the coverage (see Figure 2). We note that, as the coverage and the sample size increase, PP2 d converges to the estimator NC 83 [START_REF] Nei | Estimation of fixation indices and gene diversities[END_REF] computed from individual data (see Figure S1B). This argument was used by Kofler et al. (2011) to validate the approach, even though the estimates PP2 d depart from the true value of the parameter (Figure S1B-C).

The second of the two estimators of F ST implemented in PoPoolation2, that we refer to as PP2 a , is also biased (see Figure 2). We note that the bias decreases as the sample size increases. However, the bias does not depend on the coverage (only the variance over independent replicates does). The estimator developed by [START_REF] Ferretti | Population genomics from pool sequencing[END_REF], that we refer to as FRP 13 , is also biased (see Figure 2). However, the bias does neither depend on the pool size, nor on the coverage (only the variance over independent replicates does).

FRP 13 converges to the estimator NC 83 , computed from individual data (see Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

Figure 2). At high coverage, the mean and RMSE over independent replicates are virtually indistinguishable from that of the NC 83 estimator.

Last, we stress out that our estimator F pool ST provides estimates for multiple populations, and is therefore not restricted to pairwise analyses, contrary to

PoPoolation2's estimators. We show that, even at low sample size and low coverage, Pool-seq estimates of differentiation are virtually indistinguishable from classical estimates for Ind-seq data (see Table 3).

Robustness to unbalanced pool sizes and variable sequencing coverage

We evaluated the accuracy and the precision of the estimator F pool ST when sample sizes differ across pools, and when the coverage varies across pools and loci (see Figure 4). We found that, at low coverage, unequal sampling or variable coverage causes a negligible departure from the median of WC 84 estimates computed on individual data, which vanishes as the coverage increases. At

100X coverage, the distribution of F pool ST estimates is almost indistinguishable from that of WC 84 (see Figure 4 and Tables S2-S3).

Robustness to sequencing and experimental errors

Figure 5 shows that sequencing errors cause a negligible negative bias for F pool ST estimates. Filtering (using a minimum read count of 4) improves estimation slightly, but only at high coverage (Figure 6B). It must be noted, though, that filtering increases the bias in the absence of sequencing error, especially at low coverage (Figure 6A). With experimental error, i.e., when individuals do not contribute evenly to the final set of reads, we observed a positive bias for F pool ST estimates (Figure 5). We note that the bias decreases

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 as the size of the pools increases. Figure S2 shows the RMSE of F ST estimates for a wider range of pool sizes, coverage and experimental error rate ( ). For ≥ 0.25, increasing the coverage cannot improve the quality of the inference, if the pool size is too small. When Pool-seq experiments are prone to large experimental error rates, increasing the size of pools is the only way to improve the estimation of F ST . Filtering (using a minimum read count of 4) does not improve estimation (Figure 6C).

Application example

The reanalysis of the prickly sculpin data revealed larger pairwise estimates of multilocus F ST using PP2 d estimator, as compared to F pool ST (see Figure 7A).

Furthermore, we found that F pool ST estimates are smaller for within-ecotype pairwise comparisons as compared to between-ecotype comparisons. Therefore, the inferred relationships between samples based on pairwise F pool ST estimates show a clear-cut structure, separating the two estuarine samples from the freshwater ones (see Figure 7C). We did not recover the same structure using PP2 d estimates (see Figure 7B). Supportingly, the scaled covariance matrix of allele frequencies across samples is consistent with the structure inferred from F pool ST estimates (see Figure 7D).

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

DISCUSSION

Whole-genome sequencing of pools of individuals is increasingly popular for population genomic research on both model and non-model species (Schlötterer et al. 2014). The development of dedicated software packages (reviewed in Schlötterer et al. 2014) has undoubtedly something to do with the breadth of research questions that have been tackled using pool-sequencing. Yet, the analysis of population structure from Pool-seq data is complicated by the double sampling process of genes from the pool and sequence reads from those genes [START_REF] Ferretti | Population genomics from pool sequencing[END_REF].

The naive approach that consists in computing F ST from read counts, as if they were allele counts (e.g., as in [START_REF] Chen | Identifying genetic signatures of natural selection using pooled populations sequencing in Picea abies[END_REF], ignores the extra variance brought by the random sampling of reads from the gene pool during Pool-seq experiments. Furthermore, such computation fails to consider the actual number of lineages in the pool (haploid pool size). Altogether, these limits may result in severely biased estimates of differentiation when the pool size is low (see Figure S3). A possible alternative is to compute F ST from allele counts imputed from read counts using a maximum-likelihood approach conditional on the haploid size of the pools (e.g., as in Smadja et al. 2012;[START_REF] Leblois | Deciphering the demographic history of allochronic differentiation in the pine processionary moth Thaumetopoea pityocampa[END_REF], or from allele frequencies estimated using a model-based method that accounts for the sampling effects and the sequencing error probabilities inherent to pooled NGS experiments (see [START_REF] Fariello | Accounting for Linkage Disequilibrium in genome scans for selection without individual genotypes : the local score approach[END_REF]). However, these latter approaches may only be accurate in situations where the coverage is much larger than pool size, allowing to reduce sampling variance of reads (see Figure S3). Here, we therefore developed a new estimator of the parameter F ST for Pool-seq data, in an analysis-of-variance

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 framework [START_REF] Cockerham | Variance of gene frequencies[END_REF][START_REF] Cockerham | Analyses of gene frequencies[END_REF]. The accuracy of this estimator is barely distinguishable from that of the [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF] estimator for individual data. Furthermore, it does neither depend on the pool size nor on the coverage, and is robust to unequal pool sizes and varying coverage across demes and loci.

In our analysis, the frequency of reads within pools is a weighted average of the sample frequencies, with weights equal to the pool coverage. Therefore, our approach follows [START_REF] Cockerham | Analyses of gene frequencies[END_REF] one, which he referred to as a weighted analysis-of-variance (see also [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF][START_REF] Weir | Genetic Data Analysis II[END_REF]Weir and Hill 2002;[START_REF] Weir | An unified characterization of population structure and relatedness[END_REF]. With unequal pool sizes, weighted and unweighted analyses differ. As discussed recently in [START_REF] Weir | An unified characterization of population structure and relatedness[END_REF], the unweighted approach seems appropriate when the between component exceeds the within component, i.e. when F ST is large [START_REF] Tukey | Variances of variance components: II. The unbalanced single classification[END_REF]. It turns out that optimal weighting depends upon the parameter to be estimated [START_REF] Cockerham | Analyses of gene frequencies[END_REF] and is only efficient at lower levels of differentiation [START_REF] Robertson | Weighting in the estimation of variance components in the unbalanced single classification[END_REF]. In a likelihood analysis of the island model, Rousset (2007) derived asymptotically efficient weights that are proportional to n 2 i for the sum of squares of different samples (see also [START_REF] Robertson | Weighting in the estimation of variance components in the unbalanced single classification[END_REF]. To the best of our knowledge, such optimal weighting has never been considered in the literature.

Analysis of variance and probabilities of identity

In the analysis-of-variance framework, F ST is defined in Equation 1 as an intraclass correlation for the probability of identity in state [START_REF] Cockerham | Analyses of gene frequencies[END_REF]Rousset 1996). Extensive statistical literature is available on estimators of intraclass correlations. Beside analysis-of-variance estimators,

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 introduced in population genetics by [START_REF] Cockerham | Variance of gene frequencies[END_REF][START_REF] Cockerham | Analyses of gene frequencies[END_REF], estimators based on the computation of probabilities of identical response within and between groups have been proposed (see, e.g., [START_REF] Fleiss | Measuring nominal scale agreement among many raters[END_REF][START_REF] Fleiss | The reliability of dichotomous judgements: Unequal numbers of judges per subject[END_REF][START_REF] Mak | Analysing intraclass correlation for dichotomous variables[END_REF][START_REF] Ridout | Estimating intraclass correlation for binary data[END_REF]Wu et al. 2012), which were originally referred to as kappa-type statistics [START_REF] Fleiss | Measuring nominal scale agreement among many raters[END_REF][START_REF] Landis | A one-way components of variance model for categorical data[END_REF].

These estimators have later been endorsed in population genetics, where the "probability of identical response" was then interpreted as the frequency with which the genes are alike [START_REF] Cockerham | Analyses of gene frequencies[END_REF][START_REF] Cockerham | Analyses of gene frequencies[END_REF][START_REF] Weir | Genetic Data Analysis II[END_REF]Rousset 2007;[START_REF] Weir | An unified characterization of population structure and relatedness[END_REF]. This suggests that, with Pool-seq data, another strategy could consist in computing F ST from IIS probabilities between (unobserved) pairs of genes, which requires that unbiased estimates of such quantities are derived from read count data. We have done so in the second section of the Supplemental File S1, and we provide alternative estimators of F ST for Pool-seq data (see Equations A44 and A48 in Supplemental File S1). These estimators (denoted by F pool-PID ST and F pool-PID

ST

) have exactly the same form as the analysis-ofvariance estimator if the pools have all the same size and if the number of reads per pool is constant (Equation A33). This echoes the derivations by Rousset (2007) for Ind-seq data, who showed that the analysis-of-variance approach [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF] and the simple strategy of estimating IIS probabilities by counting identical pairs of genes provide identical estimates when sample sizes are equal (see Equation A28 and also [START_REF] Cockerham | Analyses of gene frequencies[END_REF][START_REF] Weir | Genetic Data Analysis II[END_REF]Karlsson et al. 2007). With unbalanced samples, we found that analysis-of-variance estimates have better precision and accuracy than IIS-based estimates, particularly for low levels of differ-Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 entiation (see Figure S4). Interestingly, we found that IIS-based estimates of F ST for Pool-seq data have generally lower bias and variance if the overall estimates of IIS probabilities within and between pools are computed as unweighted averages of population-specific or pairwise estimates (see Equations A39 and A43), as compared to weighted averages (Equations A46-A47).

Equation A28 further shows that our estimator may be rewritten as a function close to Q1 -Q2 / 1 -Q2 , except that it also depends on the sum i Q1i -Q1 in both the numerator and the denominator. This suggests that if the Q 1i 's differ among subpopulations, then our estimator provides an estimate of an average of population-specific F ST (Weir and Hill 2002;[START_REF] Weir | An unified characterization of population structure and relatedness[END_REF].

It follows from the derivations in the Supplemental File S1 that the estimator PP2 a (Equation 19) is biased because the IIS probability between pairs of reads within a pool Qr 1 is a biased estimator of the IIS probability between pairs of distinct genes in that pool (see Equations A34-A36 in Supplemental File S1). This is so, because the former confounds pairs of reads that are identical because they were sequenced from a single gene, from pairs of reads that are identical because they were sequenced from distinct, yet IIS genes.

A more justified estimator of F ST has been proposed by [START_REF] Ferretti | Population genomics from pool sequencing[END_REF], based on previous developments by [START_REF] Futschik | The next generation of molecular markers from massively parallel sequencing of pooled DNA samples[END_REF].

Note that, although they defined F ST as a ratio of functions of heterozygosities, they actually worked with IIS probabilities (see Equations 20 and 21).

However, although Equation 20 is strictly identical to Equation A39 in Supplemental File S1, we note that they computed the total heterozygosity by

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 integrating over pairs of genes sampled both within and between subpopulations (compare Equation 21 with A43), which may explain the observed bias (see Figure 2).

Comparison with alternative estimators

An alternative framework to [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF] analysis-of-variance has been developed by Masatoshi Nei and coworkers to estimate F ST from gene diversities [START_REF] Nei | Analysis of gene diversity in subdivided populations[END_REF][START_REF] Nei | F -statistics and analysis of gene diversity in subdivided populations[END_REF][START_REF] Nei | Estimation of fixation indices and gene diversities[END_REF][START_REF] Nei | Definition and estimation of fixation indices[END_REF]). The estimator PP2 d (see Equations 16-18) implemented in the software package PoPoolation2 (Kofler et al. 2011) follows this logic. However, it has long been recognized that both frameworks are fundamentally different in that the analysis-of-variance approach considers both statistical and genetic (or evolutionary) sampling, whereas Nei and coworkers' approach do not [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF][START_REF] Excoffier | Analysis of population subdivision[END_REF][START_REF] Holsinger | Genetics in geographically structured populations: defining, estimating and interpreting F ST[END_REF]. Furthermore, the expectation of Nei and coworkers' estimators depend upon the number of sampled populations, with a larger bias for lower numbers of sampled populations [START_REF] Goudet | The genetics of geographically structured populations[END_REF][START_REF] Excoffier | Analysis of population subdivision[END_REF][START_REF] Weir | An unified characterization of population structure and relatedness[END_REF]. This is so, because the computation of the total diversity in Equations 18 and 21 includes the comparison of pairs of genes from the same subpopulation, whereas the computation of IIS probabilities between subpopulations do not (see, e.g., [START_REF] Excoffier | Analysis of population subdivision[END_REF]. Therefore, we do not recommend using the estimator PP2 d implemented in the software package PoPoolation2 (Kofler et al. 2011).

Applications in evolutionary ecology studies

Pool-seq is being increasingly used in many application domains (Schlötterer et al. 2014), such as conservation genetics (see, e.g., Fuentes-Pardo and Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

Ruzzente 2017), invasion biology (see, e.g., [START_REF] Dexter | A genetic reconstruction of the invasion of the calanoid copepod Pseudodiaptomus inopinus across the North American Pacific Coast[END_REF]) and evolutionary biology in a broader sense (see, e.g., [START_REF] Collet | Rapid evolution of the intersexual genetic correlation for fitness in Drosophila melanogaster[END_REF]. These studies use a large range of methods, which aim at characterizing fine-scaled population structure (see, e.g., [START_REF] Fischer | Estimating genomic diversity and population differentiation -an empirical comparison of microsatellite and SNP variation in Arabidopsis halleri[END_REF], reconstructing past demography (see, e.g., [START_REF] Chen | Identifying genetic signatures of natural selection using pooled populations sequencing in Picea abies[END_REF][START_REF] Leblois | Deciphering the demographic history of allochronic differentiation in the pine processionary moth Thaumetopoea pityocampa[END_REF], or identifying footprints of natural or artificial selection (see, e.g., [START_REF] Chen | Identifying genetic signatures of natural selection using pooled populations sequencing in Picea abies[END_REF][START_REF] Fariello | Accounting for Linkage Disequilibrium in genome scans for selection without individual genotypes : the local score approach[END_REF][START_REF] Leblois | Deciphering the demographic history of allochronic differentiation in the pine processionary moth Thaumetopoea pityocampa[END_REF]).

Here, we reanalyzed the Pool-seq data produced by [START_REF] Dennenmoser | Adaptive genomic divergence under high gene flow between freshwater and brackish-water ecotypes of prickly sculpin (Cottus asper ) revealed by Pool-Seq[END_REF], who investigated the adaptive genomic divergence between freshwater and brackish-water ecotypes of the prickly sculpin C. asper, an abundant euryhaline fish in northwestern North America. Measuring pairwise genetic differentiation between samples using F pool ST , we found a clear-cut structure separating the freshwater from the brackish-water ecotypes. Such genetic strucure supports the hypothesis that populations are locally adaptated to osmotic conditions in these two contrasted habitats, as discussed in [START_REF] Dennenmoser | Adaptive genomic divergence under high gene flow between freshwater and brackish-water ecotypes of prickly sculpin (Cottus asper ) revealed by Pool-Seq[END_REF]. This structure, which is at odds with that inferred from PP2 d estimates, is not only supported by the scaled covariance matrix of allele frequencies, but also by previous microsatellite-based studies, who showed that populations were genetically more differentiated between ecotypes than within ecotypes [START_REF] Dennenmoser | Genetic population structure in prickly sculpin (Cottus asper ) reflects isolation-byenvironment between two life-history ecotypes[END_REF][START_REF] Dennenmoser | Phylogeography of the prickly sculpin (Cottus asper ) in north-western North America reveals parallel phenotypic evolution across multiple coastalinland colonizations[END_REF].

Limits of the model and perspectives

We have shown that the stronger source of bias for the F pool ST estimate is unequal contributions of individuals in pools. This is so, because we assume in our model that the read counts are multinomially distributed, which supposes that all genes contribute equally to the pool of reads [START_REF] Gautier | Estimation of population allele frequencies from next-generation sequencing data: pool-versus individual-based genotyping[END_REF],

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

i.e. that there is no variation in DNA yield across individuals and that all genes have equal sequencing coverage [START_REF] Rode | How to optimize the precision of allele and haplotype frequency estimates using pooled-sequencing data[END_REF]. Because the effect of unequal contribution is expected to be stronger with small pool sizes, it has been recommended to use pool-seq with at least 50 diploid individuals per pool [START_REF] Lynch | Population-genetic inference from pooled-sequencing data[END_REF]Schlötterer et al. 2014). However, this limit may be overly conservative for allele frequency estimates [START_REF] Rode | How to optimize the precision of allele and haplotype frequency estimates using pooled-sequencing data[END_REF]), and we have shown here that we can achieve very good precision and accuracy of F ST estimates with smaller pool sizes. Furthermore, because genotypic information is lost during Pool-seq experiments, we assume in our derivations that pools are haploid (and therefore that F IS is nil). Analyzing non-random mating populations (e.g., in selfing species) is therefore problematic.

Finally, our model, as in [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF], formally assumes that all populations provide independent replicates of some evolutionary process [START_REF] Excoffier | Analysis of population subdivision[END_REF][START_REF] Holsinger | Genetics in geographically structured populations: defining, estimating and interpreting F ST[END_REF]. This may be unrealistic in many natural populations, which motivated Weir and Hill (2002) to derive a population-specific estimator of F ST for Ind-seq data (see also [START_REF] Vitalis | Interpretation of variation across marker loci as evidence of selection[END_REF]. Even though the use of Weir and Hill's (2002) estimator is still scarce in the literature (but see [START_REF] Weir | Measures of human population structure show heterogeneity among genomic regions[END_REF][START_REF] Vitalis | DetSel: An R-Package to detect marker loci responding to selection[END_REF]), [START_REF] Weir | An unified characterization of population structure and relatedness[END_REF] recently proposed a re-interpretation of population-specific estimates of F ST in terms of allelic matching proportions, which are strictly equivalent to IIS probabilities between pairs of genes. It would therefore be straightforward to extend [START_REF] Weir | An unified characterization of population structure and relatedness[END_REF] estimator of population-specific F ST for the analysis of Pool-seq data, using the unbiased estimates of IIS probabilies provided in the Supplemental File S1.

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics,210 (1),315316317318319320321322323324325326327328329330 Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

Table 1 Summary of main notations

Notation

Parameter definition X ijr:k Indicator variable: X ijr:k = 1 if the rth read from the jth individual in the ith pool is of type k, and X ijr:k = 0 otherwise

r i:k = j r X ijr:k Number of reads of type k in the ith pool c ij Number of reads sequenced from individual j in sub- population i (unobserved individual coverage) C 1i ≡ j c ij
Total number of reads in the ith pool (pool coverage)

C 1 ≡ i C 1i
Total number of reads in the full sample (total coverage)

C 2 ≡ i C 2 1i
Squared number of reads in the full sample n i Total number of genes the ith pool (haploid pool size)

y i:k (Unobserved) number of genes of type k in the ith pool π k ≡ E(X ijr:k ) Expected frequency of reads of type k in the full sample πij:k ≡ X ij•:k (Unobserved) average frequency of reads of type k for individual j in the ith pool πi:k ≡ X i••:k Average frequency of reads of type k in the ith pool πk ≡ X •••:k Average frequency of reads of type k in the full sam- ple Q 1 (resp. Q 2 )
IIS probability for two genes sampled within (resp. between) pools

Q r 1 (resp. Q r 2 )
IIS probability for two reads sampled within (resp. between) pools Qpool

1 (resp. Qpool 2 )
Unbiased estimator of the IIS probability for genes sampled within (resp. between) populations

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics,210 (1),315316317318319320321322323324325326327328329330 19WC 84 [START_REF] Weir | Estimating F -statistics for the analysis of population structure[END_REF] Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 Multilocus F pool ST estimates were computed for various conditions of expected F ST , pool size (n) and coverage (Cov.) in an island model with n d = 8 subpopulations (pools). The mean (RMSE) is over 50 independent simulated datasets, each made of 5,000 loci. For comparison, we computed multilocus WC 84 estimates from individual genotypes (Ind-seq).

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.0 0.1 0.2 0.3

0.0 0.1 0.2 0.3 WC 84

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q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 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q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q B. F ST = 0.2 20X 50X 100X

Coverage

Figure 1 Single-locus estimats of F ST . We compared single-locus estimates of F ST based on allele count data inferred from individual genotypes (Indseq), using the WC 84 estimator, to F pool ST estimates from Pool-seq data. We simulated 5,000 SNPs using ms in an island model with n d = 8 demes. We used two migration rates corresponding to F ST = 0.05 (A) and F ST = 0.20 (B). The size of each pool was fixed to 100. We show the results for different coverages (20X, 50X and 100X). In each graph, the cross indicates the simulated value of F ST .

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Genetic differentiation (F ST A. F ST = 0.05 (n = 10) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 W C q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.10 C. F ST = 0.2 (n = 10) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.10 Each density plot, which represents the root mean squared error (RMSE) of the estimator F pool ST , was obtained using simple linear interpolation from a set of 44 × 44 pairs of pool size and coverage values. For each pool size and coverage, 500 replicates of 5,000 markers were simulated from an island model with n d = 8 demes. Plain white isolines represent the RMSE of the WC 84 estimator computed from Ind-seq data, for various sample sizes (n = 5, 10, 20, and 50). Each isoline was fitted using a thin plate spline regression with smoothing parameter λ = 0.005, implemented in the fields package for R [START_REF] Nychka | fields: Tools for spatial data[END_REF].

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018) Figure 4 Precision and accuracy of F ST estimates with varying pool size or varying coverage. Our estimator F pool ST was calculated from Pool-seq data over all demes and loci and compared to the estimator WC 84 , computed from individual genotypes (Ind-seq). Each boxplot represents the distribution of multilocus F ST estimates across 50 independent replicates of the ms simulations. We used two migration rates, corresponding to F ST = 0.05 (A and C) and F ST = 0.20 (B and D). In A-B the pool size was variable across demes, with haploid sample size n drawn independently for each deme from a Gaussian distribution with mean 100 and standard deviation 30; n was rounded up to the nearest integer, with min. 20 and max. 300 haploids per deme. In C-D, the pool size was fixed (n = 100), and the coverage (δ i ) was varying across demes and loci, with δ i ∼ Pois(∆) where ∆ ∈ {20, 50, 100}. For Poolseq data, we show the results for different coverages (20X, 50X and 100X). In each graph, the dashed line indicates the simulated value of F ST and the dotted line indicates the median of the distribution of WC 84 estimates.

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

  PP2 a : this is the alternative estimator of F ST provided in the software package PoPoolation2. It is based on an interpretation by Kofler et al. (2011) of Karlsson et al.'s (2007) estimator of F ST , as:

  . , DOI : 10.1534/genetics.118.300900Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M.,Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 R.,Nusbaum, C., and Jaffe, D. B. (2013). Characterizing and measuring bias in sequence data. Genome Biol., 14:R51.Rousset, F. (1996). Equilibrium values of measures of population subdivision for stepwise mutation processes. Genetics, 142:1357-1362. Rousset, F. (1997). Genetic differentiation and estimation of gene flow from F -statistics under isolation by distance. Genetics, 145:1219-1228. Rousset, F. (2007). Inferences from spatial population genetics. In Balding, D. J., Bishop, M., and Cannings, C., editors, Handbook of Statistical Genetics, pages 945-979, Chichester. John Wiley & Sons, Ltd. Rousset, F. (2008). genepop'007: a complete re-implementation of the genepop software for Windows and Linux. Mol. Ecol. Resour., 8:103-106. Schlötterer, C., Tobler, R., Kofler, R., and Nolte, V. (2014). Sequencing pools of individuals -mining genome-wide polymorphism data without big funding. Nat. Rev. Genet., 15:749-763. Slatkin, M. (1993). Isolation by distance in equilibrium and non-equilibrium populations. Evolution, 47:264-279. Smadja, C. M., Canbäck, B., Vitalis, R., Gautier, M., Ferrari, J., Zhou, J.-J., and Butlin, R. K. (2012). Large-scale candidate gene scan reveals the role of chemoreceptor genes in host plant specialization and speciation in the pea aphid. Evolution, 66:2723-2738. The International HapMap Consortium (2005). A haplotype map of the human genome. Nature, 437:1299-1320. Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 Weir, B. S. and Hill, W. G. (2002). Estimating F -statistics. Annu. Rev. Genet., 36:721-750. Whitlock, M. C. and Lotterhos, K. E. (2015). Reliable detection of loci responsible for local adaptation: inference of a null model through trimming the distribution of F ST . Am. Nat., 186:S24-S36. Wright, S. (1931). Evolution in Mendelian populations. Genetics, 16:97-159. Wright, S. (1951). The genetical structure of populations. Ann. Eugen., 15:323-354. Wu, S., Crespi, C. M., and Wong, W. K. (2012). Comparison of methods for estimating the intraclass correlation coefficient for binary responses in cancer prevention cluster randomized trials. Contemp. Clin. Trials, 33:869-880.

  . , DOI : 10.1534/genetics.118.300900

F

  ST = 0.05 (n = 100)

Figure 2 Figure 3

 23 Figure2Precision and accuracy of pairwise estimators of F ST . We considered two estimators based on allele count data inferred from individual genotypes (Ind-seq): WC 84 and NC 83 . For pooled data, we computed the two estimators implemented in the software package PoPoolation2, that we refer to as PP2 d and PP2 a , as well as the FRP 13 estimator and our estimator F pool ST . Each boxplot represents the distribution of multilocus F ST estimates across all pairwise comparisons in an island model with n d = 8 demes, and across 50 independent replicates of the ms simulations. We used two migration rates, corresponding to F ST = 0.05 (A-B) and F ST = 0.20 (C-D). The size of each pool was either fixed to 10 (A and C) or to 100 (B and D). For Pool-seq data, we show the results for different coverages (20X, 50X and 100X). In each graph, the dashed line indicates the simulated value of F ST and the dotted line indicates the median of the distribution of NC 83 estimates.

Figure 7

 7 Figure7Reanalysis of the prickly sculpin (Cottus asper ) Pool-seq data. In (A) we compare the pairwise F ST estimates PP2 d , and F pool ST for all pairs of populations from the estuarine (CR and FE) and freshwater samples (PI and HZ). Within-ecotype comparisons are depicted as blue dots, and betweenecotype comparisons as red triangles. In (B-C) we show UPGMA hierarchical cluster analyses based on PP2 d (B) and F pool ST (C) pairwise estimates. In (D), we show a heatmap representation of the scaled covariance matrix among the four C. asper populations, inferred from the Bayesian hierarchical model implemented in the software package BayPass.

Table 2

 2 Definition of the F ST estimators used in the text

	Notation Definition
	F pool ST	Equation 12
	FRP 13	Ferretti et al. (2013) and Equations 16,20-21
	NC 83	Nei and Chesser (1983)
	PP2

d Kofler et al. (2011) and Equations 16-18 PP2 a Kofler et al. (2011) and Equation

Table 3

 3 Overall F ST estimates from multiple pools

	Pool-seq	Ind-seq

  . Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900

			A. Var. pool size (F ST = 0.05)	B. Var. pool size (F ST = 0.2)
	Genetic differentiation (F ST )	0.0490 0.0500 0.0510 0.0520	WC 84	F ST pool	0.196 0.198 0.200 0.202 0.204 0.206	WC 84	F ST pool
			C. Var. coverage (F ST = 0.05)	D. Var. coverage (F ST = 0.2)
		0.052			0.206
	Genetic differentiation (F ST )	0.049 0.050 0.051			0.196 0.198 0.200 0.202 0.204
			WC 84	F ST pool		WC 84	F ST pool
				Ind-seq Pool-seq (coverage)
				20X	50X	100X

ACKNOWLEDGEMENTS

We thank Alexandre Dehne-Garcia for his assistance in using computer farms.

Analyses were performed on the genotoul bioinformatics platform Toulouse Midi-Pyrénées (bioinfo.genotoul.fr) and the CBGP HPC computational platform. This work is part of Valentin Hivert's Ph.D., who was supported by a grant from the INRA's Plant Health and Environment (SPE) Division, and by the BiodivERsA project EXOTIC (ANR-13-EBID-0001). Part of this work was supported by the ANR project SWING (ANR-16-CE02-0015) of the French National Research Agency, and by the CORBAM project of the French region Hauts-de-France. We thank two anonymous reviewers for their positive comments and suggestions.

Comment citer ce document : Hivert, V., Leblois, R., Petit, E., Gautier, M., Vitalis, R. (2018). Measuring Genetic Differentiation from Pool-seq Data. Genetics, 210 (1), 315-330. , DOI : 10.1534/genetics.118.300900 q q q q q q q q 0.05 A. F ST = 0.05 (n = 10) q q q q q q q q 0.05 0.06 0.07 0.08 no error seq. error exp. error B. F ST = 0.05 (n = 100) q q q q q q q 0.19 Precision and accuracy of F ST estimates with and without filtering. Our estimator F pool ST was computed from Pool-seq data over all demes and loci without error (A), with sequencing error (B) and with experimental error (C) (see the legend of Figure 5 for further details). For each case, we computed F ST without filtering (no MRC) and with filtering (using a minimum read count MRC = 4). Each boxplot represents the distribution of multilocus F ST estimates across 50 independent replicates of the ms simulations. We used a migration rate corresponding to F ST = 0.20, and pool size n = 10. We show the results for different coverages (20X, 50X and 100X). In each graph, the dashed line indicates the simulated value of F ST .