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f ‐Statistics estimation and admixture graph construction with Pool‐Seq or allele count data using the R package poolfstat

November 2021
Molecular Ecology Resources 22(8)

Authors:

French National Institute for Agricultural Research and Environment (INRAE)

By capturing various patterns of the structuring of genetic variation across populations, f ‐statistics have proved highly effective for the inference of demographic history. Such statistics are defined as covariance of SNP allele frequency differences among sets of populations without requiring haplotype information and are hence particularly relevant for the analysis of pooled sequencing (Pool‐Seq) data. We here propose a reinterpretation of the F (and D) parameters in terms of probability of gene identity and derive from this unified definition unbiased estimators for both Pool‐Seq data and standard allele count data obtained from individual genotypes. We implemented these estimators in a new version of the R package poolfstat, which now includes a wide range of inference methods: (i) three‐population test of admixture; (ii) four‐population test of treeness; (iii) F4‐ratio estimation of admixture rates; and (iv) fitting, visualization and (semi‐automatic) construction of admixture graphs. A comprehensive evaluation of the methods implemented in poolfstat on both simulated Pool‐Seq (with various sequencing coverages and error rates) and allele count data confirmed the accuracy of these approaches, even for the most cost‐effective Pool‐Seq design involving relatively low sequencing coverages. We further analyzed a real Pool‐Seq data made of 14 populations of the invasive species Drosophila suzukii which allowed refining both the demographic history of native populations and the invasion routes followed by this emblematic pest. Our new package poolfstat provides the community with a user‐friendly and efficient all‐in‐one tool to unravel complex population genetic histories from large‐size Pool‐Seq or allele count SNP data.

Estimated f 3 statistics with their 95% confidence intervals for the allele count and 30X Pool-Seq data sets

…

Admixture graphs resulting from the positioning of BR-Pal onto the scaffold tree of native and Hawaiian populations (Figure 7a) with BIC less than 6 units higher than the BIC with the best fitting graph (within red box and represented in Figure 9B). For each population, the graph (as obtained with the function add.leaf) is displayed together with i) the worst fitted f-statistics and its associated Z-score; and ii) the difference of BIC of the graph with the graphs displaying the best fitting graph (∆ BIC ) as a measure of support. For all the graphs, the fitted edge lengths are in drift units (x1,000) since drift.scaling argument was set to TRUE. 25

…

Admixture graphs resulting from the positioning of US-Col, US-Nca, US-Sdi, US-Wat and US-Wis onto the am.scaf scaffold graph that relates the scaffold tree of native populations and BR-Pal and US-Sok (red frame). The best fitting graphs (as obtained with the function add.leaf) are displayed together with i) the worst fitted f-statistics and their associated Z-score; and ii) the difference of their BIC with respect to the graphs displaying the second lowest BIC (∆ BIC ) as a measure of support. The target populations are highlighted in yellow. For all the graphs, the fitted edge lengths are in drift units (x1,000) since drift.scaling argument was set to TRUE.

…

Figures - uploaded by Arnaud Estoup

Content may be subject to copyright.

Content uploaded by Arnaud Estoup

Content may be subject to copyright.

Mol Ecol Resour. 2021;00:1–23. wileyonlinelibrary.com/journal/men

1 | INTRODUCTION

In their seminal paper, Reich et al. (2009) introduced a new popu-

lation genetics framework to decipher the history of Indian human

populations. This inference approach relied on a set of so- called

f- statistics that are aimed at capturing various patterns of the

structuring of genetic diversity across- population based on single

nucleotide polymorphism (SNP) assayed on a genome- wide scale

(see also Patterson et al., 2012). The parameters underlying these

statistics, denoted F, are defined as covariances in allele frequency

differences among sets of two (F2), three (F3) or four (F4) popula-

tions and were demonstrated to be highly informative about pop-

ulations demographic history when modelled as admixture graphs

corresponding to population trees possibly including admixture

events (Patterson et al., 2012). Hence, formal tests of admixture,

called three- population tests, between a target population and two

source population surrogates can be derived from estimates of F3.

Conversely, via the so- called four- population test, estimating F4

Received: 28 May 2021

Revised: 16 September 2021

Accepted: 8 Nove mber 2021

DOI : 10.1111/1755-0998.13557

RESOURCE ARTICLE

f- Statistics estimation and admixture graph construction with

Pool- Seq or allele count data using the R package poolfstat

Mathieu Gautier1 | Renaud Vitalis1 | Laurence Flori2 | Arnaud Estoup1

1CBGP, INRAE , CIRAD, IRD, Montpelli er

SupAg ro, Univ Montpellier, Montp ellier,

France

2SelMet , INRAE , CIRAD, Montpellier

SupAg ro, Montpel lier, France

Correspondence

Mathieu G autier, INR AE- CBGP (Centre de

Biologi e pour la Ges tion des Populations),

755 avenue du campus Agropolis,

CS30 016, 34988 M ontferrier sur lez

cedex, France.

Email: mathieu.gautier@inrae.fr

Funding information

Agence N ationale de l a Recherche , Grant/

Award Number: ANR- 16- CE02- 0 015- 01

a n d A N R - 2 0 - C E 0 2 - 0 0 1 8

Abstract

By capturing various patterns of the structuring of genetic variation across popula-

tions,

- statistics have proved highly effective for the inference of demographic his-

tory. Such statistics are defined as covariances of SNP allele frequency differences

among sets of populations without requiring haplotype information and are hence

particularly relevant for the analysis of pooled sequencing (Pool- Seq) data. We here

propose a reinterpretation of the

(and

) parameters in terms of probability of gene

identity and derive from this unified definition unbiased estimators for both Pool- Seq

data and standard allele count data obtained from individual genotypes. We imple-

mented these estimators in a new version of the R package poolfstat, which now

includes a wide range of inference methods: (i) three- population test of admixture; (ii)

four- population test of treeness; (iii)

- ratio estimation of admixture rates; and (iv)

fitting, visualization and (semi- automatic) construction of admixture graphs. A com-

prehensive evaluation of the methods implemented in poolfstat on both simulated

Pool- Seq (with various sequencing coverages and error rates) and allele count data

confirmed the accuracy of these approaches, even for the most cost- effective Pool-

Seq design involving relatively low sequencing coverages. We further analysed a real

Pool- Seq data made of 14 populations of the invasive species Drosophila suzukii, which

allowed refining both the demographic history of native populations and the invasion

routes followed by this emblematic pest. Our new package poolfstat provides the

community with a user- friendly and efficient all- in- one tool to unravel complex popu-

lation genetic histories from large- size Pool- Seq or allele count SNP data.

KEYWORDS

admixture graph, demographic inference, Drosophila suzukii, f- statistics, Pool- Seq

G AUTIER ET Al.

among quadruplets of populations allows to test for their treeness,

that is, if their joint history can be modelled as a simple (unrooted)

bifur catin g tree. Und er ce rt ain re stric tive assu mptio ns abou t th e un-

derlying phylogeny, accurate estimates of the relative contributions

of the ancestral sources of an admixed population may be obtained

from ratios of F4 involving some of its related sampled populations.

A normalized version of the F4 parameter, called Patterson's D, was

also developed by Green et al. (2010) and has become quite popular

to characterize introgression in phylogenies of closely related spe-

cies (Durand et al., 2011). Finally, f- statistics can directly be used to

fit admixture graphs (i.e. estimate branch leng ths and/or admixture

propor tions) and to rigorously assess their suppor t (Lipson, 2020;

Lipson et al., 2013; Patterson et al., 2012).

A critical advantage of F and D parameters is that they only de-

pend on population allele frequencies and their estimation does not

require haplotype information. The non- independence of neigh-

bouring SNPs (Linkage Disequilibrium or LD) can be accurately ac-

counted for with block- jackknife statistical techniques (Busing et al.,

1999; Kunsch, 1989; Patterson et al., 2012; Reich et al., 2009) when

computing standard errors of the estimated f- statistics, which are

noticeably required for three- and four- population tests and also to

assess the residuals of fitted admixture graphs. These character-

istics make the f- statistics- based inference framework particularly

attractive for the analysis of Pool- Seq dat a that result from the mas-

sive sequencing of pools of individual DNA and have become quite

popular, most particularly in non- model species (Schlötterer et al.,

2014). Indeed, although LD information is generally lost in Pool- Seq

experiments (but see Long et al., 2011; or Feder et al., 2012), they

lead to accurate and cost- effective assessment of allele frequencies

across populations on a whole genome basis (Gautier et al., 2013;

Schlötterer et al., 2014). If the derivation of unbiased estimates of al-

lele frequencies from Pool- Seq data is straightforward, estimation of

more elaborated population genetics parameters characterizing the

structuring of genetic diversity within or across populations is more

challenging (Ferretti et al., 2013; Gautier et al., 2013; Hivert et al.,

2018). As the individual origin of the sequencing reads is not iden-

tifiable within pools, it is not possible to assess whether reads are

identical because they are sequenced copies of the same individual

chromosome or because they are copies of different chromosomes

carrying the same allele. The resulting additional level of variation

thus needs to be accounted for in the estimation of population ge-

netics parameters (Ferretti et al., 2013; Gautier et al., 2013; Hivert

et al., 2018).

In the present paper, we first propose a (re)interpretation of the

F (and D) parameters in terms of probability of Identity In State (IIS

or AIS for Alike- In- State) of pairs of genes sampled either within the

same population (Q1) or between two different populations (Q2), ex-

tending results we introduced in some earlier studies (Hivert et al.,

2018; Leblois et al., 2018; Collin et al., 2021). This unified definition

simplified the derivation of the unbiased estimators for both allele-

count and Pool- Seq read count data that we implemented in a new

version of our R package poolfstat (Hivert et al., 2018) together

with methods that rely on the estimated f- statistics for historical and

demographic inference. These methods include (i) three- population

test of admixture; (ii) four- population test of treeness; (iii) F4- ratio

estimation of admixture proportion; and (iv) fitting, visualization and

(semi- automatic) construction of admixture graphs. For completion,

we briefly present the underlying methods as implemented in the

package. We then carried out a comprehensive evaluation of the

whole package on both simulated allele count and Pool- Seq read

count data, considering for the latter various sequencing coverages

and the presence or not of sequencing errors. Finally, we illustrate

the power and limitations of poolfstat by analysing real Pool- Seq

data available from a previous study (Olazcuaga et al., 2020) for 14

populations of the invasive species Drosophila suzukii. This example

illustrates how f- st atistics based inference and admixture graph

construction may confirm previous inferences and provide new

insights into both the history of populations from the native area

and the invasion routes followed by an emblematic invasive species.

We provide as Supplementary Materials, a first vignette (Appendix

S2: Vignette V1) designed as a detailed hands- on manual to outline

the main functionalities of poolfstat. This vignette may also be

viewed as a practical introduction to f- statistics- based demographic

inference although we strongly encourage unfamiliar readers (and

also others) to carefully read the original methodological papers (in

particular, Lipson et al., 2013; Patterson et al., 2012; Peter, 2016)

and the recent review by Lipson (2020). Similarly, we provide a sec-

ond vignette (Appendix S2: Vignette V2) that details the analysis of

the D. suzukii data making it fully reproducible. Both vignettes fur-

ther include practical recommendations for an efficient use and in-

terpretation of f- statistics based methods and outputs.

2 | MATERIAL AND METHODS

2.1 | Definition- , estimation- and f- statistics- based

inference methods

2.1.1 | A unified definition of F2, F3 and F4

parameter and their scaled version FST,

F∗

and D in

terms of Q1 and Q2 probabilities

Let

and

the allele frequency of an arbitrarily chosen allele

at a random SNP segregating in populations

and

, respec-

tively. The parameters

and

were originally defined in terms

of covariance in allele frequencies difference among different sets

of populations as follows (Patterson et al., 2012; Reich et al., 2009):

In total, with

populations, there are

⎛

⎜

⎝

⎞

⎟

⎠

2n(n−1

)

possible

;

3⎛

⎜

⎝

⎞

⎟

⎠

2n(n−1)(

n−2

)

possible

; and

3⎛

⎜

⎝

⎞

⎟

⎠

8n(n−1)(

n−2)(

n−3

)

pos-

(1)

2(A;B)≡𝔼



pA−pB

2

3(A;B,C)≡𝔼pA−pBpA−pC=1

2F2(A;B)+F2(A;C)−F2(B;C)

4(A,B;C,D)≡𝔼



pA−pB



pC−pD



2

F2(A;D)+F2(B;C)−F2(A;C)−F2(B;D)



GAUTIER E T Al.

sible

(excluding equivalent configurations obtained by permuting

populations within pairs; see section 4.1 of Appendix S2: Vignet te

V1). Due to the linear dependency of all these parameters (eq. 1), the

n(n−1)

(

n2−n+2

)

parameters actually span a vector space of

dimension

n(n−1

)

the basis of which may be specif ied by the set of

all the

⎛

⎜

⎝

⎞

⎟

⎠

possible

or, given a reference population

(randomly

chosen among the

ones) the set of all the

n−1

of the form

F2(i;j)

(with

≠

) and all the

⎛

⎜

⎝

n−1

⎞

⎟

⎠

of the form

F3(i;j,k)

(with

≠

;

k≠i

and

≠

) (Lipson, 2020; Patterson et al., 2012). It is important to notice

that these definitions are invariant in the choice of the reference

SNP allele (Patterson et al., 2012) leading to:

where

(resp.

) is the probability of sampling two genes (or alleles)

identical in state (IIS) within population

(resp.

) and

QA,B

is the prob-

ability of sampling two IIS genes from

and

. Similar expressions for

F3(A;B,C)

and

F4(A,B;C,D)

directly follows from Equations 1 And 2

(see Eq. 3 below).

The

and

probabilities, and hence the

and

parameters

depend on both demographic parameters (i.e. population sizes, diver-

gence times and other historical events) and marker polymorphism (i.e.

their mutation rates and ascertainment process). For instance, under a

simple pure- drift model with no mutation, if

denotes the allele fre-

quency of the ancestral population

of two isolated populations A and B,

then

−Q

A,B

=𝔼

[

=2p

1−p

, which is the heterozygosity in

R. Similarly,

−Q

=2p

1−p

−𝜏A

(resp.,

−Q

=2p

1−p

−𝜏B

)

where

𝜏A

(resp.,

𝜏B

) is the divergence time separating R and A (resp., B) on

a diffusion timescale (i.e. in drift units of

2Ne

where

is the effective

population along the branch). As a consequence, the resulting estimates

and

strongly depend on the underlying set of genetic markers

and may not be compared across different data sets, even from the same

populations. Various scaling procedure may actually help in reducing this

dependence. Scaling the

with respect to the across population hetero-

zygosity

1−Q2

leads to the standard definition of pairwise- population

FST

in terms of IIS probabilities (Hivert et al., 2018; Rousset, 2007), which

is also concordant with its original definition as the numerator of

FST

(Peter, 2016; Reich et al., 2009):

where

+QB

is the overall probability of sampling two

IIS genes within the same population (i.e. averaged over populations

and

). Similarly, the scaled versions of the

and

parameters

named

F∗

and

, respectively (Durand et al., 2011; Green et al., 2010;

Patterson et al., 2012), can also be expressed as a function of

and

probabilities leading to the following expression for all the

(and

) parameters:

2.1.2 | Unbiased parameter estimators from Pool-

Seq read count and standard allele count data

Let

yij

be the allele count for an arbitrarily chosen reference allele

and

nij

the total number of sampled alleles (e.g. twice the number of

genotyped individuals for a diploid species) at SNP

in population

For Pool- Seq read count data, the

yij

s are not observed, and for a

given pool

, it is assumed that

nij =nj

(the haploid sample size) for

each and every SNP. We further defined

rij

as the read counts for the

reference allele and

cij

the overall coverage obser ved at SNP

population

. If allele count data are directly obser ved, unbiased es-

timators of the IIS probability within population

(

1,i

) and between

a pair of populations

and

(

Qj,k

2,i)

for a given SNP

are:

Likewise, for Pool- Seq read count, unbiased estimators of

Q1,i

and

Q2,i

are defined as (Hivert et al., 2018, eqns A37 and A40):

Genome- wide estimates of all the

(and

) parameters (eq. 3) are

then simply obtained from averages of these unbiased estimators of

IIS probabilities over all the

SNPs. Importantly, for the three scaled

parameters (

FST

F∗

and

), multi- locus estimators consist of ratios

of the numerator and denominator averages and not average of ra-

tios (Bhatia et al., 2013; Hivert et al., 2018; Patterson et al., 2012;

Rousset, 2007; Weir & Goudet, 2017) as, for example:

(2)

2(A;B)≡𝔼



pA−pB





=𝔼



1−pA



−



1−pB







𝔼



pA−pB





+𝔼



1−pA



−



1−pB





2𝔼p2

A+𝔼(1−pA)2+1

2𝔼p2

B+𝔼(1−pB)2−𝔼pApB+𝔼1−pA1−pB

2

1+QB



−QA,B

ST (A;B)≡

Q1−Q

A,B

1−QA,B

=F2(A;B)

1−QA,B

(3)

2(A;B)≡

1+QB

2−QA,B

2and FST (A;B)≡F2(A;B)

1−QA,B

=QA

1+QB

1−2Q

A,B

21−QA,B

2

3(A;B,C)≡

1+QB,C

2−QA,B

2−QA,C

2and F∗

3(A;B,C)≡F3(A;B,C)

1−QA

=QA

1+QB,C

2−QA,B

2−QA,C

21−QA

1

4(A,B;C,D)≡

QA,C

2+QB,D

2−QA,D

2−QB,C

2and D(A,B;C,D)≡F4(A,B;C,D)



1−QA,B



1−QC,D



=QA,C

2+QB,D

2−QA,D

2−QB,C



1−QA,B



1−QC,D



(4)



1,i=1−2yij

(

nij −yij

)

ij (

−1

)

and 

Qj,k

2,i=yijyik +

(

nij −

)(

nik −yik

)

nijnik

(5)



1,i=1−2nj

nj−1

rij

(

cij −rij

)

ij (

−1

)

and 

Qj,k

2,i=rijrik +

(

cij −rij

)(

cik −rik

)

cijcik

G AUTIER ET Al.

the resulting pairwise

FST

estimator being similar to the one described

in Rousset (2007) for allele count data and identical to the ‘PID’ esti-

mator described in Hiver t et al. (2018) for Pool- Seq read count data

(‘Identity’ method of the poolfstat function co mputeFST). Fin ally,

the within- population heterozygosity (

≡1−Q

) of a given popula-

tion

is simply estimated as:

2.1.3 | Block- Jackknife estimation of

standard errors

Following Reich et al. (2009), standard errors of genome- wide esti-

mates of the different statistics are computed using block- jackknife

(Busing et al., 1999; Kunsch, 1989), which consists of dividing the ge-

nome into contiguous chunks of a predefined number of SNPs and

then removing each block in turn to quantify the variability of the es-

timator. For a given parameter

, if

blocks are available and



is the

estimated statistics when removing all SNPs belonging to block

, the

standard error

𝜎 F

of the genome- wide estimator



is computed as:

where

𝜇

nb∑n

b=1



, which may be slightly different than the esti-

mator obtained with all the

markers since the latter may include

SNPs that are not eligible for block- jackknife sampling (e.g. those at

the chromosome or scaf folds boundaries). Finally, block- jackknife

sampling may also be used to obtain estimates of the error covari-

ance between two estimates



and



as:

For convenience, we here chose to specify the same number

of SNPs for each block instead of a block size in genetic distance

(Patterson et al., 2012; Reich et al., 2009). We therefore do not re-

course to a weighted block- jackknife (Busing et al., 1999). In prac-

tice, this has little impact providing the distribution of markers is

homogeneous along the genome and the amount of missing data is

negligible.

2.1.4 | Admixture graph fitting

The approach implemented in the new version of poolfstat to fit

admixture graphs from f- statistics is directly inspired from the one

proposed by Patterson et al. (2012) and implemented in the qp-

Graph soft ware (see also Lipson, 2020). Briefly, let



the vector (of

length

(

nl−1

)

where

is the number of graph leaves) of the esti-

mated

and

statistics forming the basis of all the f- statistics (see

2.1.1). Similarly, let

g(e;a)=X(a)×e

the vector of their expected

values given the graph edge lengths vector

and an incidence matrix

X(a)

, which summarize the structure of the graph given the vector

of propor tions of all admixture events (for a tree- topology,

X(a)

only consists of 0 or 1). In poolfstat,

X(a)

is derived using simple

operations from another

matrix (where

is the number of

graph edges) that specifies the weights of each edge along all the

paths connecting the graph leaves to the root. It should be noticed

that an admixture event is modelled as an instantaneous mixing of

two populations

and

into a population

directly ancestral to a

child population

, while assuming a null length for the edges con-

necting

and

to the rest of graph (Appendix S2: Vignette V1,

section 5.1). We further define

as the

(

nl−1

)

(

nl−1

)

covariance

matrix of the basis f- statistics estimated by block- jackknife. Graph

fitting then consists of finding the graph parameter values (



and



)

that minimize a cost (score of the model) defined as:

where

𝚪

results from the Cholesky decomposition of

Q−1

(i.e.

Q−1=Γ

�

). Given admixture rates

S(e;a)

is quadratic in the edge

lengths

(Patterson et al., 2012) leading us to rely on the Lawson-

Hanson non- negative linear least squares algorithm implemented in

the R package nnls (Lawson & Hanson, 1995)) to estimate the vec-

tor



that minimizes

S(e;a)

(subject to the constraint of positive edge

lengths). Full minimization of

S(e;a)

is thus reduced to the identifica-

tion of the admixture rates

, which is performed using the L- BFGS- B

algorithm implemented in the optim function of the R package stats

(Nocedal & Wright, 1999).

2.1.5 | Confidence intervals and model

fit assessment

Using above notations and assuming



f∼N

(



e;

)

, t he op ti mi ze d

score

S(



)

verifies

S(



)

=−2log (L)−

where

is the likeli-

hood of the f itted graph a nd the constant

K=nlog (2𝜋)+log (|Q|)

. A s

detailed in Appendix S2: Vignette V1 (section 5.2), a

BIC

(Bayesian

Information Criterion) may then be derived from

S(



)

to compare

alternative admixture graph topologies at least when specified

with the same number of parameters (Lipson, 2020, section 3.3).

Also, the likelihood interpretation of the optimized score allows

constructing confidence intervals (CI) for the fitted parameters of

∗

3(A;B,C)=



i=1





1,i+

B,C

2,i−

A,B

2,i−

A,C

2,i





I−



i=1

1,i



and D(A,B;C,D)=



i=1





A,C

2,i+

B,D

2,i−

A,D

2,i−

B,C

2,i





i=1



1−

QA,B

2,i



1−

QC,D

2,i





j=1−1

∑

i=1



𝜎







nb−1



b=1





Fb−𝜇 F





ℂ

(



Fu;

)

=nb−1

∑

b=1(



b−𝜇 Fu

)(



b−𝜇 Fv

)

(6)

(e;a)=

(



f−g(e;a)

)�

Q−1

(



f−g(e;a)

)

(

Γ

f−ΓX(a)e

)�(

Γ

f−ΓX(a)e

)

GAUTIER E T Al.

a given graph (i.e. elements of the

and

vectors). Indeed, for a

given parameter

𝜈

(either an edge length or an admixture rate), the

difference

S𝜈

(x)−S

(



)

(where

S𝜈(x)

is the score when

𝜈=x

and

all the other parameters are set to their best fitted values) can be

interpreted as a likelihood- ratio test statistics following a

𝜒2

dis-

tribution with one degree of freedom. Lower and upper bounda-

ries

𝜈min

and

𝜈max

of the 95% CI (such that

S𝜈

(x)−S

(�

�

)

3.84

for

all

𝜈min <x<𝜈

max

) may then simply be computed using a bisection

method, as implemented in poolfstat.

Finally, a straightforward (but highly informative and recom-

mended) approach to assess the fit of an admixture graph is to

evaluate to which extent the f- statistics derived from the fitted

admixture graph parameters (g

(



)

) depart from the estimated

ones (Lipson, 2020; Pat terson et al., 2012). This can be summa-

rized via a Z- score of residuals computed as

=f−



𝜎2

where



is a

given fitted f- statistics; f is its corresponding estimated values; and

𝜎2

the block- jackknife standard error. The presence of outlying Z-

scores for at least one f- statistics (e.g.

|Z|>1.96

at a 95% signifi-

cance threshold) may suggest poor model fit while also providing

insights into the leaves or graph edges that are the most problem-

atic (Lipson, 2020).

2.1.6 | Scaling of branch lengths in drift units

Admixture graph fitting results in estimated edge lengths on the

same scale as

, which limits their interpretation, because they de-

pend both on the overall level of SNP polymorphism and on their

distance to the root (Patterson et al., 2012). Lipson et al. (2013) pro-

posed an empirical approach to rescale edge lengths on a diffusion

timescale using estimates of overall marker heterozygosities within

(i.e.

1−Q1

) or across (i.e.

1−Q2

) populations. The argument echoes

the aforementioned interpretation of pairwise

FST

as a scaled

. If

and

are the reference allele frequencies in a child node

and its

direct parent node

and their divergence time (on a diffusion time-

scale) is

𝜏

C,P=

(where

is the branch length in generations), then

conditional on

(C;P)=(1−e

−𝜏

C,P)p

1−p

and

QC,P

=1−2p

1−p

leading to

FC,P

ST =

(C;P)

1−QC,P

2(1−e−𝜏C,P

)

(i.e.

FC,P

ST ≃

2Ne

when

𝜏C,P≪1

). Hence, the estimated graph edges length



F2(C;P)=



eP↔C

are scaled in units of drift by a factor equal to



where



is the estimated heterozygosity (i.e.

−



) in the (parent)

node

. Rearranging equation 2 and using

QC,P

(conditional on

) shows that

hP=F2(C;P)+hC

, where

(

1−Q

is the hete-

rozygosit y of the child node

. Hence, all the node heteroz ygosities

can be inferred iteratively from the leaves to the root along the ad-

mixture graph using the leave heterozygosities (directly estimated

from the data) and the fit ted edge lengths (Lipson et al., 2013).

2.1.7 | Admixture graph construction

Comprehensive exploration of the space of possible admixture

graphs rapidly becomes impossible even for a moderate number

of populations. We implemented in poolfstat different heuris-

tics to facilitate admixture graph construction based on a super-

vised approach (see Appendix S2: Vignette V1 for details). First,

the ad d.lea f function allows exploring all the possible connec-

tions of a new population to an existing admixture graph. If

is the number of edges of the admixture graphs,

ne+1

possible

graphs connecting the new leaf with a non- admixed edges (i.e. in-

cluding a new rooting with the candidate leaf as an outgroup) and

(

ne−1

)

−

connecting the new leaf with a two- way admix-

ture event are then tested. Note that an admixture between the

two root edges is excluded from the exploration since it results

in a singular model. More generally, the different possible graphs

are always checked for singularity by empirically verifying that the

rank of the model incidence matrix

X(a)

is equal to the number of

edges to fit. The different fitted graph can then be ranked accord-

ing to their

BIC

, the graph with the lowest

BIC

having the strongest

support. The grap h.bu ild er function allows a larger explora-

tion of the graph space by successively adding several leaves in

a given order to an existing admixture graph. At each step of the

process, a heap stores the best resulting graph together with some

intermediary suboptimal graphs based on their

BIC

. After initial-

izing the heap with some graph (or a list of graphs), the a dd .le af

function is called to evaluate, for each candidate leaf in turn, all

its possible connections (with non- admixed or admixed edges) to

all the graphs stored in the heap. Among the obtained graphs, the

one with the lowest

BIC

together with those with a BIC within a

given

ΔBIC

(

ΔBIC =6

by default) are included in a newly generated

heap. If the resulting heap contains more than a predefined num-

ber of graphs

nmax

(

nmax

g=25

by default), only the

nmax

graphs with

the lowest BIC are finally kept in the heap of graphs to be used

for the addition of the next leaf. Although helpful, such heuristic

should be used cautiously and we recommend to only tr y adding

a small number of populations (i.e.

≤5

) to an existing graph. One

also needs to evaluate different orders of population inclusion

(Appendix S2: Vignette V1 and Appendix S3: Vignette V2). More

generally, exploring all possible leaf addition orders may become

impractical even with moderately complex admixture graphs (e.g.

>10 populations and/or more than 3 admixture events), and other

approaches are needed including promising ones in the currently

developed Admixtools 2.0 package (https://uqrma ie1.github.io/

admix tools/ artic les/graphs.html).

It is also critical to start these supervised procedures with graphs

that are representative of the whole history of the populations

under study and not too unbalanced with respect to the candidate

leaves. In particular, starting with a small tree of closely related pop-

ulations, which are distantly related to the candidate leaves must

be avoided. When prior knowledge about the history of the inves-

tigated population is limited (which is usually the case), Lipson et al.

(2013) proposed to start admixture graph construction with a scaf-

fold tree of populations displaying no evidence of admix ture. As de-

tailed in Appendix S2: Vignette V1 (section 6.1), we implemented in

poolfstat two functions that allow (i) identifying candidate sets of

unadmixed populations among all the genotyped ones (find.tree.

G AUTIER ET Al.

popset); and (ii) building rooted neighbour- joining tree (rooted.

njtree.builder).

2.2 | Overview of the new poolfstat package

Tables 1 and 2 describe the main objects and functions imple-

mented in our new version (v2.0.0) of the R package poolfstat

publicly available from the CRAN repository (https://cran.r- proje

ct.org/web/packa ges/poolf stat/index.html). In- depth analyses

of two Pool- Seq and allele count simulated data sets (see below)

are described for illustration purposes in the package vignette

provided as Appendix S2: Vignette V1. Detailed documentation

page of the dif ferent objects and functions can also be directly ac-

cessed from an R terminal with poolfstat loaded using the help

function (or the

operator).

The package includes several functions to parse allele count

(e.g. genotreemix2countdata) or Pool- Seq (e.g. vcf2p ooldata)

input data stored in various formats commonly used in population

genomics studies (Table 2). These functions allow to clearly distin-

guish these two different types of data by producing objects of ei-

ther the so- called countdata (for allele count) or pooldata (for

Pool- Seq data) classes (Table 1). This step is critical to further rely

on the appropriate unbiased estimators for the F and D parameters.

Some functions allow to perform subsequent manipulation of the

input data, for instance to only consider some of the populations or

to remove SNPs according to various criteria ( Table 2).

The three functions c omputeFST, compute.pairwiseFST and

compute.fstats implement the unbiased estimators of the differ-

ent f- , D- and within- population heterozygosities (based on allele

IIS probabilities within and between pairs of populations) together

with block- jackknife estimation of their s tandard errors. Import antly,

these three functions automatically detect the appropriate estima-

tors given the type of data (either allele or Pool- Seq read counts)

according to the input object class (either countdata or pool-

data). For the estimation of FST, the computeFST and com pute.

pairwiseFST also implement (by default) estimators based on an

Analysis of Variance framework that correspond to those developed

by Weir (1996) for allele count data and by Hivert et al. (2018) for

Pool- Seq dat a.

The fit.g ra ph function implements the approach described

above to estimate the parameters (i.e. edge lengths and admixture

rates) of an admixture graph that is stored in a graph.params

object ( Table 1). Such objects can be generated with the gener-

ate.graph.params function (Table 2) to include the target basis

f- statistics and the error covariance matrix (denoted above



and

, respectively) estimated with compute.fstats (stored in an

fstats object) and to specify the topology and the parameters of

the admixture graph. Note that the graph.params2symbolic.

fstats function allows exploring in details the properties of an ad-

mixture graph specified by a graph.params object by deriving a

symbolic representation of all the

and the model equations

(see above) using functionalities of the Ryacas package for symbolic

computation (Andersen & Højsgaard, 2019).

The fit.graph function then produced an object of class fit-

ted.graph that includes the estimated edge lengths (in

and also

optionally in drift units) and admixture proportions together with

(optionally) their 95% CI. For model fit assessment purposes, fit-

ted.graph objects also include the

BIC

and Z- score of the residuals

of the fitted basis f- statistics. Such a comparison can (and should) be

S4 object Description

countdata Standard allele count data (i.e. obtained from individual genotyping or

sequencing data)

poold ataaPool- Seq read count data

pairwiseFST Store pairwise

FST

estimates. This obje ct is generated by the com put e.

pairwiseFST function. Estimates can be conveniently visualized

with the heatmap or plot functions, the latter interfacing the

plot _ fstats function of poolfstat

fstats Store

, pairwise

FST

F∗

and

estimation result s. This object is

generated by the compute.fstats function. Estimates can be

conveniently visualized with the heat map or plot functions, the

latter interfacing the plot _ fstats function of poolfstat

graph.params Represent a population tree or an admixture graph and its parameter.

This object is generated by the generate.graph.params

function. The graph can be visualized with the plot function that

interf aces the grV iz function from the DiagrammeR package

(Iannone, 2020)

fit te d.g ra ph Represent a population tree or an admixture graph and it s underlying

fitted parameters as obtained from the fit.graph or other fitting

functions. The graph can be visualized with the plot function

that inter faces the grViz function from the DiagrammeR package

(Iannone, 2020)

aObject already existing in the first poolfstat version.

TABLE 1 Description of the main S4

object s of the poolfstat package

GAUTIER E T Al.

TABLE 2 Description of the main poolfstat functions

Function Typ e Detail

genotreemix2countdata data import Generate a coundata object (Table 1) from allele count data stored in TreeMix format

(Pickrell & Pritchard, 2012)

genobaypass2countdata data import Generate a coundata object (Table 1) from allele count data stored in BayPass format

(Gautier, 2015)

vcf2p ooldat aadata import Generate a poold ata object (Table 1) from Pool- Seq read count data stored in a vc f

file generated by commonly used calling software as VarScan (Koboldt et al., 2012),

bcftools (Li et al., 2009), GATK (McKenna et al., 2010) or FreeBayes (Garrison &

Marth, 2012). Parsing of vcf files uses C++ routines inspired by the vcfR package

(Knaus and Grünwald, 2017)

popsync2pooldatabdata import G enerate a pooldata object ( Table 1) from Pool- Seq read count data stored in the sy nc

Popoolation format (Kofler et al., 2011)

genobaypass2pooldatabdata import Generate a poold ata objec t (Table 1) from Pool- S eq read count data stored in BayPass

format (Gautier, 2015)

countdata.subset data handling Subset s a coundata object (Table 1) according to various criteria (e.g. population or SNP

indexes, marker polymorphism, call rate)

po old ata.su bsetbdata handling Subset a coundata object (Table 1) according to various criteria (e.g. population or SNP

indexes, marker polymorphism, pool coverage)

pooldata2genobaypassbdata handling Export Pool- Seq read count data stored in a pooldata object in BayPass format (Gautier,

2015)

computeFSTaestimation Estimate genome- wide

FST

over all the populations

compute.pairwiseFSTaestimation Estimate pairwise- population population

FST

. The func tion gener ates a pairwiseFST

object (Table 1)

compute.fstats estimation Estimate

f−

(

, pairwise

FST

F∗

and

- statistics and within- population

heterozygosities together with their standard errors via block- jackknife. The function

generates an fstats object (Table 1)

com pute.f4ratio estimation Estimate admixture r ates and their standard errors using ratios of f4- statistics from an

fstats object (Table 1)

generate.graph.params graph fitting Generate a graph.param s object synthesizing the structure and properties of an

admixture graph ( Table 1)

fit.g raph graph fitting Estimate parameters (edge lengths and admixture rates) of an admixture graph. The

function generates a fitt ed.gr ap h object (Table 1)

co m par e.fi tt ed.fst at s graph fitting Compare all the fitted

and

derived from a fitted graph and stored in a fitt ed.

graph objec t (Table 1) to the estimated ones (s tored in a fstats object). This function

allows graph fitting assessment

find.tree.popset graph building Find set s of populations that may use as scaffold tree based on the estimated

and

stored in a fstats object (Table 1)

rooted.njtree.builder graph building Construct and root a Neighbour- Joining tree of presumably unadmixed scaf fold

populations. This function generates a fitted.gr ap h object (Table 1)

ad d.lea f graph building Evaluate all possible connection of a new leaf (i.e. genotyped population) to an existing

graph (stored in either a graph.params or fitte d.gr ap h object) with both non-

admixed and admixed edges. This functions generates a list of f itt ed.gr ap h objects

including other information (e.g.

BIC

of all the possible graph, index of the best fitted

graph)

graph.b uilde r graph building Implement a graph builder heuristic by successively adding leaves (i.e. genotyped

populations) to an existing graph or tree (stored in either a graph.params or fitted.

graph objec t) by successively calling the ad d.lea f functions and keeping sub- optimal

graphs in a heap at each step. This func tions gener ates a list of f itt ed.g raph objec ts

including other information (e.g.

BIC

of all the possible graph, index of the best fitted

graph)

plot _ fstats visualization Plot

- statistics with their Confidence Intervals (can be called directly using plot)

graph.params2symbolic.

fstats

Other utilities Derive a symbolic representation of

and

(and the graph model system of equation)

as a function of admixture graph parameters specified in a graph.params object

(Table 1) using functionalities of the Ryacas package (Andersen & Højsgaard , 2019)

Note: a,bObjects existing or significantly improved since the first poolfstat version, respectively.

G AUTIER ET Al.

generalized to all the

and

statistics (not just the ones forming the

basis) using compare.fitted.fstats jointly applied to a fitted.

graph and a fstats objects. Notice that we developed for comparison

purposes a function named graph.params2qpGraphFiles to ex-

port admixture graph specification and their underlying estimated basis

f- statistics (both stored in a graph.params object) into qpGraph for-

mat (Patterson et al., 2012), allowing independent fitting based on the

same estimated statistics to be carried out with this later program.

The poolfstat package includes several functions to assist

construction of admixture graphs. As mentioned in the previous

section, the find.tree.popset and rooted.njtree.builder

functions allow to identify and build rooted tree(s) of scaffold of

(presumably) unadmixed populations that may be used as starting

graph(s). Besides, the ad d.le af and g raph.bu ilder functions

implement the above described heuristic to extent an existing graph

(or tree) by adding one or several leaves (i.e. genot yped populations).

These functions generate a list of fitt ed.gr ap h objects together

with other information that may be helpful for graph comparison

(e.g.

BIC

of all the graphs or index of the best fitted graph).

Finally, as detailed and exemplified in the Appendix S2: Vignette

V1, fitted graphs (stored in fit te d.g ra ph object s) and non- fitted

graphs (stored in graph.params objects) can be directly and con-

veniently plotted with the plot function, which internally interfaces

the grViz functio n from the DiagrammeR package (Iannone, 2020).

2.3 | Data analyses

2.3.1 | Simulation study

Genetic data for a total of 150 diploid individuals belonging to six

different populations (n = 25 individuals per populations) related by

the demographic scenario depicted in Figure 1 were simulated using

the msprime coalescent simulator (Kelleher et al., 2016) with the

following command:

mspm s 300 20 -t 4000 -I 6 50 50 50 50 50 50 0 -r

4000 100000000 -p 8 -es 0.0125 6 0.25 -ej 0.0125 6 2

-ej 0.0125 7 3 -ej 0.025 2 1 -ej 0.05 3 1 -ej 0.075

5 4 -ej 0.1 4 1

Each genome thus consisted of 20 independent chromosomes

L=100

Mb assuming a scaled chromosome- wide recombination

rate of

𝜌=4LNer=4, 000

as expected for instance in a population

of constant diploid effective size of

Ne=103

when the per- base

and per- generation recombination rate is

r=10

−

(i.e. one cM

per Mb). The scaled chromosome- wide mutation rate was set to

𝜃=4LNe𝜇=4, 000

which is also the expected nucleotide diversity

in a population with

Ne=103

at mutation– drift equilibrium when the

per- base mutation rate is

𝜇=10

−

. A total of 250 independent ge-

notyping data sets were simulated and each was subsequently pro-

cessed to generate 32 different types of data sets corresponding to:

• Two standard allele count data sets (namely ACm>1% and ACm>5%)

obtained by simple counting of the simulated individual (haploid)

genotypes for each population (i.e. assuming Hardy– Weinberg

equilibrium within population) and removing SNPs with a Minor

Allele Frequency (MAF) computed over all the individuals lower

than 1% (for ACm>1% data sets) or 5% (for ACm>5% data sets)

• Thirty Pool- Seq dat a sets (coded as

PS𝜆𝜀=𝜀

m>mt%

) for (i) five differ-

ent average sequencing coverages

𝜆

(equal to 30, 50, 75, 100 or

200 reads; a 30X Pool- Seq coverage representing a lower limit for

population genomics studies); (ii) two different MAF thresholds

of 1% and 5% (MAF being estimated on the read counts over all

FIGURE 1 Simulated scenario relating six sampled populations. The population

derived from a population

, which is admixed between

two ancestral sources (

and

) directly related to populations

and

and contributing to

𝛼=25%

and

1−𝛼=75%

of its genome,

respectively. The branch lengths are in a diffusion timescale, that is with

𝜏

≃

2Ne

under a pure- drift model of divergence (where

is the

number of non- overlapping generations and

the average diploid effective population sizes along the branch). The names of the internal

node populations (not sampled) are represented in grey

0.00

0.05

0.10

0.15

0.20

S1 S2

α=25% 1−α=75%

P1 P2 P6 P3 P4 P5

GAUTIER E T Al.

the pools); and (iii) three different sequencing error rates

𝜀

of 0 (no

error), 1 and 2.5 the two latter being representative of Illumina

sequencers (Glenn, 2011).

Pool- Seq data sets were simulated from the ACm≥1% allele count

data sets following a procedure similar to that described in Hivert

et al. (2018). Briefly, the vector

rij

{

ijk}

of read count s at SNP po-

sition

in population

for the nucleotide

(where by convention

k=1

and

k=2

for the derived and ancestral alleles respectively) was

sampled from a Multinomial distribution parameterized as:

where

yij

is the derived allele count for SNP i in population j (from the

corresponding ACm>1% data set);

is the haploid sample size of popula-

tion j (here

nj=50

for all j); and

cij

∑k=4

k=1

ijk

is the overall read cover-

age. To introduce variation in read coverages across pools and SNPs,

each

cij

was sampled from a Poisson distribution with a parameter

𝜆

(the

target Pool- Seq mean coverage). When

𝜀=0

, only reads for the derived

(

k=1

) or ancestral (

k=2

) alleles can be generated and the above

Multinomial sampling actually reduces to a Binomial sampling following

ij1∼Bin

;cij

)

(and

rij2=cij −rij1

). However, when

𝜀>0

, sequencing

errors might lead to non- null read counts for the two other alleles lead-

ing to tri- or tetra- allelic SNPs. Moreover, sequencing errors may also

introduce spurious additional variation by generating false SNPs at

monomorphic sites. To account for the latter, read count vectors

′

for

all the

2×109−I

monomorphic positions

i′

(where

is the number of

SNPs observed in the considered ACm>1% data set) were sampled as

i�j∼Multin

({

1−𝜀

;𝜀

;ci�j

)

with coverages

′

sampled from a

Poisson distribution (as

cij

for polymorphic positions). Yet, as usually

done with empirical data sets, we applied a minimum read count filtering

step consisting of disregarding all the alleles with less than 2 observed

reads (over all the populations). Only bi- allelic SNPs passing the overall

MAF threshold

were finally retained in the final PS

𝜆𝜀=𝜀

m≥mt%

data sets.

Analyses of the simulated data were carried out with poolf-

stat (Appendix S2: Vignette V1). Briefly, each msms simulated data

set was conver ted into an ACm>1% data set in TreeMix format

(Pickrell & Pritchard, 2012) further imported into a countdata ob-

ject with genotreemix2countdata (Tables 1 and 2) and used to

generate each corresponding ACm>5% data set using countdata.

subset. To improve comput ational effi ciency, the differ ent PS

𝜆𝜀=𝜀

m>mt%

Pool- Seq data sets were generated from the ACm>1% countdata

object s in the form of p ooldat a object using custom functions (not

included in the package) coded in C++ and integrated within R using

Rcpp (Eddelbuettel, 2013). In ad dition, to evaluate the impac t of the

(bad) practice consisting of analysing Pool- Seq data as if they were

allele count data (i.e. overlooking the sampling of reads from individ-

ual genes of the pool), we also created ‘fake’ countdata objects

from the different pooldata object s. We then used default options

(unless otherwise stated) of (i) co mputeFST to estimate genome-

wide

FST

over all the populations; (ii) compute.fstats to estimate

all the f- and D- statistics; (iii) co m pute.f4r at io to estimate

admixture proportions; and iv) f it.g ra ph to estimate the admix-

ture graph parameters (Table 2). As the number of SNPs was variable

across the different simulated data sets, we adjusted the number of

successive SNPs defining a block for block- jackknife estimation of

standard errors by dividing the total number of available SNPs by

500. This thus resulted on average in 490 blocks of 4.1 Mb over the

genome for all the analysed data sets (the simulated genomes con-

sisting of 20 chromosomes). Note that the parameter estimates were

always taken as the block- jackknife mean values rather than esti-

mates over all SNPs (i.e. including those in the chromosome ends). In

practice, the differences between the two are insignificant (e.g.

Appendix S2: Vignette V1).

For validation purposes, we also analysed the 250 ACm>1% data

sets with programs from the Ad mixTools suite (Patterson et al.,

2012) after conversion to the appropriate input format using custom

awk script s. More specifically, we ran q pfstat s (v. 200) to estimate

the 15 basis f- statistics, that is taking

as the reference population,

the five

of the form (P1, Px) and the ten

of the form (P1; Px, Py)

(where

x=2, …,6

and

y=3, …,6

with

y≠x

) and their corresponding

error covariance matrix. Default options were considered except for

the disabling of the scaling of estimated values (using option - l 1) to

facilitate their comparison with poolfstat estimates. We also ran

with default options q p3Po p (v. 65 0) to est ima te

f∗

for all th e 60 pos-

sible triplet configurations and qpDstat (v. 970) to estimate the D-

statistics for all the 45 possible quadruplet configurations together

with their associated Z- scores. By default, these three programs

define blocks of 5 cM to implement the (weighted) block- jackknife

procedure. As we here converted the simulated SNP positions from

Mb to cM assuming one cM per Mb (see above), the sizes of the 400

blocks were thus about 20% larger than for poolfstat analyses.

2.3.2 | Analysis of a real Drosophila suzukii Pool-

Seq data

The spotted wing drosophila, D. suzukii, represents an attractive model

to study biological invasion and hence recent historical and demographic

history. Native to South East Asia, this pest species was first observed

outside its native range in Hawaii in 1980, and later rapidly invaded

America and Europe simultaneously between 2008 and 2013 (Fraimout

et al., 2017). Using DNA sequences and microsatellite markers, Adrion

et al. (2014) and Fraimout et al. (2017) deciphered the routes taken by D.

suzukii during its worldwide invasion. Both studies showed that America

and European populations globally represent separate invasion routes

with different native source populations. Olazcuaga et al. (2020) recently

generated Pool- Seq genomic data from 22 worldwide population sam-

ples to detect genetic variants associated with the historical status (i.e.

invasive versus native) of the sampled populations. We here focused our

illustration on 14 Pool- Seq data from this study (with 50– 100 diploid

ij ∼Multin

y

j

1−𝜀





1−

j

𝜀



1−

j

1−𝜀



𝜀

3;𝜀



;cij



G AUTIER ET Al.

individuals per pool) for populations representative of the Asian native

area (six populations), Hawaii (one population) and the invaded continen-

tal America (seven populations), where the species was first observed

in 2008 on the Western coast of the USA (around Watsonville, CA;

Figure 2a). Besides native populations, we have restricted our analysis

to the American continent because the invasion of this area is character-

ized by multiple admixture events between different source populations

(Fraimout et al., 2017), which makes it an appealing situation to evalu-

ate the power and the limitation of poolfstat analyses. Moreover, 13

of our 14 population samples consist of individuals originating from the

same sites (albeit sometimes collected at different dates for some pools;

Table 2 in Appendix S3: vignette V2) as those genotyped at 25 microsat-

ellite markers and analysed with an Approximate Bayesian Computation

Random Forest (ABC- RF) approach to infer the routes of invasion on a

worldwide scale by Fraimout et al. (2017).

To allow for complete reproduction (and exploration) of our anal-

yses, all the command lines used to analyse the D. suzukii Pool- Seq

data set are described in the Appendix S3: Vignette V2. Briefly, we

combined the 14 (bam) files, obtained by Olazcuaga et al. (2020) after

aligning the 14 Pool- Seq data onto the latest D. suzukii assembly (Paris

et al., 2020), into an mpileup file using SAMtools 1.9 with options

- q 20 - Q20 (Li et al., 2009). Variant calling was then performed using

VarScan mpileup2snp v2.3.4 Koboldt et al. (2012) run with options

- - min- coverage 10 - - min- avg- qual 25 - - min- var- freq 0.005

- - p- value 0.5 (i.e. with very loose criteria). After discarding positions

mapping to non- autosomal contigs, the resulting vcf file was parsed

with the vcf2pooldata function of poolfstat with default options

except for i) the overall MAF threshold (computed from read counts)

that was set to 5%; and ii) the minimal read coverage for each pool that

was set to 50. The resulting pooldata object was further filtered with

pooldata.subset to discard (i) all positions with a coverage higher

than the 99th coverage percentile within at least one pool; and (ii) dis-

card all SNPs with MAF < 5% over all the populations from the na-

tive area to favour ancestral SNPs. The final data set then consisted of

read counts for 1,588,569 bi- allelic SNPs with a median read coverage

varying from 64 (US- Sok) to 95 (CN- Bei and US- Haw) among the 14

pools (Table 2 in the Appendix S3: Vignette V2). We defined blocks of

10,000 consecutive SNPs for block- jackknife estimation of standard er-

rors leading to a total of 145 blocks of 698 kb on average (varying from

414 kb to 2.03 Mb). Hence, most analyses actually relied on 1,450,000

SNPs that mapped to the 15 largest contigs of the assembly (totalling

116 Mb). In other words, SNPs mapping to the smallest (and less reli-

able) contigs were discarded in addition to the few ones mapping to the

end of the 15 retained contigs.

3 | RESULTS

3.1 | Evaluation of poolfstat on simulated data

Histori cal and demogra phic inference bas ed on f- and D- statistics has

already been extensively evaluated in previous studies (Lipson et al.,

2013; Patterson et al., 2012; Peter, 2016). Therefore, the purpose

of our simulation study was essentially threefold: (i) to validate the

estimators implemented in poolfstat by comparing, for allele

count data, with those obtained with the reference Adm ixTools

suite (Patterson et al., 2012); (ii) to evaluate the performance of the

estimators for Pool- Seq data as a function of read coverage and se-

quencing errors; and (iii) to provide example data sets with known

ground truth for illustration purposes.

3.1.1 | Description of the simulated data sets

We simulated 250 genetic data sets for six populations (named

) each consisting of 25 diploid individuals and that were histori-

cally related by the admixture graph represented in Figure 1 (see

2.3.1). Each of these data sets was further used as template to gen-

erate two allele count data sets (applying 1% or 5% threshold on the

overall MAF for ACm>1% and ACm>5% data sets, respectively) and to

simulate 30 Pool- Seq data sets with five different mean read cover-

ages (

𝜆∈{30;50;75;100;200}

); three sequencing error rates

(

𝜀

∈

{

0;10−3;2.5 ×10−3

}

) and two MAF (computed over all read

counts) thresholds (referred to as PS

𝜆𝜀=𝜀

m>1%

and PS

𝜆𝜀=𝜀

m>5%

for 1% and

5% MAF thresholds, respectively). This thus leads to a total of

8,00 0 simulated data sets. The average number of available SNPs

and false SNPs (for PS

𝜆𝜀=1

m≥mt%

and PS

𝜆𝜀=2.5

m>mt%

data sets) is given in

Appendix S1: Table S1 for each of the 32 different types of data sets

and represented as a func tion of the mean coverages

𝜆

and MAF

thresholds in Appendix S1: Figure S1.

Overall, 471,919 SNPs and 240,369 SNPs were available on aver-

ag e for al le le co unt dat a se ts at the 1% (ACm>1% ) an d 5% (A Cm>5%) MAF

thresholds, respectively, consistent with the L- shaped distribution

of allele frequencies (Appendix S1: Figure S2A). As expected from

binomial sampling (Appendix S1: Figure S2B), for Pool- Seq data sets

generated with no sequencing error, the number of SNPs remained

always lower than the ACm>1% data sets at the 1% MAF threshold

although increasing with coverages from 13.8% for PS

30𝜀

m>1%

2.01% for PS

200m>1%𝜀=0

data sets (see Appendix S1: Table S1 leg-

end for details). Conversely, at the 5% MAF threshold, the number of

SNPs was slightly higher than the ACm>5% data sets (from 2.58% for

30𝜀

m>5%

to 1.51% for PS

200𝜀

m>5%

), which is related to i) the shape of

the allele frequency spectrum (stochastic variation in read sampling

leading to include more SNPs with

0.01 <MAF <0.05

than exclude

SNPs with

MAF >0.05

from the simulated genotyping data because

they are more numerous); and (ii) variation in the simulated read cov-

erages that explains the decreasing trend with

𝜆

With sequencing errors, our filtering steps proved efficient to

remove false SNPs except at the 1% MAF threshold when

𝜀=2.5

when

𝜀=1

at the lowest coverage (

𝜆=30

and

𝜆=50

). These con-

figurations displayed substantial to very high proportions of false

SNPs (up to 93.8% for PS

50𝜀

2.5

m>1%

) although decreasing with coverage

(Appendix S1: Figure S1B). A 5% MAF threshold always resulted in

the complete removal of all the false SNPs for all the investigated

scenarios (Appendix S1: Table S1). Note that for the highest cover-

ages, sequencing errors lead to a relative reduction of the number

GAUTIER E T Al.

FIGURE 2 Historical and demographic inferences about native and invasive Drosophila suzukii populations from Pool- Seq data based

f−

statistics. (a) Geographic location of the 14 population samples (Olazcuaga et al., 2020). Names are coloured according to their

area of origin. The (invasive) Hawaiian population, which was considered as intermediate between the Asian native and the continental

America invasive area, was first observed in 1980, that is ca. 300 generations before the invasion of the American continent assuming

10 generations per year. Solid points indicate the 13 population sampling sites in common with Fraimout et al. (2017). (b) Best fitting

admixture graph connecting five populations of the native areas and the Hawaiian population with two inferred admixture events. (c)

Best fitting admixture graph connecting three invasive populations from continental America with populations from the native area (and

Hawaii). In (b) and (c), estimates of branch lengths (×103, in drif t units of

2Ne

) and admix ture rates (and their 95% CI into bracket) are indicated

next to the corresponding edges. The worst fitted

- statistics is written in red for each of the two graphs

(a)

CN−Bei

CN−Lia

CN−Nin

CN−Shi

JP−Sap

JP−Tok

BR−Pal

US−Col

US−Haw

US−Nca

US−Sdi

US−Sok

US−Wat

US−Wis

Native

Hawaiian (>1980)

American (>2008)

(b) (c)

G AUTIER ET Al.

of SNPs due to the generation of spurious tri- or tetra- allelic SNPs

from the simulated bi- allelic SNPs (compare, e.g., PS

200𝜀

2.5

m>5%

and

100𝜀

2.5

m>5%

on Appendix S1: Figure S1A).

3.1.2 | Comparison of poolfstat and

Admixtools estimates for allele count data

We first analysed the 250 simulated ACm>1% data sets to estimate with

both p oolfstat and Admixtools programs (i ) the 15 basis f- statistics

(taking

as the reference population) consisting of five

and ten

(Figure 3a) and their corresponding error covariance matrix (Figure 3b);

(ii) the 60

f∗

(Fig ure 3c) and their associated Z- s cores (Figure 3d); and iii)

the 45 D- statistics (Figure 3e) and their associated Z- scores (Figure 3f).

The estimates were all found in almost perfect agreement between

the two implementations with Mean Absolute Differences (MAD)

negligible when compared to the range of variation of the underlying

values. For f- and D- statistics, slight differences were mostly due to

the plotted poolfstat estimates corresponded to block- jackknife

means (i.e. excluding SNPs outside blocks as those from chromosome

ends). Using poolfstat estimates based on all the SNPs indeed re-

sulted in almost null MAD (MAD’ in Figure 3a, c and e), up to round-

ing errors due to lower decimal precision in the printed output of the

Admixtools programs. Note that the differences in block- jackknife

implementation among the two programs (Material and Methods) had

very minor impact on the estimation of error variance and covariance

of the estimates (Figure 3b). Accordingly, the MAD computed on Z-

scores remained very small (although inflated for higher values) and

Z- score based decision for the underlying three- population admixture

(Figure 3d) or four- population treeness test s (Figure 3f) were highly

consistent (with a propor tion

𝛽=97.7%

and

𝛽=98.0%

, respectively, of

Z- scores significant with the two programs among the ones significant

with at least one program).

3.1.3 | Performance of

- and

f∗

- based tests of

admixture and

and D- based tests of treeness for

allele count and Pool- Seq data

We ran the compute.fstats function on all the simulated allele

count and Pool- Seq data sets to estimate all f- and D- statistics. To

further evaluate the impact of (improperly) treating read counts as

allele counts when analysing Pool- Seq data, we also analysed the

simulated Pool- Seq data sets (focusing only on PS

𝜆𝜀=0

m>mt%

data sets,

i.e. simulated without sequencing error) as if they were allele count

data. Overall, 42 different configurations were thus investigated

each originating from the 250 allele count data sets simulated under

the demographic scenario represented in Figure 1, leading to a total

of 42×250 = 10,500 analyses.

Table 3 and Appendix S1: Table S2 provide the estimated power

(true positive rate, TPR) and false positive rate (FPR) of the

- and

f∗

- based test of admixture for each configurations. As P6 was the only

admixed population, each TPR was estimated as the proportion of

(

f∗

, respectively) with an associated Z- score <- 1.65 (95% significance

threshold) for the (P6;P2,P3) population triplet (i.e. among 250 es-

timates). Conversely, the FPR was computed as the proportion of

(

f∗

, respectively) with an associated Z- score <−1.65 among all the

50 population triplets that do not involve P6 as a target (i.e. among

12,250=250 × 50 estimates). Consistent with Pat terson et al. (2012),

the performance of

- and

f∗

- based test of admixture was vir tually

the same for all the configurations. When the same MAF threshold

was appli ed, the perfor mance of Pool- Seq dat a generated with no s e-

quencing error was very close to that obtained with allele count data

although the power tended to slightly decrease with decreasing se-

quencing coverage. Interestingly, increasing the MAF threshold from

1% to 5% increased the power by more than 10% and in all cases,

no false positive signal of admixture was detected. Surprisingly, se-

quencing errors in Pool- Seq data also tended to increase the power

from a negligible amount (less or close to 1%) at 5% MAF threshold

to a quite substantial amount at 1% MAF threshold (decreasing with

coverage and increasing with sequencing error rate). At the extreme,

a power of 100% was even observed when

𝜆≤50

and

𝜀≥1

. This

trend was actually directly related to the propor tion of false SNPs

introduced by sequencing error (Appendix S1: Figure S1B) that re-

sulted in a downward bias of

and

f∗

estimates, although the un-

derlying tests remained robust as all the estimated FPR were null

except for PS

50𝜀

2.5

m>1%

data (FPR = 6.47%), which displayed the highest

propor tion of false SNPs (>90%, Appendix S1: Figure S1B). However,

this observed apparent robustness of the three- population tests to

false SNPs should be interpreted cautiously since it may rather result

from the moderate to high expected

and

f∗

values in our simulated

scenario for the population triplets that do not involve P6 as a target.

Overall, applying a 5% MAF threshold on Pool- Seq data (even with

𝜀

= 2.5) to remove false SNPs (see above) allowed recovering the

performances similar to that obtained when analysing data sets with

no sequencing error. Finally, it is worth stressing that analysing Pool-

Seq data as allele counts, whatever the coverage or MAF threshold

considered, lead to no power in detecting admixture event with

- or

f∗

- based tests due to a strong upward estimation bias.

Table 4 and Appendix S1: Table S3 similarly provide the es-

timated power (TPR) and FPR of the

- and D- based tests of tree-

ness for the 42 configurations investigated in the simulation study.

Given the simulated scenario, eight of the 45 different population

quadruplets (namely (P1,P2;P3,P4); (P1,P2;P3,P5); (P1,P2;P4,P5);

(P1,P3;P4,P5); (P1,P6;P4,P5); (P2,P3;P4,P5); (P2,P6;P4,P5); and

(P3,P6;P4,P5)) have a null expected

(and D) value. Note that this

may easily be shown with the symbolic calculus derivation imple-

mented in graph.params2symbolic.fstats (Table 2). For each

configuration, the TPR of the treeness test was then estimated as the

proportion of

(D, respectively) with an associated absolute Z- score

<1.96 (95% significance threshold) for these eight population qua-

druplets ((P1,P2;P3,P4); (P1,P2;P3,P5); (P1,P2;P4,P5); (P1,P3;P4,P5);

(P1,P6;P4,P5); (P2,P3;P4,P5); (P2,P6;P4,P5); (P3,P6;P4,P5)) over all

the 250 different underlying analyses (i.e. among 2,000=250 × 8 es-

timates). Conversely, the FPR was estimated as the proportion of

(respectively D) with an associated absolute Z- score <1.96 among all

GAUTIER E T Al.

the 37 remaining population quadruplets (i.e. among 9250 = 250 × 37

estimates). The power for both F4- and D- based tests were remarkably

consistent across all the different configurations. In addition, the tests

were all found almost perfectly calibrated since the estimated power

were close to 95%, the probability of rejecting the null hypothesis at

the chosen 95% significance threshold for Z- scores. Likewise, all FPR

remained low (

≤0.15%

), although increasing with MAF thresholds

(more than twice higher for a given type of data when increasing the

MAF threshold from 1% to 5%). Overall, sequencing errors and cover-

age had no impact on the performance of the

- and

- based test of

FIGURE 3 Comparison of poolfstat and Admixtools estimates across 250 simulated allele count data sets (ACm≥1% ). (a) All estimates

of the 15 basis f- statistics taking

as the reference population and corresponding to 5

of the form (

) and the 10

of the form (

;

) (with

x=2,..,6

;

y=3,..,6

and

y>x

). (b) All Block- jackknife estimates of the covariance matrix

of the 15 basis f- statistics (15 error

variances and 105 error covariances). (c) All estimates of the 60

f∗

(scaled

) and their associated Z- scores (d). (e) All estimates of the 45

D- statistics (scaled

) and their associated Z- scores (f). For each comparison, the mean absolute difference (MAD) between the parameter

estimates of the two programs are given on the upper left corner of the plots. In (a), (c) and (e), poolfstat estimates correspond to block-

jackknife means (i.e. they only include SNPs eligible for block- jackknife). The given MAD’ value is the MAD between Admixtools and

poolfstat estimates that include all SNPs (see documentation for the compute.fstats function). In (d), a consistency score

𝛽

is also given

and was computed as the proportion of Z- scores <−1.65 (i.e. significant three- population test of admixture at a 5% threshold) with both

programs among the

n=216

ones significant in at least one of the two programs. Similarly, in f), the given consistency score

𝛽

is computed

as the proportion of absolute Z- scores <1.96 (i.e. passing the four- population treeness at a 5% threshold) with both programs among the

n = 1912 ones with an absolute Z- scores <1.96 in at least one of the t wo programs)

G AUTIER ET Al.

treeness. As expected, analysing read counts as allele count data did

not affect the performance of these tests (see Discussion).

3.1.4 | Precision of the

- ratio- based

estimation of the admixture rate α

Given the simulated scenario, two different ratios of

estimates could

be used to estimate the admixture proportion

𝛼=0.25

(Figure 1),

namely

𝛼

(P1,P4;P3,P6)

f4(P1,P4;P2,P3)

and

𝛼

(P1,P5;P3,P6)

f4(P1,P5;P2,P3)

(Patterson et al., 2012).

The graph.params2symbolic.fstats function (Table 2) may also

prove useful to identify appropriate quadruplets (Appendix S3:

Vignette V2). We used the c omp ute.f4 ra ti o function to obtain

these t wo estimates from all the simulated data sets together with their

95% CI (defined as

𝛼 ±1.96𝜎 𝛼

where

𝜎 𝛼

is the block- jackknife standard-

error estimate). Table 5 and Appendix S1: Table S4 provide the mean of

the estimated

𝛼 1

and

𝛼 2

, respectively, over the 250 analysed data sets

for each of the 42 investigated configurations. As expected from the

above evaluation of treeness tests, estimates of

𝛼

were highly consist-

ent among all the investigated configurations and similar for the two

considered

- ratio with a mean value varying bet ween 0.245 and

0.248. Yet, a slight downward bias (always

<2%

) could be noticed, but

the estimated 95% CIs were almost always optimal (or close to) since

they cont ained the true sim ulated value (

𝛼=0.25

) from 90.0 % to 95.2%

of the time (Table 5 and Appendix S1: Table S4).

3.1.5 | Evaluation of graph fitting

We further estimated for all the simulated data sets branch lengths

in drift units and admixture proportion

𝛼

with their 95% CIs by fit-

ting the simulated graph with fit.grap h. As for the

- ratio- based

estimation, estimates of

𝛼

were virtually unbiased and consistent

across all the 42 different investigated configurations (Appendix

S1: Figure S3). Nevertheless, the 95% CIs were always too narrow

since they contained the actual value (

𝛼=0.25

) from only 4 0.8%

to 74.4% of the time (Appendix S1: Table S5) as expected from the

𝜒2

approximation of the LRT underlying the computation of these

CIs. Figure 4 and Appendix S1: Table S4 plot the distributions of the

estimated lengths for the ten branches of the simulated admixture

graph branches (over the 250 estimates per configuration) when ap-

plying 5% and 1% MAF threshold, respectively. The corresponding

mean estimates and proportions of 95% CI's including the true value

are provided in Appendix S1: Tables S6– S15. Note that the branches

P8↔R

and

P9↔R

that are connected to the root

(Figure 1) can

only be estimated jointly (as

𝜏P8↔P9=𝜏P8↔R+𝜏P9↔R

being arbitrar-

ily set in its middle).

At the 5% MAF threshold, very similar performance was ob-

tained for the allele count and the different Pool- Seq data sets

whatever the simulated read coverage or sequencing error rates

(Figure 4a– c). Hence, mostly unbiased branch lengths were esti-

mated for the four leaves (terminal branches)

𝜏P1↔P7

𝜏P2↔S1

𝜏P6↔S

and

𝜏P3↔S2

. As previously observed with

𝛼

, the estimated 95% CIs

remained too narrow particularly for

𝜏P2↔S1

for which <50% of the

CI's contained the true value (Appendix S1: Table S7) compared

to more than 80% for

𝜏P1↔P7

(Appendix S1: Table S6). A s expected

from the drift- scaling approximation, the estimated branch lengths

tended to be slightly downwardly biased (ca. 2%) for the two other

leaves (

𝜏P4↔P9

and

𝜏P5↔P9

), but the estimated 95% CI displayed similar

characteristics since from 48.0% to 87.6% contained the true val-

ues, the proportion increasing in Pool- Seq data sets when coverage

and sequencing error decreased (Appendix S1: Tables S10 and S11).

Conversely, the internal branch lengths tended to be upwardly bi-

ased from a slight (ca. 2%) for

𝜏S1↔P7

𝜏P7↔P8

and

𝜏S2↔P8

(Appendix

TABLE 3 Comparison of the performance of

- based tests of admixture for different types of dat a simulated under the Figure 1 scenario

processing poolfstat analyses

MAF threshold Seq. error

𝜺

Pool- Seq (read counts) dat a

Allele

count data

𝝀=30

𝝀=50

𝝀=75

𝝀=100

𝝀=200

>1% 082.0 (0.00) 84.4 (0.00) 86.0 (0.00) 86.0 (0.00) 85.2 (0.00) 85.6 (0.00)

1100 (0.00) 100 (0.00) 86.8 (0.00) 87.2 (0.00) 86.4 (0.00)

2.5 100 (0.00) 100 (6.47) 99.6 (0.00) 92.8 (0.00) 88.4 (0.00)

0 0.00 (0.00)*0.00 (0.00)*0.00 (0.00)* 0.00 (0.00)* 0.00 (0.00)*

>5% 093.6 (0.00) 95.2 (0.00) 96.4 (0.0 0) 96.0 (0.00) 96.0 (0.00) 96.8 (0.00)

194.0 (0.00) 96.8 (0.00) 96.4 (0.0 0) 97.2 (0.00) 96.8 (0.00)

2.5 94.0 (0.00) 96.0 (0.00) 96.0 (0.00) 97.2 (0.00) 96.8 (0.00)

00.00 (0.00)* 0.00 (0.00)* 0.00 (0.00)* 0.0 0 (0.00)* 0.00 (0.00)*

Note: For each MAF threshold (MAF > 1% or MAF > 5%), the table gives true and false (in parenthesis) positive rates (in %) for 21 different types of

analyses relying on (i) allele count data; (ii) 15 dif ferent Pool- Seq read count data (five mean cover ages

𝜆

and three sequencing error rates

𝜀

); and (iii)

Pool- Seq read count data simulated with

𝜀=0

treated as allele count s (corresponding result s of this bad practice are highlighted in italics and *). Each

TPR was computed from the analysis of 250 independent data sets ( generated from the data simulated under Figure 1 demographic scenario) as the

proportion of f3 with an associated Z- score <−1.65 (95% significance threshold) for the (P6; P2, P3) population triplet (n = 250 estimates). The FPR

was similarly computed as the proportion of f3 with an associated Z- score <−1.65 among all the 50 population triplets that do not involve P6 as target

population (n = 250 × 50=12,250 estimates).

GAUTIER E T Al.

S1: Tables S12 to S14), to a moderate amount (ca. 20%) for the root

including branch

𝜏P8↔P9

, the true value being then always outside

the estimated 95% CIs (Appendix S1: Table S15). Yet, when analysing

data with a lower MAF threshold of 1%, this bias almost completely

vanished (Appendix S1: Figure S4 and Table S15).

For the other branches, the estimates had similar characteris-

tics (yet with a slightly decreased performance for the

𝜏P4↔P9

and

𝜏P5↔P9

leaves) for allele count data or Pool- Seq data simulated with-

out sequencing error (Appendix S1: Figure S 4A). In agreement with

previous observations, at the 1% MAF threshold, sequencing errors

lead to strong downward bias at the lowest simulated coverages,

that is when the percentage of false SNPs became non- negligible

(Appendix S1: Figures S4B and S4C). Finally, whatever the chosen

MAF threshold, improperly analysing read counts as allele count

data always lead to a substantial upward bias of the lengths of all

the leaves (Figure 4d). Notice, however, that this had no or limited

impact on the estimation of internal branch lengths.

3.1.6 | Evaluation of graph construction

To provide insights into the reliability of graph construction, we

evaluated the performance of the a dd .le af function in position-

ing the admixed population P6 on the underlying (((P1,P2),P3),(

TABLE 4 Comparison of the performance of f4- based test of treeness for different types of dat a simulated under the Figure 1 scenario

processing poolfstat analyses

MAF threshold Seq. error

𝜺

Pool- Seq (read counts) dat a

Allele

count data

𝝀=30

𝝀=50

𝝀=75

𝝀=100

𝝀=200

> 1% 094.0 (0.05) 94.4 (0.06) 94.1 (0.04) 94.5 (0.05) 94.3 (0.02) 94.2 (0.02)

194.3 (0.04) 94.2 (0.03) 94.3 (0.03) 94.3 (0.03) 94.1 (0.05)

2.5 94.8 (0.06) 94.5 (0.05) 94.8 (0.03) 94.5 (0.06) 94.3 (0.04)

094.0 (0.05)* 94.4 (0.06)* 94.1 (0.04)*94.5 (0.05)* 94.3 (0.02)*

> 5% 094. 5 (0.14) 94.3 (0.11) 94.1 (0.14) 94.9 (0.09) 94.3 (0.08) 94. 3 (0.11)

194.5 (0.09) 94. 5 (0.11) 94.5 (0 .13) 94.2 (0.09) 94.1 ( 0.15)

2.5 95.2 (0.13) 93. 8 (0.11) 94.2 (0.12) 94. 5 (0.11) 94.3 (0.13 )

094. 5 (0. 14)* 94.3 (0.11)* 94.1 ( 0.14)* 94.9 (0 .09)* 94.3 (0.08)*

Note: For each MAF threshold (MAF > 1% or MAF > 5%), the table gives true and false (in parenthesis) positive rates (in %) for 21 different types of

analyses relying on (i) allele count data; (ii) 15 dif ferent Pool- Seq read count data (five mean cover ages

𝜆

and three sequencing error rates

𝜀

); and (iii)

Pool- Seq read count data simulated with

𝜀=0

treated as allele count s (corresponding result s of this bad practice are highlighted in italics and *). Each

TPR was computed from the analysis of 250 independent data sets ( generated from the data simulated under Figure 1 demographic scenario) as the

proportion of

with an associated absolute Z- score < 1.96 (95% signific ance threshold) among all the eight population quadruplets ((P1,P2;P3,P4);

(P1,P2;P3,P5); (P1,P2;P4,P5); (P1,P3;P4,P5); (P1,P6;P4,P5); (P2,P3;P4,P5); (P2,P6;P4,P5); (P3,P6;P4,P5)) with a null expected

(n = 250 × 8=2000

estimates). The FPR was similarly computed as the proportion of

with an associated absolute Z- score < 1.96 among all the 37 remaining population

quadruplets (n = 250 × 37=9250 estimates).

TABLE 5 Comparison of

- ratio- based estimation of the simulated admixture proportion

in Figure 1 scenario for different types of data

processing poolfstat analyses

MAF Threshold seq. error

𝜺

Pool- Seq (read counts) dat a

Allele

count data

𝝀=30

𝝀=50

𝝀=75

𝝀=100

𝝀=200

>1% 00.247 (92 .4) 0.247 (92.8) 0.247 (94 .4) 0.247 (93.6) 0.247 (92.0) 0.247 (92.8)

10.248 (91.6) 0.247 (91.6) 0. 247 (92.4) 0.247 (93.2) 0.247 (92.0)

2.5 0.247 (91.6) 0.246 (94.0) 0.248 (93.2) 0.247 (91.2) 0.248 (92.0)

00.247 (92. 4) 0. 247 (92.8 )* 0 .247 (94. 4)* 0 .247 (93.6)* 0. 247 (92.0 )*

>5% 00.247 (92.4) 0.248 (93.2) 0.247 (93.6) 0.247 (93.2) 0.247 (92.4) 0 .247 (92.4)

10.248 (93.2) 0.247 (91.6) 0.247 (91.6) 0.247 (92.8) 0.247 (91.2)

2.5 0.247 (91.6) 0.246 (95.2) 0.248 (93.2) 0.247 (90.0) 0.248 (93.2)

00.247 (92. 4)* 0.248 (93.2)* 0. 247 (93.6)* 0.247 (93.2)* 0.247 (92.4)*

Note: For each MAF threshold (MAF > 1% or MAF > 5%), the table gives the mean of t he estimated

𝛼

(P1,P4;P3,P6)

f4(P1,P4;P2,P3)

(across 250 independent

simulated data sets) for 21 different t ypes of analyses relying on (i) allele count data; (ii) 15 different Pool- Seq read count data (five mean coverages

𝜆

and three sequencing error rates

𝜀

); and (iii) Pool- Seq read count data simulated with

𝜀=0

treated as allele count s (corresponding result s of this bad

practice are highlighted in italics and *). The proportion (in %) of the 250 es timated 95% CIs that contain the true simulated value (

𝛼=0.25

) is given in

parenthesis.

G AUTIER ET Al.

P4,P5)) tree (Figure 1) for the dif ferent types of simulated data

sets. Appendix S1: Table S16 gives the proportion of correctly in-

ferred admixture graphs (i.e. corresponding to the simulated sce-

nario) with a

ΔBIC >6

support with all other tested graphs over the

250 analysed data sets for each of the 42 investigated configura-

tions. As the reference tree with rooted topology (((P1,P2),P3),(P4

,P5)) consists of eight branches, P6 may be connected with either

(i) nine non- admixed edges (connection to either one of the eight

branches or as an outgroup) or ii)

⎛

⎜

⎝

⎞

⎟

⎠

−1=

admixed edges from

two- way admixture events. Except for the PS

50𝜀

2.5

m>1%

data set (the

one with the highest percentage of false SNPs), the correct

FIGURE 4 Distribution of the estimated drift- scaled lengths for all the branches in Figure 1 simulated scenario using admixture graph

fitting (as implemented in the fit.gr ap h function of poolfstat) for different t ypes of data with a

threshold on the overall SNP

MAF. Each box plot summarize the distribution of the 250 estimated lengths of each of the ten branches obtained from the analysis of

either allele count data set (‘Counts’) or one of the five different simulated Pool- Seq read count data sets (‘PSλX’) with different mean

coverages (

𝜆=30;50;75;100;

and 200) as generated from the genotyping data simulated under the scenario depicted in Figure 1. Pool-

Seq read count data were generated with no sequencing errors (

𝜀=0

) in (a) and (d) and with a sequencing error rate of

𝜀=1

and

𝜀=2.5

in panel (b) and (c) respectively (Appendix S1: Table S1). In (d), the read count data were analysed as allele counts, which corresponds to a

bad prac tice. Note that the two branches coming from the root are combined since the position of the root is not identifiable by the model

(i.e.

𝜏P8↔P9=𝜏P8↔R+𝜏P9↔R

). Note that the box plots obtained from the analysis of count data are replicated in each panel for comparison

purposes. For each branch, a red dotted line indicates the underlying simulated value. For Pool- Seq data, the overall MAF was estimated

from read counts

GAUTIER E T Al.

admixture graph was always retrieved with a fairly high support

(

ΔBIC >15

3.2 | Analysis of real Drosophila suzukii Pool-

Seq data

We here sketched the main findings from the analyses using p oolf-

stat of a subset of the Pool- Seq data previously generated by

Olazcuaga et al. (2020) focusing on 14 population samples of the

invasive species D. suzukii. For more details, we encourage readers

to consult the Appendix S3: Vignette V2.

3.2.1 | Structuring of genetic diversity across the 14

populations

Overall , the estimated gl obal

FST

across th e 14 populati ons was 7.03%

(95% CI;

[

6.90%;7.32%

]

). Estimates of all the pairwise- population FST

confirmed that populations tended to cluster according to their geo-

graphic area of origin (i.e. Asia, America and Hawaii; Figure 2A), with

some geographically close populations showing low level of differ-

entiation. For instance, in the American invaded area the US- Nca,

US- Col and US- Nca populations all displayed pairwise FST signifi-

cantly lower than 1%. Likewise, in the native area, the three popula-

tions CN- Bei, CN- Nin and CN- Lia originating from North- Western

China were all found very closely related (all pairwise m. being lower

or very close to 1%). Conversely, the Hawaiian sample (US- Haw) was

found the most highly dif ferentiated population, all pairwise

FST

in-

cluding US- Haw ranging from 11.7% (with US- Sok) to 17.0% (with

US- Col) suggesting strong drift in this population as confirmed by its

lowest estimated heterozygosity (Appendix S3: Vignette V2).

3.2.2 | f3- based tests of admixture suggest

pervasive admixture in the invaded area

Out of the 14 sampled populations, t wo (CN- Lia and JP- Tok) from

the Asian native area and four (US- Col, US- Nca, US- Wat and US-

Wis) from the continental American invaded areas showed at least

one significantly negative

at the 95% significance threshold (i.e. Z-

score

<−1.65

). Table 6 summarizes for each of these 6 populations

the number of significantly negative f3 together with the triplet with

the lowest Z- score giving insights into the pair of populations that

branch the closest to the two original sources (assuming a two- way

admixture event). Except for CN- Lia, all the detected signals were

significant at a far more stringent threshold (e.g. Z- score

<−2.33

at 99% significance threshold). The

and

f∗

statistics gave almost

exactly the same results (Appendix S3: Vignette V2).

In the native area, JP- Tok showed clear evidence of admixture

with 11 significant tests that all involved JP- Sap (from Northern

Japan) as a source proxy. The three lowest

values were obtained

with three Chinese populations (CN- Nin, CN- Bei and CN- Shi in in-

creasing order of

). Assuming an admixture- graph like history, this

suggests that the two populations branching the closest to the two

sources of JP- Tok were JP- Sap and CN- Nin. The remaining Chinese

population, CN- Lia showed some weak evidence for admixture with

only one test barely significant at the 95% threshold for the triplet

involving CN- Shi and JP- Sap as source proxies (Table 6).

Out of the seven invasive populations from continental America,

the four populations US- Col, US- Wis, US- Nca and US- Wat showed

strong evidence of admixture. Although it has up to now been con-

sidered as the closest to the first invading population of Continental

America (based on historical records), the Western American US- Wat

population displayed the strongest signals with 11 (strongly) signifi-

cant tests. Interestingly, the three signals supported by the lowest

and hence more significant Z- score all involved pairs of source pop-

ulation proxies originating from the continental American invaded

area namely, in order of increasing Z- score (i.e. decreasing evidence),

the (US- Sdi,US- Sok), (BR- Pal,US- Sok) and (US- Col,US- Sok) pairs. As

the underlying

CI's did not overlap with those of the other triplet

configurations, these three pairs of populations may be considered as

the closest (among the sampled populations) to the original US- Wat

source populations. It is worth noting that the Western American US-

Sok population was involved in nine of the 11 significant negative

statistic with US- Wat as a target. The three others populations, US-

Col, US- Wis and US- Nca only had a moderate number of significant

tests (compared to others). Such tests always involved at least one of

the two other populations and overlapping

CI's. This suggests com-

plex patterns of recurrent admixture event among US- Col, US- Wis

and US- Nca, a feature consistent with their low level of differentiation

and close geographic origins.

TABLE 6 Results of the

- based tests of admixture on

populations from the D. suzukii invasive species

Population Origin

nb. of signif.

tests (f3

Z<−1.65

)

Triplet with the

lowest f3 Z- score

CN- Lia Native 1CN- Lia;CN- Shi,JP-

Sap (Z = −1 .66 )

JP- Tok Native 11 JP- Tok;CN- Nin,JP-

Sap (Z = −7. 11)

U S - C o l Invasive (AM) 2US- Col;BR- Pal,US-

Wis (Z = −3.31)

US- Nca Invasive (AM) 6 US- Nca;JP- Sap,US-

Col (Z = −3.89)

U S - W a t Invasive (AM) 13 US- Wat;US- Sdi,US-

Sok (Z = −2 3. 6)

US- Wis Invasive (AM) 4US- Wis;JP- Sap,US-

Col (Z = −5.02)

Note: For all the population displaying at least one significant signal of

admixture at the 95% signif icance threshold (f3

Z<−1.65

), the table

gives the number of significant tests (out of the

C13

performed

per population) and the triplet displaying the lowest Z- score (i.e. most

significant test).

G AUTIER ET Al.

3.2.3 | Exploring invasion scenarios with admixture

graph construction and fitting

To provide further insights into the relationships of the sur veyed pop-

ulations and the probable scenarios of invasion of D. suzukii in the

American area, we relied on admixture graph construction. Our pur-

pose was not to build a comprehensive admixture graph for the 14

populations, which may be elusive given the close relationships of

some populations and the pervasiveness of recent and presumably

recurrent admixture events among the different populations, but

rather to identify key regional event that occurred at early time of the

invasion history of the species. From our extensive analyses (Appendix

S3: Vignette V2), we were able to build and estimate the parameters

of two admixture graphs represented in Figure 2b,c. The first admix-

ture graph described the somewhat complex and so far non-

investigated relationships among the populations of the native area

(including the early invasive population established in Hawaii since

1980) with a ver y good fit since the Z- score of the residuals for the

worst fitted f- statistics was 1.06 (Figure 2b). In agreement with previ-

ous findings (and geographic proximity), the Hawaiian population was

found more closely related to the Japane se population JP- Sap than to

the other Chinese populations, but it experienced a strong differenti-

ation from their common ancestor (named JP in Figure 2b) with an

estimated branch length of 0.255 drift units (

2Ne

). Yet, it was not pos-

sible from our data to definitively conclude that US- Haw originates

from a Japanese population since we have no element to claim that

the (ancestral) node population JP was located in Japan. To that end,

additional sampling of Japanese populations would be required. The

inferred graph also confirmed above

- based test results of an ad-

mixed origins of JP- Tok between a population closely related to JP-

Sap (the main contributor) and a second source likely of Chinese origin

although the same caution as for JP are needed regarding the geo-

graphic origins of this internal node population. Similarly, CN- Lia was

found admixed with a contributing source of Chinese ancestry related

to CN- Shi largely predominant (

𝛼 C=96.0%

; 95% CI [95.7; 96.3]), and a

second minor contributing source of presumably Japanese origin (re-

lated to JP- Sap). This may explain why the corresponding

- based test

was only barely significant (Table 6). Interestingly, the graph topology

also allowed estimating the Chinese ancestry of CN- Lia based on

ratio resulting in consistent but larger 95% CI (

𝛼 C=95.6

; 95% CI

[94.4;96.8]) as expected from above simulation study. CN- Nin, the

remaining population from the native area, could not be positioned

with reasonable accuracy onto the admixture graph of Figure 2b, the

resulting worst fitted

- statistics associated to the best fitting graph

having a Z- score = 3.43. However, both its genetic proximit y with CN-

Lia and the best fitting admixture graph resulting from its positioning

onto the scaffold tree including US- Haw, JP- Sap, CN- Bei and CN- Shi

suggested a small amount of Japanese introgression (see Appendix

S3: Vignette V2 for more details).

The second admixture graph represented in Figure 2c allowed

providing insights into the history of introduction of D. suzukii into

the American continent. It related the three continental American

population, US- Sok, US- Wat and BR- Pal, to a scaffold including the

four unadmixed populations US- Haw, JP- Sap, CN- Shi and CN- Bei

with a good fit (the worst fitted f- statistics had a Z- score = −1.83).

The underlying scenario suggested that continental American popu-

lations originated from at least two major and successive admixture

events. The first admixture event leads to the internal node popula-

tion named Am1 and occurred in balanced proportions between two

sources, a Japanese one closely related to JP- Sap and a Hawaiian one

relatively distantly related to US- Haw (according to the estimated

branch lengths). The US- Sok population was the sampled continental

American population the closest to Am1 and may thus be assumed

the most closely related to the first invading population (in agreement

with

- based test results). Yet, US- Sok remained separated by about

0.0816 drift units from Am1, which may explain why no significantly

negative

were found for triplets with US- Sok as a target.

The second major admixture events occurred between an internal

node population named Am2 and a Chinese population closely related

to the common ancestor of CN- Bei and CN- Shi, with CN- Shi contrib-

uting slightly more (58.5% against 41.5% for the other Am1- related

ancestor). Interestingly, the closest Am2 representatives among the

sampled populations were BR- Pal and US- Sdi (also in agreement with

- based tests) suggesting a more Southern geographical origin for Am2.

We found that some additional ancestry from a ghost population or

recurrent admixture events (e.g. related to Hawaiian populations) may

also have contributed to US- Sdi, but this lead to a poor fit (worst fit-

ted

- statistics Z- score=−5.87 for the best fitting graph resulting from

the positioning of US- Sdi onto the graph, see Appendix S3: Vignette

V2). Therefore, US- Sdi is not represented in Figure 2c. Although geo-

graphically distant, the Brazilian population BR- Pal thus appeared as the

best proxy for Am2, thereby suggesting a rapid spread of D. suzukii in

South America from this population without any subsequent admixture

events. Additional (and preferably ancient) sample from South- American

populations would help refining this scenario. Finally, according to the

inferred graph, US- Wat was found to originate from a recent admixture

between a population very closely related to US- Sok (and thus Am1)

and a population deriving from Am2 with similar contributions of both.

In agreement with

- based admixture tests that suggested com-

plex admixture histories among the closely related US- Col, US- Wis

and US- Nca populations, no satisfactory admixture graph could be

found when trying to position each of these onto the Figure 2c graph.

Nevertheless, their resulting best fitted graphs all suggested a high

contribution of the Am2 admixed source, a second contributing source

being related to Japanese populations (Appendix S3: Vignette V2).

4 | DISCUSSION

4.1 | A new version of poolfstat for f- statistics

estimation and associated inference from both Pool-

Seq and allele count data

The R package poolfstat was originally developed by Hivert et al.

(2018) to implement an unbiased estimator of FST for PoolSeq data

and provide utilities to facilitate manipulation of such data. We

GAUTIER E T Al.

here proposed a substantially improved version that implements

unbiased estimators of

and

parameters together with their

scaled versions (i.e. pairwise FST,

F∗

and

, respectively). Although

we primarily focused on the analysis of Pool- Seq data, we extended

the package to analyse standard allele count (as obtained from indi-

vidual genotyping or sequencing dat a) and to implement unbiased

estimators equivalent to those available in the Admixtools suite

(Patterson et al., 2012) allowing us in turn to validate our estima-

tion procedure. Recently, the admixr package was developed to

interface most of the Ad mixtools programs with R for the estima-

tion of

- statistics (only from allele count data), with the noticeable

exception of the admixture graph fitting program qpGraph (Petr

et al., 2019). We implemented in poolfstat our own functions for

fitting, building, visualizing and performing quality assessment of

admixture graphs based on the estimated f- statistics. The underly-

ing procedures shared strong similarities with those implemented in

qpGraph (Patterson et al., 2012) resulting on the same fitting on

some examples (e.g. Appendix S2: Vignette V1) or also MixMapper

(Lipson et al., 2013, 2014) programs. As recognized by the devel-

opers themselves, the latter program specifically developed for

admixture inference from allele count data was written in C++ and

MATLAB making it ‘cumbersome to use’ for users, as ourselves,

with no MATLAB license. Moreover, to facilitate local exploration

of the admixture graphs space, we also implemented in poolfstat

efficient semi- automated building utilities (ad d.le af and g raph.

build er functions). It should be noticed that although it does not

include functions for the estimation of f- statistics, the admix-

turegraph R package (Leppälä et al., 2017) also provides several

alternative valuable utilities for the fitting (based on a slightly differ-

ent approach), the manipulation, and the visualization of admixture

graphs together with utilities for the plotting of the statistics with

their confidence intervals or the symbolic derivation of f- statistics

(as poolfstat). Overall, our effort of developing with poolfstat

a self- contained, efficient and user- friendly R package capable of

performing the entire workflow for f- statistics- based demographic

inference from both standard allele count and Pool- Seq read count

data will hopefully make such a powerful framework accessible to a

wider range of researchers and biological models.

4.2 | A unified definition of the F parameters in

terms of probability of gene identities

To derive our unbiased estimators, we proposed to recapitulate and

unify the different definitions of the

and D parameters in terms

of probability of gene identity within population (

) or between

pairs of populations (

) as summarized in Equation (1). This formu-

lation offers a complementary perspective to the original descrip-

tion of these parameters in terms of covariance of allele frequencies

(Patterson et al., 2012) although in practice, a little algebra shows

that the unbiased estimators derived from these two alternative for-

mulations for allele count data are strictly equivalent. Formally, the

and

probabilities can be viewed as expected identity (in state)

of genes across independent replicates of the (stochastic) evolution-

ary process (Rousset, 2007) that may be expressed as a function of

other demo- genetic population parameters. Hence, the obtained

expressions for

and

in terms of

and

probabilities can

be directly related to those by Peter (2016) in terms of coalescent

times which allowed him to provide an in- depth exploration of their

theoretical properties under a wide range of demographic models

other than admix ture graphs (see, e.g., Figure 7 in Peter, 2016). More

precisely, under an infinite- site mutation model with constant per-

generation mutation rate

𝜇

, the probability that t wo genes are iden-

tical in state is

∞

t=1

(1−𝜇)

=1−2𝜇𝔼







𝜇

2

, where

is the probability that the two genes coalesced t generations in

the past and

𝔼



≡

∞

t=1

is the expected coalescence time of

two genes (see Slatkin, 1991; Rousset, 20 07, pp.58– 59). Using

Q(1)

=1−2𝜇𝔼

[

11]

and

Q(2)

=1−2𝜇𝔼

[

22]

as the IIS probabilities

within populations

and

, respectively, and

=1−2𝜇𝔼

[

12]

the IIS probability between

and

allows recovering equations

16 (after fixing a typo into it), 20c and 24 by Peter (2016) for

and

, respectively. Likewise, the estimators derived from (unbi-

ased) estimators of

and

are equivalent to those expressed in

terms of average pairwise differences between and within popula-

tions which are natural estimators for

𝜇𝔼

[

]

terms as proposed by

Peter (2016, eq. 17) for

estimator based on allele count data (e.g.

noting that

𝜋 11

=1−



𝜋 22

=1−



and

𝜋 12 =1−

following his

notations). For Pool- Seq data, replacing the latter estimators of nu-

cleotide diversities by the unbiased estimators described in Ferretti

et al. (2013, eqs. 3 and 10) would also result in the same estimator

for

(and other parameters) as those we derived from

and

estimators.

In practice, estimators are obtained by averaging over (a high)

number of SNPs, which amounts assuming that each represent

an independent outcome of a common demographic processes

that shaped the genome- wide patterns of genetic diversity. This

generally allows to provide accurate estimations and LD between

markers (i.e. violation of the marker independence assumption) can

be accounted for with block- jackknife estimation of standard er-

rors (Patterson et al., 2012). Importantly, as originally noticed by

Patterson et al. (2012), expressions of

and

in terms of co-

alescent times (Peter, 2016) show explicitly that they both depend

on the demography (via

𝔼[

]

) and the marker mutation rate (

𝜇

). In

the scaled versions of

and

(

FST

and

F∗

, respectively), the param-

eter

𝜇

cancels out making them presumably more comparable across

different data sets. It should however be noticed, that for demo-

graphic inference purposes, scaling of the

- statistics is not needed.

Indeed, the three- population test of admixture is informed by the

sign of

, which is not af fected by the denominator of

F∗

. Similarly,

the four- population test evaluates departure of

(i.e. the numerator

) from a null value expected under the hypothesis of treeness.

Patterson et al. (2012) also showed both analytically and using sim-

ulations that

and

estimates remained mostly robust to various

realistic SNP ascert ainment scheme. It is finally worth stressing that

admixture graph inference only requires additivity of

(Patterson

et al., 2012), a feature not fulfilled by

FST

(or

F∗

and

G AUTIER ET Al.

4.3 | Estimation of f- statistics and inference from

Pool- Seq data

Our simulation study showed that accurate estimates of

and

parameters could be obtained from Pool- Seq data from the unbi-

ased estimators we developed, thereby extending our findings for

the Pool- Seq

FST

estimator (Hivert et al., 2018). With no sequenc-

ing error, this remained true even at a read coverage as low as 30X,

which was here lower than our simulated haploid sample size of 50.

Increasing the coverage only provided marginal gain. When intro-

ducing sequencing errors, the per formance of the estimators tended

to decrease for the lowest investigated read coverages (up to 50X)

and MAF filtering threshold. This was however essentially due to

the presence of spurious SNPs that were not completely filtered out

when considering too loose criteria. As a result, simply increasing

the threshold on the overall MAF (computed from read counts over

all the pool samples) to 5% allowed to remove all the spurious SNPs

and recover accurate estimates of the parameters at the lowest read

coverages. In agreement with original observations made for allele

count data (Patterson et al., 2012), all the f- statistics based analy-

ses (i.e. three- population test of admixture, four- population test of

treeness,

- ratio estimation of admixture proportions or admixture

graph fitting) remained remarkably robust to a MAF- based ascer-

tainment process. From our simulation study, discarding lowly poly-

morphic SNPs was only found to increase the bias of the drif t- scaled

length estimates of internal branch in admixture graph. In practice,

cost- effective designs consisting of sequencing pools of 30 to 50

individuals at a 50– 100× coverage and applying MAF threshold of

5% to filter the called SNPs are expected to provide good perfor-

mance for all the different f- statistics- based inference methods we

presented here.

For Pool- Seq data, all the above conclusions were neverthe-

less only valid for the analyses based on the unbiased estimator

that accounts for the additional level of variation introduced by the

sampling of the DNA of pooled individuals (nonidentifiable) at the

sequencing step. We found that improperly analysing Pool- Seq read

counts as standard allele counts had high detrimental consequence

on the estimation of all the

parameters that involved

proba-

bilities (within population probability of identity) in their definition,

that is

FST

(as previously observed by Hivert et al., 2018, see also

Figure S5),

and

F∗

leading to a complete loss of power of the as-

sociated three- population test in our simulation. When processing

admixture graph fitting, this also resulted in a strong upward bias in

the estimation of branch lengths, including the external ones that

were accurately estimated when relying on unbiased estimators.

Loosely speaking, not accounting for the extra- variance introduced

by the sampling of reads has the same effect of adding a (substan-

tial) amount of extra drift explaining the two aforementioned ob-

servations. Although not investigated here (and of little interest

since we should definitely rely on unbiased estimators), the amount

of extra variance may be inversely proportional to the pool haploid

sample size (i.e. bias may decrease with increasing pool sample size).

Conversely, analysing Pool- Seq read counts as standard allele counts

did not affect the performance of the

(and

)- based test of tree-

ness or the estimation of admixture proportion from

- ratio. This

was expected from the properties of the underlying parameters that

only depends on the

probabilities across the different pairs of

population involved in the quadruplet of interest (Eq. 3) resulting in

the same estimators (see Eqs. 4 and 5) for both allele count and Pool-

Seq data. More generally, analysing Pool- Seq read count data with

popular programs that were developed for standard allele count

data such as those from the Admixtools (Patterson et al., 2012) or

TreeMix (Pickrell & Pritchard, 2012) suites should be avoided and,

if not, results should be carefully interpreted.

4.4 | Insights into the history of the invasive

species D. suzukii from Pool- Seq data analysis

To illustrate both the power and limitations of f- statistics based

methods for historical and demographic inference as implemented

in poolfstat, we analysed Pool- Seq dat a available for 14 popu-

lations of the invasive species D. suzukii (Olazcuaga et al., 2020).

These population samples were representative of both the Asian

native area and the recently invaded American area. Most of them

consisted of individuals originating from the same sites as those

genotyped in Fraimout et al. (2017) at microsatellite markers and

analysed under an ABC- RF framework. The results remained con-

sistent between the two studies, both of them pointing to com-

plex invasion pathways including multiple introductions leading

to admixed origins of the continental American populations. The

main source contributions were from Hawaii, where D. suzukii was

described about 30 years earlier and the native area (China and

Japan). However, some inferred scenarios appeared somewhat

conflicting. First , for the Hawaiian population that played a key

role in American invasion route, both poolfstat and Fraimout

et al. (2017) suggested a Japanese origin. However, we here found

that the sample the closest to the source (internal node population

JP in Figure 2) was JP- Sap (sampled in Sapporo) while Fraimout

et al. (2017) rather concluded it was JP- Tok (sampled in Tokyo),

which was not found to be directly contributing to US- Haw in

poolfstat analyses and was even found to be admixed by native

populations from Japan and China. In the ABC- RF treatments by

Fraimout et al. (2017), all populations from the native area were

assumed to be non- admixed and no ‘ghost’ (i.e. unsampled) popu-

lations were included in the model, whereas such populations are

present in admixture graphs through internal nodes. Moreover,

the samples from Hawaii and Tokyo both differed in their exact

location and collection date (2013 and 2016 for Hawaii, 2014 and

2016, for Tokyo) between the two studies, which may further ex-

plain the observed discrepancies and more generally promotes the

sequencing of additional samples in this area to better resolve the

origin of the Hawaiian population(s).

Interestingly, poolfstat results challenged the initial view

about the pioneering origin of the Californian population US-

Wat in the invasion of continental America (as suggested by

GAUTIER E T Al.

historical records) suggesting it rather originates from an admix-

ture between two already established but unsampled continen-

tal American populations, one presumably Northern (related to

Am1 and here represented by US- Sok) and the other presumably

more Southern (related to Am2 and here represented by BR- Pal

from Brazil and US- Sdi from South- California). This discrepancy

between Fraimout et al. (2017) and poolfstat results points

to three key issues. First , a too strong reliance on the reported

date of first observation of the species in the invasive area when

formalizing the scenario to be compared in ABC modelling may

actually mislead inference and this especially since D. suzukii was

first observed at very close dates in the US- Wat, US- Sdi and US-

Sok sampled locations (i.e. 2008, 2009 and 2009, respectively).

As a matter of fact, Fraimout et al. (2017) only considered sce-

narios in which US- Wat was the first population introduced in

continent al America. Second, in ABC, scenarios are defined by

hand justifying the use of dates of first observation to minimize

their number (Estoup & Guillemaud, 2010). The functions imple-

mented in poolfstat circumvent this constraint by facilitating

a quick and automatic exploration of the admixture graph space

to identif y key historical events relating the populations of inter-

est. Third, our finding reinforces the concern that the formaliza-

tion of invasion scenarios including the possibility of unsampled

populations is crucial. This possibility is by construction included

in admixture graph construction but is also possible in ABC mod-

elling (Guillemaud et al., 2010). Similarly, Fraimout et al. (2017)

argued for an admixed origin of the Brazilian BR- Pal population

(first observed in 2013) between undefined North- Western and

North- Eastern American sources, while we here found that this

population was the best prox y for the ancestral ‘ghost’ American

population Am2 (Figure 2c), which may be viewed as one of the

main contributor of all the sampled North- American populations

(but US- Sok). Again, this results underline advantages of not re-

lying on historical dates as for poolfstat analyses and promote

the sequencing of additional samples in South and North- Western

America areas to more thoroughly decipher the invasion routes

followed by continental American populations.

If Pool- Seq data analysed with poolfstat allowed to refine

historical and demographic scenarios in both the native and in-

vaded areas, the D. suzukii Pool- Seq data analysis also illustrated

some inherent constraints imposed to the modelled demographic

history when fitting admixture graph models. In particular, more

complex histories involving recurrent admixture events turned out

to be difficult or even impossible to fit unless a number of source

key samples are included, as observed here for the North- Eastern

American populations. In real- life applications involving a large

number of invasive populations characterized by numerous and

recurrent introduction events, summarizing precisely and with a

good fit the history of all surveyed populations with a comprehen-

sive admixture graph may remain elusive. However, as previously

underlined (Lipson, 2020; Lipson & Reich, 2017; Patterson et al.,

2012), in addition to providing robust formal tests of admixture or

treeness, a decisive advantage of f- based inference methods is to

allow straightforward assessment of the fitted admixture graph by

carefully inspecting and reporting Z- score of the residuals of the

fitted statistic s, an option not available in other related methods

such as TreeMix (Pickrell & Pritchard, 2012). Beyond modelling

the history of populations as admixture graphs (via formal tests

of admixture of treeness or graph fitting), Peter (2016) provided

valuable theoretical insights to interpret the estimated f- statistics

under alternative demographic models such as island, stepping-

stone or serial founder models. This suggests in turn that these

statistics should be informative to estimate the parameters of

demographic scenarios more complex than admixture graphs (e.g.

under an ABC framework as in Collin et al., 2021).

ACKNOWLEDGEMENTS

This work was suppor ted by the French National Research Agency

(ANR) for the projects SWING (ANR- 16- CE02- 0015- 01) and

G A N D H I ( A N R - 2 0 - C E 0 2 - 0 0 1 8 ) .

DATA AVAIL ABILI TY STATEMENT

The vcf file generated for the D. suzukii Pool Seq data is publicly

available from the Zenodo repository (http://doi.org/10.5281/

zenodo.4709080). Further details are provided in Appendix S2:

Vignette V1 and Appendix S3: Vignet te V2.

ORCID

Mathieu Gautier https://orcid.org/0000-0001-7257-5880

Renaud Vitalis https://orcid.org/0000-0001-7096-3089

Laurence Flori https://orcid.org/0000-0002-7529-8521

Arnaud Estoup https://orcid.org/0000-0002-4357-6144

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SUPPORTING INFORMATION

Additional supporting information may be found in the online ver-

sion of the ar ticle at the publisher’s website.

How to cite this article: Gautier, M., Vitalis, R., Flori, L., &

Estoup, A . (2021). f- Statistics estimation and admixture graph

construction with Pool- Seq or allele count data using the R

package poolfstat. Molecular Ecology Resources, 00, 1– 23.

https: //doi .or g/10.1111/1755- 0998.13557

Code Count data type

MAF Coverage Seq. error Nb. of SNPs Nb. of false SNPs

threshold (λ) () Mean ±s.d. Mean ±s.d.

ACm>1% Alleles 1% - - 471,919 ±1,474 -

ACm>5% Alleles 5% - - 240,369 ±1,118 -

PS30=0

m>1% Reads 1%?30 0 406,742 ±1,362 -

PS50=0

m>1% Reads 1% 50 0 442,426 ±1,419 -

PS75=0

m>1% Reads 1% 75 0 449,663 ±1,433 -

PS100=0

m>1% Reads 1% 100 0 454,056 ±1,434 -

PS200=0

m>1% Reads 1% 200 0 462,429 ±1,466 -

PS30=0

m>5% Reads 5% 30 0 246,560 ±1,111 -

PS50=0

m>5% Reads 5% 50 0 245,296 ±1,108 -

PS75=0

m>5% Reads 5% 75 0 244,706 ±1,126 -

PS100=0

m>5% Reads 5% 100 0 244,432 ±1,140 -

PS200=0

m>5% Reads 5% 200 0 243,993 ±1,131 -

PS30=1h

m>1% Reads 1% 30 1h613,255 ±1,471 206,051 ±443

PS50=1h

m>1% Reads 1% 50 1h871,953 ±1,605 432038 ±654

PS75=1h

m>1% Reads 1% 75 1h445,922 ±1,417 3,866 ±58

PS100=1h

m>1% Reads 1% 100 1h440,012 ±1,399 196 ±14

PS200=1h

m>1% Reads 1% 200 1h408,432 ±1,300 0 ±0

PS30=1h

m>5% Reads 5% 30 1h246,442 ±1,123 0 ±0

PS50=1h

m>5% Reads 5% 50 1h243,699 ±1,113 0 ±0

PS75=1h

m>5% Reads 5% 75 1h240,422 ±1,111 0 ±0

PS100=1h

m>5% Reads 5% 100 1h236,566 ±1,089 0 ±0

PS200=1h

m>5% Reads 5% 200 1h215,471 ±1,007 0 ±0

PS30=2.5h

m>1% Reads 1% 30 2.5h3,402,213 ±2,147 2,999,100 ±1,773

PS50=2.5h

m>1% Reads 1% 50 2.5h6,803,455 ±2,847 6,380,412 ±2,512

PS75=2.5h

m>1% Reads 1% 75 2.5h686,967 ±1,464 282,076 ±530

PS100=2.5h

m>1% Reads 1% 100 2.5h417,951 ±1,243 39,043 ±189

PS200=2.5h

m>1% Reads 1% 200 2.5h251,968 ±863 2 ±1

PS30=2.5h

m>5% Reads 5% 30 2.5h243,414 ±1,088 0 ±0

PS50=2.5h

m>5% Reads 5% 50 2.5h234,158 ±1,100 0 ±0

PS75=2.5h

m>5% Reads 5% 75 2.5h220,095 ±1,050 0 ±0

PS100=2.5h

m>5% Reads 5% 100 2.5h203,702 ±967 0 ±0

PS200=2.5h

m>5% Reads 5% 200 2.5h132,980 ±669 0 ±0

Table S1. Description of the simulated datasets. A total of 250 independent genotyping datasets were simulated

according to the demographic scenario represented in Figure 1. For each simulated genotyping datasets, we generated

two allele count datasets applying 1% or 5% threshold (column 3) on the overall MAF (computed from the allele counts);

and simulated ﬁfteen Pool-Seq read count datasets with ﬁve diﬀerent mean read coverages λ(column 4) and three diﬀerent

sequencing error rates (column 5) and applied 1% or 5% threshold (column 3) on the overall MAF (computed from read

counts) for each of them (resulting in a total of 30 Pool-Seq read count dataset per simulated genotyping data). As highlighted

by the asterisk (?), for Pool-Seq data with λ=30, the actual MAF threshold was slightly higher than indicated due the additional

ﬁltering criterion on the minimal read count (MRC>2, see Material and Methods). Indeed, when λ=30, the overall coverage

(that follows a Poisson distribution with parameter 6 ×30 =180) is ≤200 for on expected proportion of 93.5% of the SNPs

resulting to a more stringent MAF threshold of 2

6×30 =1.11% when requiring MRC>2. The last two columns give the average

number (and standard deviation) of SNPs and false SNPs (for read count data simulated with  > 0) computed over the 250

independent simulated datasets available for each of the 32 conﬁgurations.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 82.0 (0.00) 84.4 (0.00) 86.0 (0.00) 86.0 (0.00) 85.2 (0.00)

85.6 (0.00)

1h100 (0.00) 100 (0.00) 86.8 (0.00) 87.2 (0.00) 86.4 (0.00)

2.5h100 (0.00) 100 (6.47) 99.6 (0.00) 92.8 (0.00) 88.4 (0.00)

0 0.00 (0.00)?0.00 (0.00)?0.00 (0.00)?0.00 (0.00)?0.00 (0.00)?

>5%

0 93.6 (0.00) 95.2 (0.00) 96.4 (0.00) 96.0 (0.00) 96.0 (0.00)

96.8 (0.00)

1h94.0 (0.00) 96.8 (0.00) 96.4 (0.00) 97.2 (0.00) 96.8 (0.00)

2.5h94.0 (0.00) 96.0 (0.00) 96.0 (0.00) 97.2 (0.00) 96.8 (0.00)

0 0.00 (0.00)?0.00 (0.00)?0.00 (0.00)?0.00 (0.00)?0.00 (0.00)?

Table S2. Comparison of the performance of f?

3-based tests of admixture for diﬀerent types of data sim-

ulated under the Figure 1 scenario processing poolfstat analyses. For each MAF threshold (MAF>1% or

MAF>5%), the table gives True and False (in parenthesis) Positive Rates (in %) for 21 diﬀerent types of analyses relying on i)

allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand three sequencing error rates ); and iii)

Pool-Seq read count data simulated with =0 treated as allele counts (corresponding results of this bad practice are highlighted

in italics and ?). Each TPR was computed from the analysis of 250 independent datasets (generated from the data simulated

under Figure 1 demographic scenario) as the proportion of f?

3with an associated Z-score <−1.65 (95% signiﬁcance threshold)

for the (P6;P2,P3) population triplet (n=250 estimates). The FPR was similarly computed as the proportion of f?

3with an

associated Z-score<−1.65 among all the 50 population triplets that do not involve P6 as target population (n=250×50=12,250

estimates).

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 94.0 (0.05) 94.4 (0.06) 94.1 (0.04) 94.5 (0.05) 94.3 (0.02)

94.2 (0.02)

1h94.3 (0.04) 94.2 (0.03) 94.3 (0.03) 94.3 (0.03) 94.2 (0.05)

2.5h94.8 (0.06) 94.5 (0.05) 94.8 (0.03) 94.5 (0.06) 94.3 (0.04)

0 94.0 (0.05)?94.4 (0.06)?94.1 (0.04)?94.5 (0.05)?94.3 (0.02)?

>5%

0 94.5 (0.14) 94.3 (0.11) 94.1 (0.14) 94.8 (0.09) 94.3 (0.08)

94.3 (0.11)

1h94.5 (0.09) 94.5 (0.11) 94.5 (0.13) 94.2 (0.09) 94.0 (0.15)

2.5h95.2 (0.13) 93.8 (0.11) 94.2 (0.12) 94.5 (0.11) 94.3 (0.13)

0 94.5 (0.14)?94.3 (0.11)?94.1 (0.14)?94.9 (0.09)?94.3 (0.08)?

Table S3. Comparison of the performance of D-based test of treeness for diﬀerent types of data simulated

under the Figure 1 scenario processing poolfstat analyses. For each MAF threshold (MAF>1% or MAF>5%),

the table gives True and False (in parenthesis) Positive Rates (in %) for 21 diﬀerent types of analyses relying on i) allele count

data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand three sequencing error rates ); and iii) Pool-Seq read

count data simulated with =0 treated as allele counts (corresponding results of this bad practice are highlighted in italics and

?). Each TPR was computed from the analysis of 250 independent datasets (generated from the data simulated under Figure 1

demographic scenario) as the proportion of D-statistics with an associated absolute Z-score <1.96 (95% signiﬁcance thresh-

old) among all the eight population quadruplets ((P1,P2;P3,P4); (P1,P2;P3,P5); (P1,P2;P4,P5); (P1,P3;P4,P5); (P1,P6;P4,P5);

(P2,P3;P4,P5); (P2,P6;P4,P5); (P3,P6;P4,P5)) with a null expected F4(n=250×8=2,000 estimates). The FPR was similarly

computed as the proportion of D-statistics with an associated absolute Z-score <1.96 among all the 37 remaining population

quadruplets (n=250×37=9,250 estimates).

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 0.246 (94.0) 0.246 (93.2) 0.246 (92.0) 0.245 (93.6) 0.246 (93.2)

0.246 (92.8)

1h0.246 (92.0) 0.247 (96.0) 0.246 (92.8) 0.246 (93.2) 0.245 (91.6)

2.5h0.245 (91.2) 0.245 (92.8) 0.246 (92.0) 0.246 (92.4) 0.246 (94.0)

0 0.246 (94.0)?0.246 (93.2)?0.246 (92.0)?0.245 (93.6)?0.246 (93.2)?

>5%

0 0.246 (93.2) 0.246 (93.2) 0.246 (92.4) 0.245 (94.0) 0.246 (93.6)

0.246 (93.2)

1h0.246 (91.6) 0.247 (94.8) 0.246 (92.4) 0.246 (93.2) 0.246 (92.0)

2.5h0.245 (91.2) 0.245 (93.2) 0.246 (91.2) 0.246 (92.0) 0.246 (92.8)

0 0.246 (93.2)?0.246 (93.2)?0.246 (92.4)?0.245 (94.0)?0.246 (93.6)?

Table S4. Comparison of F4-ratio based estimation of the simulated admixture proportion αin Figure

1 scenario for diﬀerent types of data processing poolfstat analyses. For each MAF threshold (MAF>1% or

MAF>5%), the table gives the mean of the estimated ˆα=f4(P1,P5;P3,P6)

f4(P1,P5;P2,P3)) (across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value (α=0.25) is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 0.247 (44.8) 0.247 (44.0) 0.247 (43.6) 0.246 (42.0) 0.247 (42.4)

0.246 (42.8)

1h0.247 (41.6) 0.247 (43.2) 0.246 (44.4) 0.247 (43.2) 0.246 (44.4)

2.5h0.251 (74.4) 0.252 (56.8) 0.247 (42.8) 0.246 (40.4) 0.247 (42.0)

0 0.247 (44.8)?0.247 (44.0)?0.247 (43.6)?0.246 (42.0)?0.247 (42.4)?

>5%

0 0.247 (45.6) 0.247 (43.2) 0.247 (43.2) 0.246 (41.6) 0.247 (42.0)

0.246 (42.8)

1h0.247 (42.8) 0.247 (40.8) 0.246 (44.0) 0.247 (44.4) 0.247 (42.8)

2.5h0.246 (44.0) 0.246 (43.6) 0.247 (42.4) 0.247 (43.2) 0.247 (41.6)

0 0.247 (45.2)?0.247 (43.2)?0.247 (43.2)?0.246 (41.6)?0.247 (41.6)?

Table S5. Comparison of the estimation of the simulated admixture proportion αin Figure 1 scenario using

admixture graph ﬁtting (as implemented in the fit.graph function of poolfstat) for diﬀerent types of

data processing poolfstat analyses. For each MAF threshold (MAF>1% or MAF>5%), the table gives the mean of

the estimated ˆα(across 250 independent simulated datasets) for 21 diﬀerent types of analyses relying on i) allele count data; ii)

15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand three sequencing error rates ); and iii) Pool-Seq read count

data simulated with =0 treated as allele counts (corresponding results of this bad practice are highlighted in italics and ?).

The proportion (in %) of the 250 estimated 95% conﬁdence intervals that contain the true simulated value (α=0.25) is also

given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 5.1×10−2(86.0) 5.1×10−2(91.2) 5.1×10−2(91.2) 5.1×10−2(92.0) 5.1×10−2(92.4)

5.1×10−2(91.6)

1h4.4×10−2(0.40) 4.1×10−2(0.00) 5.1×10−2(94.4) 5.1×10−2(94.0) 5.1×10−2(95.6)

2.5h6.0×10−3(0.00) 1.1e-03 (0.00) 4.3×10−2(0.80) 4.9×10−2(90.0) 5.0×10−2(97.2)

0 7.2×10−2(0.00)?7.2×10−2(0.00)?7.2×10−2(0.00)?7.2×10−2(0.00)?7.2×10−2(0.00)?

>5%

0 5.1×10−2(81.6) 5.1×10−2(85.6) 5.1×10−2(89.6) 5.1×10−2(89.2) 5.1×10−2(91.2)

5.0×10−2(90.4)

1h5.1×10−2(84.0) 5.1×10−2(88.8) 5.1×10−2(89.6) 5.1×10−2(91.6) 5.0×10−2(92.0)

2.5h5.1×10−2(87.2) 5.0×10−2(90.0) 5.0×10−2(92.4) 5.0×10−2(90.0) 5.0×10−2(90.8)

0 7.2×10−2(0.00)?7.2×10−2(0.00)?7.2×10−2(0.00)?7.1×10−2(0.00)?7.1×10−2(0.00)?

Table S6. Comparison of the estimation of the simulated length (in drift units) for the P7↔P1 branch

τP7↔P1= 5 ×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP7↔P1(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 2.5×10−2(44.8) 2.5×10−2(49.2) 2.5×10−2(51.2) 2.5×10−2(50.8) 2.5×10−2(52.0)

2.5×10−2(50.8)

1h2.0×10−2(32.4) 1.9×10−2(12.0) 2.4×10−2(48.8) 2.5×10−2(47.6) 2.4×10−2(51.6)

2.5h0.00 (0.00) 0.00 (0.00) 2.0×10−2(25.6) 2.4×10−2(49.6) 2.4×10−2(48.8)

0 4.5×10−2(0.40)?4.4×10−2(0.40)?4.4×10−2(0.00)?4.4×10−2(0.40)?4.4×10−2(0.40)?

>5%

0 2.5×10−2(47.2) 2.5×10−2(48.8) 2.4×10−2(46.4) 2.4×10−2(45.6) 2.4×10−2(46.8)

2.4×10−2(49.2)

1h2.5×10−2(44.0) 2.5×10−2(46.4) 2.4×10−2(47.2) 2.4×10−2(47.2) 2.4×10−2(48.4)

2.5h2.4×10−2(44.8) 2.4×10−2(47.6) 2.4×10−2(46.0) 2.4×10−2(46.8) 2.4×10−2(46.4)

0 4.4×10−2(0.80)?4.4×10−2(0.40)?4.4×10−2(0.00)?4.4×10−2(0.00)?4.4×10−2(0.00)?

Table S7. Comparison of the estimation of the simulated length (in drift units) for the P2↔S1 branch

τP2↔S1= 2.5×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP2↔S1(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 2.5×10−2(65.6) 2.5×10−2(70.8) 2.5×10−2(70.8) 2.5×10−2(68.8) 2.5×10−2(68.8)

2.5×10−2(70.0)

0.10% 2.1×10−2(4.80) 1.9×10−2(0.00) 2.5×10−2(70.0) 2.5×10−2(72.8) 2.5×10−2(70.0)

0.25% 0.0 (0.00) 0.0 (0.00) 2.0×10−2(5.6) 2.4×10−2(57.2) 2.5×10−2(71.2)

0 4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?

>5%

0 2.5×10−2(62.0) 2.5×10−2(62.4) 2.5×10−2(62.0) 2.5×10−2(62.4) 2.5×10−2(62.8)

2.5×10−2(60.8)

0.10% 2.5×10−2(66.4) 2.5×10−2(61.2) 2.5×10−2(61.2) 2.5×10−2(59.6) 2.5×10−2(61.6)

0.25% 2.5×10−2(66.4) 2.5×10−2(63.6) 2.5×10−2(61.6) 2.5×10−2(61.2) 2.5×10−2(61.2)

0 4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.4×10−2(0.00)?4.4×10−2(0.00)?

Table S8. Comparison of the estimation of the simulated length (in drift units) for the P6↔S branch

τP6↔S= 2.5×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP6↔S(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 2.6×10−2(54.4) 2.6×10−2(56.0) 2.6×10−2(59.6) 2.6×10−2(59.2) 2.6×10−2(57.6)

2.6×10−2(56.4)

1h2.1×10−2(28.8) 1.9×10−2(10.0) 2.6×10−2(57.6) 2.6×10−2(61.6) 2.6×10−2(59.6)

2.5h0.00 (0.00) 0.00 (0.00) 2.1×10−2(25.2) 2.5×10−2(55.2) 2.5×10−2(58.8)

0 4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?

>5%

0 2.6×10−2(52.0) 2.6×10−2(52.8) 2.5×10−2(54.0) 2.6×10−2(56.4) 2.5×10−2(54.0)

2.5×10−2(54.8)

1h2.6×10−2(53.2) 2.5×10−2(51.2) 2.5×10−2(54.0) 2.5×10−2(54.4) 2.5×10−2(54.4)

2.5h2.6×10−2(56.8) 2.5×10−2(54.4) 2.5×10−2(54.0) 2.5×10−2(50.8) 2.5×10−2(53.6)

0 4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?4.5×10−2(0.00)?

Table S9. Comparison of the estimation of the simulated length (in drift units) for the P3↔S2 branch

τP3↔S2= 2.5×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP3↔S2(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 0.143 (20.0) 0.141 (6.00) 0.141 (5.60) 0.141 (4.80) 0.141 (4.00)

0.140 (3.20)

1h0.126 (0.00) 0.120 (0.00) 0.140 (2.40) 0.140 (3.20) 0.140 (2.00)

2.5h4.2×10−2(0.00) 3.0×10−2(0.00) 0.125 (0.00) 0.137 (0.00) 0.139 (0.00)

0 0.160 (2.80)?0.158 (10.0)?0.158 (9.20)?0.158 (10.0)?0.158 (12.4)?

>5%

0 0.148 (85.6) 0.147 (77.6) 0.146 (74.0) 0.146 (70.4) 0.146 (65.6)

0.145 (60.0)

1h0.147 (80.4) 0.146 (73.6) 0.146 (69.2) 0.146 (67.2) 0.145 (62.4)

2.5h0.147 (77.6) 0.146 (67.6) 0.145 (62.4) 0.145 (59.6) 0.145 (56.4)

0 0.165 (0.00)?0.164 (0.00)?0.163 (0.00)?0.163 (0.80)?0.163 (0.80)?

Table S10. Comparison of the estimation of the simulated length (in drift units) for the P4↔P9 branch

(τP4↔P9= 0.150)in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP4↔P9(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 0.142 (12.0) 0.141 (2.80) 0.140 (1.60) 0.140 (1.20) 0.140 (0.80)

0.140 (0.40)

1h0.126 (0.00) 0.120 (0.00) 0.140 (0.40) 0.140 (0.40) 0.139 (0.40)

2.5h4.2×10−2(0.00) 3.0×10−2(0.00) 0.125 (0.00) 0.137 (0.00) 0.139 (0.00)

0 0.159 (1.60)?0.158 (7.20)?0.158 (8.00)?0.158 (9.60)?0.157 (11.2)?

>5%

0 0.147 (87.6) 0.146 (77.6) 0.146 (70.0) 0.146 (64.4) 0.145 (59.2)

0.145 (53.2)

1h0.147 (82.0) 0.146 (72.0) 0.145 (60.4) 0.145 (58.4) 0.145 (54.0)

2.5h0.146 (75.6) 0.145 (61.6) 0.145 (53.6) 0.145 (52.8) 0.144 (48.0)

0 0.164 (0.00)?0.163 (0.00)?0.163 (0.00)?0.163 (0.00)?0.162 (0.00)?

Table S11. Comparison of the estimation of the simulated length (in drift units) for the P5↔P9 branch

(τP5↔P9= 0.150)in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP5↔P9(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 2.7×10−2(48.0) 2.6×10−2(54.0) 2.6×10−2(50.0) 2.6×10−2(50.4) 2.6×10−2(50.0)

2.6×10−2(52.8)

1h2.4×10−2(42.0) 2.3×10−2(36.4) 2.6×10−2(48.8) 2.6×10−2(51.6) 2.6×10−2(50.8)

2.5h5.8×10−3(0.00) 8.9×10−4(0.00) 2.3×10−2(51.2) 2.6×10−2(51.2) 2.6×10−2(49.2)

0 2.7×10−2(46.4)?2.7×10−2(54.0)?2.7×10−2(50.4)?2.7×10−2(50.0)?2.7×10−2(52.0)?

>5%

0 2.7×10−2(44.0) 2.7×10−2(49.2) 2.7×10−2(44.8) 2.7×10−2(48.0) 2.7×10−2(46.8)

2.7×10−2(50.8)

1h2.7×10−2(46.4) 2.7×10−2(52.8) 2.7×10−2(47.6) 2.7×10−2(50.0) 2.7×10−2(46.8)

2.5h2.7×10−2(46.4) 2.7×10−2(44.4) 2.6×10−2(47.2) 2.6×10−2(48.8) 2.6×10−2(46.8)

0 2.8×10−2(45.2)?2.7×10−2(49.2)?2.7×10−2(46.0)?2.7×10−2(48.8)?2.7×10−2(46.8)?

Table S12. Comparison of the estimation of the simulated length (in drift units) for the S1↔P7 branch

τS1↔P7= 2.5×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτS1↔P7(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 5.0×10−2(95.2) 4.9×10−2(93.2) 4.9×10−2(94.0) 4.9×10−2(94.4) 4.9×10−2(94.4)

4.9×10−2(94.4)

1h4.4×10−2(7.60) 4.3×10−2(0.80) 4.9×10−2(92.8) 4.9×10−2(94.8) 4.9×10−2(93.2)

2.5h1.9×10−2(0.00) 1.5×10−2(0.00) 4.4×10−2(3.60) 4.8×10−2(81.6) 4.9×10−2(92.0)

0 5.1×10−2(93.2)?5.0×10−2(94.8)?5.0×10−2(94.4)?5.0×10−2(94.4)?5.0×10−2(95.2)?

>5%

0 5.3×10−2(67.6) 5.2×10−2(72.4) 5.2×10−2(72.8) 5.2×10−2(72.0) 5.2×10−2(74.0)

5.2×10−2(74.8)

1h5.2×10−2(72.8) 5.2×10−2(73.6) 5.2×10−2(73.6) 5.2×10−2(73.6) 5.2×10−2(76.0)

2.5h5.2×10−2(72.8) 5.2×10−2(76.4) 5.2×10−2(76.4) 5.2×10−2(76.4) 5.2×10−2(80.0)

0 5.4×10−2(50.4)?5.3×10−2(54.0)?5.3×10−2(55.2)?5.3×10−2(56.0)?5.3×10−2(58.0)?

Table S13. Comparison of the estimation of the simulated length (in drift units) for the P7↔P8

branch τP7↔P8= 5.0×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the

fit.graph function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF

threshold (MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP7↔P8(across 250 independent simulated

datasets) for 21 diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean

coverages λand three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts

(corresponding results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95%

conﬁdence intervals that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 7.4×10−2(71.6) 7.4×10−2(68.0) 7.4×10−2(69.6) 7.4×10−2(68.8) 7.4×10−2(69.2)

7.4×10−2(67.6)

1h6.6×10−2(5.60) 6.4×10−2(0.00) 7.3×10−2(66.0) 7.3×10−2(68.0) 7.3×10−2(66.8)

2.5h2.5×10−2(0.00) 1.8×10−2(0.00) 6.6×10−2(4.80) 7.2×10−2(54.0) 7.3×10−2(62.0)

0 7.6×10−2(76.0)?7.5×10−2(77.2)?7.5×10−2(77.6)?7.5×10−2(76.0)?7.5×10−2(76.4)?

>5%

0 7.7×10−2(65.6) 7.7×10−2(67.6) 7.7×10−2(69.6) 7.7×10−2(70.4) 7.7×10−2(71.6)

7.7×10−2(71.6)

1h7.7×10−2(67.6) 7.7×10−2(69.2) 7.7×10−2(72.8) 7.7×10−2(71.6) 7.6×10−2(72.0)

2.5h7.7×10−2(70.8) 7.6×10−2(69.6) 7.6×10−2(73.6) 7.6×10−2(70.0) 7.6×10−2(74.4)

0 7.9×10−2(52.4)?7.9×10−2(54.4)?7.8×10−2(58.4)?7.8×10−2(60.8)?7.8×10−2(60.8)?

Table S14. Comparison of the estimation of the simulated length (in drift units) for the S2↔P8 branch

τS2↔P8= 7.5×10−2in Figure 1 scenario using admixture graph ﬁtting (as implemented in the fit.graph

function of poolfstat) for diﬀerent types of data processing poolfstat analyses. For each MAF threshold

(MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτS2↔P8(across 250 independent simulated datasets) for 21

diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand

three sequencing error rates ); and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding

results of this bad practice are highlighted in italics and ?). The proportion (in %) of the 250 estimated 95% conﬁdence intervals

that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 0.155 (59.2) 0.153 (81.2) 0.153 (84.4) 0.153 (84.0) 0.153 (87.2)

0.152 (90.0)

1h0.138 (0.40) 0.132 (0.00) 0.152 (88.4) 0.152 (88.4) 0.152 (92.0)

2.5h5.62×10−2(0.00) 4.49×10−2(0.00) 0.137 (0.00) 0.149 (94.8) 0.151 (96.4)

0 0.158 (19.2)?0.156 (43.6)?0.156 (45.6)?0.156 (47.6)?0.156 (54.8)?

>5%

0 0.177 (0.00) 0.177 (0.00) 0.177 (0.00) 0.177 (0.00) 0.177 (0.00)

0.178 (0.00)

1h0.176 (0.00) 0.176 (0.00) 0.177 (0.00) 0.177 (0.00) 0.177 (0.00)

2.5h0.175 (0.00) 0.176 (0.00) 0.176 (0.00) 0.176 (0.00) 0.176 (0.00)

0 0.18 (0.00)?0.18 (0.00)?0.18 (0.00)?0.181 (0.00)?0.181 (0.00)?

Table S15. Comparison of the estimation of the simulated length (in drift units) for the P8↔P9 branch

(τP8↔P9= 0.150)that combines the two branches from the root Rin Figure 1 scenario using admixture

graph ﬁtting (as implemented in the fit.graph function of poolfstat) for diﬀerent types of data processing

poolfstat analyses. For each MAF threshold (MAF>1% or MAF>5%), the table gives the mean of the estimated ˆτP8↔P9

(across 250 independent simulated datasets) for 21 diﬀerent types of analyses relying on i) allele count data; ii) 15 diﬀerent

Pool-Seq read count data (ﬁve mean coverages λand three sequencing error rates ); and iii) Pool-Seq read count data simulated

with =0 treated as allele counts (corresponding results of this bad practice are highlighted in italics and ?). The proportion

(in %) of the 250 estimated 95% conﬁdence intervals that contain the true simulated value is given in parenthesis.

MAF seq. error Pool-Seq (read counts) data allele count

threshold  λ =30 λ=50 λ=75 λ=100 λ=200 data

>1%

0 100 (23.5) 100 (28.0) 100 (29.7) 100 (28.8) 100 (33.8)

100 (33.2)

1h100 (28.5) 100 (31.4) 100 (31.7) 100 (31.9) 100 (33.9)

2.5h100 (29.9) 42.4 (0.00) 100 (31.2) 100 (26.9) 100 (26.4)

0 100 (23.5)?100 (28.0)?100 (29.7)?100 (28.8)?100 (33.8)?

>5%

0 100 (17.3) 100 (19.6) 100 (24.5) 100 (19.7) 100 (23.3)

100 (23.1)

1h100 (22.7) 100 (24.8) 100 (23.2) 100 (22.4) 100 (23.7)

2.5h100 (19.8) 100 (20.5) 100 (23.2) 100 (18.7) 100 (19.3)

0 100 (17.3)?100 (19.6)?100 (24.5)?100 (19.7)?100 (23.3)?

Table S16. Performance of the add.leaf function in positioning the simulated population P6 on the un-

derlying (((P1,P2),P3),(P4,P5)) tree (Figure 1) for diﬀerent types of simulated data processing poolfstat

analyses. For each MAF threshold (MAF>1% or MAF>5%), the table gives the proportion of correctly inferred admixture

graph (i.e., positioning of the P6 population as deriving from an admixture events between two populations directly ancestral

to P2 and P3 with a ∆BIC >6 support) across 250 independent simulated datasets for 21 diﬀerent types of analyses relying

on i) allele count data; ii) 15 diﬀerent Pool-Seq read count data (ﬁve mean coverages λand three sequencing error rates );

and iii) Pool-Seq read count data simulated with =0 treated as allele counts (corresponding results of this bad practice are

highlighted in italics and ?). Note that a total of 36 diﬀerent positioning of P6 on the (((P1,P2),P3),(P4,P5)) rooted tree are

evaluated for each call of add.leaf function. Indeed as the reference tree consists of eight branches, P6 may be connected

with i) 9 non-admixed edges (connection to either one of the 8 branches or as an outgroup) or, ii) 27 admixed edges from

two-way admixture events (27=C8

2-1, an admixture event between the two branches from the root being not identiﬁable). The

lowest ∆BIC between the true graph and the 35 other possible graphs over the 250 diﬀerent datasets is given in parenthesis for

each type of analyses.

Figure S1. Average number of SNPs (A) and percentage of false SNPs (B) in the simulated datasets as a

function of read coverage and MAF ﬁltering thresholds (see Table S1). As highlighted with parentheses around the

points in A and B, for Pool-Seq data with λ=30, the actual MAF threshold is slightly higher than indicated due the additional

ﬁltering criterion imposed on the minimal read count (MRC>2; see Material and Methods). This is because for λ=30, the

overall coverage (that follows a Poisson distribution with parameter 6 ×30 =180) is ≤200 for on expected proportion of

93.5% of the SNPs resulting to a more stringent MAF threshold of 2

6×30 =1.11% when requiring MRC>2. This feature also

explains the observed increased in the percentage of false observed from 30X to 50X coverage for the red (MAF>1% and

=10−3) and green (MAF>1% and =2.5×10−3) straight lines in B). Note also that the red (MAF>5% and =10−3) and

green (MAF>5% and =2.5×10−3) dashed lines are perfectly superposed in B) since no false SNP are included in all the

corresponding conﬁgurations. 10

Figure S2. Allele Frequency Spectrum of the simulated genotyping data (A) and read binomial sampling

properties (B).A) Combined distribution of the overall SNP minor allele frequencies over all the 250 simulated genotyping

datasets. B) Expected proportion of SNPs passing MAF ﬁltering steps after binomial sampling of the minor read count r(i.e.,

with r

c>MAF, where cis the overall read coverage) as a function of the SNP allele frequency y

n(where yis the minor allele

count and nis the haploid number of individuals) for two diﬀerent coverage (c=30 ×6=180 and c=200 ×6=1200

corresponding to the two extremes of the simulated mean read coverage) and MAF thresholds (1% and 5%). The 5% MAF

value is represented by a vertical dotted line in the two graphs.

Figure S3. Distribution of the estimated admixture proportion α=0.25 in the Figure 1 simulated scenario

using admixture graph ﬁtting (as implemented in the fit.graph function of poolfstat) for diﬀerent types

of data processing poolfstat analyses. Each box plot summarizes the distribution of the 250 estimated values obtained

from the analysis of either allele count dataset (the leftmost box named “Counts” of each group) or one of the ﬁve diﬀerent

simulated Pool-Seq read count datasets (“PSλX”) with diﬀerent mean coverages (λ=30; 50; 75; 100; and 200) as generated

from the genotyping data simulated under the scenario depicted in Figure 1. Pool-Seq read count data were generated with

no sequencing errors (=0) in A) and D) and with a sequencing error rate of =1hand =2.5hin panel B) and C),

respectively (Table S1). In D), the read count data were analyzed as allele counts which corresponds to a bad practice. In each

of the four panels, analyses performed after discarding SNPs with an overall MAF (estimated from read counts in Pool-Seq

data) ≤1% and ≤5%) are grouped in the left-hand and right-hand sides, respectively. Note that the two box plots obtained from

the analysis of count data (MAF>1% and MAF>5%) are replicated in each panel for comparison purposes. The red dotted line

indicates the simulated value of α.

Figure S4. Distribution of the estimated drift-scaled lengths for all the branches in Figure 1 simulated sce-

nario using admixture graph ﬁtting (as implemented in the fit.graph function of poolfstat) for diﬀerent

types of data with a 1% threshold on the overall SNP MAF. Each box plot summarize the distribution of

the 250 estimated lengths of each of the ten branches obtained from the analysis of either allele count dataset

(“Counts”) or one of the ﬁve diﬀerent simulated Pool-Seq read count datasets (“PSλX”) with diﬀerent mean cov-

erages (λ=30; 50;75; 100; and 200) as generated from the genotyping data simulated under the scenario depicted

in Figure 1. Pool-Seq read count data were generated with no sequencing errors (=0) in A) and D) and with

a sequencing error rate of =1hand =2.5hin panel B) and C), respectively (Table S1). In D), the read

count data were analyzed as allele counts which corresponds to a bad practice. The two branches coming from the

root are combined since the position of the root is not identiﬁable by the model (i.e., τP8↔P9=τP8↔R+τP9↔R).

The box plots obtained from the analysis of count data are replicated in each panel for comparison purposes. For

each branch, a red dotted line indicates the underlying simulated value. For Pool-Seq data, the overall MAF was

estimated from read counts.

Figure S5. Distribution of the estimated global FS T for the data simulated under the Figure 1 simulated

scenario (using the computeFST function of poolfstat run with default option) for diﬀerent types of data.

Each box plot summarizes the distribution of the 250 estimated values obtained from the analysis of either allele count dataset

(“Counts”) or one of the ﬁve diﬀerent simulated Pool-Seq read count datasets (“PSλX”) with diﬀerent mean coverages (λ=

30; 50; 75; 100; and 200) as generated from the genotyping data simulated under the scenario depicted in Figure 1. Pool-Seq

read count data were generated with no sequencing errors (=0) in A) and D) and with a sequencing error rate of =1h

and =2.5hin panel B) and C) respectively (Table S1). In D), the read count data were analyzed as allele counts which

corresponds to a bad practice. In each of the four panels, analyses performed after discarding SNPs with on the overall MAF

(estimated from read counts in Pool-Seq data) ≤1% (respectively ≤5%) are grouped in the left-hand (respectively right-hand)

side. The two box plots obtained from the analysis of count data (MAF>1% and MAF>5%) are replicated in each panel for

comparison purposes and the mean estimated values for the allele count data with MAF>1% is indicated by a dotted line.

Vignette for the package poolfstat (version 2+)

Mathieu Gautier

2021-09-14

Contents

1 Preamble: presentation of the working example data set 2

2 Reading and manipulating input data 3

2.1 Generating a countdata object for allele count data . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Generating a pooldata object for Pool-Seq read count data . . . . . . . . . . . . . . . . . . . 4

2.3 Manipulating countdata and pooldata objects ........................... 5

3 Estimating FS T 6

3.1 Estimating genome-wide FST across all the populations . . . . . . . . . . . . . . . . . . . . . 6

3.2 Estimating and visualizing pairwise-population FST ....................... 8

4 Estimating and visualizing f-statistics (f2,f3,f?

3,f4and D) 10

4.1 The compute.fstats function and fstats objects........................... 11

4.2

The plot_fstats function for visualization of

(and pairwise

FST

(and

) and

(and

D) estimates and their conﬁdence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Estimating admixture proportions with f4-ratios ......................... 18

5 Using f-statistics to estimate parameters of admixture graphs 21

5.1 Generating a graph.params object with the generate.graph.params function . . . . . . . . . . 21

5.2 Fitting a graph with the ﬁt.graph function............................. 25

5.3 Adding a new leaf to an existing graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Building admixture graph from scratch 32

6.1 Building scaﬀold trees of unadmixed populations . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.2 Extending an initial tree (or graph) with the graph.builder function............... 35

7 Other utilities 37

7.1

Symbolic representation of the

parameters, admixture graph equations and the scaled

covariance matrix Ωwith graph.params2symbolic.fstats ...................... 37

7.2 Generating ﬁles for the qpGraph software with graph.params2qpGraphFiles ........... 38

8 References 40

A Apprendix 42

A.1 Block-Jackknife estimation of standard errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

This vignette describes how the R package poolfstat can be used to compute various f- and D-statistics

(estimation of

FST

, Patterson’s

and

parameters

) in population genomics studies from

allele count or Pool-Seq read count data. The package also includes functions to ﬁt and construct admixture

graphs to infer the demographic history of populations based on the estimated

-statistics together with

their visualization. This document is conceived as a hands-on tutorial providing users with an overview

of the package functions with working example analyzing the Pool-Seq and allele count simulated data

sets described in section 1 and publicly available for download from the Zenodo repository

. Details and

(numerous) reference for the all the underlying methods are available in Gautier et al. (2021).

The poolfstat package is currently available for most platforms (Linux, MS Windows and MacOSX) from the

CRAN repository (http://cran.r-project.org/) and may be installed using a standard procedure. Once the

package has been successfully installed on your system, it can be loaded by:

library(poolfstat)

1 Preamble: presentation of the working example data set

Genetic data were simulated using the coalescent simulator msprime (Kelleher et al. 2016) for a total of 150

diploid individuals belonging to 6 diﬀerent populations (n=25 per populations) that were historically related

by the admixture graph represented in Figure 1. Each genome consisted of 20 independent chromosomes of

= 100 Mb assuming a scaled chromosome-wide recombination rate of

= 4

LNer

= 4

000

. Similarly, the

scaled chromosome-wide mutation

= 4

LNeµ

= 4

000

. More precisely, the following msprime command

was used:

mspms 300 20 -t 4000 -I 6 50 50 50 50 50 50 0 -es 0.0125 6 0.25 -ej 0.0125 6 2 -ej 0.0125

7 3 -ej 0.025 2 1 -ej 0.05 3 1 -ej 0.075 5 4 -ej 0.1 4 1 -r 4000 100000000 -p 8

0.00

0.05

0.10

0.15

0.20

S1 S2

α=25% 1−α=75%

P1 P2 P6 P3 P4 P5

Figure 1: Simulated scenario relating the 6 populations of the working example. Names of the internal node

populations for which no data is available are written in grey.

1Following Patterson et al. (2012), we use Fto refer to the parameter and fto the statistics estimated from the data

2See http://doi.org/10.5281/zenodo.4709728

For instance,

= 4

000 if the per-base and per-generation recombination rate

= 10

−8

(i.e., it the cM per Mb ratio is equal

to 1) in a population of constant diploid eﬀective size Ne= 103

For instance, a nucleotide diversity of

= 4

000 is expected at mutation-drift equilibrium in a population of constant diploid

eﬀective size Ne= 103if the per-base mutation rate is µ= 10−8

The simulation output was further parsed to remove all variants displaying a Minor Allele Frequency (MAF)

lower than 1% which lead to a total of 472,410 remaining SNPs (from 23,246 to 24,237 per chromosome) with

position information stored in the ﬁle snp6p.snpinfo.gz

. From the resulting genotyping data, allele counts

for both the ancestral and the derived alleles (which was taken as reference) were easily obtained by simple

counting

. A Pool-Seq data set with no sequencing error was subsequently simulated from the allele count

data as described in Gautier et al. (2021). Brieﬂy, for each SNP

in each population

, we sampled a read

count

rij

for the reference allele from a Binomial distribution following

rij ∼Bin yij

nj;cij 

where

yij

is the

derived allele count for SNP

in population

is the (haploid) sample size of population

(here

= 50

for all

) and

cij

is the overall read coverage at SNP

position

. Varying overall read coverages across pools

and SNPs were simulated by sampling the

cij

’s from a Poisson distribution with mean

= 30, i.e., assuming

a read coverage of 30X for the diﬀerent pools

. Two ﬁles representative of “real-life” data format were ﬁnally

generated to store the simulate allele count and read count data:

•

a ﬁle named sim6p.genotreemix.gz

containing allele count data for each SNP and each population in

the same format as the one used in the population program Treemix (Pickrell and Pritchard 2012)

•

a ﬁle named sim6p.poolseq30X.vcf.gz

containing read count data for each SNP and each population in

avcf format similar to the one obtained with the software VarScan (Koboldt et al. 2012)

2 Reading and manipulating input data

The package poolfstat includes several utilities to parse allele count or Pool-Seq read count input data stored

in various standard formats. It is worth stressing that

distinguishing standard allele count data9from

Pool-Seq read count data is critical

to rely on the appropriate

-statistics estimator (Gautier et al.

2021; Hivert et al. 2018). Hence, in the poolfstat package, two diﬀerent S4 object classes are deﬁned to

store the data:

•

the countdata S4 class for allele count data with elements (slots) detailed in the documentation page

that can be accessed with the help function (or ?operator) as follows:

help(countdata)

•

the pooldata S4 class for (PoolSeq) read count data (PoolSeq data) with elements (slots) detailed in the

documentation page that can be accessed with the help function (or ?operator) as follows:

help(pooldata)

These classes characterize the type and origin of the data and are automatically detected by the computeFST

(section 3.1.1), compute.pairwiseFST (section 3.2) and compute.fstats (section 4) functions that implement

the diﬀerent unbiased estimators thereby ensuring that the appropriate estimation procedure is used.

2.1 Generating a countdata object for allele count data

Acountdata object can be generated from allele count data stored in two diﬀerent input ﬁle format:

•

The input ﬁle format required by the popular Treemix program (Pickrell and Pritchard 2012)

using the function genotreemix2countdata

•

The input ﬁle format required by the program BayPass (Gautier 2015) for allele count data using the

function genobaypass2countdata

The following example shows how to generate a countdata object (here named sim6p.allelecount) from the

sim6p.genotreemix.gz example ﬁle that contains the allele count data (in a Treemix ﬁle format) for the

simulated example data (section 1):

5MAF was estimated over all the 300 haploid individuals

6The simulated data being haploid this implicitly assumes Hardy-Weinberg equilibrium for the diﬀerent populations

7Note that the read count for the alternate allele is simply cij −rij

8Such a coverage is actually in the lower limit of what is usually recommended for Pool-Seq experiment

9i.e., obtained from individual genotyping data

sim6p.allelecount<-genotreemix2countdata(genotreemix.file = "sim6p.genotreemix.gz")

Information about marker positions (chromosome or scaﬀold of origin of the markers and position on the

chromosome) may be provided using the snp.pos argument of the genotreemix2countdata and genobay-

pass2countdata functions. For instance, to include the mapping information stored in the snp6p.snpinfo.gz

example ﬁle (section 1), one may use the following commands10:

positions<-read.table("sim6p.snpinfo.gz",header=TRUE,row.names=1,stringsAsFactors = FALSE)

sim6p.allelecount<-genotreemix2countdata(genotreemix.file = "sim6p.genotreemix.gz",

snp.pos=positions,verbose=FALSE)

sim6p.allelecount #display a summary of the object

* * * Countdata Object * * *

* Number of SNPs = 472410

* Number of Pops = 6

* Pop Names :

P1; P2; P3; P4; P5; P6

**************

Notice

For operations requiring marker position information i.e., block-jackknife estimation of standard

error (Appendix A.1) and estimation of multi-locus

FST

on sliding windows over the genome (section

3.1.1), markers are always assumed to be ordered in the genome as their relative position in the input

ﬁles. If no map information is provided, a default position is given to all the SNPs and they are all

assumed to map to the same chromosome. In this case, some windows or blocks may then extend

over two consecutive contigsa.

Providing the reference assembly used to order the markers is not too fragmented, the block-jackknife estimates of

standard errors may only be marginally aﬀected since the blocks are deﬁned by a number of consecutive markers (and

not by a physical length)

Additional arguments may allow ﬁltering the data for low marker polymorphism levels (min.maf ) or

genotyping call rate (min.indgeno.per.pop). More details are available in the documentation pages

of the genotreemix2countdata and genobaypass2countdata functions accessible with the commands

?genotreemix2pooldata and ?genobaypasstreemix2pooldata.

2.2 Generating a pooldata object for Pool-Seq read count data

pooldata

object can be generated from Pool-Seq read count data stored in either of the four following

formats:

•

vcf ﬁle generated by most SNP calling software commonly used to analyze Pool-Seq data including

VarScan (Koboldt et al. 2012), bcftools/SAMtools (Li et al. 2009), GATK (McKenna et al. 2010) or

FreeBayes (Garrison and Marth 2012) using the vcf2pooldata function11

•

rsync ﬁles generated by the PoPoolation software (Kofler et al. 2011) using the popsync2pooldata

function

•

The two input ﬁles (pool read count and pool haploid sizes) required by the program BayPass (Gautier

2015) to analyze Pool-Seq data using the genobaypass2pooldata function

•

The input ﬁle required by the SelEstim program (Vitalis et al. 2014) to analyze Pool-Seq data using

the genoselestim2pooldata function

If the number of markers in the snp.pos object is not consistent with that of allele count ﬁle or the given matrix is not

2-column, default value for marker positions are given and a warning message is printed in the console

The format of the vcf ﬁle is automatically detected from the genotype format that includes i) both an AD and RD ﬁelds for

VarScan vcf ﬁles; or ii) only an AD ﬁeld (with comma-separated read counts for the diﬀerent allele) other than VarScan vcf.

Parsing of vcf ﬁles has been substantially improved since poolfstat version 1.2 with computationally intensive text manipulation

now implemented in C++ routines inspired by those of the vcfR package (Knaus and Grünwald 2017)

The following example shows how to generate a pooldata object for the simulated Pool-Seq data contained in

the sim6p.poolseq30X.vcf.gz vcf ﬁle (section 1).

sim6p.readcount30X <-vcf2pooldata(vcf.file="sim6p.poolseq30X.vcf.gz",poolsizes=rep(50,6))

Reading Header lines

Parsing allele counts

VarScan like format detected for allele count data: the AD field contains allele depth

for the alternate allele and RD field for the reference allele

(N.B., positions with more than one alternate allele will be ignored)

472410 lines processed in 0 h 0 m 3 s : 472410 SNPs found

Data consists of 472410 SNPs for 6 Pools

sim6p.readcount30X #display a summary of the resulting pooldata object

* * * PoolData Object * * *

* Number of SNPs = 472410

* Number of Pools = 6

* Pool Names :

Pool1; Pool2; Pool3; Pool4; Pool5; Pool6

**************

Additional arguments may allow ﬁltering the data according to the read coverage of the pool (e.g., min.maf,

min.rc,min.cov.per.pool or max.cov.per.pool ). More details are available in the in the documentation

pages of these diﬀerent functions accessible with the commands

?vcf2pooldata

?popsync2pooldata

?genobaypass2pooldata and ?genoselestim2pooldata respectively.

2.3 Manipulating countdata and pooldata objects

The function pooldata.subset (respectively countdata.subset) allows subsetting pooldata (resp. countdata) to

retrieve just parts (e.g., some SNPs and/or population samples) of the original data sets according to various

criteria (see

?pooldata.subset

?countdata.subset

for more details). In the example below, a pooldata

object containing only information for populations P2,P3 and P6 for SNP with a MAF>0.05 is created

from the sim6p.readcount object previously created:

sim6p.readcount30X.subset<-pooldata.subset(sim6p.readcount30X,pool.index=c(2,3,6),

min.maf=0.05,verbose=FALSE)

sim6p.readcount30X.subset #display a summary of the resulting pooldata object

* * * PoolData Object * * *

* Number of SNPs = 241280

* Number of Pools = 3

* Pool Names :

Pool2; Pool3; Pool6

**************

Note that indexes of the retained SNPs from the original data set may be obtained by setting the return.snp.idx

argument to TRUE in the functions pooldata.subset (or countdata.subset). In this case the row of the matrix

stored in the snpinfo slot of the pooldata (or countdata) output object are named “rs”snp.idx (where snp.idx

is the SNP index for the original object) which makes it straightforward to obtain the indexes of the selected

SNP as shown below:

sim6p.readcount30X.subset<-pooldata.subset(sim6p.readcount30X,pool.index=c(2,3,6),

min.maf=0.05,return.snp.idx=TRUE,verbose=FALSE)

selected.snps.idx <- as.numeric(sub("rs","",rownames(sim6p.readcount30X.subset@snp.info)))

head(selected.snps.idx)

[1]134568

For pooldata objects, the functions pooldata2genobaypass and pooldata2selestim may also be used to generate

input ﬁles for the aforementioned BayPass and SelEstim programs, including the generation of sub-samples

of the original data (see ?pooldata2genobaypass and ?pooldata2selestim for more details).

3 Estimating FST

The

FST

parameter is commonly used to quantify the structuring of the genetic diversity among populations

(see e.g. Hivert et al. 2018 and references therein). It may be deﬁned as:

FST ≡Q1−Q2

1−Q2

where

is the Identity In State (IIS) probability for genes sampled within populations (or pools), and

the IIS probability for genes sampled between populations (or pools).

The computeFST and compute.pairwiseFST (for all pairs of populations) functions implement two diﬀerent

FST estimators relying on:

•

a decomposition of the total variance of allele or read count frequencies in an analysis-of-variance

framework (Weir and Cockerham 1984) which is the default procedure of the functions (as speciﬁed

by the argument method=“Anova”). The implemented estimators are derived in Weir (1996) (eq. 5.2)

(see also Akey et al. 2002) for allele count data (i.e., countdata objects, see 2.1); and in Hivert et al.

(2018) (eq. 9) for (Pool-Seq) read count data (i.e., pooldata objects, see 2.2).

•

unbiased estimators

and

of the IIS probabilities

and

(as speciﬁed by the argument

method=“Identity”). For allele count data (i.e., countdata objects, see 2.1) this estimator actually

correspond to the one used by Karlsson et al. (2007). For Pool-Seq read count data (i.e., pooldata

objects, see 2.2), the

of the

estimators are described in equations A39 and A43 respectively in

Hivert et al. (2018) Supplementary Materials.

Note that multi-locus estimates (i.e., genome-wide estimates or sliding windows estimates) are derived as the

sum of locus-speciﬁc numerators over the sum of locus-speciﬁc denominators of the diﬀerent quantities (see,

e.g., Hivert et al. 2018 and references therein).

3.1 Estimating genome-wide FST across all the populations

3.1.1 The computeFST function

The computeFST function automatically uses the appropriate estimator given the input object class (either

allele count for countdata objects or Pool-Seq read count data for pooldata objects). For instance with the

simulated example data, we obtain the following estimates of FST with:

•allele count data:

sim6p.allelecount.fst<-computeFST(sim6p.allelecount)

sim6p.allelecount.fst$FST #genome-wide Fst over all populations

[1] 0.132319

•Pool-Seq read count data:

sim6p.readcount30X.fst<-computeFST(sim6p.readcount30X)

sim6p.readcount30X.fst$FST #genome-wide Fst over all populations

[1] 0.1324199

Note that by default, the method Anova is implemented in the computeFST function which may be changed

with the method argument of the function (see section 3).

3.1.2 Block-Jackknife estimation of ˆ

FST standard-error and conﬁdence intervals:

Standard-error of the

FST

estimates can be estimated using a block-jackknife sampling approach (see Appendix

A.1) by specifying the number of consecutive SNPs deﬁning a block with the argument nsnp.per.bjack.block

(by default nsnp.per.bjack.block=0, i.e., no block-jackknife is carried out) as illustrated below for:

•allele count data:

sim6p.allelecount.fst<-computeFST(sim6p.allelecount,nsnp.per.bjack.block = 1000)

Starting Block-Jackknife sampling

462 Jackknife blocks identified with 462000 SNPs (out of 472410 ).

SNPs map to 20 different chrom/scaffolds

Average (min-max) Block Sizes: 4.232 ( 3.515 - 4.975 ) Mb

The resulting genome-wide

FST

estimated as the mean over block-jackknife sample is stored in the mean.fst

element of the output list. Note that mean.fst may slightly diﬀer from the default genome-wide estimate of

FST

(stored in the FST element of the output list, as above with no block-jackknife sampling is carried out)

as it is only computed from the SNPs eligible for block-jackknife (see Appendix A.1):

sim6p.allelecount.fst$FST #genome-wide Fst over all populations

[1] 0.132319

sim6p.allelecount.fst$mean.fst #block-jacknife estimate of s.e.

[1] 0.1324889

The standard-error of the genome-wide

FST

estimate is stored in the se.fst element of the output list and

may be used to construct 95% conﬁdence intervals (CI) intervals of the estimated values:

sim6p.allelecount.fst$se.fst #s.e. of the genome-wide Fst estimate

[1] 0.0007603234

#95% c.i. of the estimated genome-wide Fst

sim6p.allelecount.fst$mean.fst+c(-1.96,1.96)*sim6p.allelecount.fst$se.fst

[1] 0.1309987 0.1339791

•Pool-Seq read count data:

sim6p.readcount30X.fst<-computeFST(sim6p.readcount30X,nsnp.per.bjack.block = 1000,

verbose=FALSE)

sim6p.readcount30X.fst$FST #genome-wide Fst over all populations

[1] 0.1324199

sim6p.readcount30X.fst$mean.fst #block-jacknife estimate of s.e.

[1] 0.1326089

sim6p.readcount30X.fst$se.fst #s.e. of the genome-wide Fst estimate

[1] 0.0007620463

#95% c.i. of the estimated genome-wide Fst

sim6p.readcount30X.fst$mean.fst+c(-1.96,1.96)*sim6p.readcount30X.fst$se.fst

[1] 0.1311153 0.1341025

3.1.3 Computing multi-locus FST to scan the genome over sliding-windows of SNPs

The sliding.window.size argument allows computing multi-locus

FST

for sliding windows over the diﬀerent

chromosomes (or scaﬀolds/contigs), e.g., to carry out genome-scans for adaptive diﬀerentiation. Each sliding

window includes a number of consecutive SNPs speciﬁed by the sliding.window.size argument. This is

illustrated below for the Pool-Seq read count example data (similar results would be obtained with allele

count data):

sim6p.readcount30X.fst<-computeFST(sim6p.readcount30X,sliding.window.size=50)

Start sliding-window scan

20 chromosomes scanned (with more than 50 SNPs)

Average (min-max) Window Sizes 207.4 ( 90.1 - 453.2 ) kb

plot(sim6p.readcount30X.fst$sliding.windows.fst$CumulatedPosition/1e6,

sim6p.readcount30X.fst$sliding.windows.fst$MultiLocusFst,

xlab="Cumulated Position (in Mb)",ylab="Muli-locus Fst",

col=as.numeric(sim6p.readcount30X.fst$sliding.windows.fst$Chr),pch=16)

abline(h=sim6p.readcount30X.fst$FST,lty=2)

Figure 2: Manhattan plot of the multi-locus

FST

computed over sliding-windows of 50 SNPs on the Pool-

Seq example data. The dashed line indicates the estimated overall genome-wide

FST

. The 20 simulated

chromosomes are represented by alternate colors.

As expected (since the data set was simulated under neutrality), no clear outlier signal of adaptive diﬀerentia-

tion (like e.g., a tower of overly diﬀerentiated windows) shows up (Figure 2).

3.2 Estimating and visualizing pairwise-population FST

3.2.1 The compute.pairwiseFST and the heatmap functions

The compute.pairwiseFST function allows to estimate the genome-wide

FST

for all the

npop(npop−1)

pairs of

populations from data stored in either a countdata or a pooldata object. As for the computeFST function

(section 3), the compute.pairwiseFST function automatically uses the appropriate estimation procedure for

the type of the input data (either allele count for countdata objects or Pool-Seq read count data for pooldata

objects). The function returns an S4 object of class pairwisefst whose elements (slots) are detailed in the

documentation page that can be accessed with the following command (or with the ?operator):

help(pairwisefst)

The pairwise-population

FST

may then be visualized using the generic heatmap function directly applied on

the obtained pairwisefst object as illustrated below for Pool-Seq example results (similar results are obtained

with the allele count data):

sim6p.pairwisefst<-compute.pairwiseFST(sim6p.readcount30X,verbose=FALSE)

Overall Analysis Time: 0 h 0 m 3 s

heatmap(sim6p.pairwisefst)

Pool5

Pool4

Pool6

Pool3

Pool2

Pool1

Pool5

Pool4

Pool6

Pool3

Pool2

Pool1

Figure 3: Heatmap representing the pairwise-population

FST

matrix of the six populations of the 30X

Pool-Seq example data set

The resulting heat map (Figure 3) is consistent with the simulated scenario (Figure 1). Note that the

population P3 is the closest to the admixed population P6 (leading to their early clustering in the binary

tree representation) as expected from the high contribution of the P3 ancestor (1

−α

= 75%) to the admixed

ancestor of P6 and the short timing of admixture (τ=t

2Ne= 0.0125).

3.2.2 Block-Jackknife estimation of ˆ

FST standard-error and visualisation of conﬁdence inter-

vals

As with the computeFST function, standard-error of the pairwise-population

FST

estimates may be estimated

using a block-jackknife sampling approach (see Appendix A.1) by specifying the number of consecutive

SNPs forming each block with the argument nsnp.per.bjack.block (by default nsnp.per.bjack.block=0, i.e.,

no block-jackknife is carried out). The resulting estimated standard-errors may directly be used to derive

conﬁdence intervals (see above) that can also be plotted with the plot_fstats function (or directly using the

plot command that calls plot_fstats for pairwisefst objects). This is illustrated below with the allele count

example data (similar results are obtained with the Pool-Seq read count data):

sim6p.pairwisefst<-compute.pairwiseFST(sim6p.allelecount,

nsnp.per.bjack.block = 1000,verbose=FALSE)

Overall Analysis Time: 0 h 0 m 7 s

#Estimated pairwise Fst are stored in the slot values:

#5 first estimated pairwise

head(sim6p.pairwisefst@values)

Fst Estimate Fst bjack mean Fst bjack s.e. Q2 Estimate Q2 bjack mean Q2 bjack s.e. Nsnp

P1;P2 0.04946710 0.04936227 0.0007453263 0.8323244 0.8323799 0.0005895541 472410

P1;P3 0.09478083 0.09499692 0.0010947422 0.8243978 0.8244107 0.0005931841 472410

P1;P4 0.17816331 0.17843286 0.0015418662 0.8095896 0.8096142 0.0006251644 472410

P1;P5 0.17716282 0.17716008 0.0015559078 0.8094495 0.8096229 0.0006214764 472410

P1;P6 0.07039455 0.07055328 0.0009233910 0.8265288 0.8265487 0.0005869646 472410

P2;P3 0.09543866 0.09560477 0.0011336466 0.8236410 0.8236723 0.0005878641 472410

plot(sim6p.pairwisefst)

0.05 0.10 0.15

Fst

P3;P4

P1;P4

P2;P4

P3;P5

P1;P5

P2;P5

P4;P6

P5;P6

P4;P5

P2;P3

P1;P3

P1;P6

P2;P6

P1;P2

P3;P6

Figure 4: Estimated pairwise-population

FST

with their 95% conﬁdence intervals for allele count example

data set

The resulting estimated pairwise-population

FST

displayed in Figure 4 are consistent with the simulated

scenario (Figure 1). The lowest level of diﬀerentiation is observed for the P3 and P6 population pair as

expected from the high contribution of the P3 ancestor (1

−α

= 75%) to the admixed ancestor of P6 and

the short timing of admixture (τ=t

2Ne= 0.0125).

4 Estimating and visualizing f-statistics (f2,f3,f?

3,f4and D)

The

and

statistics were introduced in a seminal paper by Reich and co-workers (Reich et al. 2009)

retracing the history of Indian human population and forms the core components of a general framework for

demographic history inference detailed in Patterson et al. (2012; see also Lipson et al. 2013; Peter 2016;

Lipson 2020). These statistics measure (expected) covariance in allele frequencies among sets of two (

three (

) or four (

) populations and are formally deﬁned as follows (denoting

the SNP reference allele

frequency in population i):

•F2(A;B)≡Eh(pA−pB)2i

•F3(A;B, C )≡E[(pA−pB) (pA−pC)] ≡1

2(F2(A;B)+F2(A;C)−F2(B;C))

•F4(A, B;C, D )≡E[(pA−pB) (pC−pD)] ≡1

2(F2(A;D)+F2(B;C)−F2(A;C)−F2(B;D))

The deﬁnitions of the F parameters are not depending on the reference allele choice since

((1 −pA)−(1 −pB))2

(pB−pA)2

(pA−pB)2

. As a consequence,

and all the other

parameters may also be deﬁned in terms

of IIS within and between pairs of population as

(

;

) =

Q1−Q2

(see section 3) which allows deriving

unbiased estimators for both Pool-Seq read count and standard allele count data (Gautier et al. 2021).

Notice

With Ipopulations, there are

I

2

(

I−

1) possible

(i.e., 15 with

= 6 populations);

I

3

(

I−

1)(

I−

2) possible

(i.e., 60 with

= 6 populations); and 3

I

4

(

I−

1)(

I−

2)(

I−

possible

(i.e., 45 with

= 6 populations). However, due to their underlying linear dependency

(see the above deﬁnitions), these

(

I−

1)(

I2−I

+ 2) F-statistics form a vector space of dimension

(

I−

1) the basis of which may be speciﬁed by the set of all the

I

2

possible

statistics or, given

a reference population

(randomly chosen among the I ones) the set of all the

I−

statistics of

the form

(

;

)(with

) and the

I−1

2F3

statistics of the form

(

;

j, k

)(with

;

and

) (Patterson et al. 2012; Lipson 2020). The resulting basis is informative about population

history and may be used to ﬁt admixture graph (see section 5).

Moreover, although

statistics are diﬃcult to interpret or to compare across pairs of populations (see the

notice below), formal tests of population admixture (“3-populations” test) and tests of treeness of population

quadruplets (“4-populations” test) can directly be performed using

(see e.g., section 4.1.2) and

(see e.g.,

section 4.1.3) statistics respectively (Patterson et al. 2012). Indeed, if

(

;

B, C

)

0, we can conclude

that population A originates from a population that is admixed between two source populations related to

populations B and C respectively (although the signal may vanish if population A has drifted too much

drifted since admixture or the admixture rates are too close to 0 or 1). Conversely, if

(

A, B

;

C, D

)=0, the

populations A, B, C and D are related by a bifurcating tree with the unrooted topology (A,B;C,D) although

some may be admixed (if the paths connecting the (A,B) and (C,D) population pairs are not overlapping, see

section 4.1.3 for an example). Finally, under certain circumstances, proportions of ancestry that contributed

to a given admixed population can be estimated with ratios of

statistics (see e.g., 4.3) for set of carefully

related populations (Patterson et al. 2012).

Notice

The parameters

and

are not scaled with respect to the distribution of marker information

content (i.e., heterozygosities). As a consequence, their resulting estimates may strongly depend on

the chosen set of genetic markers (Patterson et al. 2012). The well-known

FST

parameter and the

two parameters

and

introduced by Patterson et al. (2012) correspond to scaled versions of

and

expected to be less sensitive to the SNP ascertainment and thus more comparable across

data sets. As shown in Gautier et al. (2021), these can be deﬁned in terms of IIS probabilities as:

•FST(A;B)≡F2(A;B)

1−QA,B

=QA

1+QB

1−2QA,B

21−QA,B

2

•F?

3(A;B, C )≡F3(A;B, C)

1−QA

=QA

1+QB,C

2−QA,B

2−QA,C

21−QA

1

•D(A, B;C, D )≡F4(A, B;C, D)

1−QA,B

21−QC,D

2=QA,C

2+QB,D

2−QA,D

2−QB,C

21−QA,B

21−QC,D

2

Three-population and Four-population tests naturally extend to

and

statistics respectively. An

advantage of

over

is that it is constrained to the [

−

1] interval and may thus be interpreted

as the magnitude of deviation to treeness of the tested quadruplet (Patterson et al. 2012).

4.1 The compute.fstats function and fstats objects

The compute.fstats function implements unbiased estimators of the parameters

(and

FST

(and

and

deﬁned above for allele count data (stored in countdata objects, see section 2.1) or Pool-Seq read

count data (stored in pooldata objects, see section 2.2) as described in Gautier et al. (2021)

. The function

also allows estimating within-population heterozygosities (deﬁned as 1

−Q1

) which is needed to scale branch

lengths of admixture graphs in drift units (see section 5) or for rooting neighbor-joining trees of unadmixed

populations (see section 6.1). Block-jackknife estimates of standard errors of the diﬀerent estimators (needed

for “Three-populations” and “Four-populations” tests, see below; and to ﬁt admixture graph, see section 5)

and their covariance (needed to ﬁt admixture graph, see section 5) may also be performed.

As for the computeFST (section 3.1.1) and compute.pairwiseFST (section 3.2), the compute.fstats functions

automatically detects which estimator to implement according to the class of the input object (either

countdata or pooldata). The function estimates by default all the within population heterozygosities, the

(and its scaled version

FST 13

(and its scaled version

)

statistics (and the scaled version

FST

)

Computation of

statistics (i.e., scaled

) is not carried out by default (as speciﬁed with the computeDstat

argument set to FALSE by default) since this may add some non negligible additional computation time for

data with a large number of populations due to the extra computation of the

scaling factor

although in

the example below (with only 6 populations) the diﬀerence in running time is negligible.

The results are then stored in an object of class fstats whose elements (slots) are detailed in the documentation

page accessible with the following command (or the ?operator):

help(fstats)

The underlying f-statistics may then be easily accessible or visualized with the plot_fstats function (or directly

using the plot command that calls plot_fstats for fstats objects) as illustrated below for the allele count

(sim6p.allelecount) and Pool-Seq read count (sim6p.readcount30X) example data16.

##Estimation of f-statistics on count data

sim6p.allelecount.fstats<-compute.fstats(sim6p.allelecount,nsnp.per.bjack.block = 1000,

computeDstat = TRUE)

Estimating Q1

Estimating Q2

Estimating within-population heterozygosities

Estimating F2

Estimating F3

Estimating F4

Computing Dstat

Starting Block-Jackknife sampling

462 Jackknife blocks identified with 462000 SNPs (out of 472410 ).

SNPs map to 20 different chrom/scaffolds

Average (min-max) Block Sizes: 4.232 ( 3.515 - 4.975 ) Mb

computing Q1 averages per blocks

computing Q2 averages per blocks

computing F2 averages per blocks

Starting computation of estimators s.e.

within-pop heterozygosity s.e. estimation done

F2 s.e. estimation done

F3 and F3* s.e. estimation done

estimating F4 and Dstat s.e. (may be long since require denominator averages per blocks)

F4 and D s.e. estimation done

Overall Analysis Time: 0 h 0 m 3 s

Although not deﬁned in the same way, estimator for allele count data are strictly equivalent to those by Patterson et al.

(2012)

The estimator is then actually exactly the same as the one implemented in the compute.pairwiseFST or computeFST

functions when the argument method=“Identity”

The compute.fstats function is optimized in such a way that the computational cost for the estimation of pairwise

FST

3and F4from the F2-statistics and Q2estimates is negligible

For instance on a tested allele count real data set consisting of 640,000 SNPs genotyped on 24 populations, compute.fstats

ran in 2 m 56 s (6 m 50 if nsnp.per.bjack.block = 5000 ) with computeDstat=TRUE and only 7 s (16 s) if computeDstat=FALSE

Note that estimates of the diﬀerent statistics are highly similar between the allele count and the Pool-Seq read count data.

sim6p.allelecount.fstats

* * * fstats Object * * *

Example of useful visualization functions are plot.fstats

##Estimation of f-statistics on Pool-Seq data (without computation of Dstat)

sim6p.readcount30X.fstats<-compute.fstats(sim6p.readcount30X,nsnp.per.bjack.block = 1000,

verbose=FALSE)

##Estimation of f-statistics on Pool-Seq data (with computation of Dstat)

sim6p.readcount30X.fstats<-compute.fstats(sim6p.readcount30X,nsnp.per.bjack.block = 1000,

computeDstat = TRUE,verbose=FALSE)

4.1.1 f2and FST estimates (f2.values and fst.values slots of the fstat object)

# count data (3 first f2)

head(sim6p.allelecount.fstats@f2.values,3)

Estimate bjack mean bjack s.e.

P1,P2 0.008294424 0.008274108 0.0001298376

P1,P3 0.016643720 0.016680443 0.0002108642

P1,P4 0.033924154 0.033971079 0.0003456714

# 30X Pool-Seq data (3 first f2)

head(sim6p.readcount30X.fstats@f2.values,3)

Estimate bjack mean bjack s.e.

Pool1,Pool2 0.00829429 0.008275487 0.0001351787

Pool1,Pool3 0.01665818 0.016696323 0.0002159221

Pool1,Pool4 0.03398383 0.034045968 0.0003525136

# count data (3 first Fst)

head(sim6p.allelecount.fstats@fst.values,3)

Estimate bjack mean bjack s.e.

P1,P2 0.04946710 0.04936227 0.0007453263

P1,P3 0.09478083 0.09499692 0.0010947422

P1,P4 0.17816331 0.17843286 0.0015418662

# 30X Pool-Seq data (3 first Fst)

head(sim6p.readcount30X.fstats@fst.values,3)

Estimate bjack mean bjack s.e.

Pool1,Pool2 0.04947320 0.04937253 0.000774369

Pool1,Pool3 0.09487048 0.09508626 0.001119595

Pool1,Pool4 0.17839940 0.17872795 0.001572758

Notice that the pairwise

FST

estimates are the same as those obtained with the compute.pairwiseFST function

(see section 3.2) run with method=Identity (which are here actually equal with estimates obtained by the

default Anova method because there is no variation in the total allele counts for all SNPs).

Notice

By construction

(

;

) =

(

B, A

)(and

FST

(

;

) =

FST

(

B, A

)). If

is the index of population

in the popnames or poolnames slots of the countdata or pooldata objects (i.e., the column order in

the corresponding allele or read count data matrices) used to obtain the fstats object, the

(

A, B

)

(resp. FST (B, A)) conﬁgurations reported in the slot f2.values (resp. f st.values) satisfy iA< iB.

4.1.2 f3and f?

3estimates (f3.values and f3star.values slots of the fstat object) and

3-Population tests:

# count data (3 first f3)

head(sim6p.allelecount.fstats@f3.values,3)

Estimate bjack mean bjack s.e. Z-score

P1;P2,P3 0.004053338 0.004048392 0.0001176972 34.39668

P1;P2,P4 0.004089298 0.004087474 0.0001326605 30.81155

P1;P2,P5 0.004155216 0.004149524 0.0001341405 30.93416

# 30X Pool-Seq data (3 first f3)

head(sim6p.readcount30X.fstats@f3.values,3)

Estimate bjack mean bjack s.e. Z-score

Pool1;Pool2,Pool3 0.004057570 0.004053794 0.0001247034 32.50750

Pool1;Pool2,Pool4 0.004114142 0.004114670 0.0001365234 30.13893

Pool1;Pool2,Pool5 0.004157049 0.004154896 0.0001411765 29.43051

# count data (3 first f3*)

head(sim6p.allelecount.fstats@f3star.values,3)

Estimate bjack mean bjack s.e. Z-score

P1;P2,P3 0.02552286 0.02549608 0.0007467871 34.14102

P1;P2,P4 0.02574929 0.02574221 0.0008480235 30.35554

P1;P2,P5 0.02616436 0.02613299 0.0008581050 30.45430

# 30X Pool-Seq data (3 first f3*)

head(sim6p.readcount30X.fstats@f3star.values,3)

Estimate bjack mean bjack s.e. Z-score

Pool1;Pool2,Pool3 0.02555404 0.02553297 0.0007888059 32.36915

Pool1;Pool2,Pool4 0.02591032 0.02591640 0.0008722134 29.71337

Pool1;Pool2,Pool5 0.02618054 0.02616977 0.0009009973 29.04533

Notice

By construction

(

;

B, C

) =

(

;

C, B

)(and

(

;

B, C

) =

(

;

C, B

)). If

is the index of

population

in the popnames or poolnames slots of the countdata or pooldata objects (i.e., the

column order in the corresponding allele or read count data matrices) used to obtain the fstats object,

the F3(A;B, C )(and F?

3(A;B, C )) conﬁgurations reported in the slot f3.values satisfy iB< iC.

As shown in the above example, activation of block-jackknife estimation of standard errors (i.e., argument

nsnp.per.bjack.block>0) results in the computation of Z-scores (i.e., ratio of the block-jackknife estimated

mean and standard-error) which quantiﬁes the deviation of the estimated

-statistics from 0 (in units of s.e.).

This gives a simple decision criterion for three-population tests of admixture (i.e., negative

or negative

For instance a Z-score

<−

65 provides evidence for admixture (i.e., signiﬁcantly negative

) at the 95%

signiﬁcance threshold:

# count data (F3-based 3-pop test)

tst.sel<-sim6p.allelecount.fstats@f3.values$`Z-score`<-1.65

sim6p.allelecount.fstats@f3.values[tst.sel,]

Estimate bjack mean bjack s.e. Z-score

P6;P2,P3 -0.0002671143 -0.0002623528 8.638581e-05 -3.036989

# 30X Pool-Seq data (F3-based 3-pop test)

tst.sel<-sim6p.readcount30X.fstats@f3.values$`Z-score`<-1.65

sim6p.readcount30X.fstats@f3.values[tst.sel,]

Estimate bjack mean bjack s.e. Z-score

Pool6;Pool2,Pool3 -0.0002417509 -0.0002304611 8.851744e-05 -2.603567

# count data (F3*-based 3-pop test)

tst.sel<-sim6p.allelecount.fstats@f3star.values$`Z-score`<-1.65

sim6p.allelecount.fstats@f3star.values[tst.sel,]

Estimate bjack mean bjack s.e. Z-score

P6;P2,P3 -0.001631657 -0.001603205 0.0005274779 -3.039379

# 30X Pool-Seq data (F3*-based 3-pop test)

tst.sel<-sim6p.readcount30X.fstats@f3star.values$`Z-score`<-1.65

sim6p.readcount30X.fstats@f3star.values[tst.sel,]

Estimate bjack mean bjack s.e. Z-score

Pool6;Pool2,Pool3 -0.001475866 -0.001407435 0.0005402774 -2.605024

In agreement with the simulated scenario (Figure 1), both the allele count and Pool-Seq read count data

support an admixed origin for population P6 with ancestral sources related to P2 and P3.

4.1.3 f4and Destimates (f4.values and Dstat.values slots of the fstat object) and 4-Population

tests:

# count data (3 first f4)

head(sim6p.allelecount.fstats@f4.values,3)

Estimate bjack mean bjack s.e. Z-score

P1,P2;P3,P4 3.595944e-05 3.908225e-05 1.154193e-04 0.3386111

P1,P2;P3,P5 1.018776e-04 1.011325e-04 1.197917e-04 0.8442359

P1,P2;P3,P6 5.594310e-04 5.622762e-04 6.138658e-05 9.1595941

# 30X Pool-Seq data (3 first f4)

head(sim6p.readcount30X.fstats@f4.values,3)

Estimate bjack mean bjack s.e. Z-score

Pool1,Pool2;Pool3,Pool4 5.657166e-05 6.087576e-05 1.194004e-04 0.5098456

Pool1,Pool2;Pool3,Pool5 9.947887e-05 1.011014e-04 1.253135e-04 0.8067878

Pool1,Pool2;Pool3,Pool6 5.653400e-04 5.621051e-04 6.409453e-05 8.7699380

# count data (3 first D)

head(sim6p.allelecount.fstats@Dstat.values,3)

Estimate bjack mean bjack s.e. Z-score

P1,P2;P3,P4 0.0006651386 0.0007230774 0.002135688 0.3385688

P1,P2;P3,P5 0.0018826436 0.0018705234 0.002216171 0.8440338

P1,P2;P3,P6 0.0107508025 0.0108152267 0.001182842 9.1434253

# 30X Pool-Seq data (3 first D)

head(sim6p.readcount30X.fstats@Dstat.values,3)

Estimate bjack mean bjack s.e. Z-score

Pool1,Pool2;Pool3,Pool4 0.001045743 0.001125524 0.002208046 0.5097373

Pool1,Pool2;Pool3,Pool5 0.001837431 0.001868847 0.002316956 0.8065961

Pool1,Pool2;Pool3,Pool6 0.010853673 0.010799873 0.001235136 8.7438717

Notice

When comparing two pairs of populations (

A, B

)and (

C, D

), the

statistics for the 8 quadruplets

(A,B;C,D); (B,A;C,D); (A,B;D,C); (B,A;D,C); (C,D;A,B); (C,D;B,A); (D,C;A,B) and (D,C;B,A) have

the same absolute value by deﬁnition of the F4parameter:

F4(A, B;C, D ) = F4(B, A;D, C) = F4(C, D;A, B) = F4(D, C;A, B)

−F4(A, B;C, D ) = F4(B, A;C, D) = F4(C, D;B , A) = F4(D, C;B , A) = F4(A, B;D, C)

and similarly

D(A, B;C, D ) = D(B, A;D, C) = D(C, D;A, B) = D(D, C;A, B)

−D(A, B;C, D ) = D(B, A;C, D) = D(C, D;B , A) = D(D, C;B , A) = D(A, B;D, C)

is the index of population

in the popnames or poolnames slots of the countdata or pooldata

objects (i.e., the column order in the corresponding allele or read count data matrices) used to

obtain the fstats object, the

(

A, B

;

C, D

)(and

(

A, B

;

C, D

)) conﬁgurations reported in the slot

f4.values (and Dstat.values) satisfy iA< iB;iA< iCand iC< iD.

As for

and

(section 4.1.2), activating block-jackknife estimation of standard errors (i.e., the argument

nsnp.per.bjack.block>0) results in the computation of Z-scores (i.e., ratio of the block-jackknife estimated

mean and standard-error) which quantiﬁes the deviation of the estimated

-statistics from 0 (in units of

s.e.). This gives a simple decision criterion for four-population tests of treeness (i.e., non null

For instance a Z-score lower than 1.96 in absolute value provides no evidence against the null-hypothesis of

treeness for the tested population conﬁguration at the 95% signiﬁcance threshold:

# count data

tst.sel<-abs(sim6p.allelecount.fstats@f4.values$`Z-score`)<1.96

sim6p.allelecount.fstats@f4.values[tst.sel,]

Estimate bjack mean bjack s.e. Z-score

P1,P2;P3,P4 3.595944e-05 3.908225e-05 1.154193e-04 0.3386111

P1,P2;P3,P5 1.018776e-04 1.011325e-04 1.197917e-04 0.8442359

P1,P2;P4,P5 6.591816e-05 6.205022e-05 9.313675e-05 0.6662270

P1,P3;P4,P5 1.309202e-05 1.464589e-05 1.268061e-04 0.1154983

P1,P6;P4,P5 4.031413e-05 4.458874e-05 1.058348e-04 0.4213050

P2,P3;P4,P5 -5.282615e-05 -4.740433e-05 1.302898e-04 -0.3638376

P2,P6;P4,P5 -2.560403e-05 -1.746147e-05 1.085089e-04 -0.1609221

P3,P6;P4,P5 2.722212e-05 2.994286e-05 7.650662e-05 0.3913760

# 30X Pool-Seq data

tst.sel<-abs(sim6p.readcount30X.fstats@f4.values$`Z-score`)<1.96

as.data.frame(sim6p.readcount30X.fstats@f4.values)[tst.sel,]

Estimate bjack mean bjack s.e. Z-score

Pool1,Pool2;Pool3,Pool4 5.657166e-05 6.087576e-05 1.194004e-04 0.50984558

Pool1,Pool2;Pool3,Pool5 9.947887e-05 1.011014e-04 1.253135e-04 0.80678780

Pool1,Pool2;Pool4,Pool5 4.290721e-05 4.022566e-05 9.502690e-05 0.42330817

Pool1,Pool3;Pool4,Pool5 3.769747e-07 -3.302134e-06 1.277902e-04 -0.02584028

Pool1,Pool6;Pool4,Pool5 4.073843e-05 4.734750e-05 1.053345e-04 0.44949661

Pool2,Pool3;Pool4,Pool5 -4.253023e-05 -4.352780e-05 1.336022e-04 -0.32580156

Pool2,Pool6;Pool4,Pool5 -2.168779e-06 7.121841e-06 1.102031e-04 0.06462467

Pool3,Pool6;Pool4,Pool5 4.036145e-05 5.064964e-05 8.000836e-05 0.63305432

In other words, both the allele count and Pool-Seq read count data only provide no evidence against the null

hypothesis of treeness at the 95% threshold for quadruplets involving i) non-admixed populations (P1,P2,

P3,P4 and P5 ) for conﬁgurations consistent with the simulated scenario; and ii) the admixed population P6

for conﬁgurations of the form (P6,X;P4,P5) where P4 and P5 are the two outgroup populations and X=P1,

P2 or P3. This is actually expected since for these latter quadruplets, the path connecting P4 and P5 is not

overlapping with either of the paths connecting P6 to P1,P2 or P3 in the simulated graph (Figure 1).

4.1.4 Population heterozygostity estimates (heterozygosities slot of the fstat object)

# count data (3 first populations)

head(sim6p.allelecount.fstats@heterozygosities,3)

Estimate bjack mean bjack s.e.

P1 0.1588121 0.1587849 0.0005972556

P2 0.1599502 0.1599070 0.0006057583

P3 0.1591048 0.1590328 0.0005900926

# 30X Pool-Seq data (3 first populations)

head(sim6p.readcount30X.fstats@heterozygosities,3)

Estimate bjack mean bjack s.e.

Pool1 0.1587839 0.1587670 0.0006012812

Pool2 0.1599319 0.1599083 0.0006096762

Pool3 0.1590771 0.1590230 0.0006024449

4.2 The plot_fstats function for visualization of f2(and pairwise FST ), f3(and

3) and f4(and D) estimates and their conﬁdence intervals

The plot_fstats function (that may be called directly using plot on fstats objects) allows plotting all or only

some (using the pop.sel,pop.f3.target or value.range arguments) of the estimated

fST

statistics from a fstats object. In addition, for

and

statistics, the highlight.signif argument allows

highlighting in red signiﬁcant (as deﬁned with the ci.perc argument) three-population or four-population

tests. Some example plots are shown in Figures 5, 6 and 7 which were generated with the following codes:

4.2.1 Example of f2statistics plot (Figure 5)

layout(matrix(1:2,2,1,byrow=T))

plot(sim6p.allelecount.fstats,main="F2 (Allele Count)")

plot(sim6p.readcount30X.fstats,main="F2 (30X Pool-Seq)")

Similar plots may be obtained for the scaled

(i.e., pairwise

fST

) by specifying stat.name=“Fst” (see also

the compute.pairwiseFST functions described in section 3.2).

4.2.2 Example of f3statistics plot (Figure 6)

layout(matrix(1:4,2,2,byrow=T))

plot(sim6p.allelecount.fstats,stat.name="F3",main="F3 (Allele Count)")

plot(sim6p.readcount30X.fstats,stat.name="F3",main="F3 (30X Pool-Seq)")

plot(sim6p.readcount30X.fstats,stat.name="F3",pop.f3.target=c("Pool6","Pool1"),

main="30X Pool-Seq (only F3 with P6 or P1 as target pops)")

plot(sim6p.readcount30X.fstats,stat.name="F3",value.range=c(NA,5e-3),

main="30X Pool-Seq (only F3 < 5e-3)")

Similar plots may be obtained for the scaled f3(i.e., f?

3) by specifying stat.name=“F3star”.

4.2.3 Example of f4and Dstatistics plot (Figure 7)

layout(matrix(1:6,3,2,byrow=T))

plot(sim6p.allelecount.fstats,stat.name="Dstat",main="D (Allele Count)")

plot(sim6p.readcount30X.fstats,stat.name="Dstat",main="D (30X Pool-Seq)")

plot(sim6p.allelecount.fstats,stat.name="F4",main="F4 (Allele Count)")

0.005 0.010 0.015 0.020 0.025 0.030 0.035

F2 (Allele Count)

P3,P4

P2,P4

P1,P4

P3,P5

P1,P5

P2,P5

P4,P6

P5,P6

P4,P5

P2,P3

P1,P3

P1,P6

P2,P6

P1,P2

P3,P6

0.005 0.010 0.015 0.020 0.025 0.030 0.035

F2 (30X Pool−Seq)

Pool3,Pool4

Pool2,Pool4

Pool1,Pool4

Pool3,Pool5

Pool1,Pool5

Pool2,Pool5

Pool4,Pool6

Pool5,Pool6

Pool4,Pool5

Pool2,Pool3

Pool1,Pool3

Pool1,Pool6

Pool2,Pool6

Pool1,Pool2

Pool3,Pool6

Figure 5: Estimated

statistics with their 95% conﬁdence intervals for the allele count and 30X Pool-Seq

data sets

plot(sim6p.readcount30X.fstats,stat.name="F4",main="F4 (30X Pool-Seq)")

plot(sim6p.readcount30X.fstats,stat.name="F4",pop.sel=c("Pool1","Pool2"),

main="30X Pool-Seq (only F4 with both P1 and P2)")

plot(sim6p.readcount30X.fstats,stat.name="F4",value.range=c(-2e-3,2e-3),

main="30X Pool-Seq (only -2e-3 < F4 < 2e-3)")

4.3 Estimating admixture proportions with f4-ratios

Given an admixture graph (assumed to be correct), ratios of

-statistics (Patterson et al. 2012) may

provide estimates of the relative contributions of the ancestral sources of a (two-way) admixed populations

(P6 in our example) if outgroups (P1 and P4 or P5 in our example) for the two source population proxies

(e.g., P2 and P3 in our example) have been sampled. For instance, the proportion

2-related ancestry

in population P6(Figure 1) is equal to:

α=F4(P1, P 4; P3, P 6)

F4(P1, P 4; P2, P 3) =F4(P1, P 5; P3, P 6)

F4(P1, P 5; P2, P 3)

0.000 0.005 0.010 0.015 0.020 0.025 0.030

F3 (Allele Count)

P4;P3,P6

P5;P3,P6

P4;P1,P2

P5;P1,P2

P4;P2,P6

P5;P2,P6

P4;P1,P6

P5;P1,P6

P4;P1,P3

P4;P2,P3

P5;P1,P3

P5;P2,P3

P3;P4,P5

P2;P4,P5

P1;P4,P5

P6;P4,P5

P4;P2,P5

P4;P5,P6

P4;P3,P5

P4;P1,P5

P3;P1,P2

P5;P1,P4

P5;P3,P4

P5;P4,P6

P5;P2,P4

P1;P3,P6

P2;P3,P6

P3;P2,P5

P3;P2,P4

P3;P1,P4

P3;P1,P5

P2;P3,P4

P2;P3,P5

P1;P3,P5

P1;P3,P4

P6;P1,P2

P1;P5,P6

P1;P4,P6

P2;P4,P6

P2;P5,P6

P3;P2,P6

P6;P1,P4

P6;P1,P5

P3;P1,P6

P1;P2,P6

P6;P2,P5

P6;P2,P4

P2;P1,P3

P2;P1,P4

P1;P2,P5

P2;P1,P5

P1;P2,P4

P1;P2,P3

P3;P5,P6

P3;P4,P6

P2;P1,P6

P6;P3,P4

P6;P3,P5

P6;P1,P3

P6;P2,P3

0.000 0.005 0.010 0.015 0.020 0.025 0.030

F3 (30X Pool−Seq)

Pool4;Pool3,Pool6

Pool5;Pool3,Pool6

Pool4;Pool1,Pool2

Pool5;Pool1,Pool2

Pool4;Pool2,Pool6

Pool5;Pool2,Pool6

Pool4;Pool1,Pool6

Pool5;Pool1,Pool6

Pool4;Pool1,Pool3

Pool4;Pool2,Pool3

Pool5;Pool1,Pool3

Pool5;Pool2,Pool3

Pool3;Pool4,Pool5

Pool2;Pool4,Pool5

Pool1;Pool4,Pool5

Pool6;Pool4,Pool5

Pool4;Pool5,Pool6

Pool4;Pool2,Pool5

Pool4;Pool1,Pool5

Pool4;Pool3,Pool5

Pool3;Pool1,Pool2

Pool5;Pool3,Pool4

Pool5;Pool1,Pool4

Pool5;Pool2,Pool4

Pool5;Pool4,Pool6

Pool1;Pool3,Pool6

Pool2;Pool3,Pool6

Pool3;Pool2,Pool5

Pool3;Pool2,Pool4

Pool3;Pool1,Pool5

Pool3;Pool1,Pool4

Pool2;Pool3,Pool4

Pool2;Pool3,Pool5

Pool1;Pool3,Pool4

Pool1;Pool3,Pool5

Pool6;Pool1,Pool2

Pool1;Pool5,Pool6

Pool1;Pool4,Pool6

Pool2;Pool5,Pool6

Pool2;Pool4,Pool6

Pool3;Pool2,Pool6

Pool6;Pool1,Pool4

Pool6;Pool1,Pool5

Pool3;Pool1,Pool6

Pool1;Pool2,Pool6

Pool6;Pool2,Pool4

Pool6;Pool2,Pool5

Pool2;Pool1,Pool3

Pool2;Pool1,Pool4

Pool1;Pool2,Pool5

Pool2;Pool1,Pool5

Pool1;Pool2,Pool4

Pool1;Pool2,Pool3

Pool3;Pool5,Pool6

Pool3;Pool4,Pool6

Pool2;Pool1,Pool6

Pool6;Pool3,Pool4

Pool6;Pool3,Pool5

Pool6;Pool1,Pool3

Pool6;Pool2,Pool3

0.000 0.005 0.010 0.015 0.020

30X Pool−Seq (only F3 with P6 or P1 as target pops)

Pool1;Pool4,Pool5

Pool6;Pool4,Pool5

Pool1;Pool3,Pool6

Pool1;Pool3,Pool4

Pool1;Pool3,Pool5

Pool6;Pool1,Pool2

Pool1;Pool5,Pool6

Pool1;Pool4,Pool6

Pool6;Pool1,Pool4

Pool6;Pool1,Pool5

Pool1;Pool2,Pool6

Pool6;Pool2,Pool4

Pool6;Pool2,Pool5

Pool1;Pool2,Pool5

Pool1;Pool2,Pool4

Pool1;Pool2,Pool3

Pool6;Pool3,Pool4

Pool6;Pool3,Pool5

Pool6;Pool1,Pool3

Pool6;Pool2,Pool3

0.000 0.001 0.002 0.003 0.004 0.005

30X Pool−Seq (only F3 < 5e−3)

Pool1;Pool2,Pool6

Pool6;Pool2,Pool4

Pool6;Pool2,Pool5

Pool2;Pool1,Pool3

Pool2;Pool1,Pool4

Pool1;Pool2,Pool5

Pool2;Pool1,Pool5

Pool1;Pool2,Pool4

Pool1;Pool2,Pool3

Pool3;Pool5,Pool6

Pool3;Pool4,Pool6

Pool2;Pool1,Pool6

Pool6;Pool3,Pool4

Pool6;Pool3,Pool5

Pool6;Pool1,Pool3

Pool6;Pool2,Pool3

Figure 6: Estimated

statistics with their 95% conﬁdence intervals for the allele count and 30X Pool-Seq

data sets

The compute.f4ratio function implements

-ratio based estimators of admixture proportion from an fs-

tats object. It requires to specify both the numerator (num.quadruplet argument) and the denominator

(den.quadruplet)

quadruplets

. The following example illustrates how to use the function to the admixture

proportion

(

αsimulated

= 0

25) from the P2-related source that contributed to the P6 ancestral population

(Figure 1):

In practice, the required

may not be directly available from the f4.values slot of the fstats since, as mentioned in section

4.1.3, only one out of the 8 equivalent quadruplet conﬁgurations is reported (the chosen one only depending on the ordering of

the populations in the countdata or pooldata objects used for estimation). However, the function automatically computes the

for the numerator and denominator conﬁgurations speciﬁed by the user from the estimator available in the input fstats object

−0.3 −0.2 −0.1 0.0 0.1 0.2

D (Allele Count)

P1,P4;P2,P5

P1,P5;P2,P4

P1,P4;P3,P5

P1,P5;P3,P4

P2,P5;P3,P4

P2,P4;P3,P5

P1,P3;P2,P6

P1,P6;P2,P3

P1,P3;P2,P5

P1,P3;P2,P4

P1,P4;P2,P3

P1,P5;P2,P3

P1,P3;P4,P6

P1,P3;P5,P6

P2,P3;P5,P6

P2,P3;P4,P6

P1,P4;P2,P6

P1,P5;P2,P6

P1,P6;P2,P5

P1,P6;P2,P4

P1,P2;P3,P6

P1,P2;P4,P6

P1,P2;P5,P6

P1,P2;P3,P5

P1,P2;P4,P5

P1,P6;P4,P5

P1,P2;P3,P4

P3,P6;P4,P5

P1,P3;P4,P5

P2,P6;P4,P5

P2,P3;P4,P5

P1,P5;P3,P6

P1,P4;P3,P6

P2,P5;P3,P6

P2,P4;P3,P6

P1,P6;P3,P5

P1,P6;P3,P4

P2,P6;P3,P4

P2,P6;P3,P5

P1,P5;P4,P6

P1,P4;P5,P6

P2,P4;P5,P6

P2,P5;P4,P6

P3,P5;P4,P6

P3,P4;P5,P6

−0.3 −0.2 −0.1 0.0 0.1 0.2

D (30X Pool−Seq)

Pool1,Pool4;Pool2,Pool5

Pool1,Pool5;Pool2,Pool4

Pool1,Pool5;Pool3,Pool4

Pool1,Pool4;Pool3,Pool5

Pool2,Pool5;Pool3,Pool4

Pool2,Pool4;Pool3,Pool5

Pool1,Pool3;Pool2,Pool6

Pool1,Pool6;Pool2,Pool3

Pool1,Pool3;Pool2,Pool4

Pool1,Pool3;Pool2,Pool5

Pool1,Pool4;Pool2,Pool3

Pool1,Pool5;Pool2,Pool3

Pool1,Pool3;Pool5,Pool6

Pool1,Pool3;Pool4,Pool6

Pool2,Pool3;Pool5,Pool6

Pool1,Pool5;Pool2,Pool6

Pool1,Pool4;Pool2,Pool6

Pool2,Pool3;Pool4,Pool6

Pool1,Pool6;Pool2,Pool5

Pool1,Pool6;Pool2,Pool4

Pool1,Pool2;Pool3,Pool6

Pool1,Pool2;Pool4,Pool6

Pool1,Pool2;Pool5,Pool6

Pool1,Pool2;Pool3,Pool5

Pool1,Pool2;Pool3,Pool4

Pool3,Pool6;Pool4,Pool5

Pool1,Pool6;Pool4,Pool5

Pool1,Pool2;Pool4,Pool5

Pool2,Pool6;Pool4,Pool5

Pool1,Pool3;Pool4,Pool5

Pool2,Pool3;Pool4,Pool5

Pool1,Pool5;Pool3,Pool6

Pool1,Pool4;Pool3,Pool6

Pool2,Pool5;Pool3,Pool6

Pool2,Pool4;Pool3,Pool6

Pool1,Pool6;Pool3,Pool5

Pool1,Pool6;Pool3,Pool4

Pool2,Pool6;Pool3,Pool5

Pool2,Pool6;Pool3,Pool4

Pool1,Pool5;Pool4,Pool6

Pool1,Pool4;Pool5,Pool6

Pool2,Pool5;Pool4,Pool6

Pool2,Pool4;Pool5,Pool6

Pool3,Pool5;Pool4,Pool6

Pool3,Pool4;Pool5,Pool6

−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015

F4 (Allele Count)

P1,P4;P2,P5

P1,P5;P2,P4

P1,P4;P3,P5

P1,P5;P3,P4

P2,P5;P3,P4

P2,P4;P3,P5

P1,P3;P2,P6

P1,P6;P2,P3

P1,P3;P2,P5

P1,P3;P2,P4

P1,P4;P2,P3

P1,P5;P2,P3

P1,P3;P4,P6

P1,P3;P5,P6

P2,P3;P5,P6

P2,P3;P4,P6

P1,P4;P2,P6

P1,P5;P2,P6

P1,P6;P2,P5

P1,P6;P2,P4

P1,P2;P3,P6

P1,P2;P4,P6

P1,P2;P5,P6

P1,P2;P3,P5

P1,P2;P4,P5

P1,P6;P4,P5

P1,P2;P3,P4

P3,P6;P4,P5

P1,P3;P4,P5

P2,P6;P4,P5

P2,P3;P4,P5

P1,P5;P3,P6

P1,P4;P3,P6

P2,P5;P3,P6

P2,P4;P3,P6

P1,P6;P3,P5

P1,P6;P3,P4

P2,P6;P3,P4

P2,P6;P3,P5

P1,P5;P4,P6

P1,P4;P5,P6

P2,P4;P5,P6

P2,P5;P4,P6

P3,P5;P4,P6

P3,P4;P5,P6

−0.015 −0.010 −0.005 0.000 0.005 0.010 0.015

F4 (30X Pool−Seq)

Pool1,Pool4;Pool2,Pool5

Pool1,Pool5;Pool2,Pool4

Pool1,Pool5;Pool3,Pool4

Pool1,Pool4;Pool3,Pool5

Pool2,Pool5;Pool3,Pool4

Pool2,Pool4;Pool3,Pool5

Pool1,Pool3;Pool2,Pool6

Pool1,Pool6;Pool2,Pool3

Pool1,Pool3;Pool2,Pool4

Pool1,Pool3;Pool2,Pool5

Pool1,Pool4;Pool2,Pool3

Pool1,Pool5;Pool2,Pool3

Pool1,Pool3;Pool5,Pool6

Pool1,Pool3;Pool4,Pool6

Pool2,Pool3;Pool5,Pool6

Pool2,Pool3;Pool4,Pool6

Pool1,Pool5;Pool2,Pool6

Pool1,Pool4;Pool2,Pool6

Pool1,Pool6;Pool2,Pool5

Pool1,Pool6;Pool2,Pool4

Pool1,Pool2;Pool3,Pool6

Pool1,Pool2;Pool4,Pool6

Pool1,Pool2;Pool5,Pool6

Pool1,Pool2;Pool3,Pool5

Pool1,Pool2;Pool3,Pool4

Pool3,Pool6;Pool4,Pool5

Pool1,Pool6;Pool4,Pool5

Pool1,Pool2;Pool4,Pool5

Pool2,Pool6;Pool4,Pool5

Pool1,Pool3;Pool4,Pool5

Pool2,Pool3;Pool4,Pool5

Pool1,Pool5;Pool3,Pool6

Pool1,Pool4;Pool3,Pool6

Pool2,Pool5;Pool3,Pool6

Pool2,Pool4;Pool3,Pool6

Pool1,Pool6;Pool3,Pool5

Pool1,Pool6;Pool3,Pool4

Pool2,Pool6;Pool3,Pool5

Pool2,Pool6;Pool3,Pool4

Pool1,Pool5;Pool4,Pool6

Pool1,Pool4;Pool5,Pool6

Pool2,Pool5;Pool4,Pool6

Pool2,Pool4;Pool5,Pool6

Pool3,Pool5;Pool4,Pool6

Pool3,Pool4;Pool5,Pool6

0.000 0.005 0.010 0.015

30X Pool−Seq (only F4 with both P1 and P2)

Pool1,Pool4;Pool2,Pool5

Pool1,Pool5;Pool2,Pool4

Pool1,Pool3;Pool2,Pool6

Pool1,Pool6;Pool2,Pool3

Pool1,Pool3;Pool2,Pool4

Pool1,Pool3;Pool2,Pool5

Pool1,Pool4;Pool2,Pool3

Pool1,Pool5;Pool2,Pool3

Pool1,Pool5;Pool2,Pool6

Pool1,Pool4;Pool2,Pool6

Pool1,Pool6;Pool2,Pool5

Pool1,Pool6;Pool2,Pool4

Pool1,Pool2;Pool3,Pool6

Pool1,Pool2;Pool4,Pool6

Pool1,Pool2;Pool5,Pool6

Pool1,Pool2;Pool3,Pool5

Pool1,Pool2;Pool3,Pool4

Pool1,Pool2;Pool4,Pool5

−0.0015 −0.0010 −0.0005 0.0000 0.0005

30X Pool−Seq (only −2e−3 < F4 < 2e−3)

Pool1,Pool2;Pool3,Pool6

Pool1,Pool2;Pool4,Pool6

Pool1,Pool2;Pool5,Pool6

Pool1,Pool2;Pool3,Pool5

Pool1,Pool2;Pool3,Pool4

Pool3,Pool6;Pool4,Pool5

Pool1,Pool6;Pool4,Pool5

Pool1,Pool2;Pool4,Pool5

Pool2,Pool6;Pool4,Pool5

Pool1,Pool3;Pool4,Pool5

Pool2,Pool3;Pool4,Pool5

Pool1,Pool5;Pool3,Pool6

Pool1,Pool4;Pool3,Pool6

Pool2,Pool5;Pool3,Pool6

Pool2,Pool4;Pool3,Pool6

Figure 7: Estimated

and

statistics with their 95% conﬁdence intervals for the allele count and 30X

Pool-Seq data sets

# count data (two possible estimates)

compute.f4ratio(sim6p.allelecount.fstats,num.quadruplet = "P1,P4;P3,P6",

den.quadruplet="P1,P4;P2,P3")

[1] -0.2497209

compute.f4ratio(sim6p.allelecount.fstats,num.quadruplet = "P1,P5;P3,P6",

den.quadruplet="P1,P5;P2,P3")

[1] -0.2462923

# 30X Pool-Seq data (two possible estimates)

compute.f4ratio(sim6p.readcount30X.fstats,num.quadruplet = "Pool1,Pool4;Pool3,Pool6",

den.quadruplet="Pool1,Pool4;Pool2,Pool3")

[1] -0.25285

compute.f4ratio(sim6p.readcount30X.fstats,num.quadruplet = "Pool1,Pool5;Pool3,Pool6",

den.quadruplet="Pool1,Pool5;Pool2,Pool3")

[1] -0.24559

Standard errors of the estimated admixture proportions are automatically computed if the input fstats object

was obtained by running the compute.fstats function with return.F4.blockjackknife.samples = TRUE (and

nsnp.per.bjack.block>0 ) to activate block-jackknife estimation of standrd errors) as illustrated below for the

Pool-Seq read count data:

sim6p.readcount30X.fstats<-compute.fstats(sim6p.readcount30X,nsnp.per.bjack.block = 1000,

verbose=FALSE,return.F4.blockjackknife.samples = TRUE)

alpha.est=compute.f4ratio(sim6p.readcount30X.fstats,num.quadruplet = "Pool1,Pool4;Pool3,Pool6",

den.quadruplet="Pool1,Pool4;Pool2,Pool3")

#95% c.i. of alpha derived from the se

alpha.est[2]+c(-1.96,1.96)*alpha.est[3]

[1] -0.3013463 -0.2100506

#prediction error in units of s.e.

as.numeric(abs(0.25-alpha.est[2])/alpha.est[3])

[1] 21.71337

Note that the simulated value (

= 0

25) is within the conﬁdence interval and actually less than 0

25 standard

error higher than the F4-ratio estimate.

5 Using f-statistics to estimate parameters of admixture graphs

The f-statistics can be used to estimate the parameters (branch lengths and/or admixture proportions) of

trees or admixture graphs (i.e., trees including admixture edges) that summarize the demographic history of

the surveyed populations. The approach implemented in poolfstat to ﬁt admixture graphs from f-statistics is

directly inspired (and actually highly similar) to the one used in the qpGraph software originally described by

Patterson et al. (2012; see also Lipson 2020). The core functions used for admixture graph ﬁtting consist

of:

•

the generate.graph.params function to deﬁne graph parameters specifying the candidate graph to ﬁt

and the underlying f-statistics provided as an fstats object (section 4)

•the ﬁt.graph function to estimate graph parameters using an optimization algorithm

•

the compare.ﬁtted.fstats function to assess model ﬁt by comparing estimated and ﬁtted f-statistics

(Patterson et al. 2012; Lipson 2020)

•

the add.leave function to evaluate all the possible admixture graphs (or trees) resulting from the addition

of a new leave to an existing graph (connected with either non-admixed or admixed edges).

5.1 Generating a graph.params object with the generate.graph.params function

5.1.1 Specifying the structure of the admixture graph in a graph.params object

Admixture graph speciﬁcation including the structure of the graph (i.e., topology consisting of edges and, if

any admixture proportions) are deﬁned in an object of class graph.params detailed in the documentation

page accessible with the following command (or the ?operator):

help(graph.params)

The graph.params objects may be constructed with the generate.graph.params function from a user-deﬁned

(character) matrix specifying the structure of the admixture graph (or tree if no admixture edges are included)

and consisting of three columns deﬁning for each edge (whether admixed or not) i) the child node; ii) the

parent node; iii) the admixture proportion (blank for non-admixed edges). As a result, in the input matrix,

each admixture event is speciﬁed by two rows for the two admixture edges corresponding to the same

admixed child node and two diﬀerent source nodes as the parent node (i.e., source populations). Their third

column elements contain the two underlying admixture proportions coded as aand (1-a) (the parentheses

are mandatory and an error message is printed if absent) where ais the name of the admixture proportion

(names of admixture proportions should not include space). The example below shows the construction of

agraph.params object specifying the admixture graph for the scenario used to simulate the example data

(Figure 1):

sim.graph<-rbind(c("P1","P7",""),c("P2","s1",""),c("P3","s2",""),c("P6","S",""),

c("S","s1","a"),c("S","s2","(1-a)"),c("s2","P8",""),c("s1","P7",""),

c("P4","P9",""),c("P5","P9",""),c("P7","P8",""),

c("P8","R",""),c("P9","R",""))

sim.graph.params<-generate.graph.params(sim.graph)

sim.graph.params

* * * graph.params Object * * *

Example of useful functions are:

plot() to visualize the graph (interface for grViz() from the DiagrammeR package)

fit.graph() to estimate graph parameter values

****************

The root is automatically identiﬁed by the generate.graph.params function (as the node only present in the

parent node column) and several checks are made within the function. It is however recommended to check

the graph by plotting using the plot function. The following code shows how to plot the graph stored in the

example sim.graph.params object that was generated above (Figure 8):

plot(sim.graph.params)

P4 P5P7

(1-a)

P8 P9

Figure 8: Plot of the admixture graph specifying the simulated scenario

Notice

The plot function applied to graph.params objects internally calls the grViz function from the

DiagrammeR package (Iannone 2020) which actually generates an object of class htmlwidget that

will print itself into HTML in a browser. Also, if run within the Rstudio IDE, the graph will be

plotted in the View pane

. Hence, graph plots may not be easily exported from R into a pdf (or

other) device although the following trick may be useful to that end:

require(webshot,htmlwidgets,imager)

tmp<-plot(sim.graph.params) #plot the graph

saveWidget(tmp,"tmp.html") ; webshot("tmp.html","tmp.png")

load.image("tmp.png")%>% autocrop() %>% plot(axes=F)

To insert a graph plot into an external document or edit it (e.g., to change node or edge colors, node

names, etc.), one may also directly rely on its dot coded deﬁnition

stored in the dot.graph slot of the

graph.params (or ﬁtted.graph, see section 5.2) object outside R. A graph dot ﬁle may also be generated

by specifying an output ﬁle name preﬁx with the outﬁlepreﬁx argument of the generate.graph.params

functionc.

aRegularly clearing the View items using the broom icon is recommended

From the open source graph visualization software graphviz (https://graphviz.org/). For instance, dot ﬁles

can be converted into png ﬁle using the command dot -Tpng inputgraph.dot in a Linux terminal. Several online

user-friendly implementations also allow very convenient manipulation of dot ﬁles from a web-browser, see e.g.,

https://dreampuf.github.io/GraphvizOnline

Alternatively, one may also use the command writeLines(x@dot.g raph,con=outﬁle) where xis the graph.params

object and outﬁle is the desired name of the dot output ﬁle (e.g., “out.dot”)

The graph.params object produced by the generate.graph.params function includes a symbolic representation

of the graph incidence matrix (slot graph.matrix) which consists of a

leaves by

edges matrix containing

the edges weight for the paths from each leave to the root

. Examples for the sim.graph.params object that

speciﬁes the simulation scenario are given below:

#names of the edges (automatically given)

sim.graph.params@edges.names

[1] "P7<->P1" "s1<->P2" "s2<->P3" "S<->P6" "P8<->s2" "P7<->s1" "P9<->P4" "P9<->P5" "P8<->P7" "R<->P8" "R<->P9"

#names of the admixture proportions (automatically given)

sim.graph.params@adm.params.names

[1] "a"

#graph incidence matrix

sim.graph.params@graph.matrix

P7<->P1 s1<->P2 s2<->P3 S<->P6 P8<->s2 P7<->s1 P9<->P4 P9<->P5 P8<->P7 R<->P8 R<->P9

P1 "1" "0" "0" "0" "0" "0" "0" "0" "1" "1" "0"

P2 "0" "1" "0" "0" "0" "1" "0" "0" "1" "1" "0"

P3 "0" "0" "1" "0" "1" "0" "0" "0" "0" "1" "0"

P6 "0" "0" "0" "1" "1-a" "a" "0" "0" "a" "1" "0"

P4 "0" "0" "0" "0" "0" "0" "1" "0" "0" "0" "1"

P5 "0" "0" "0" "0" "0" "0" "0" "1" "0" "0" "1"

In the symbolic representation, the names for the graph edges and admixture proportions correspond to those stored in the

edges.names and adm.params.names slots of the graph.params object respectively

Notice

As mentioned by Pickrell and Pritchard (2012), Patterson et al. (2012) and Lipson (2020),

the three branch lengths surrounding an admixture event (e.g., edges

s1 ↔S

s2 ↔S

and

S↔P6

connecting s1 to S;s2 to S; and Sto P6 respectively for the admixture event in Figure 8) are

not identiﬁable and can only be estimated jointly in a single compound parameter (e.g.,

a2×es1↔S

+ (1

−a

)

2×es2↔S

eS↔P6

in Figure 8) unless samples from the source populations (s1,s2

and/or S) or samples from diﬀerent populations deriving from the same admixed source are available

(Lipson 2020). Following Patterson et al. (2012) and Lipson (2020), this identiﬁability issue

is solved by nullifying the length of admixture edges (i.e., setting

es1↔S

es2↔S

= 0 in the above

example) which may lead to some overestimation of the divergence (branch length) of the admixed

population (here P6 ) from its direct admixed ancestor (here S) if the two source populations (here s1

and s2 ) have experienced strong divergence since their separation from the graph connecting the

other populations (which is not the case in the simulated example in which both

es1↔S

and

es2->S

are

equal to 0). This diﬀers from the choice made by Pickrell and Pritchard (2012) in the Treemix

model consisting of setting eS↔P6 = 0 and es2↔S= 0 (if a > 0.5) or es1↔S= 0 (if a < 0.5).

The graph incidence matrix plays a pivotal role for graph ﬁtting since the ﬁt.graph function (see section 5.1.2)

use it to build the model equations. A (simpliﬁed) symbolic representation of these model equations together

with expression for the parameters

and

can also be generated from a graph.params object with the

graph.params2symbolic.fstats function (see section 7.1).

5.1.2 Generating a graph.params object with f-statistics estimates for admixture graph ﬁtting

The f-statistics estimates (

and

) need to be included within the graph.params object to allow further

ﬁtting of the admixture graph (i.e., estimation of its edge lengths and, if any, admixture proportions) with the

ﬁt.graph function (see section 5.2). This can be done by providing the generate.graph.params function with

an fstats object including all the

and

statistics involving the graph leaves (which is usually the case)

and block-jackknife estimates of their standard errors and error covariance matrix (i.e., the compute.fstats

function used to generate the fstats object must have been run with nsnp.per.bjack.block>0), otherwise an

error message is returned

. If

the number of leaves of the admixture graph and

a “reference” population

among the

ones

, the generate.graph.params function selects the estimates (block-jackknife means) for

the

nl−

statistics of the form

(

;

)(with

a leave population other than

) and the

nl−1

2f3

statistics of the form F3(A;B, C )(with Band Ctwo leave populations other than A) which form the basis

of the f-statistics vector (see section 4). The basis f-statistics are stored in the f2.target and f3.target slots

of the resulting graph.params object (with their corresponding names available in the f2.target.pops and

f3.target.pops slots respectively). In addition, as required to further ﬁt the admixture graph (see section

5.2), the

nl(nl−1)

covariance matrix of the basis f-statistics is stored in the f.Qmat slot of the

graph.params object. Optionally, if available in the fstats object, estimates of the leaves heterozygoties (needed

to scale ﬁtted branch lengths in drift units, see 5.2.1) are stored in the Het slot of the graph.params object.

The following code shows how to generate a graph.params object for the example (allele count) data:

sim.graph.params<-generate.graph.params(sim.graph,fstats = sim6p.allelecount.fstats)

Total Number of Parameters: 11 (10 edges lengths + 1 adm. coeff.)

Total Number of Statistics: 15 (5 F2 and 10 F3)

As shown above, the functions returns the number of parameters

npar

= 2

+ 2

na−

3of the admixture

graph where

and

are the number of admixture events

and the number of leaves

, respectively

(Lipson 2020). Note that the plotting properties of the graph.params object remain the same whether fstats

information is included or not (i.e., the plot function may be used as above to generate the representation

Conversely, the fstats object may include f-statistics involving populations other than the graph leaves, the gener-

ate.graph.params function selecting only the f-statistics relevant for the ﬁtting of the input graph

by default,

is the ﬁrst population in the vector of leaves but, although of limited interest, another reference population

may be speciﬁed with the popref argument

displayed in Figure 8).

5.2 Fitting a graph with the ﬁt.graph function

The ﬁt.graph functions provides estimate of the parameters (i.e., edge length and admixture proportions) of an

admixture graph stored in a graph.params object as detailed in Gautier et al. (2021) and directly inspired by

Patterson et al. (2012). Brieﬂy, let

represent the vector of length

nl(nl−1)

(where

is the number of graph

leaves) of the estimated

and

basis f-statistics

and

g(e;a)

X(a)×e

the vector of these expected basis

f-statistics values given the vector of graph edges lengths

and the incidence matrix

X(a)

that depends on the

structure of the graph and the admixture rates

a22

. Let further

represent the

nl(nl−1)

covariance

matrix of the basis F-statistics estimates estimated by block-jackknife and stored in the slot f.Qmat of the

input graph.params object (see 5.1.2)

. The function attempts to ﬁnd the graph parameter values (

ˆe

and

ˆa

)

that minimize a cost (score of the model) deﬁned as

(

;

) =

b

f−g(e;a)0

Q−1b

f−g(e;a)

. As mentioned

by Patterson et al. (2012), given admixture rates

(

;

)is actually quadratic in the edge lengths

allow

the ﬁt.graph function to rely on the Lawson-Hanson non-negative linear least squares algorithm implemented

in the nnls function (nnls package) to estimate the vector

ˆe

that minimizes

(

;

)(subject to the constraint of

positive edge lengths). Full minimization of

(

;

)is thus reduced to the identiﬁcation of the admixture rates

which is performed using the L-BFGS-B method

implemented in the optim function (stats package). The

eps.admix.prop argument allows specifying the lower and upper bound of the admixture rates to eps.admix.prop

and 1-eps.admix.prop respectively. In addition, assuming

f∼

(g(ˆe;ˆa),Q)25

(

ˆe

;

ˆa

) =

−

log

(

)

−K

where

is the likelihood of the ﬁtted graph and

nlog

) +

log

(

|Q|

)allowing to almost directly derive a

BI C

(Bayesian Information Criterion) of the ﬁtted graph from the optimized score

(

ˆe

;

ˆa

)

. The BIC may be

used for comparison of diﬀerent admixture graphs (see section 5.3) providing they were all ﬁtted based on

the same vector of f-statistics (i.e., they include the same set of populations). Indeed, when comparing two

graphs

and

with BIC equal to

BI C1

and

BI C2

,∆

BI C2−BI C1'

log (BF12 )

where

BF12

the Bayes Factor associated to

and

graph comparison (eq. 9, Kass and Raftery 1995). The (slightly)

modiﬁed Jeﬀreys’ rule proposed by Kass and Raftery (1995) might further be used to evaluate to which

extent the data support

with e.g., ∆

12 >

6(respectively ∆

12 >

10) providing “strong” (respectively

“very strong”) evidence in favor of G127.

The ﬁt.graph function returns an object of class ﬁtted.graph detailed in the documentation page accessible

with the following command (or the ?operator):

help(fitted.graph)

The following code shows how to ﬁt the example graph stored in the graph.params object sim.graph.params

generated above and some features of the resulting ﬁtted.graph object:

sim.fittedgraph<-fit.graph(sim.graph.params)

Starting estimation of admixture rates (LBFGS score optimisation)

Initial score= 316.0469

Estimation ended in 0 m 0 s

Final Score: 1.004106

BIC: 276.6086

21stored in the f2.target and f3.target slots of the input graph.params object, see section 5.1.2

22If there is no admixture in the graph, X(a)only contains only 0 or 1

The argument Q.lambda of the ﬁt.graph function may be used to add a small constant (e.g., 10

−4

) to all to the diagonal

elements of

(i.e., the variance of the basis f-statistics estimates) as proposed by Patterson et al. (2012; see also Lipson 2020)

to avoid numerical problems. Note that Qmat.diag.adjust is always set 0 in the ﬁnal estimate of the score Sor the BIC

24Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm with box constraints

i.e., the observed vector of f-statistics is multivariate normal distributed around the expected

(

;

)vector speciﬁed by the

admixture graph and the covariance structure empirically estimated

26BIC =S+npar log 1

2nl(nl−1)−1

2nl(nl−1) log(2π)−log(|Q|)

The two thresholds of 6and 10 on ∆

BIC

corresponds to thresholds of 13 and 21 deciban (units of 10

log10

scale) on BF. The

original Jeﬀreys’ rule considered

BF12

thresholds of 10,15 and 20 decibans (corresponding to ∆

BIC

thresholds of 4

6,6

9and

9.2) as “strong”, “very strong” and “decisive” evidence in favor of model 1.

#Estimated edge lengths

sim.fittedgraph@edges.length

P7<->P1 s1<->P2 s2<->P3 S<->P6 P8<->s2 P7<->s1 P9<->P4 P9<->P5 P8<->P7 R<->P8 R<->P9

0.004079551 0.001970516 0.002148837 0.002111818 0.006334944 0.002228311 0.012858490 0.012529775 0.004160468 0.006445943 0.006445943

#Estimated admixture proportion

sim.fittedgraph@admix.prop

[1] 0.2478947

#Final Score

sim.fittedgraph@score

[1] 1.004106

#BIC

sim.fittedgraph@bic

[1] 276.6086

The ﬁtted.graph object also stores the output results from the optim optimization function in the slot op-

tim.results. For instance, in the example below, convergence was reached without any issue (convergence=0 )

#optim function results (list)

sim.fittedgraph@optim.results

$par

[1] 0.2478947

$value

[1] 1.004106

$counts

function gradient

8 8

$convergence

[1] 0

$message

[1] "CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH"

As for graph.params objects (see section 5.1.1), the plot function may be directly applied to ﬁtted.graph objects

to plot the admixture graph with estimated parameter values

. The admix.fact and edge.fact argument

of the ﬁt.graph function allow to apply a multiplying factor to the printed branch lengths and admixture

proportions (by default admix.fact=100, i.e., admixture proportions are printed in %; and edge.fact =1000,

i.e. edge lengths are printed in

). Figure 9 shows the ﬁtted example graph and was generated with the

following command:

plot(sim.fittedgraph)

In case of convergence problem (i.e., convergence not equal to 0), a message detailing execution error in the optimization

algorithm is stored in the optim.results component named message. For more details, see the documentation for the optim

function with the command ?optim providing the package optim is loaded

the graph is also coded in dot format with a character vector stored in the dot.graph slot of the resulting ﬁtted.graph object.

As for generate.graph.params, a dot ﬁle may also be printed out by specifying an output ﬁle name preﬁx with the outﬁlepreﬁx

argument of the ﬁt.graph function (or using the command writeLines(x@dot.graph,con=outﬁle) where xis the ﬁtted.graph object

and outﬁle is the desired named of the dot output ﬁle). See 5.1.1 for more details on how to externally export and customize dot

ﬁles

P4 P5P7

25%

75%

4 6

13 13

6 6

Figure 9: Plot of the admixture graph specifying the simulated scenario with ﬁtted parameter values (x1000

for edge lengths)

Notice

The two edges from the root node of the graph are not identiﬁable and only their joint lengths can

be estimated. The root position is arbitrarily set in the mid position (i.e., the two root edges have

the same length by construction as shown in Figure 9).

5.2.1 Scaling of branch lengths in drift units

The estimated edge lengths are in the same scale as the other f-statistics which limits their interpretation

since they strongly depend on the SNP ascertainment process (see section 4). Lipson et al. (2013) showed

however that the lengths may be approximately scaled in genetic drift units (i.e., in units of

2Ne

where

is a number of generations and

is the diploid eﬀective population along the branch) using estimates

of overall marker heterozygosities within (i.e., 1

−Q1

) or across (i.e., 1

−Q2

) population (Gautier et al.

2021). Brieﬂy, for a given edge

P↔C

relating a child node

to its parent node

with an (unscaled)

estimated branch length

ˆeP↔C

ˆτP↔C

= 2

ˆeP↔C

where

ˆτP↔C

is the estimated branch length scaled in drift

units and

is the estimated heterozygosity in the (parent) node

. The parent node heterozygosities can be

estimated from leaves to root by using the property

+ 2

eP↔C

= 0 that relate the child

and parent

node heterozygosities (

and

respectively) and

eP↔C

(Lipson et al. 2013). When the drift.scaling

argument is set to TRUE

, the ﬁt.graph function returns the edge lengths scaled in drift units in a slot

named edges.length.scaled together with the estimated node heterozygosities (nodes.het slot) as shown below

with the example data:

sim.fittedgraph.scaled<-fit.graph(sim.graph.params,drift.scaling = TRUE,verbose=FALSE)

Note that the obtained results are the same as above with no drift scaling since the latter is a post-processing

step independent of admixture graph parameters estimation

#Estimated edge lengths

sim.fittedgraph.scaled@edges.length.scaled

P7<->P1 s1<->P2 s2<->P3 S<->P6 P8<->s2 P7<->s1 P9<->P4 P9<->P5 P8<->P7 R<->P8 R<->P9

0.04964054 0.02405296 0.02631275 0.02516072 0.07378969 0.02711439 0.14321071 0.13954967 0.04846131 0.07700936 0.07700936

When plotting the resulting ﬁtted.graph objects, branch lengths are displayed in drift scaled units as shown in

the Figure 10 below obtained with the following command:

30providing the input graph.params object includes estimates of within-population heterozygosities (see section 5.1.2)

plot(sim.fittedgraph.scaled)

P4 P5P7

25%

75%

48 74

143 140

77 77

Figure 10: Plot of the admixture graph specifying the simulated scenario with ﬁtted edge lengths scaled in

drift units (x1,000)

The estimated branch lengths are close to the simulated ones (Figure 1) as represented below (see Gautier

et al. 2021 for a more in-depth exploration of the accuracy of the estimates):

P7<->P1 s1<->P2 s2<->P3 S<->P6 P8<->s2 P7<->s1 P9<->P4 P9<->P5 P8<->P7 R<->P8+R<->P9

Estimated 0.04964054 0.02405296 0.02631275 0.02516072 0.07378969 0.02711439 0.1432107 0.1395497 0.04846131 0.1540187

Simulated 0.05000000 0.02500000 0.02500000 0.02500000 0.07500000 0.02500000 0.1500000 0.1500000 0.05000000 0.1500000

5.2.2 Estimating 95% conﬁdence intervals of the estimated parameters values

Calling the ﬁt.graph function with the compute.ci argument set to TRUE allows deriving (rough) 95%

conﬁdence intervals for the admixture graph parameter estimates. The procedure considers each parameter in

turn (the other parameters being set to their estimated values) and consists of interpreting the score diﬀerence

Sν−S?

(where

is the ﬁtted graph score associated with estimated parameter value

ν?

and

Sν

is the score

associated with a parameter value

ν6

ν?

) as a likelihood-ratio test statistics following a

χ2

distribution with

one degree of freedom (see Gautier et al. 2021 for details). The lower and upper boundaries

νmin

and

νmax

of the 95% CI (such

Sν−S?<

84 for all

νmin < ν < νmax

) are then simply computed using a bisection

method (with a 10−4precision threshold).

sim.fittedgraph.with.ci<-fit.graph(sim.graph.params,compute.ci=TRUE,

drift.scaling = TRUE,verbose = FALSE)

#95% CI for the admixture proportion

sim.fittedgraph.with.ci@admix.prop.ci

95% Inf. 95% Sup.

a 0.2354879 0.2604726

#95% CI for edge length (including drift scaled as drift.scaling=TRUE)

sim.fittedgraph.with.ci@edges.length.ci

95% Inf. 95% Sup. 95% Inf. (drift scaled) 95% Sup. (drift scaled)

P7<->P1 0.003888322 0.004322695 0.04731364 0.05259916

s1<->P2 0.001785780 0.002153260 0.02179800 0.02628362

s2<->P3 0.001947384 0.002392453 0.02384593 0.02929586

S<->P6 0.001913835 0.002294537 0.02280190 0.02733768

P8<->s2 0.006037994 0.006638187 0.07033080 0.07732187

P7<->s1 0.002019406 0.002471907 0.02457242 0.03007851

P9<->P4 0.012406433 0.013340492 0.13817596 0.14857899

P9<->P5 0.012040331 0.013011938 0.13409851 0.14491974

P8<->P7 0.003835432 0.004464375 0.04467527 0.05200122

R<->P8 0.006219328 0.006686937 0.07430200 0.07988849

R<->P9 0.006219328 0.006686937 0.07430200 0.07988849

The simulated values are all within the estimated 95% CI of the estimated value except for the long

eP9↔P4

and

eP9↔P5

branches that are slightly higher than the upper boundary. Note also that if the 95% CI for the

admixture proportion

contains the simulated value of 0.25, it is narrower than the one that may be derived

from the block-jackknife estimate of the

-ratio standard error (see section 4.3 and Gautier et al. 2021 for

a more thorough evaluation of the estimated 95% CI).

5.2.3 Assessing the ﬁt of the graph with the compare.ﬁtted.fstats function

As outlined by Patterson et al. (2012) and Lipson (2020), a straightforward but highly informative

approach to assess the ﬁt of the graph is to compare the f-statistics derived from the ﬁtted admixture graph

parameters to the estimated ones, i.e., to evaluate to which extent the ﬁtted F-statistics lie within the

conﬁdence intervals of the estimated ones. This may be summarized by computing a Z-score of the residuals

for each f-statistics as

f−g

σ2

where

and

stand for the ﬁtted and estimated values respectively and

σ2

is the standard error of

. This information is available for the basis f-statistics in the ﬁtted.outstats slot of

the ﬁtted.graph object generated by the ﬁt.graph functions, as illustrated below:

#Fitted basis F-stats

sim.fittedgraph.scaled@fitted.outstats

Stat. value Fitted Value Z-score

P1,P2 0.008274108 0.008278377 0.03287705

P1,P3 0.016680443 0.016723801 0.20561876

P1,P4 0.033971079 0.033990395 0.05587989

P1,P5 0.033727224 0.033661681 -0.18890298

P1,P6 0.012237561 0.012265163 0.16197672

P1;P2,P3 0.004048392 0.004079551 0.26473869

P1;P2,P4 0.004087474 0.004079551 -0.05972586

P1;P2,P5 0.004149524 0.004079551 -0.52164304

P1;P2,P6 0.004610668 0.004631937 0.19288766

P1;P3,P4 0.008230291 0.008240019 0.05210483

P1;P3,P5 0.008244937 0.008240019 -0.02672913

P1;P3,P6 0.011937638 0.011973206 0.20637614

P1;P4,P5 0.021148713 0.021131906 -0.06164584

P1;P4,P6 0.007189623 0.007208661 0.11673856

P1;P5,P6 0.007234212 0.007208661 -0.16250690

However, the ﬁt should be evaluated for all the f-statistics (not only those forming the f-statistics vector-space

basis used to ﬁt the admixture graph) with the compare.ﬁtted.fstats function. This may in turn provide

insights into those populations (or graph edges) leading to poor ﬁt (Lipson 2020). As shown below, the

function requires the original fstats object (that may contain f-statistics for additional populations not

represented in the admixture graph) and the ﬁtted.graph object. It then produces a matrix with information

on all the ﬁtted stats and the n.worst.stats (by default n.worst.stats=5) f-statistics, i.e. with the lowest

absolute Z-score, are directly printed in the console:

sim.fitted.fstats.comp<-compare.fitted.fstats(sim6p.allelecount.fstats,sim.fittedgraph)

5 Worst fit for:

Estimated Fitted Z–score

P1,P2;P3,P5 1.011325e-04 0.000000e+00 -0.8442359

P1,P2;P5,P6 4.611437e-04 5.523863e-04 0.7459512

P1,P2;P4,P5 6.205022e-05 3.469447e-18 -0.6662270

P2;P1,P5 4.124584e-03 4.198826e-03 0.5401737

P1;P2,P5 4.149524e-03 4.079551e-03 -0.5216430

#Information on the last five fitted F-statistics

tail(sim.fitted.fstats.comp)

Estimated Fitted Z–score

P2,P6;P3,P4 -4.787097e-03 -4.764545e-03 0.16513037

P2,P6;P3,P5 -4.804558e-03 -4.764545e-03 0.27257790

P2,P6;P4,P5 -1.746147e-05 0.000000e+00 0.16092207

P3,P4;P5,P6 -1.765179e-02 -1.765643e-02 -0.01671976

P3,P5;P4,P6 -1.762185e-02 -1.765643e-02 -0.12418834

P3,P6;P4,P5 2.994286e-05 6.938894e-18 -0.39137603

As shown above, no outlying ﬁtted f-statistics (e.g., with

|Z|>

2) is observed on the example providing strong

support for the ﬁtted admixture graph.

5.3 Adding a new leaf to an existing graph

The add.leaf function allows to perform iterative calls to the ﬁt.graph function in order to evaluate all possible

connections of a given leaf (population) to an existing graph with non-admixed and/or admixed edges. Three

input arguments are required:

•

a graph speciﬁed within a graph.params object (obtained with the generate.graph.params function, see

section 5.1.1) or a ﬁtted.graph object (obtained with the ﬁt.graph,add.leaf or graph.builder functions,

see sections 5.2 and 6.2 below allowing more convenient exploration of the admixture graph space via

recursive calls)

•the name of the leaf to add (leaf.to.add argument)

•

an fstats object (see 4) containing a minima estimates of all the

and

statistics (and their standard

errors) involving the leaves of the input graph and the leaf to add

By default the function tests all the possible positions of the candidate leaf (leaf.to.add) to the graph with

non-admixed (including a new rooting with the candidate leaf as an outgroup) or admixed edges. If

is the number of non-admixed edges in the original graph, the number of tested graphs equals

+ 1 for

non-admixed candidate edges

and

2ne

(

ne−

1) for admixed candidate edges

. Optional arguments may

allow disabling the evaluation of non-admixed (by setting the only.test.non.admixed.edges argument to TRUE)

or admixed (by setting the only.test.admixed.edges argument to TRUE) candidate edges.

The object returned by the add.leaf function is a list consisting of:

•an element named n.graphs corresponding to the number of tested graphs

•

an element named ﬁtted.graphs.list which consists of a list of ﬁtted.graph objects (indexed from 1 to

n.graphs and in the same order as the list “graphs”) containing the ﬁt.graph function results for each

candidate graph

•

an element named best.ﬁtted.graph which is the ﬁtted.graph object associated to the candidate graph

with minimal BIC (see 5.2) among all the n.graphs graphs within ﬁtted.graphs.list.

•

an element named bic which is a vector consisting of the n.graphs BIC (indexed from 1 to n.graphs and

in the same order as the ﬁtted.graphs.list list).

Use of the add.leaf function is illustrated below on the example data by evaluating the connection of the

P6 population on the graph (actually tree) connecting the ﬁve other populations (P1,P2,P3,P4 and P5 )

speciﬁed according to the simulated topology in a graph.params object (5.1.1) named sim5p.tree.params and

plotted in Figure 11:

sim5p.tree<-sim.graph<-rbind(c("P1","P7",""),c("P2","P7",""),c("P3","P8",""),

c("P7","P8",""),c("P4","P9",""),c("P5","P9",""),

c("P8","R",""),c("P9","R",""))

sim5p.tree.params<-generate.graph.params(sim5p.tree)

#Note: fstats object is not necessary at this stage

plot(sim5p.tree.params)

31The newly added node is named “N-”leaf.to.add

The three added nodes are named “S-”leaf.to.add, “S1-”leaf.to.add and “S2-”leaf.to.add and the admixture proportions are

named with a letter (A to Z depending on the number of admixed nodes already present in the graph)

P1 P2

P3 P4 P5P7

P8 P9

Figure 11: Plot of the ﬁve-population graph on which to add the P6population with the add.leaf function

All the possible positions of the P6 population on the sim5p.tree graph (here using the f-statistics estimated

on the simulated allele count data, see section 4), are tested as follows:

add.P6<-add.leaf(sim5p.tree.params,leaf.to.add="P6",

fstats=sim6p.allelecount.fstats,

verbose=FALSE,drift.scaling=TRUE)

Note that the verbose option set to FALSE allows disabling the printing of the progress and timing of each

analysis (which may be useful in practice) and the drift.scaling option set to TRUE allows passing it to each

ﬁt.graph call to obtained estimates of branch lengths in drift units (see section 5.2).

The graph with the lowest BIC among all the tested graphs may then be plotted by calling the plot function

on the corresponding ﬁtted.graph object stored in the best.ﬁtted.graph element of the add.P6 output list with

the following command that generates Figure 12:

plot(add.P6$best.fitted.graph)

P3P1

P4 P5

S-P6

S1-P6

24 25%

S2-P6

75%

143 140

77 77

Figure 12: Plot of the graph with the lowest BIC among all the possible graphs connecting

6to the

ﬁve-population tree tested by the add.leaf. The ﬁtted edge lengths are in drift units (x1000) since drift.scaling

argument was set to TRUE when calling add.leaf.

The best ﬁtting graph based on

BI C

criterion (stored in the best.ﬁtted.graph slot of the output list) is in

perfect agreement with the simulated scenario (Figure 1). It actually corresponds to the one directly ﬁtted

using the simulated scenario which was represented in Figure 10 above (only the names of the nodes involved

in the admixture events diﬀer since automatically given by the add.leaf function). In addition, comparisons

of the BIC of the diﬀerent graphs provide strong support in favor of this best ﬁtting graph, the second lowest

BIC being more than 86 units larger i.e., far above the threshold of 10 for very strong evidence (see section

5.2):

#D_BIC w.r.t. best fitted BIC

D_BIC=add.P6$bic-add.P6$best.fitted.graph@bic

#5 First lowest DeltaBIC (the first value of zero corresponding to the best fitted graph)

head(sort(D_BIC))

[1] 0.00000 86.31675 86.31675 279.48967 284.90577 284.90577

6 Building admixture graph from scratch

Lipson et al. (2013) proposed a two-step approach (implemented in the MixMapper software

) to build

admixture graph when prior knowledge about history and relationships of investigated population is limited

(which is usually the case). It consists of ﬁrst building a scaﬀold tree of unadmixed populations and then

adding the remaining populations successively on the graph. Such a supervised approach nevertheless requires

to carefully assess at each step the graph ﬁt and possibly try diﬀerent ordering in the inclusion of populations

(or removal of some populations). The poolfstat package provides functions to help building scaﬀold trees that

may further be used as input tree for the add.leaf function previously described above (5.3) to implement the

Lipson et al. (2013) two-step approach.

6.1 Building scaﬀold trees of unadmixed populations

In the absence of admixture, the

statistics among all pairs of populations are expected to be additive

along the paths of the (binary) tree summarizing the history of the populations (Lipson et al. 2013). As a

result, the (unrooted) tree topology and branch lengths connecting unadmixed populations may be inferred

with a neighbor-joining algorithm to derive a scaﬀold tree for further admixture graph construction. Based

on the estimated f-statistics stored in a fstats object, the functions described below allows to i) identify

candidate sets of unadmixed populations among the genotyped ones (ﬁnd.tree.popset function); ii) infer a

neighbor-joining scaﬀold tree from a candidate set of unadmixed populations (rooted.njtree.builder function);

and iii) to infer root position based on the consistency of within population heterozygosities between the two

resulting partitions of rooted trees (see p1799 in Lipson et al. 2013).

6.1.1 The ﬁnd.tree.popset function to identify sets of candidate scaﬀold populations

The ﬁnd.tree.popset function selects maximal sets of unadmixed populations from an fstats object

. The

procedure involves i) discarding all the populations showing a signiﬁcantly negative

at a signiﬁcance

theshold speciﬁed with the f3.zcore.threshold argument (equal to

−

65 by default, i.e., 95% signiﬁcance

threshold, see section 4.1.2); and ii) keeping only sets of populations with all possible quadruplets passing

the

-based test of treeness i.e., with an absolute

Z-score lower than a threshold speciﬁed with the

f4.zcore.threshold argument (equal 2by default, i.e., 95% signiﬁcance threshold, see section 4.1.3). The latter

step is implemented via a greedy algorithm (that may be run in parallel by specifying a number of threads

with the nthreads argument) consisting of trying to extend the size of the population sets from all sets of

four populations after adding candidate populations one at a time. If the number of populations is large,

this algorithm may take some times. Note that increasing (respectively decreasing) f3.zcore.threshold toward

value closer to 0 may allow decreasing (respectively increasing) the number of initial candidate populations

to be tested for inclusion in a set. Similarly, increasing (respectively decreasing) f4.zcore.threshold may allow

increasing (respectively decreasing) the size of the sets. When applied to the example allele count and read

count data, a single set of 5 unadmixed populations (P1,P2,P3,P4 and P5 ) is retrieved as expected from

the simulated scenario (Figure 1):

# count data

scaf.pops<-find.tree.popset(sim6p.allelecount.fstats,verbose=FALSE)

scaf.pops$pop.sets

33http://cb.csail.mit.edu/cb/mixmapper/

providing it was Z-scores were estimated for

and

statistics, i.e., that block-jackknife estimates of s.e. were carried out,

see section 4

[,1] [,2] [,3] [,4] [,5]

PopSet1 "P1" "P2" "P3" "P4" "P5"

# 30X Pool-Seq data

scaf.pops<-find.tree.popset(sim6p.readcount30X.fstats,verbose=FALSE)

scaf.pops$pop.sets

[,1] [,2] [,3] [,4] [,5]

PopSet1 "Pool1" "Pool2" "Pool3" "Pool4" "Pool5"

As previously mentioned (section 4), for a given set consisting of n populations, a total of 3

n

4

(

n−

1)(

n−

2)(

n−

3) quadruplets can be formed. In other words, a given set of four populations A, B, C and D is

actually represented by only three quadruplets representative of the three possible unrooted tree topologies

i) (A,B;C,D); ii) (A,C;B,D); and iii) (A,D;B,C). Among these, only a single quadruplet is expected to pass

the treeness test (i.e., if the correct unrooted tree topology is (A,C;B,D), then the absolute value of the

Z-scores associated to f4(A,B;C,D) and f4(A,D;B,C) or their equivalent will be high). For each of identiﬁed

sets of presumably unadmixed populations, the list of the

n

4

quadruplets passing the treeness test is given

in the passing.quadruplets element of the output list as illustrated below:

# list of the 15 quadruplets passing the treeness test for the identified set

scaf.pops$passing.quaduplets

[,1] [,2] [,3]

PopSet1 "Pool1,Pool3;Pool4,Pool5" "Pool2,Pool3;Pool4,Pool5" "Pool1,Pool2;Pool4,Pool5"

[,4] [,5]

PopSet1 "Pool1,Pool2;Pool3,Pool4" "Pool1,Pool2;Pool3,Pool5"

In addition, for each of the identiﬁed sets, the range of variation of the passing quadruplets is given in the

Z_f4.range element of the output list:

scaf.pops$Z_f4.range

Min. |Zscore| Max. |Zscore|

PopSet1 0.02584028 0.8067878

When several sets are identiﬁed, this information may be useful to prioritize the sets of unadmixed populations

(e.g., via a minimax criterion consisting of choosing the set of populations that has the lowest maximal

absolute Z-score for its underlying quadruplets that pass the treeness test).

6.1.2 The rooted.njtree.builder to building (and root) a tree of candidate scaﬀold populations

The rooted.njtree.builder allows ﬁrst building a Neighbor-Joining

of a set of presumably unadmixed

populations (as obtained e.g., from the ﬁnd.tree.popset functions) given as a vector of population names in

the pop.sel argument based on the matrix of their pairwise

stored in the provided fstats object (fstats

argument). The resulting (unrooted) tree is then rooted by relying on the property that root Rheterozygosity

= 1

−QA,B

estimated from all the possible pairs of populations Aand Bthat satisﬁes the property of

being only connected through R in the tree (i.e., Aand Beach belong to one of the two tree partitions

deﬁned by the

) should be consistent (Lipson et al. 2013). In other words, the most likely rooted tree

among the (2

nl−

3) possible ones should be the one displaying the narrower range of variation of the

estimates. Note that the root position is always placed in the mid-position of the candidate branch.

The object returned by the rooted.njtree.builder function is a list consisting of:

•

an element named n.rooted.trees corresponding to the number of possible rooted binary trees that were

evaluated

•

an element named ﬁtted.rooted.trees.list which consists of a list of ﬁtted.graph objects (indexed from 1

to n.rooted.trees).

for each of these quadruplets, seven other equivalent combinations formed by permuting populations within each pair can be

deﬁned as mentioned in the notice p16

36relying on the nj function from the popular package ape (Paradis et al. 2004)

•

an element named best.rooted.tree which corresponds to the ﬁtted.graph object associated with the most

likely rooted tree (among all the ﬁtted.rooted.trees.list ones) identiﬁed as the one displaying the minimal

standard deviation over the hRestimates

•

an element named root.het.est.var consisting of a matrix of n.rooted.trees rows and 4 columns with i) the

average estimated root heterozygosity

across all the pairs of leaves that are relevant for estimation

(see above); ii) the size of the range of variation; iii) the standard deviation of the

estimates, and iv)

the number of population pairs relevant for estimation

•

if n.edges>3, an element named nj.tree.eval that gives for each evaluated rooted tree, the ﬁve

f−

statistics

conﬁguration displaying the worst ﬁt, i.e., with the ﬁve highest absolute Z-score for the predicted value

(obtained by internally calling the compare.ﬁtted.fstats function). Note that these are independent of

the rooting (so cannot be used to infer the root position).

Use of the rooted.njtree.builder function is illustrated below to build the scaﬀold tree (using Pool-Seq read

count data) based on the set of unadmixed populations (identiﬁed with ﬁnd.tree.popset) and plotted with

the plot function applied to the ﬁtted.graph object of the resulting list (best.rooted.tree element) to generate

Figure 13:

scaf.tree<-rooted.njtree.builder(fstats=sim6p.readcount30X.fstats,

pop.sel=scaf.pops$pop.sets[1,],plot.nj=FALSE)

Score of the NJ tree: 0.6884542

Construction of all the 7 possible rooted tree from the NJ tree

(stored as graph in the rooted.graph object of the output list)

plot(scaf.tree$best.rooted.tree)

Pool1 Pool2

Pool3 Pool4 Pool5

Root

47 144 140

49 50

Figure 13: Plot of the rooted scaﬀold tree of unadmixed populations inferred by rooted.njtree.builder. The

ﬁtted edge lengths are in a drift scale (x1000).

The ﬁt of the Neighbor-Joining tree can be checked by inspecting the nj.tree.eval element:

scaf.tree$nj.tree.eval

Estimated Fitted Z–score

Pool1,Pool2;Pool3,Pool5 1.011014e-04 0.000000000 -0.8067878

Pool1,Pool2;Pool3,Pool4 6.087576e-05 0.000000000 -0.5098456

Pool2;Pool1,Pool5 4.120591e-03 0.004181198 0.4330143

Pool1;Pool2,Pool5 4.154896e-03 0.004094289 -0.4293005

Pool1,Pool2;Pool4,Pool5 4.022566e-05 0.000000000 -0.4233082

The following range of variation of the hRestimates were obtained for the diﬀerent possible root position:

scaf.tree$root.het.est.var

Mean Range sd ncomps

Tree1 0.1814754 0.0234940763 0.0115371318 4

Tree2 0.1810025 0.0228772762 0.0113327886 4

Tree3 0.1857808 0.0155159278 0.0076153034 6

Tree4 0.1907099 0.0007923822 0.0002825614 6

Tree5 0.1880369 0.0113894828 0.0055518382 4

Tree6 0.1878868 0.0111334494 0.0054505348 4

Tree7 0.1833542 0.0152408946 0.0085429545 4

The “best” inferred rooted scaﬀold tree (i.e., the fourth one with both lowest range of

variation) is

consistent with the simulated scenario. It may further be used as a reference graph to construct the complete

admixture graph after adding Pool6 population using the add.leaf function (see 5.3). The obtained graph

plotted in Figure 14 with the commands below is very close to the one previously inferred with allele count

data (Figure 12) and very strongly supported by the data:

add.pool6<-add.leaf(scaf.tree$best.rooted.tree,leaf.to.add="Pool6",

fstats=sim6p.readcount30X.fstats,verbose=FALSE,drift.scaling=TRUE)

plot(add.pool6$best.fitted.graph)

Pool6

Pool3

Pool2

Pool4 Pool5

Pool1

S-Pool6

S1-Pool6

75%

S2-Pool6

2425%

Root

144 140

Figure 14: Plot of the graph with the lowest BIC among all the possible graphs connecting

P ool

6to the

scaﬀold tree of unadmixed population tested by the add.leaf. The ﬁtted edge lengths are in drift units (x1000)

since drift.scaling argument was set to TRUE when calling add.leaf.

#D_BIC w.r.t. best fitted BIC

D_BIC=add.pool6$bic-add.pool6$best.fitted.graph@bic

#5 First lowest DeltaBIC (the first value of zero corresponding to the best fitted graph)

head(sort(D_BIC))

[1] 0.00000 78.75611 78.75611 254.27229 259.68839 259.68839

Notice

In practice, the rooted.njtree.builder function should be used with caution since both the Neighbor-

Joining tree construction and the heterozygosity-based rooting of the tree may be sensitive to

long-branch attraction (most particularly if some highly diverged populations are included). The

inferred topology may even violate treeness test for some of the quadruplets (see e.g., the empirical

example detailed in the Supplementary Vignette V2 by Gautier et al. 2021).

6.2 Extending an initial tree (or graph) with the graph.builder function

The graph.builder function implements an heuristic to carry out a larger exploration of the space of possible

graphs (but usually still not exhaustive) obtained from the joint addition of several populations (leaves) given

as an input vector (leaves.to.add argument) to an existing graph (as generated using the rooted.njtree.builder

function described above or included in a graph.params or ﬁtted.graph object) or a list of graphs. The

algorithm consists of adding the leaves in the order of the input vector to each of the graphs stored in a heap

via successive calls to the add.leaf function (section 5.3). More precisely, the heap ﬁrst consists of the initial

input graph (or list of graph) and at each iteration, the function add.leaf is used to evaluate all the possible

connections of each candidate leaf (with non-admixed or admixed edges) to all the graphs of the heap. For

each of the latter, the newly ﬁtted graphs displaying a BIC less than heap.dbic units (set to 6 by default)

away from the best ﬁtting graph (i.e., the one with the lowest BIC) included in the new heap. Once all

the graphs have been evaluated, if the heap contains more than max.heap.size (set to 25 graphs by default)

graphs, only the max.heap.size graphs with the lowest BIC are kept in the heap. Finally, after testing the

latest candidate leaf, only the graphs with a BIC less than heap.dbic units away from the graph with lowest

BIC in the heap are retained in the ﬁnal list of graphs.

The object returned by the function is a list consisting of:

•an element named n.graphs corresponding to the ﬁnal number of graphs

•

an element named ﬁtted.graphs.list which consists of a list of ﬁtted.graph objects (indexed from 1 to

n.graphs) containing the ﬁt.graph function results for each graph

•

an element named best.ﬁtted.graph which is the ﬁtted.graph object associated to the graph with the

lowest BIC among all the n.graphs graphs included in ﬁtted.graphs.list.

•

an element named bic which is a vector containing the BIC of the n.graphs BIC (indexed from 1 to

n.graphs and in the same order as ﬁtted.graphs.list).

Use of the graph.builder function is illustrated below on the PoolSeq example data by starting from an initial

rooted tree constructed with the rooted.njtree.builder for the three populations Pool1,Pool3,Pool4 and Pool5.

This tree is extended by successively adding the two remaining populations Pool2 and Pool6 :

#build an initial 3 population trees with "Pool1","Pool3","Pool4" and "Pool5"

init.tree<-rooted.njtree.builder(fstats=sim6p.readcount30X.fstats,

pop.sel=c("Pool1","Pool3","Pool4","Pool5"),plot.nj=FALSE)

Score of the NJ tree: 0.0006790273

Construction of all the 5 possible rooted tree from the NJ tree

(stored as graph in the rooted.graph object of the output list)

#adding the three remaining pops

final.graphs<-graph.builder(x=init.tree$best.rooted.tree,leaves.to.add=c("Pool2","Pool6"),

fstats=sim6p.readcount30X.fstats)

####################

Adding: Pool2

21 graphs evaluated in 0 h 0 m 1 s

6 graphs stored in the heap

####################

Adding: Pool6

261 graphs evaluated in 0 h 0 m 5 s

7 graphs stored in the heap

Final Number of graphs: 7

(min. BIC= 275.8408 )

Overall Analysis Time: 0 h 0 m 6 s ( 282 graphs evaluated)

#D_BIC w.r.t. to the "true" graph as identified previously (object add.pool6$best.fitted.graph)

D_BIC=final.graphs$bic-add.pool6$best.fitted.graph@bic

#5 First lowest DeltaBIC (the first value of zero corresponding to the best fitted graph)

head(sort(D_BIC))

[1] 0.000000 4.667688 4.923348 4.923348 4.923348 5.416100

Among the 7 ﬁnal graphs, the one with the lowest BIC is exactly the same as the one plotted in Figure 14 and

corresponds to the simulated scenario. It should however be noticed that other alternative graphs are also

identiﬁed with a good support. Moreover, starting with other population trees (e.g., a three population tree

consisting of Pool1,Pool2 and Pool5) could result in several graphs with the same support (i.e., ∆

BI C

= 0)

but with a diﬀerent positioning of the root (not shown). In practice, it may be important to start with

scaﬀold trees that are as large as possible and representative of the structuring of diversity of the represented

populations (i.e., not too unbalanced with respect to the leaves to be added). Some prior knowledge about

the relationships of some of the populations may also be helpful to that respect. As exampliﬁed in the

Supplementary Vignette V2 of Gautier et al. (2021), it is also highly recommended to test diﬀerent orders

of inclusion (possibly all) of the leaves (as speciﬁed in the vector leaves.to.add).

7 Other utilities

7.1 Symbolic representation of the Fparameters, admixture graph equations

and the scaled covariance matrix Ωwith graph.params2symbolic.fstats

Given a graph topology relating the populations stored in a graph.params object, the graph.params2symbolic.fstats

functions provide symbolic representation of the model equations used to ﬁt the underlying admixture

graph and all the

and

parameters together with the scaled covariance matrix of population allele

frequencies called

Ω

after Gautier (2015). Such representation may be useful for a closer examination of

graph properties (or education purposes). The output objects consists of a list with the following elements:

•

a character matrix named model.matrix consisting of the matrix

relating the parameters underlying

the basis f-statistics and graph edge lengths in the model equations deﬁned as f=Mb where fis the

vector of the basis f-statistics (row names of the model.matrix

) and

is the vector of graph edges

(column names of model.matrix M).

•

a character matrix named omega consisting of the the scaled covariance matrix of population allele

frequencies Ω(see e.g., Gautier 2015).

•

a character vector F2.equations consisting of the symbolic representations of all the

2nl

(

nl−

parameters (with edge and admixture proportion parameter names as deﬁned in the graph.params

object)

•

a character vector F3.equations consisting of the symbolic representations of all the

2nl

(

nl−

1)(

nl−

parameters (with edge and admixture proportion parameter names as deﬁned in the graph.params

object)

•

a character vector F4.equations consisting of the symbolic representations of all the

8nl

(

nl−

1)(

nl−

2)(

nl−

parameters (with edge and admixture proportion parameter names as deﬁned in the

graph.params object)

These diﬀerent equations can also be printed in an output text ﬁle (with name speciﬁed by the outﬁle

argument). The following example shows results obtained using the graph.params object sim.graph.params

generated in 5.1.1 (Figure 8) that speciﬁes the simulation scenario (Figure 1):

sim.fstats.sym<-graph.params2symbolic.fstats(sim.graph.params,outfile = "Fstats_equations")

#Model equations matrix

sim.fstats.sym$model.matrix

P7<->P1 s1<->P2 s2<->P3 S<->P6 P8<->s2 P7<->s1 P9<->P4 P9<->P5 P8<->P7 R<->P8 R<->P9

F2(P1,P2) "1" "1" "0" "0" "0" "1" "0" "0" "0" "0" "0"

F2(P1,P3) "1" "0" "1" "0" "1" "0" "0" "0" "1" "0" "0"

F2(P1,P4) "1" "0" "0" "0" "0" "0" "1" "0" "1" "1" "1"

F2(P1,P5) "1" "0" "0" "0" "0" "0" "0" "1" "1" "1" "1"

F2(P1,P6) "1" "0" "0" "1" "a^2-2*a+1" "a^2" "0" "0" "a^2-2*a+1" "0" "0"

F3(P1;P2,P3) "1" "0" "0" "0" "0" "0" "0" "0" "0" "0" "0"

F3(P1;P2,P4) "1" "0" "0" "0" "0" "0" "0" "0" "0" "0" "0"

F3(P1;P2,P5) "1" "0" "0" "0" "0" "0" "0" "0" "0" "0" "0"

F3(P1;P2,P6) "1" "0" "0" "0" "0" "a" "0" "0" "0" "0" "0"

F3(P1;P3,P4) "1" "0" "0" "0" "0" "0" "0" "0" "1" "0" "0"

F3(P1;P3,P5) "1" "0" "0" "0" "0" "0" "0" "0" "1" "0" "0"

F3(P1;P3,P6) "1" "0" "0" "0" "1-a" "0" "0" "0" "1-a" "0" "0"

F3(P1;P4,P5) "1" "0" "0" "0" "0" "0" "0" "0" "1" "1" "1"

F3(P1;P4,P6) "1" "0" "0" "0" "0" "0" "0" "0" "1-a" "0" "0"

F3(P1;P5,P6) "1" "0" "0" "0" "0" "0" "0" "0" "1-a" "0" "0"

#scaled covariance matrix of allele frequencies (Omega)

sim.fstats.sym$omega

P1 P2 P3

P1 "P7<->P1+P8<->P7+R<->P8" "P8<->P7+R<->P8" "R<->P8"

P2 "P8<->P7+R<->P8" "s1<->P2+P7<->s1+P8<->P7+R<->P8" "R<->P8"

P3 "R<->P8" "R<->P8" "s2<->P3+P8<->s2+R<->P8"

P6 "P8<->P7*a+R<->P8" "(P7<->s1+P8<->P7)*a+R<->P8" "P8<->s2+R<->P8-P8<->s2*a"

P4 "0" "0" "0"

P5 "0" "0" "0"

P6 P4 P5

P1 "P8<->P7*a+R<->P8" "0" "0"

P2 "(P7<->s1+P8<->P7)*a+R<->P8" "0" "0"

P3 "P8<->s2+R<->P8-P8<->s2*a" "0" "0"

P6 "S<->P6+(P8<->s2+P7<->s1+P8<->P7)*a^2-2*P8<->s2*a+P8<->s2+R<->P8" "0" "0"

P4 "0" "P9<->P4+R<->P9" "R<->P9"

P5 "0" "R<->P9" "P9<->P5+R<->P9"

#F2 statistics (first five)

head(sim.fstats.sym$F2.equations)

[1] "F2(P1,P2) = P7<->P1+s1<->P2+P7<->s1"

[2] "F2(P1,P3) = P7<->P1+P8<->P7+s2<->P3+P8<->s2"

[3] "F2(P1,P6) = P7<->P1+(a^2-2*a+1)*P8<->P7+S<->P6+(a^2-2*a+1)*P8<->s2+a^2*P7<->s1"

[4] "F2(P1,P4) = P7<->P1+P8<->P7+R<->P8+P9<->P4+R<->P9"

[5] "F2(P1,P5) = P7<->P1+P8<->P7+R<->P8+P9<->P5+R<->P9"

[6] "F2(P2,P3) = s1<->P2+P7<->s1+P8<->P7+s2<->P3+P8<->s2"

#F3 statistics (first five)

head(sim.fstats.sym$F3.equations)

[1] "F3(P1;P2,P3) = P7<->P1"

[2] "F3(P1;P2,P6) = P7<->P1+a*P7<->s1"

[3] "F3(P1;P2,P4) = P7<->P1"

[4] "F3(P1;P2,P5) = P7<->P1"

[5] "F3(P1;P3,P6) = P7<->P1+(1-a)*P8<->P7+(1-a)*P8<->s2"

[6] "F3(P1;P3,P4) = P7<->P1+P8<->P7"

#F4 statistics (first five)

head(sim.fstats.sym$F4.equations)

[1] "F4(P1,P2;P3,P6) = P7<->s1*a"

[2] "F4(P1,P2;P3,P4) = 0"

[3] "F4(P1,P2;P3,P5) = 0"

[4] "F4(P1,P2;P6,P4) = -P7<->s1*a"

[5] "F4(P1,P2;P6,P5) = -P7<->s1*a"

[6] "F4(P1,P2;P4,P5) = 0"

7.2 Generating ﬁles for the qpGraph software with graph.params2qpGraphFiles

The graph.params2qpGraphFiles function allows generating the ﬁles required by the qpGraph software

(Patterson et al. 2012) from a graph.params object that includes estimates of f-statistics (see section 5.1.2).

If fis the preﬁx character speciﬁed with the outﬁlepreﬁx argument of the function (by default f=out), these

are:

•a ﬁle named “f.graph” that speciﬁes the graph in qpGraph format

•

a ﬁle named “f.fstats” with estimates of F-statistics (and their covariance) included in the input

graph.params object

•

a parameter ﬁle named “f.parqpGraph” to run qpGraph (this ﬁle may be edited by hand if other options

are needed).

The qpGraph software (v7365 and above to allow f-statistics estimates to be provided as input) may then be

run on a terminal using the following options:

qpGraph -p f.parqpGraph -g f.graph -o out.ggg -d out.dot.input

The “f.fstats” f-statistics ﬁle must be in the same directory or its PATH should be explicitly speciﬁed by

editing the “f.parqpGraph” parameter ﬁle. The following example runs qpGraph (providing appropriate

install of the software) on the sim.graph.params object generated in 5.2 (see Figure 9 representing the ﬁtted

graph obtained with the ﬁt.graph function):

graph.params2qpGraphFiles(sim.graph.params,outfileprefix = "sim.graph")

Fstats input file for qpGraph written in sim.graph.fstats

Graph input file for qpGraph written in sim.graph.graph

Parameter File for qpGraph with some default parameters written in sim.graph.parqpGraph

#running qpGraph (installed locally) outside R

system("qpGraph -p sim.graph.parqpGraph -g sim.graph.graph -o sim.graph.g -d sim.graph.dot")

#plotting the dot file generated by qpGraph with grViz (as done internally by poolfstat)

require(DiagrammeR)

grViz("sim.graph.dot")

sim.graph.graph::P1P2P3P50.0000000.0001010.0001010.0001200.840

P4 P5

P1P3

4 13 13

75%

225%

Figure 15: Fitting results obtained by qpGraph on the same data as the one used to generate Figure 9.

Comparison of Figures 15 and 9 shows that the same results are obtained with the two ﬁtting methods (note

that edge lengths are not scaled in drift units on the two ﬁgures).

8 References

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A Apprendix

A.1 Block-Jackknife estimation of standard errors

Standard-error of genome-wide estimates of

FST

and other

-statistics can estimated using a block-jackknife

sampling approach (see Patterson et al. 2012 and references therein). The algorithm implemented in

poolfstat consists of dividing the genome into contiguous chunks of a pre-deﬁned number of SNPs (speciﬁed

by the argument nsnp.per.bjack.block of the computeFST,compute.pairwiseFST or compute.fstats functions,

see sections 3.1.1, 3.2 and 4 respectively) and then removing each block in turn to quantify the variability of

the estimates. If nbblocks are available and b

fiis the estimate of the statistics when removing all the SNPs

belonging to block i, the standard error cσfof the estimator b

fof the statistics of interest is computed as:

cσf=v

tnb−1

i=nb

i=1 b

fi−cµf2

where

cµf

i=nb

i=1 b

which may be slightly diﬀerent from the estimator obtained with all the markers since

the latter may include SNPs that are not eligible for block-jackknife sampling (e.g., those at the chromo-

some/scaﬀolds boundaries or those belonging to chromosome/scaﬀolds with less than nsnp.per.bjack.block

SNPs). Finally, block-jackknife sampling may also be used to obtain estimates of the covariance between the

estimates c

faand b

fbas (using similar notations):

Cov c

fa;b

fb=nb−1

i=nb

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Analysis of a D. suzukii Pool-Seq data set with the R package

poolfstat

2021-05-27

Contents

1 Data preparation and SNP ﬁltering 2

2 Estimation of global FST and all f−and D- statistics 4

3 Overview of the within-population genetic diversity from estimates of heterozygosities 6

4 Overview of the structuring of genetic diversity across populations from pairwise-

population FST estimates 6

5 Insights from f3-based admixture tests 8

6 Exploring invasion scenarios with admixture graph construction 12

6.1 Scaﬀold trees built (naively) from scratch are unreliable . . . . . . . . . . . . . . . . . . . . . 12

6.2

Investigating the relationships among the populations from the native area (and the Hawaiian

population) .............................................. 14

6.3

Investigating the origins of the populations from the American invasive area using admixture

graphs................................................. 22

References 34

This vignette details the analysis with the R package poolfstat

of the Pool-Seq data previously generated

by Olazcuaga et al. (2020) for 14 population samples of the invasive species Drosophila suzukii (Figure

1 and Table 1). The objective of this analysis was to provide an illustration of the main poolfstat package

functionalities on real data while providing insights into the history of both native populations and the

recent invasion of this species into America (Fraimout et al. 2017). As illustrated in Figure 1, the pool

samples represented 6 populations from the Asian native area, one from Hawaii (see Figure 1 legend) and 7

populations from the American continent. Out of the 14 population samples, all but CN-Bei originate from

the same site as in Fraimout et al. (2017) who inferred the routes of invasion on a worldwide scale under

an Approximate Bayesian Computation Random Forest (ABC-RF) approach using a data set consisting of

685 individuals belonging to 23 populations (from 15 to 44 individuals per sample) and genotyped at 25

autosomal microsatellite loci. As detailed in Table 1, for nine (CN-Lia, CN-Nin, CN-Shi, JP-Sap, BR-Pal,

US-Col, US-Sdi, US-Sok and US-Wat) of the 14 population samples, all or most (30 out 50 of and 40 out 50

for CN-Lia and CN-Nin, respectively) of the individuals included in the sequenced pool were in common

among the two studies. This thus makes the outcome of the Pool-Seq data analysis based on poolfstat directly

comparable with the analysis performed by Fraimout et al. (2017).

CN−Bei CN−Lia

CN−Nin

CN−Shi

JP−Sap

JP−Tok

BR−Pal

US−Col

US−Haw

US−Nca

US−Sdi

US−Sok

US−Wat US−Wis

Native

Hawaiian (>1980)

American (>2008)

Figure 1: Origin of the 14 population samples of D. suzukii. Population names are colored according to their

area of origin, i.e., whether they originate from the Asian native, Hawaii or continental America where the

species was ﬁrst observed in 2008. As done in previous studies, we distinguished the Hawaiian population

from the American invasive area because the presence of D. suzukii was ﬁrst described in 1980 in Hawaii.

Hence, this Paciﬁc Island population may be considered as intermediate between the Asian native and most

recently invaded areas (approximately 300 generations later assuming 10 generations per year). The 13

population samples originating from the same site as in Fraimout et al. (2017) are indicated by solid points.

1 Data preparation and SNP ﬁltering

The read alignment (bam) ﬁles from Olazcuaga et al. (2020) obtained after mapping Pool-Seq data

for each of the 14 population samples onto the latest near-chromosome scale assembly of the Drosophila

suzukii (Paris et al. 2020) were combined into an mpileup ﬁle using SAMtools 1.9 with options -q 20

-Q20 (Li et al. 2009). Variant calling was then performed on the resulting ﬁle using VarScan mpileup2snp

v2.3.4 (Koboldt et al. 2012) run with options –min-coverage 10 –min-avg-qual 25 –min-var-freq 0.005

–p-value 0.5 –output-vcf 1 ensuring very loose criteria to identify SNPs further SNP ﬁltering being delayed

to the parsing of the vcf ﬁle. In particular we set the p-value threshold to 0.5 leading to the identiﬁcation

All the analysis were carried on a laptop computer equipped with an intel Xeon Quad-Core 3.0 Ghz processor and 32 Gb

RAM

Sample name Status Sample Size Sampling site Sampling date SRA ID

CN-Bei (-) Native 50 (0) Beijing, China 2014 SRR10260017

CN-Lia (CN-Lia) Native 50 (50) Liaoyuan, China 2014 SRR10260016

CN-Nin (CN-Nin) Native 50 (30) Ningbo, China 2014/2016 SRR10260027

CN-Shi (CN-Shi) Native 50 (40) Shiping county, China 2014/2016 SRR10260024

JP-Sap (JP-Sap) Native 50 (50) Sapporo, Japan 2014 SRR10260023

JP-Tok (JP-Tok) Native 50 (0) Tokyo, Japan 2016 SRR10260022

BR-Pal (BR-PA) Invasive (AM) 50 (50) Porto Alegre, Brazil 2014 SRR10260033

US-Col (US-Col) Invasive (AM) 50 (50) Fort Collins, USA 2015 SRR10260032

US-Haw (US-Haw) Invasive (PA) 50 (0) Hawaii (Hilo), USA 2016 SRR10260031

US-Nca (US-NC) Invasive (AM) 100 (0) Raleigh, USA 2016 SRR10260030

US-Sdi (US-SD) Invasive (AM) 50 (50) San-Diego, USA 2014 SRR10260029

US-Sok (US-Sok) Invasive (AM) 75 (75) Dayton, USA 2014 SRR10260028

US-Wat (US-Wat) Invasive (AM) 50 (50) Watsonville, USA 2014 SRR10260026

US-Wis (US-Wis) Invasive (AM) 75 (0) Barneveld, USA 2016 SRR10260025

Table 1: Pool sample description (adapted from Table S2 in Olazcuaga et al., 2020). Population names

and the number of individuals included in the pools in common with Fraimout et al. (2017) are given in

parenthesis in the ﬁrst and third columns, respectively. Only the CN-Bei population sample is absent from

the microsatellite data set of Fraimout et al. (2017). As for the population origins (column status), we

distinguished i) the Asian native area (6 population samples); ii) the Paciﬁc islands (PA) invasive area (the

Hawaiian sample); and iii) the contiental American invasive area (7 population samples).

of SNP with only one read supporting the non-reference allele in a single pool whatever the coverage

Positions mapping to non-autosomal contigs (Paris et al. 2020) were subsequently discarded from the vcf

ﬁle to obtain the ﬁle named dsu.auto.vcf.gz which can be downloaded from the public Zenodo repository

(http://doi.org/10.5281/zenodo.4709080) together with a ﬁle named dsu.auto.info that contains names and

haploid sample sizes of each pool. As shown below, we then used the vcf2pooldata function to parse this vcf

ﬁle relying on default options except for the overall MAF threshold (computed from read counts) that was set

to 5% (min.maf argument) and the minimal read coverage for each pool that was set to 50 (min.cov.per.pool

argument). The resulting pooldata object was further ﬁltered with the pooldata.subset function to discard i)

all positions with a coverage higher than the 99% quantile coverage in at least one pool (cov.qthres.per.pool

argument); and ii) discard all SNPs with MAF<5% over all the populations from the native area to ensure

that mostly ancestral SNPs were included in the data as implicitly assumed by f-statistics based approaches

(see the main text of the manuscript and associated references).

#loading information file containing:

# i) pool names (in the same order as in the vcf file)

# ii) haploid pool size

dsu.info=read.table("dsu.auto.info",stringsAsFactors = F)

#parsing of the vcf file for autosomal contigs

dsu.dat=vcf2pooldata(vcf.file="dsu.auto.vcf.gz",

poolsizes =dsu.info[,2],poolnames = dsu.info[,1],

min.maf = 0.05,min.cov.per.pool = 50,remove.indels = T,

nlines.per.readblock = 1e7)

Reading Header lines

Parsing allele counts

VarScan like format detected for allele count data: the AD field contains allele depth

for the alternate allele and RD field for the reference allele

(N.B., positions with more than one alternate allele will be ignored)

1e+07 lines processed in 0 h 5 m 54 s : 773642 SNPs found

The Fisher exact test use for variant calling in VarScan gives

rCc

C2c

where

is the coverage and

is the number

of reads supporting the non-reference allele. Hence, p= 0.5if r= 1

2e+07 lines processed in 0 h 10 m 50 s : 1546200 SNPs found

3e+07 lines processed in 0 h 15 m 27 s : 2136045 SNPs found

30922056 lines processed in 0 h 15 m 50 s : 2161633 SNPs found

Data consists of 2161633 SNPs for 14 Pools

#Further filtering of SNPs keeping only those with MAF>0.05 on the native populations

#and coverage not exceeding the 99% pool-specific quantile of coverage in a least one pop

pop.nat.names=c("CN-Bei","CN-Lia","CN-Nin","CN-Shi","JP-Tok","JP-Sap")

pop.nat.idx=which(dsu.dat@poolnames %in% pop.nat.names)

tmp.f.nat=rowSums(dsu.dat@refallele.readcount[,pop.nat.idx])/

rowSums(dsu.dat@readcoverage[,pop.nat.idx])

maf.native=0.5-abs(0.5-tmp.f.nat)

dsu.dat=pooldata.subset(dsu.dat,snp.index = which(maf.native>0.05),

cov.qthres.per.pool = c(0,0.99))

Data consists of 1588569 SNPs for 14 Pools

The ﬁnal data sets thus consists of 1,588,569 SNPs with sequencing coverage statistics among the 14 diﬀerent

pools detailed in Table 2.

Mean Coverage Median Coverage Coverage Range

BR-Pal 76.72 76 50-135

CN-Bei 95.93 95 50-160

CN-Lia 69.68 69 50-127

CN-Nin 66.7 65 50-122

CN-Shi 68.41 67 50-138

JP-Sap 84.53 84 50-174

JP-Tok 68.05 67 50-116

US-Col 79.35 79 50-144

US-Haw 95.41 95 50-160

US-Nca 74.24 73 50-121

US-Sdi 89.64 89 50-143

US-Sok 65.29 64 50-130

US-Wat 73.34 72 50-126

US-Wis 76.7 76 50-143

Table 2: Summary of the sequencing coverage of the 14 Pool-Seq samples over the 1,588,569 selected SNPs

2 Estimation of global FST and all f−and D- statistics

To characterize the overall structuring of genetic diversity, we estimated the

FST

over all the 14 populations

using the estimator previously described by Hivert et al. (2018) and implemented in the computeFST

function. We estimated the standard error of the resulting estimate using a block-jackknife approach specifying

blocks of 10,000 SNPs. As shown below (information printed by the computeFST function), this resulted

in 145 blocks of 700 kb on average. Note that the number of SNPs eligible for black-jackknife (and thus

used for estimation) were reduced to 1,450,000 SNPs (i.e., 90% out of the 1,588,569) due to the removal of

SNPs mapping to contigs represented by less than 10,000 SNPs (i.e., small contigs which may actually be less

reliable as corresponding to the most fragmented part of the assembly) and to a lesser extent SNPs mapping

to contig boundaries.

dsu.global.fst=computeFST(dsu.dat,nsnp.per.bjack.block = 10000)

Starting Block-Jackknife sampling

145 Jackknife blocks identified with 1450000 SNPs (out of 1588569 ).

SNPs map to 15 different chrom/scaffolds

Average (min-max) Block Sizes: 0.698 ( 0.414 - 2.306 ) Mb

#Global Fst estimate

dsu.global.fst$mean.fst

[1] 0.07107105

#95% CI of the Fst estimate (based on the block-jackknife s.e.)

dsu.global.fst$mean.fst+c(-1.96,1.96)*dsu.global.fst$se.fst

[1] 0.06897769 0.07316441

We next computed all the

f−

and

D−

statistics together with their block-jackknife standard error (with

blocks of 15,000 SNPs as for the global FST ) using the compute.fstats function. These included:

•f2

for all the 91 pairs of populations together with their scaled version that correspond to the pairwise

FST

estimator based on the IIS probabilities (i.e., similar to the one implemented by the computeFST

or compute.pairwiseFST functions when specifying method=“Idendity”)

•f3(and their scaled version f?

3) for all the 1,092 possible triplets of populations

•f4(and their scaled version D) for all the 3,003 possible quadruplets of populations

dsu.fstats=compute.fstats(dsu.dat,computeDstat = TRUE,

return.F4.blockjackknife.samples = TRUE,

nsnp.per.bjack.block = 10000)

Estimating Q1

Estimating Q2

Estimating within-population heterozygosities

Estimating F2

Estimating F3

Estimating F4

Computing Dstat

Starting Block-Jackknife sampling

145 Jackknife blocks identified with 1450000 SNPs (out of 1588569 ).

SNPs map to 15 different chrom/scaffolds

Average (min-max) Block Sizes: 0.698 ( 0.414 - 2.306 ) Mb

computing Q1 averages per blocks

computing Q2 averages per blocks

computing F2 averages per blocks

Starting computation of estimators s.e.

within-pop heterozygosity s.e. estimation done

F2 s.e. estimation done

F3 and F3* s.e. estimation done

estimating F4 and Dstat s.e. (may be long since require denominator averages per blocks)

F4 and D s.e. estimation done

Overall Analysis Time: 0 h 1 m 37 s

Note that the return.F4.blockjackknife.samples argument was set to TRUE to allow for estimation of some

admixture rates using

ratios (see below). In addition, as shown below most of the computation time is

spent on estimating D-statistics which are actually not always needed (e.g., treeness tests may be based

) for most future analysis steps. In addition, due to internal optimization of the code block-jackknife

estimation of the standard errors is performed at very limited computational costs (except for D-statistics):

#computing all fstats without D-statistics

#with verbose=FALSE and hence use of sys.time to estimate computation time

tb=Sys.time()

tmp<-compute.fstats(dsu.dat,computeDstat = FALSE,

return.F4.blockjackknife.samples = TRUE,

nsnp.per.bjack.block = 10000,verbose=FALSE)

cat("Analysis (without D-statistics estimation) took",

round(difftime(Sys.time(),tb,units="secs"),1),"s")

Analysis (without D-statistics estimation) took 12.6 s

#computing all fstats without D-statistics

# and without blockjacknife estimates of s.e. (little interest)

tb=Sys.time()

tmp<-compute.fstats(dsu.dat,computeDstat = FALSE,verbose=FALSE)

cat("Analysis (without D-statistics and block-jackknife s.e. estimation)

took",round(difftime(Sys.time(),tb,units="secs"),1),"s")

Analysis (without D-statistics and block-jackknife s.e. estimation)

took 6.3 s

3 Overview of the within-population genetic diversity from esti-

mates of heterozygosities

Estimates of within-population heterozygosities (as 1

−ˆ

) provide a rough assessment of the genetic diversity

of the diﬀerent population. It should here be recalled that the SNP ascertainment process (see section 1) is

expected to favor SNP polymorphic in the native area. The estimates and their 95% CI can be plotted as

follows:

plot(dsu.fstats@heterozygosities$`bjack mean`,ylab="Heterozygosity",xaxt="n",xlab="",

pch="",las=3,ylim=c(0.21,0.29))

axis(1,1:dsu.dat@npools,rownames(dsu.fstats@heterozygosities),las=3)

#plotting CI from bloc jackknife estimate of s.e.

ci=cbind(dsu.fstats@heterozygosities$`bjack mean`-1.96*dsu.fstats@heterozygosities$`bjack s.e.`,

dsu.fstats@heterozygosities$`bjack mean`+1.96*dsu.fstats@heterozygosities$`bjack s.e.`)

tmp<-sapply(1:dsu.dat@npools,f<-function(z){

abline(v=z,lty=3,col="grey")

arrows(z,ci[z,1],z,ci[z,2],angle = 90,code=3,length=0.05,lwd=1.5,

col=tmp.col[rownames(dsu.fstats@heterozygosities)[z]])

})

legend("bottomleft",c("Native","Hawaiian (>1980)","American (>2008)"),

fill=c("#1A1237","#847CA3","#E45A5A"))

As shown in Figure 2, US-Haw was clearly depleted in diversity as previously observed based on microsatellite

data (Fraimout et al. 2017) which is also consistent with its island origin. The populations from the native

area tended to display more diversity than those of the American invasive area with the noticeable exception

of CN-Shi (that displayed signiﬁcantly lower heterozygosity than the other populations from the native area)

and US-Wat (that displayed the highest level of diversity among the populations from the invasive American

area, similar to those of the native area). Note that ﬁltering of SNPs during data preparation (see 1) may

have resulted in upwardly biasing the relative amount of diversity in the native area.

4 Overview of the structuring of genetic diversity across popula-

tions from pairwise-population FST estimates

Figure 3 plots estimates of

FST

with their 95% CI for all the populations using the plot function applied

directly onto the dsu.fstat fstats object computed above.

0.22 0.24 0.26 0.28

Heterozygosity

BR−Pal

CN−Bei

CN−Lia

CN−Nin

CN−Shi

JP−Sap

JP−Tok

US−Col

US−Haw

US−Nca

US−Sdi

US−Sok

US−Wat

US−Wis

Native

Hawaiian (>1980)

American (>2008)

Figure 2: Estimates of within-population heterozygosities with their 95% CI from the ascertained SNPs

(MAF>0.05 in the native area).

layout(matrix(1:2,1,2))

plot(dsu.fstats,stat.name="Fst",main="Fst<=5%",cex.main=1.5,value.range=c(0,0.05))

abline(v=0.01,lty=2,col="blue")

#plot(dsu.fstats,stat.name="Fst",main="5%<Fst<10%",cex.main=1.5,value.range=c(0.05,0.1))

plot(dsu.fstats,stat.name="Fst",main="Fst>5%",cex.main=1.5,value.range=c(0.05,NA))

An alternative (and complementary) view, based on the compute.pairwiseFST function (estimating by default

the

FST

estimator derived under an analysis-of-variance framework) and representing the matrix of pairwise

FST

stored in the resulting pairwisefst object as an heatmap (using the heatmap function), is provided in

Figure 4.

dsu.pairwise.fst=compute.pairwiseFST(dsu.dat)

Computation of the 91 pairwise Fst

Overall Analysis Time: 0 h 0 m 38 s

heatmap(dsu.pairwise.fst)

Note that the two diﬀerent estimates of pairwise-population FST remain very similar (Figure 5).

plot(dsu.pairwise.fst@values$`Fst Estimate`,dsu.fstats@fst.values$`bjack mean`,

xlab="compute.pairwiseFST (method=Anova)",ylab="compute.fstats")

abline(a=0,b=1,lty=2)

As expected from their recent history and as already observed with other analysis (Olazcuaga et al. 2020),

populations tended to cluster according to their area of origin (Figure 4) with some closely geographically

related populations showing very low level of diﬀerentiation. For instance, in continental America, US-Nca,

US-Col and US-Nca all displayed pairwise

FST <

01 with each others (Figure 3) and in the Asian native

area, the three populations CN-Bei, CN-Nin and CN-Lia originating for North-West China were all found

0.00 0.01 0.02 0.03 0.04

Fst<=5%

US−Nca,US−Sdi

US−Sdi,US−Wis

BR−Pal,US−Sdi

US−Sdi,US−Wat

US−Col,US−Sdi

JP−Tok,US−Wat

US−Col,US−Wat

US−Wat,US−Wis

BR−Pal,US−Wat

US−Nca,US−Wat

CN−Bei,JP−Tok

CN−Nin,JP−Tok

BR−Pal,US−Nca

CN−Lia,JP−Tok

BR−Pal,US−Wis

BR−Pal,US−Col

CN−Nin,CN−Shi

CN−Bei,CN−Shi

US−Sok,US−Wat

CN−Lia,CN−Shi

CN−Bei,CN−Nin

CN−Lia,CN−Nin

JP−Sap,JP−Tok

US−Col,US−Nca

CN−Bei,CN−Lia

US−Nca,US−Wis

US−Col,US−Wis

0.06 0.08 0.10 0.12 0.14 0.16 0.18

Fst>5%

US−Col,US−Haw

US−Haw,US−Wis

US−Haw,US−Nca

CN−Shi,US−Haw

BR−Pal,US−Haw

CN−Nin,US−Haw

CN−Bei,US−Haw

CN−Lia,US−Haw

JP−Sap,US−Haw

US−Haw,US−Sdi

JP−Tok,US−Haw

US−Haw,US−Wat

US−Haw,US−Sok

CN−Shi,US−Sok

CN−Shi,US−Sdi

US−Sdi,US−Sok

CN−Bei,US−Sok

CN−Nin,US−Sok

BR−Pal,US−Sok

US−Col,US−Sok

JP−Sap,US−Sdi

CN−Lia,US−Sok

BR−Pal,CN−Shi

BR−Pal,JP−Sap

CN−Nin,US−Sdi

US−Sok,US−Wis

CN−Bei,US−Sdi

CN−Shi,US−Nca

CN−Shi,US−Col

CN−Lia,US−Sdi

CN−Shi,US−Wis

US−Nca,US−Sok

JP−Sap,US−Col

JP−Tok,US−Sdi

BR−Pal,CN−Nin

BR−Pal,CN−Bei

JP−Sap,US−Wis

BR−Pal,JP−Tok

CN−Shi,US−Wat

CN−Nin,US−Nca

CN−Bei,US−Nca

CN−Nin,US−Col

BR−Pal,CN−Lia

CN−Bei,US−Col

CN−Nin,US−Wis

CN−Bei,US−Wis

JP−Sap,US−Nca

CN−Lia,US−Nca

JP−Sap,US−Sok

JP−Tok,US−Col

CN−Lia,US−Col

CN−Lia,US−Wis

JP−Tok,US−Wis

CN−Nin,US−Wat

JP−Tok,US−Sok

CN−Bei,US−Wat

JP−Tok,US−Nca

Figure 3: Estimates of pairwise

FST

for the population pairs of the D. suzukii data set. For ease of reading,

plots have been arranged according to the level of diﬀerentiation into small (

0.05) and moderate to high

level of diﬀerentiation (

0.05). In the leftmost plot, the blue dotted line represent the 0.01

FST

threshold

(i.e., very weak levels of diﬀerentiation)

closely related. Conversely, the Hawaiian sample (US-Haw) was found the most highly diﬀerentiated with all

the other populations, the pairwise

FST

ranging from 11.7% (with US-Sok) to 17.0% (with US-Col). Note

that this explains why US-Haw behave as an outgroup in the heatmap of Figure 4, thereby illustrating that

such representation may be misleading if interpreted in terms of demographic history (Hawaii obviously not

being the area of origin of the species).

5 Insights from f3-based admixture tests

Six of the 14 populations showed at least one signiﬁcantly negative

statistics (both statistics being

almost exactly similar) at the 95% threshold of signiﬁcance (i.e., with an associated Z-score

<−

65). Table

3 summarizes for each of these 6 target populations, the number of signiﬁcantly negative

(and

) together

with the triplet displaying the lowest Z-score (out of the

13×12

= 78 tests per population), as obtained with

US−Haw

JP−Tok

JP−Sap

CN−Shi

CN−Lia

CN−Bei

CN−Nin

US−Sok

US−Wat

US−Sdi

BR−Pal

US−Nca

US−Wis

US−Col

US−Haw

JP−Tok

JP−Sap

CN−Shi

CN−Lia

CN−Bei

CN−Nin

US−Sok

US−Wat

US−Sdi

BR−Pal

US−Nca

US−Wis

US−Col

Figure 4: Heatmap representation of the matrix of pairwise-population FST of 14 D. suzukii populations.

the following script:

f3.signif=dsu.fstats@f3.values$`Z-score`< -1.65

f3s.signif=dsu.fstats@f3star.values$`Z-score`< -1.65

nf3.signif.per.pop=table(dsu.fstats@comparisons$F3[f3.signif,1])

nf3s.signif.per.pop=table(dsu.fstats@comparisons$F3star[f3s.signif,1])

##custom function to retrieve the triplet with lowest Z-score for a given target pop.

tmp.f<-function(x,stat="F3"){

if(stat=="F3"){

x.f3=dsu.fstats@f3.values[dsu.fstats@comparisons$F3[,1]==x,]

}else{

x.f3=dsu.fstats@f3star.values[dsu.fstats@comparisons$F3star[,1]==x,]

0.00 0.05 0.10 0.15

compute.pairwiseFST (method=Anova)

compute.fstats

Figure 5: Comparison of the pairwise-population estimates

FST

computed by the compute.pairwiseFst

(method="Anova") and compute.fstats (equivalent to method="Identity" in compute.pairwiseFst). Note that

for the latter, the estimates were taken as the block-jacknife means.

}

idx=which.min(x.f3$`Z-score`)

return(paste0(rownames(x.f3)[idx]," (Z=",

round(x.f3$`Z-score`[idx],2),")"))

}

###

pop.signif=unique(c(names(nf3.signif.per.pop),names(nf3s.signif.per.pop)))

f3.mostsignif=sapply(pop.signif,tmp.f)

f3s.mostsignif=sapply(pop.signif,tmp.f,stat="F3s")

Pop. Origin #f3<0 #f3*<0 most. sign. triplet (f3) Most sign. triplet (f3*)

CN-Lia Native 1 1 CN-Lia;CN-Shi,JP-Sap (Z=-1.66) CN-Lia;CN-Shi,JP-Sap (Z=-1.65)

JP-Tok Native 11 11 JP-Tok;CN-Nin,JP-Sap (Z=-7.11) JP-Tok;CN-Nin,JP-Sap (Z=-7.09)

US-Col Invasive (AM) 2 2 US-Col;BR-Pal,US-Wis (Z=-3.31) US-Col;BR-Pal,US-Wis (Z=-3.32)

US-Nca Invasive (AM) 6 6 US-Nca;JP-Sap,US-Col (Z=-3.89) US-Nca;JP-Sap,US-Col (Z=-3.9)

US-Wat Invasive (AM) 13 13 US-Wat;US-Sdi,US-Sok (Z=-23.64) US-Wat;US-Sdi,US-Sok (Z=-23.81)

US-Wis Invasive (AM) 4 4 US-Wis;JP-Sap,US-Col (Z=-5.02) US-Wis;JP-Sap,US-Col (Z=-5.05)

Table 3: Summary of the f3 and f3* based tests for all the populations displaying at least one signiﬁcant test

at the 95% threshold (Z< -1.65)

Figure 6 plots the lowest

(including the signiﬁcantly negative ones highlighted in red) for the six populations

displaying at least one signiﬁcant three-population test.

layout(matrix(1:6,3,2))

for(i in pop.signif){

plot(dsu.fstats,stat.name="F3",main=i,cex.main=1.5,value.range=c(NA,1e-3),

pop.f3.target=i,ci.perc=90)

#ci.perc was here set to 90% to obtain the same significant

#tests (highlighted in red) as with Z<-1.65 (since the test is one-sided)

}

−2e−04 0e+00 2e−04 4e−04 6e−04 8e−04

CN−Lia

CN−Lia;CN−Nin,JP−Tok

CN−Lia;CN−Nin,CN−Shi

CN−Lia;CN−Nin,US−Sdi

CN−Lia;CN−Nin,US−Sok

CN−Lia;CN−Nin,US−Nca

CN−Lia;CN−Nin,US−Haw

CN−Lia;BR−Pal,CN−Nin

CN−Lia;CN−Nin,US−Wat

CN−Lia;CN−Nin,US−Wis

CN−Lia;CN−Nin,US−Col

CN−Lia;CN−Nin,JP−Sap

CN−Lia;CN−Bei,CN−Nin

CN−Lia;CN−Bei,CN−Shi

CN−Lia;BR−Pal,CN−Bei

CN−Lia;CN−Bei,US−Sdi

CN−Lia;CN−Bei,US−Col

CN−Lia;CN−Bei,US−Haw

CN−Lia;CN−Bei,US−Wis

CN−Lia;CN−Bei,US−Nca

CN−Lia;CN−Bei,US−Wat

CN−Lia;CN−Shi,US−Sdi

CN−Lia;BR−Pal,CN−Shi

CN−Lia;CN−Shi,US−Col

CN−Lia;CN−Bei,JP−Tok

CN−Lia;CN−Bei,US−Sok

CN−Lia;CN−Shi,US−Wis

CN−Lia;CN−Shi,US−Nca

CN−Lia;CN−Shi,US−Wat

CN−Lia;CN−Shi,US−Haw

CN−Lia;CN−Bei,JP−Sap

CN−Lia;CN−Shi,US−Sok

CN−Lia;CN−Shi,JP−Tok

CN−Lia;CN−Shi,JP−Sap

−0.0015 −0.0010 −0.0005 0.0000 0.0005

JP−Tok

JP−Tok;CN−Shi,US−Sok

JP−Tok;CN−Lia,US−Sok

JP−Tok;CN−Bei,US−Sok

JP−Tok;JP−Sap,US−Sok

JP−Tok;CN−Nin,US−Sok

JP−Tok;JP−Sap,US−Nca

JP−Tok;JP−Sap,US−Wat

JP−Tok;JP−Sap,US−Haw

JP−Tok;JP−Sap,US−Sdi

JP−Tok;JP−Sap,US−Wis

JP−Tok;JP−Sap,US−Col

JP−Tok;CN−Lia,JP−Sap

JP−Tok;BR−Pal,JP−Sap

JP−Tok;CN−Shi,JP−Sap

JP−Tok;CN−Bei,JP−Sap

JP−Tok;CN−Nin,JP−Sap

−5e−04 0e+00 5e−04 1e−03

US−Col

US−Col;US−Sok,US−Wis

US−Col;CN−Lia,US−Nca

US−Col;CN−Shi,US−Nca

US−Col;US−Nca,US−Wis

US−Col;CN−Bei,US−Nca

US−Col;US−Haw,US−Wis

US−Col;US−Nca,US−Sdi

US−Col;CN−Nin,US−Wis

US−Col;CN−Lia,US−Wis

US−Col;CN−Shi,US−Wis

US−Col;CN−Bei,US−Wis

US−Col;US−Wat,US−Wis

US−Col;BR−Pal,US−Nca

US−Col;US−Sdi,US−Wis

US−Col;BR−Pal,US−Wis

−1e−03 −5e−04 0e+00 5e−04 1e−03

US−Nca

US−Nca;CN−Nin,US−Wis

US−Nca;US−Col,US−Wat

US−Nca;US−Sdi,US−Wis

US−Nca;US−Col,US−Haw

US−Nca;US−Haw,US−Wis

US−Nca;US−Wat,US−Wis

US−Nca;JP−Tok,US−Wis

US−Nca;US−Sok,US−Wis

US−Nca;JP−Sap,US−Wis

US−Nca;US−Col,US−Sok

US−Nca;JP−Tok,US−Col

US−Nca;JP−Sap,US−Col

−0.004 −0.003 −0.002 −0.001 0.000 0.001

US−Wat

US−Wat;JP−Tok,US−Sok

US−Wat;BR−Pal,JP−Sap

US−Wat;CN−Nin,US−Sok

US−Wat;CN−Lia,US−Sok

US−Wat;CN−Bei,US−Sok

US−Wat;CN−Shi,US−Sok

US−Wat;BR−Pal,US−Haw

US−Wat;US−Haw,US−Wis

US−Wat;US−Nca,US−Sok

US−Wat;US−Haw,US−Nca

US−Wat;US−Col,US−Haw

US−Wat;US−Sok,US−Wis

US−Wat;US−Col,US−Sok

US−Wat;US−Sdi,US−Sok

US−Wat;BR−Pal,US−Sok

−5e−04 0e+00 5e−04

US−Wis

US−Wis;US−Col,US−Sdi

US−Wis;US−Nca,US−Wat

US−Wis;US−Haw,US−Nca

US−Wis;US−Nca,US−Sdi

US−Wis;US−Col,US−Wat

US−Wis;CN−Nin,US−Nca

US−Wis;CN−Lia,US−Nca

US−Wis;CN−Bei,US−Col

US−Wis;CN−Shi,US−Nca

US−Wis;CN−Shi,US−Col

US−Wis;CN−Bei,US−Nca

US−Wis;BR−Pal,US−Nca

US−Wis;CN−Lia,US−Col

US−Wis;CN−Nin,US−Col

US−Wis;US−Col,US−Haw

US−Wis;US−Col,US−Nca

US−Wis;US−Col,US−Sok

US−Wis;JP−Tok,US−Col

US−Wis;JP−Sap,US−Col

Figure 6: Estimates of the lowest

statistics with their 95% Conﬁdence Intervals for the 6 populations

displaying at least one signiﬁcant three-population test at the 95% threshold (shown in red).

In the native area, JP-Tok showed clear evidence of admixture with 11 signiﬁcant tests that all involved

JP-Sap as a source proxy. The three lowest

were obtained with the three Chinese populations (CN-Nin,

CN-Bei and CN-Shi in increasing order of

). Assuming an admixture-graph like history, this suggested

that the two populations branching the closest to the two sources of JP-Tok were JP-Sap and CN-Nin. The

remaining Chinese population, CN-Lia showed some little evidence of admixture with only one test barely

signiﬁcant at the 95% threshold for the triplet involving CN-Shi and JP-Sap as source proxies.

Out of the 7 populations from the American continent (excluding Hawaii), 4 showed evidence of admixture,

namely US-Col, US-Wis, US Nca and US-Wat. The strongest evidence (13 signiﬁcant tests) was found for

US-Wat which has up to now been considered as the closest to the ﬁrst invader of continental America

based on historical records (Fraimout et al. 2017). Moreover, the three signals with the lowest Z-score

all involved two source populations originating from the (continental) American invasive area. As their

underlying

CI did not overlap with the other tests, these three pairs of populations may be considered

as the closest (among the sampled populations) to the original US-Wat source populations. Hence, the

three most signiﬁcant signals all involved US-Sok (Northern US) as a source proxy while the other source

proxies were, in order of increasing Z-score (i.e., decreasing evidence as Z-score are here negative), BR-Pal,

US-Sdi and US-Col respectively. Challenging the initial view, this thus suggests that US-Wat derived from

an admixture between two populations already established in the continental America, one Northern (here

represented by US-Sok) and the other Southern (here represented by BR-Pal, US-Sdi and to a lesser extent

US-Col that may actually derived from US-Sdi considering historical records) as will be investigated below

via admixture graph construction. For this Southern “ghost” population, we may further speculate that

BR-Pal, although geographically distantly related, could be the closest proxy as a result of a rapid spread of

D. suzukii in Southern America although the absence of additional samples in South America makes this

hypothesis diﬃcult to test. The three other American populations, US-Col, US-Wis and US-Nca, with at

least one signiﬁcantly negative

displayed only a moderate number of signiﬁcant tests (compared to other

populations) and were all very closely related (section 4). For each of these, the signiﬁcantly negative

had

highly overlapping CI and always involved one or the two other populations

. This thus suggests complex

patterns of recurrent admixture events among these three populations (consistent with their low level of

diﬀerentiation and close geographic origins).

6 Exploring invasion scenarios with admixture graph construction

To provide further insights into the relationships of the surveyed populations and the probable scenarios

of invasion of the species in American and European area, we relied on admixture graph construction.

The purpose was no here to build a comprehensive admixture graph, which may be elusive given the

close relationships of the populations and the pervasiveness of recent admixture events among the diﬀerent

populations, but rather to identify key regional event that happened at early time of invasion. Importantly,

we pay critical attention to the support of the diﬀerent graphs to assess the validity of the proposed scenarios

by making extensive use of the compare.ﬁtted.fstats function.

6.1 Scaﬀold trees built (naively) from scratch are unreliable

We ﬁrst evaluated a naive approach consisting of building a scaﬀold tree of presumably unadmixed populations

using the ﬁnd.tree.popset and rooted.njtree.builder functions. Although two candidate sets of 5 populations

could be identiﬁed, the bifurcating trees obtained from them provided a poor ﬁt to the data, making them

obviously unreliable for further admixture graph construction. More precisely, we inferred for each of these

two sets the best rooted bifurcating tree using the rooted.njtree.builder function and evaluate its ﬁt with

the compare.ﬁtted.fstats function (note that the ﬁtted f-statistics are invariant to root positioning for such

bifurcating trees). In addition, we also used the branch length estimates obtained with the ﬁt.graph function

(instead of those relying on neighbor-joining algorithm used by rooted.njtree.builder ) of the best rooted tree

but this lead to similar conclusions.

#Identification of candidate (unadmixed) scaffold populations from scratch

scaf.pop=find.tree.popset(dsu.fstats)

Number of sets: 24 of Npops= 4 each

More precisely, the four signiﬁcant tests with US-Wis as a target all involved US-Col as a source proxy and one also involved

US-Nca (although with the lowest

); out of the six signiﬁcant tests with US-Nca as a target, three involved US-Col as a source

proxy and the three other involved US-Wis; and the two signiﬁcant tests with US-Col as a proxy involved US-Wis as a source

proxy

Number of sets: 2 of Npops= 5 each

scaf.pop$pop.sets

[,1] [,2] [,3] [,4] [,5]

PopSet1 "BR-Pal" "CN-Bei" "CN-Shi" "JP-Sap" "US-Sok"

PopSet2 "CN-Bei" "CN-Shi" "JP-Sap" "US-Sdi" "US-Sok"

#construction of scaffold trees for each set

scaf.pop.trees=list()

for(i in 1:scaf.pop$n.sets){

scaf.pop.trees[[i]]=list()

tmp.njtree<-rooted.njtree.builder(pop.sel = scaf.pop$pop.sets[i,],

fstats = dsu.fstats,verbose=FALSE)

scaf.pop.trees[[i]][["njtree"]]<-tmp.njtree$best.rooted.tree

tmp.graph=generate.graph.params(tmp.njtree$best.rooted.tree@graph,dsu.fstats,verbose=F)

scaf.pop.trees[[i]][["fitgraph"]]<-fit.graph(tmp.graph,verbose=FALSE)

rm(tmp.njtree,tmp.graph)

}

#Evaluation of scaffold trees using compare.fitted.fstats

for(i in 1:scaf.pop$n.sets){

cat("\nScaffold set:",i,"\n")

cat("\tRooted NJ tree\n")

tmp.comp<-compare.fitted.fstats(dsu.fstats,

fitted.graph = scaf.pop.trees[[i]][["njtree"]],

n.worst.stats = 1)

cat("\tRooted fit graph tree\n")

tmp.comp<-compare.fitted.fstats(dsu.fstats,

fitted.graph = scaf.pop.trees[[i]][["fitgraph"]],

n.worst.stats = 1)

}

Scaffold set: 1

Rooted NJ tree

1 Worst fit for:

Estimated Fitted Z–score

BR-Pal,CN-Shi;JP-Sap,US-Sok -0.005313611 -1.734723e-18 31.46101

Rooted fit graph tree

1 Worst fit for:

Estimated Fitted Z–score

BR-Pal,CN-Shi;JP-Sap,US-Sok -0.005313611 0 31.46101

Scaffold set: 2

Rooted NJ tree

1 Worst fit for:

Estimated Fitted Z–score

CN-Shi,US-Sdi;JP-Sap,US-Sok 0.005093941 0.002936964 -11.15586

Rooted fit graph tree

1 Worst fit for:

Estimated Fitted Z–score

CN-Shi,JP-Sap;US-Sdi,US-Sok 0.00433532 0 -6.78807

Hence, the bifurcating scaﬀold tree inferred for the ﬁrst and second candidate sets of candidate scaﬀold

populations lead to f-statistics outlying from the raw estimated f-statistics, by up to 31 and 6.8 standard

errors (as measured by the printed Z-score) respectively when considering neighbor-joining estimates of

branch lengths.

It should be noticed that for both sets, the resulting NJ tree lead to quadruplets that violate the treeness

test, i.e., that resulted in a quadruplet conﬁgurations that disagree with the one that was retained by the

ﬁnd.tree.popset function. For instance, in the ﬁrst set of candidate populations, the obtained NJ tree topology

resulted in the BR-Pal,CN-Shi;JP-Sap,US-Sok quadruplet which did not pass the

-based treeness tests as the

quadruplet conﬁguration passing the treeness test for these four populations is BR-Pal,JP-Sap;CN-Shi,US-Sok:

tmp.sel=apply(dsu.fstats@comparisons$F4,1,

ff<-function(x){sum(x %in% c("BR-Pal","CN-Shi","JP-Sap","US-Sok"))==4})

dsu.fstats@f4.values[tmp.sel,]

Estimate bjack mean bjack s.e. Z-score

BR-Pal,CN-Shi;JP-Sap,US-Sok -0.0051982044 -0.0053136113 0.0001688951 -31.461012

BR-Pal,JP-Sap;CN-Shi,US-Sok 0.0005166386 0.0002464223 0.0009666008 0.254937

BR-Pal,US-Sok;CN-Shi,JP-Sap 0.0057148430 0.0055600336 0.0008822927 6.301802

6.2 Investigating the relationships among the populations from the native area

(and the Hawaiian population)

We thus adopted an alternative and more historically informed strategy to build the starting scaﬀold graph,

relying on the known Asian origin of the species (which may safely be assumed to be correct). We ﬁrst

seek to describe the historical relationships among the 6 populations from the native area also including the

Hawaiian population that was ﬁrst observed in 1980, i.e., more than 30 years (ca. 300 generations for D.

suzukii) earlier than the other populations of invasive areas (Fraimout et al. 2017).

6.2.1 Building a scaﬀold tree of native and the Hawaiian populations

We thus used the same approach as in the above section 6.1 to build a scaﬀold tree but focusing only on the

six native and the Hawaiian populations:

#Identification of candidate (unadmixed) scaffold populations

pop.am.continent=c("BR-Pal","US-Col","US-Nca","US-Sdi","US-Sok","US-Wat","US-Wis")

scaf.pop.nat=find.tree.popset(dsu.fstats,excluded.pops = pop.am.continent)

Number of sets: 2 of Npops= 4 each

scaf.pop.nat$pop.sets

P1 P2 P3 P4

PopSet1 "CN-Bei" "CN-Nin" "CN-Shi" "US-Haw"

PopSet2 "CN-Bei" "CN-Shi" "JP-Sap" "US-Haw"

#f4 associated Z-score range

#(here the range min and max are the same since only four populations in the sets)

scaf.pop.nat$Z_f4.range

Min. |Zscore| Max. |Zscore|

PopSet1 1.086422 1.086422

PopSet2 1.060292 1.060292

Here two sets of 4 candidate populations could then be proposed. We chose to focus on the PopSet 2

(CN-Bei,CN-Shi ; JP-Sap,US-Haw) because it displayed the smallest

associated Z-score (|Z|=1.06) and

CN-Bei was more distantly related to CN-Shi than CN-Nin from both a geographical (Figure 1) and genetical

point of view (Figure 3). As this selected set only contained 4 populations, the (best) unrooted topology was

directly available from the quadruplet passing the

treeness test, i.e. ((CN-Bei,CN-Shi),(JP-Sap,US-Haw)).

Rooting the tree based on heterozygosities of leaf populations (as implemented in the rooted.njtree.builder

function) would actually be quite sensitive to long-branch attraction by the highly diverged US-Haw (Figure

3) resulting in positionning US-Haw as the outgroup. Yet, based on historical knowledge of the introduced

status of this population, we here chose to set the root of the tree on the branch relating the common ancestor

of the Chinese populations (CN-Bei and CN-Shi) and the common ancestor of JP-Sap and US-Haw. Such a

positioning of the root is consistent with a (recent) Japanese origin of the Hawaiian population (Fraimout

et al. 2017). We ﬁnally obtained the scaﬀold tree represented in Figure 7a ﬁtted as follows:

#define (manually) the rooted scaffold tree

scaf.tree=rbind(c("CN","R",""),

c("JP","R",""),

c("CN-Bei","CN",""),

c("CN-Shi","CN",""),

c("JP-Sap","JP",""),

c("US-Haw","JP",""))

#fit the scaffold tree (with edges lengths drift units and 95% CI estimated)

scaf.tree.params<-generate.graph.params(scaf.tree,fstats = dsu.fstats)

Total Number of Parameters: 5 (5 edges lengths + 0 adm. coeff.)

Total Number of Statistics: 6 (3 F2 and 3 F3)

scaf.tree.fit<-fit.graph(scaf.tree.params,drift.scaling = TRUE,compute.ci = TRUE)

Estimation started (direct algebraic solution)

Estimation ended in 0 m 0 s

Final Score: 1.124219

BIC: 97.95432

#plot the tree

plot(scaf.tree.fit)

#CI of edges lengths

scaf.tree.fit@edges.length.ci

95% Inf. 95% Sup. 95% Inf. (drift scaled) 95% Sup. (drift scaled)

R<->CN 0.0025699799 0.003649279 0.022374993 0.031771684

R<->JP 0.0025699799 0.003649279 0.022374993 0.031771684

CN<->CN-Bei 0.0009098188 0.001161736 0.006498229 0.008297507

CN<->CN-Shi 0.0038994202 0.005559202 0.027850956 0.039705673

JP<->JP-Sap 0.0052402047 0.007662313 0.034940631 0.051090761

JP<->US-Haw 0.0365762281 0.039862096 0.243882932 0.265792437

6.2.2 Positionning separately each remaining native populations on the scaﬀold tree

We then positioned the three other native populations (CN-Nin, CN-Lia and JP-Tok) on the scaﬀold tree as

follows:

for(p in c("CN-Nin","CN-Lia","JP-Tok")){

add.p=add.leaf(scaf.tree.fit,leaf.to.add = p,dsu.fstats,verbose=FALSE,

drift.scaling=TRUE,compute.ci=TRUE)

add.p.delta.bic=format(sort(add.p$bic-min(add.p$bic))[2],digits = 3)

add.p.comp.fit=compare.fitted.fstats(dsu.fstats,add.p$best.fitted.graph,

n.worst.stats = 0)

add.p.worst.fit=add.p.comp.fit[which.max(abs(add.p.comp.fit$`Z--score`)),]

plot(add.p$best.fitted.graph)

}

As shown in Figure 7, all the three populations could unambiguously be placed onto the scaﬀold tree as

admixed populations with a good ﬁt of each resulting graphs. Indeed, for each set of ﬁve populations (four

scaﬀold one and the added population), the

BI C

of the best ﬁtting graph was always far lower than all

the other possible graphs (∆

BI C >

10). Note that for JP-Tok ∆

BI C

= 0 in Figure 7d because the graph

with the second lowest BIC corresponds to the positioning of the admixed source S1-JP-Tok to the other

(a) Scaﬀold

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

∆BIC=12.1

(b) Placing CN-Nin onto scaﬀold

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

∆BIC=28.2

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

∆BIC=0

(d) Placing JP-Tok onto scaﬀold

Figure 7: Inference of admixture graph connecting the populations from the native area (extended to

Hawaii). The scaﬀold tree is displayed in a) together with the best inferred admixture graph obtained by

the (independent) positionning of the three other populations from the native area, CN-Nin (b); CN-Lia

ﬁtted f-statistics and its associated Z-score is given and the diﬀerence of BIC of the graph with the graphs

displaying the second lowest BIC is provided (∆

BI C

) as a measure of support. For all graphs, the ﬁtted edge

lengths are in drift units (x1000) since drift.scaling argument was set to TRUE.

alternative branch which is strictly equivalent to the graph displayed (as the two branches coming from R are

not identiﬁable). These two equivalent graphs are highly supported since ∆

BI C

= 20

4between either of

them and the graph with the third lowest BIC).

6.2.3 Searching for the most comprehensive admixture graph relating all native populations

We next relied on the graph.builder function to try to jointly include all the three remaining populations

(CN-Lia, CN-Nin and JP-Tok) on the graph. As a ﬁrst step, we considered simply adding two of these

populations simultaneously, leading to 6 diﬀerent runs of the function (3 pairs of populations and 2 orders of

inclusion for each pair. Importantly, we carefully evaluated the ﬁt of each of the best ﬁtting graph identiﬁed

using the compare.ﬁtted.fstats function:

#Creating the six pairs of leaves to add for graph.builder

#a custom function to remove sets (generate by expand grid) with non unique pop

cleangrid<-function(x){n=ncol(x);x[apply(x,1,f<-function(y){length(unique(y))==n}),n:1]}

pops.to.include<-expand.grid(c("CN-Nin","CN-Lia","JP-Tok"),

c("CN-Nin","CN-Lia","JP-Tok")) %>% as.matrix() %>% cleangrid()

#exploring and evaluating the graphs

for(i in 1:nrow(pops.to.include)){

cat("##########################################\n")

cat("Inclusion Order:",pops.to.include[i,],"\n")

tmp.graph<-graph.builder(scaf.tree.fit,leaves.to.add = pops.to.include[i,],

fstats=dsu.fstats,verbose=F)

#number of graphs within Dbic<6 (default)

cat("\tNumber Of Graphs:",tmp.graph$n.graphs,"\n")

#min BIC

cat("\tMin BIC:",min(tmp.graph$bic),"\n")

#worst fitted fstats (best fitting graph)

cat("\tWorst fstats fit:\n")

tmp.comp.fit=compare.fitted.fstats(dsu.fstats,tmp.graph$best.fitted.graph,

n.worst.stats = 0)

print(tmp.comp.fit[which.max(abs(tmp.comp.fit$`Z--score`)),])

}

##########################################

Inclusion Order: CN-Nin CN-Lia

Number Of Graphs: 2

Min BIC: 332.6374

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Lia;JP-Sap,US-Haw -0.0002110636 0 4.995489

##########################################

Inclusion Order: CN-Nin JP-Tok

Number Of Graphs: 2

Min BIC: 308.1103

Worst fstats fit:

Estimated Fitted Z–score

CN-Shi,US-Haw;JP-Sap,JP-Tok -0.0005025053 -0.00067728 -1.530277

##########################################

Inclusion Order: CN-Lia CN-Nin

Number Of Graphs: 2

Min BIC: 316.4586

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Nin;JP-Sap,US-Haw -0.0001769363 3.469447e-18 3.62954

##########################################

Inclusion Order: CN-Lia JP-Tok

Number Of Graphs: 6

Min BIC: 299.4749

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Shi;JP-Sap,US-Haw -5.079243e-05 -3.469447e-18 1.060292

##########################################

Inclusion Order: JP-Tok CN-Nin

Number Of Graphs: 2

Min BIC: 368.2099

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,US-Haw;CN-Shi,JP-Tok 0.005775447 6.938894e-18 -7.207447

##########################################

Inclusion Order: JP-Tok CN-Lia

Number Of Graphs: 6

Min BIC: 299.4749

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Shi;JP-Sap,US-Haw -5.079243e-05 0 1.060292

Note the beneﬁts of exploring all the possible orders of inclusion with graph.builder for a given set of

populations. For instance, the best ﬁtted graph obtained when including JP-Tok and CN-Nin displayed a

highly lower BIC if CN-Nin was the ﬁrst to be included. Conversely, in some cases, the same best ﬁtting

graph may be obtained with the diﬀerent possible inclusion orders (e.g., see the results obtained with the

CN-Lia and JP-Tok pair).

Interestingly, this exploration suggests that CN-Lia, CN-Nin and JP-Tok may not be included jointly to

the scaﬀold tree to obtain a comprehensive admixture graph for the seven native populations. Indeed, and

although these two populations are closely related, the graphs obtained when adding both CN-Nin and

CN-Lia to the scaﬀold tree displayed poor ﬁtted f-stats (some with associated |Z|>3.95). This was conﬁrmed

when trying to add the three populations simultaneously (and exploring all the 6 possible orders of inclusion)

with graph.builder as shown below:

###a custom function to list all possible orders of inclusion (permutations)

#adapted from https://www.r-bloggers.com/2019/06/learning-r-permutations-and-combinations-with-base-r/

permutations <- function(x) {

n<- length(x)

if (n == 1){out<-x}else{

out <- NULL

for (i in 1:n){out <- rbind(out, cbind(x[i], permutations(x[-i])))}

}

return(out)

}

pops.to.include<-permutations(c("CN-Nin","CN-Lia","JP-Tok"))

#exploring and evaluating the graphs

for(i in 1:nrow(pops.to.include)){

cat("##########################################\n")

cat("Inclusion Order:",pops.to.include[i,],"\n")

tmp.graph<-graph.builder(scaf.tree.fit,leaves.to.add = pops.to.include[i,],

fstats=dsu.fstats,verbose=F)

#number of graphs within Dbic<6 (default)

cat("\tNumber Of Graphs:",tmp.graph$n.graphs,"\n")

#min BIC

cat("\tMin BIC:",min(tmp.graph$bic),"\n")

#worst fitted fstats (best fitting graph)

cat("\tWorst fstats fit:\n")

tmp.comp.fit=compare.fitted.fstats(dsu.fstats,tmp.graph$best.fitted.graph,

n.worst.stats = 0)

print(tmp.comp.fit[which.max(abs(tmp.comp.fit$`Z--score`)),])

}

##########################################

Inclusion Order: CN-Nin CN-Lia JP-Tok

Number Of Graphs: 2

Min BIC: 480.5329

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Lia;JP-Sap,US-Haw -0.0002110636 0 4.995489

##########################################

Inclusion Order: CN-Nin JP-Tok CN-Lia

Number Of Graphs: 5

Min BIC: 480.5329

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Lia;JP-Sap,US-Haw -0.0002110636 -3.469447e-18 4.995489

##########################################

Inclusion Order: CN-Lia CN-Nin JP-Tok

Number Of Graphs: 2

Min BIC: 463.636

Worst fstats fit:

Estimated Fitted Z–score

CN-Lia,JP-Tok;CN-Nin,CN-Shi -0.0007377375 -0.0004944342 3.733745

##########################################

Inclusion Order: CN-Lia JP-Tok CN-Nin

Number Of Graphs: 11

Min BIC: 513.4301

Worst fstats fit:

Estimated Fitted Z–score

CN-Lia,US-Haw;CN-Shi,JP-Tok 0.005572856 0.000100214 -7.206377

##########################################

Inclusion Order: JP-Tok CN-Nin CN-Lia

Number Of Graphs: 8

Min BIC: 518.0997

Worst fstats fit:

Estimated Fitted Z–score

CN-Lia,JP-Tok;CN-Nin,US-Haw 0.004911067 -0.000220809 -7.205941

##########################################

Inclusion Order: JP-Tok CN-Lia CN-Nin

Number Of Graphs: 8

Min BIC: 518.0997

Worst fstats fit:

Estimated Fitted Z–score

CN-Lia,JP-Tok;CN-Nin,US-Haw 0.004911067 -0.000220809 -7.205941

Overall, only six out of the seven populations from the native area (extended to Hawaii) could be related

with an admixture graph with good ﬁtting properties i.e., either by adding to the scaﬀold tree (consisting of

US-Haw, JP-Sap, CN-Bei and CN-Shi) the population Jp-Tok and either CN-Lia or CN-Nin. Note that the

two resulting best ﬁtting graph cannot be compared based on the BIC criterion as the two sets of populations

diﬀer. A closer inspection of the diﬀerent graphs obtained (6 within ∆

BI C <

6with the best ﬁtting graph for

the CN-Lia, Jp-Tok included pairs and only one for the CN-Nin, Jp-Tok pair) made us chose as the most

relevant graph the graph displayed in Figure 8. As a matter of fact, this graph had the lowest absolute

Z-score for the worst ﬁtted f-stats (that was actually the same as the one for the scaﬀold tree) and it was also

quite consistent with the one obtained when adding each population individually to the scaﬀold tree (Figure

7c and 7d).

natgraph<-graph.builder(scaf.tree.fit,leaves.to.add = c("CN-Lia","JP-Tok"),dsu.fstats,

verbose=F,drift.scaling=T,compute.ci=TRUE)

plot(natgraph$best.fitted.graph)

JP-Tok

JP-Sap

CN-Lia

CN-Shi

CN-Bei

US-Haw

S-JP-Tok

S2-CN-Lia

S1-JP-Tok

S-CN-Lia

3 64%

S2-JP-Tok

36%

S1-CN-Lia

3296%

25531

Figure 8: Best ﬁtting admixture graph connecting six out of the seven populations representative of the native

area (extended to Hawaii). The ﬁtted edge lengths are in drift units (x1000) since drift.scaling argument was

set to TRUE.

The diﬃculties encountered to jointly add CN-Lia and CN-Lin may be related to the fact that they are only

very slightly diﬀerentiated (Figures 3 and 4) making it diﬃcult to simplify their joint history with simple

two-way admixture graphs (likely due to a high amount of gene ﬂow between them). It should also be noticed

that the CN-Nin sample may be slightly heterogeneous as it mixes individuals from very close locations but

sampled at two diﬀerent times (Table 1).

6.2.4 Estimates of the parameters of the admixture graph for the native and Hawaiian pop-

ulations and their conﬁdence intervals

The CI’s for branch lengths (on both unscaled and drift scaled units) of the admixture graph connecting the

6 populations Figure 8 together with the two admixture proportions are directly available from the natgraph

object (as it was generated with options drift.scaled=TRUE and compute.ci=TRUE):

#branch lengths

natgraph$best.fitted.graph@edges.length.ci

95% Inf. 95% Sup. 95% Inf. (drift scaled) 95% Sup. (drift scaled)

S-JP-Tok<->JP-Tok 6.315686e-04 0.0007827838 0.0044362902 0.005498462

S2-CN-Lia<->S1-JP-Tok 8.258283e-04 0.0018089613 0.0058431246 0.012799255

R<->S2-JP-Tok 4.876279e-04 0.0009225970 0.0043358045 0.008203386

S1-JP-Tok<->JP-Sap 3.915283e-04 0.0005084688 0.0027874828 0.003620039

S2-JP-Tok<->CN 4.100608e-03 0.0054398001 0.0287396430 0.038125541

S-CN-Lia<->CN-Lia 2.491030e-04 0.0003931522 0.0017960161 0.002834601

CN<->S1-CN-Lia 7.342082e-05 0.0002078678 0.0005283615 0.001495888

JP<->S2-CN-Lia 4.088231e-03 0.0052797637 0.0272544120 0.035197830

S1-CN-Lia<->CN-Shi 3.821499e-03 0.0051975302 0.0273294661 0.037170156

R<->JP 4.876279e-04 0.0009225970 0.0043358045 0.008203386

CN<->CN-Bei 9.842052e-04 0.0011107899 0.0070826785 0.007993626

JP<->US-Haw 3.675243e-02 0.0397726523 0.2450120628 0.265146533

#admixture proportions

#The names were automatically given by graph.builder as A (and 1-A) for ancestry proportion

#contributing to the first (CN-Lia) and B for ancestry proportion contributing to

#the second (JP-Tok) included populations

natgraph$best.fitted.graph@admix.prop.ci

95% Inf. 95% Sup.

B 0.6312323 0.6576909

A 0.9566978 0.9633939

Alternatively, given the topology of the graph, it is also possible to estimate the proportion of Chinese

ancestry in the CN-Lia population using ratios of f4estimates as:

#Prop of Chinese ancestry in CN-Lia (A parameter defined above)

#The ratio F4(JP-Sap,CN-Shi;CN-Bei,US-Haw)/F4(JP-Sap,CN-Lia;CN-Bei,US-Haw)

#estimates A as can be shown with symbolic computations of F4

natgraph.par<-generate.graph.params(natgraph$best.fitted.graph@graph,verbose=FALSE)

natgraph.symbolicfstats<-graph.params2symbolic.fstats(natgraph.par)

natgraph.symbolicfstats$F4.equations[

grep("F4\$JP-Sap,CN-Shi;CN-Bei,US-Haw\$",natgraph.symbolicfstats$F4.equations)]

[1] "F4(JP-Sap,CN-Shi;CN-Bei,US-Haw) = -(S2-JP-Tok<->CN+R<->S2-JP-Tok+R<->JP)"

natgraph.symbolicfstats$F4.equations[

grep("F4\$JP-Sap,CN-Lia;CN-Bei,US-Haw\$",natgraph.symbolicfstats$F4.equations)]

[1] "F4(JP-Sap,CN-Lia;CN-Bei,US-Haw) = -(R<->JP+S2-JP-Tok<->CN+R<->S2-JP-Tok)*A"

#estimation via ratio of F4

alpha_lia=compute.f4ratio(dsu.fstats,num.quadruplet = c("CN-Bei","US-Haw","CN-Lia","JP-Sap"),

den.quadruplet = c("CN-Bei","US-Haw","CN-Shi","JP-Sap"))

alpha_lia[2]

bjack mean

0.9559757

#95% CI

alpha_lia[2] + c(-1.96,1.96)*alpha_lia[3]

[1] 0.9435186 0.9684329

The resulting estimates of Chinese ancestry in the CN-Lia population are thus consistent. However, as

expected from the simulation study (see the main text of the manuscript), the conﬁdence intervals estimated

with the graph ﬁtting procedure are smaller (and presumably too optimistic) than the ones estimates with

the f4-ratio which may be considered as more reliable.

Interestingly, a similar proportions of Chinese ancestry could be obtained using

-ratio for CN-Nin than for

CN-Lia (assuming CN-Nin and CN-Lia have a similar ancestry with respect to the other populations). It was

also consistent with the one obtained in Figure 7b:

#Prop of Chinese ancestry in CN-Lia

alpha_nin=compute.f4ratio(dsu.fstats,num.quadruplet = c("CN-Bei","US-Haw","CN-Nin","JP-Sap"),

den.quadruplet = c("CN-Bei","US-Haw","CN-Shi","JP-Sap"))

alpha_nin[2]

bjack mean

0.9469928

#95% CI

alpha_nin[2] + c(-1.96,1.96)*alpha_nin[3]

[1] 0.9342755 0.9597100

Note that no such

-ratio based estimation of admixture proportions could be carried for the JP-Tok

population as no outgroup was available for one of its ancestry (i.e., one source connects directly to the root).

6.3 Investigating the origins of the populations from the American invasive

area using admixture graphs

The

-based tests of admixture (section 5) showed that four (US-Col, US-Nca, US-Wat and US-Wis) of the

seven American populations displayed admixture signals. In addition, all involved the most signiﬁcant ones

always involved a source related to a population from the invasive area suggesting that that were not directly

connected to the ﬁrst invader. We will thus ﬁrst attempt to position the remaining American populations

(BR-Pal, US-Sdi and US-Sok) with respect to the native and Hawaiian populations.

6.3.1 Positioning BR-Pal, US-Sdi and US-Sok with respect to the native populations

We investigated the relationships of the three American populations, BR-Pal, US-Sdi and US-Sok, with the

native populations by identifying with the add.leaf function the best positioning of each population onto

either the admixture graph (Figure 8) or the scaﬀold tree (Figure 7a) obtained above using the following

command (output not shown):

for(p in c("BR-Pal","US-Sdi","US-Sok")){

#positioning onto the admixture graph of native pops

add.p=add.leaf(natgraph$best.fitted.graph,leaf.to.add = p,dsu.fstats,verbose=FALSE,

drift.scaling=TRUE,compute.ci=TRUE)

plot(add.p$best.fitted.graph)

add.p.comp=compare.fitted.fstats(dsu.fstats,add.p$best.fitted.graph,n.worst.stats = 1)

#positioning onto the scaffold tree of native pops

add.p=add.leaf(scaf.tree.fit,leaf.to.add = p,dsu.fstats,verbose=FALSE,

drift.scaling=TRUE,compute.ci=TRUE)

plot(add.p$best.fitted.graph)

add.p.comp=compare.fitted.fstats(dsu.fstats,add.p$best.fitted.graph,n.worst.stats = 1)

}

As summarized in Figure 9, the three populations could be placed with a good resulting ﬁt to each graph of

native populations. The Z-score of the worst ﬁtted f-statistics was always

2in absolute value except for the

admixture resulting from the placing of US-Sok on the admixture graph of native populations (Z=-4.14; Figure

9E). In addition, for each population, the positioning onto the two graphs was concordant and suggested

that they were all admixed although this was not detected by the

based test of admixture (Table 3).

This is likely due to a too strong subsequent divergence from their ancestral admixed source. All the three

populations shared an Hawaiian (ancestral to US-Haw) source, the other source being of presumably Chinese

origin for both US-Sdi and BR-Pal and of Japanese origin (ancestral to JP-Sap) for US-Sok. For BR-Pal,

it should be noticed that the support for the positioning onto the scaﬀold tree (Figure 9B) remained loose

A) BR−Pal onto nat. pops. graph

∆BIC=17.6

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Lia (Z=1.57)

B) BR−Pal onto nat. pop. scaf. tree

∆BIC=0.468

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

C) US−Sdi onto nat. pops. graph

∆BIC=6.44

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

D) US−Sdi onto nat. pop. scaf. tree

∆BIC=5.04

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

E) US−Sok onto nat. pops. graph

∆BIC=7.84e−06

worst fit: CN−Bei,JP−Tok;US−Haw,US−Sok (Z=−4.14)

F) US−Sok onto nat. pop. scaf. tree

∆BIC=19.1

worst fit: CN−Bei,CN−Shi;US−Haw,US−Sok (Z=−1.16)

Figure 9: Positionning of the BR-Pal, US-Sdi and US-Sok populations from the American invasive areas onto

the admixture graph connecting the six native and Hawaiian populations (Figure 8) and the scaﬀold tree

of native (and Hawaiian) populations (Figure 7a). The best ﬁtting graphs (as obtained with the function

add.leaf) are displayed together with i) the worst ﬁtted f-statistics and their associated Z-score; and ii) the

diﬀerence of their BIC with respect to the graphs displaying the second lowest BIC (∆

BI C

) as a measure of

support. The target populations are highlighted in yellow. For all the graphs, the ﬁtted edge lengths are in

drift units (x1,000) since drift.scaling argument was set to TRUE.

(∆

BI C <

6) and a closer inspection of the corresponding add.leaf results showed that three alternative graphs

resulted in a BIC very similar to this best ﬁtting graph:

addPal=add.leaf(scaf.tree.fit,leaf.to.add = "BR-Pal",dsu.fstats,verbose=FALSE,

drift.scaling=T,compute.ci=T)

#Difference of BIC wrt the best fitting graph

addPal.dbic=addPal$bic-min(addPal$bic)

format(addPal.dbic,digits = 3)

[1] "111.581" "111.581" "246.719" "246.647" "169.198" " 27.333" "111.581" "116.186" "115.099"

[10] "116.186" " 0.468" "116.186" "115.099" "116.186" " 0.468" "251.252" "116.186" " 0.468"

[19] "115.099" " 0.000" " 31.939"

As shown in Figure 10, these corresponded to the branching of S1-BR-Pal source on i) the CN<->CN-Bei

branch (Figure 10D) with a null distance from CN which is actually the same as Figures 9B and 10A (the

estimated admixture proportions being also the same); ii) the R<->CN branch (Figure 10B) or strictly

equivalently the R<->JP branch (Figure 10C). The two latter did not lead to any null branches (which is

usually preferable) and the branching of S1-BR-Pal was always closer to CN (10

−3

and 11

−3

drift

units respectively, recalling that the positioning of the root R is arbitrarily set in the middle of the JP<->CN

unrooted branch) than JP in agreement with a Chinese origin. This is concordant with Figures 9A) in which

the presence of JP-Tok and its presumably Chinese ancestral source (automatically named S2-JP-Tok) allows

to rule out this latter positioning of S1-BR-Pal with a high support (∆

BI C

= 17

6). Note also that this

resulted in either cases in an higher estimated Chinese ancestry for BR-Pal.

For US-Sdi, the three same types of branching as for BR-Pal were within 5.04 BIC units of the best ﬁtting

graph (i.e., positioning of the S1-US-Sdi source with respect to CN in Figure 9D), i.e., the support for these

alternative graphs was clearly lower according to the BIC criterion (and the absolute Z-score for the worst

ﬁtted f-stats was also higher) as shown with the following codes:

addSdi=add.leaf(scaf.tree.fit,leaf.to.add = "US-Sdi",dsu.fstats,verbose=FALSE,

drift.scaling=T,compute.ci=T)

#Difference of BIC wrt the best fitting graph

addSdi.dbic=addSdi$bic-min(addSdi$bic)

format(addSdi.dbic,digits = 3)

[1] "185.91" "185.91" "541.90" "541.30" "206.66" " 13.04" "185.91" "190.52" "187.77" "190.52"

[11] " 5.04" "190.52" "187.77" "190.52" " 5.04" "545.90" "190.52" " 5.04" "187.77" " 0.00"

[21] " 17.65"

#worst fitted fstats for the three graphs within 6 BIC units from the best one

sel.idx=which(addSdi.dbic<6)

sel.idx=sel.idx[order(addSdi.dbic[sel.idx])]

sel.idx=sel.idx[-1]#remove the best fitting graph

for(i in 1:length(sel.idx)){

tmp.comp=compare.fitted.fstats(dsu.fstats,addSdi$fitted.graphs.list[[sel.idx[i]]],

n.worst.stats = 1)

}

1 Worst fit for:

Estimated Fitted Z–score

CN-Bei,CN-Shi;US-Haw,US-Sdi 9.460063e-05 3.469447e-18 -2.410857

1 Worst fit for:

Estimated Fitted Z–score

CN-Bei,CN-Shi;US-Haw,US-Sdi 9.460063e-05 3.469447e-18 -2.410857

1 Worst fit for:

Estimated Fitted Z–score

CN-Bei,CN-Shi;US-Haw,US-Sdi 9.460063e-05 3.469447e-18 -2.410857

A )

worst fit: CN−Bei,CN−Shi;JP−Sap,US−Haw (Z=1.06)

B )

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Shi (Z=1.19)

∆BIC=0.468

C )

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Shi (Z=1.19)

∆BIC=0.468

D )

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Shi (Z=1.19)

∆BIC=0.468

Figure 10: Admixture graphs resulting from the positioning of BR-Pal onto the scaﬀold tree of native and

Hawaiian populations (Figure 7a) with BIC less than 6 units higher than the BIC with the best ﬁtting graph

(within red box and represented in Figure 9B). For each population, the graph (as obtained with the function

add.leaf) is displayed together with i) the worst ﬁtted f-statistics and its associated Z-score; and ii) the

diﬀerence of BIC of the graph with the graphs displaying the best ﬁtting graph (∆

BI C

) as a measure of

support. For all the graphs, the ﬁtted edge lengths are in drift units (x1,000) since drift.scaling argument

was set to TRUE.25

In addition, the positions of US-Sdi on the extended graph and the scaﬀold graph of native populations were

concordant (Figures 9C and 9D) making the position of the Chinese source of US-Sdi on the CN<->CN-Shi

more likely. The distance of S1-US-Sdi from CN was also non null (as opposed to the distance separating

S1-BR-Pal from CN in Figure 9B).

Finally the position of US-Sok on the scaﬀold tree of native populations was highly supported (Figure 9F).

However, its positioning onto the extended graph of native populations was less clear (Figure 9E) with two

other alternative graphs with almost the same BIC as the best ﬁtted graph:

addSok=add.leaf(natgraph$best.fitted.graph,leaf.to.add = "US-Sok",dsu.fstats,verbose=FALSE,

drift.scaling=T,compute.ci=T)

#Difference of BIC wrt the best fitting graph

addSok.dbic=addSok$bic-min(addSok$bic)

#worst fitted fstats for the three graphs within 6 BIC units from the best one

sel.idx=which(addSok.dbic<6)

sel.idx=sel.idx[order(addSok.dbic[sel.idx])]

sel.idx=sel.idx[-1]#remove the best fitting graph

#Delta_BIC (wrt the best fitted graph)

format(addSok.dbic[sel.idx],digits = 3)

[1] "7.84e-06" "9.73e-06"

for(i in 1:length(sel.idx)){

tmp.comp=compare.fitted.fstats(dsu.fstats,addSok$fitted.graphs.list[[sel.idx[i]]],

n.worst.stats = 1)

}

1 Worst fit for:

Estimated Fitted Z–score

CN-Bei,JP-Tok;US-Haw,US-Sok 0.002670761 0.0002218372 -4.139588

1 Worst fit for:

Estimated Fitted Z–score

CN-Bei,JP-Tok;US-Haw,US-Sok 0.002670761 0.0002218628 -4.139545

These actually only diﬀered by the position of the Japanese source named S1-US-Sok in Figure 9E with

respect to the Japanese sources of CN-Lia (S2-CN-Lia) and JP-Tok (S1-JP-Tok), the estimated distances

between these three sources S1-US-Sok, S2-CN-Lia and S1-JP-Tok being null. Hence, simplifying the graph by

combining the three source into a single one lead to a similar ﬁt (but lower BIC as the number of parameters

is reduced):

tst.SokGraph=rbind(c("S-JP-Tok","S1-JP-Tok","B"),c("S-JP-Tok","S2-JP-Tok","(1-B)"),

c("JP-Tok","S-JP-Tok",""), c("S1-JP-Tok" ,"JP",""),

c("S2-JP-Tok","R",""),c("JP-Sap","S1-JP-Tok",""),

c("CN","S2-JP-Tok",""),c("S-CN-Lia","S1-CN-Lia","A"),

c("S-CN-Lia","S1-JP-Tok","(1-A)"),c("CN-Lia","S-CN-Lia",""),

c("S1-CN-Lia","CN",""),c("CN-Shi","S1-CN-Lia",""),c("JP","R",""),

c("CN-Bei","CN",""),c("US-Haw","S1-US-Sok",""),

c("US-Sok","S-US-Sok",""),c("S-US-Sok","S1-JP-Tok","C"),

c("S-US-Sok","S1-US-Sok","(1-C)"),c("S1-US-Sok","JP",""))

tst.SokGraph.par=generate.graph.params(tst.SokGraph,dsu.fstats)

Total Number of Parameters: 15 (12 edges lengths + 3 adm. coeff.)

Total Number of Statistics: 21 (6 F2 and 15 F3)

tst.SokGraph.fit=fit.graph(tst.SokGraph.par,drift.scaling = T,verbose=FALSE)

tst.SokGraph.fit@bic

[1] 435.3341

tmp.comp=compare.fitted.fstats(dsu.fstats,tst.SokGraph.fit,n.worst.stats = 1)

1 Worst fit for:

Estimated Fitted Z–score

CN-Bei,JP-Tok;US-Haw,US-Sok 0.002670761 0.000221864 -4.139543

Yet this simpliﬁed version of the Figure 9E graph should be interpreted with caution as i) the absolute Z-score

for the worst ﬁtted f-statistics remains high (Z=-4.14); ii) the estimated distance (in drift units) of JP-Tok

from its source S-JP-Tok is null; and iii) it somewhat contradicts the original graph displayed in Figure

8 (e.g., in which the two Japanese sources of CN-Lia and JP-Tok are clearly diﬀerentiated). Overall this

suggests more complex relationships among the Japanese sources of US-Sok and CN-Lia and JP-Tok from the

native area that cannot be captured by the sample at hand (or even modeled as a simple admixture graph).

To simplify the modeling of the demographic history of the American invasive populations, we further used

the scaﬀold tree of native populations to represent their ancestry. Hence, we discarded the two admixed

native populations CN-Lia and JP-Tok. Indeed, no evidence for a direct contribution of CN-Lia or JP-Tok to

the ancestry of BR-Pal, US-Sdi or US-Sok could be found from the above analyses.

6.3.2 Searching for the most comprehensive admixture graph including American invasive

populations

We relied on the graph.builder function to try to jointly include American invasive populations ﬁrst focusing

on BR-Pal, US-Sdi and US-Sok for the same reasons as mentioned above and evaluated all the possible

inclusion order by pair (as previously when analyzing the native populations). We below simpliﬁed the output

by only printing the analyses that resulted in a best ﬁtted graph with a worst ﬁtted f-statistics Z-score lower

than 2 (in absolute value):

cleangrid<-function(x){n=ncol(x);x[apply(x,1,f<-function(y){length(unique(y))==n}),n:1]}

pops.to.include<-expand.grid(c("BR-Pal","US-Sdi","US-Sok"),

c("BR-Pal","US-Sdi","US-Sok")) %>% as.matrix() %>% cleangrid()

#exploring and evaluating the graphs

for(i in 1:nrow(pops.to.include)){

tmp.graph<-graph.builder(scaf.tree.fit,leaves.to.add = pops.to.include[i,],

fstats=dsu.fstats,verbose=F)

tmp.comp.fit=compare.fitted.fstats(dsu.fstats,tmp.graph$best.fitted.graph,

n.worst.stats = 0)

if(max(abs(tmp.comp.fit$`Z--score`))<2){

cat("##########################################\n")

cat("Inclusion Order:",pops.to.include[i,],"\n")

#number of graphs within Dbic<6 (default)

cat("\tNumber Of Graphs:",tmp.graph$n.graphs,"\n")

#min BIC

cat("\tMin BIC:",min(tmp.graph$bic),"\n")

#worst fitted fstats (best fitting graph)

cat("\tWorst fstats fit:\n")

print(tmp.comp.fit[which.max(abs(tmp.comp.fit$`Z--score`)),])

}

##########################################

Inclusion Order: US-Sok BR-Pal

Number Of Graphs: 4

Min BIC: 271.6173

Worst fstats fit:

Estimated Fitted Z–score

CN-Bei,CN-Shi;US-Haw,US-Sok 4.971676e-05 -1.040834e-17 -1.160098

As shown, out the six tested (ordered) pairs of included populations, including ﬁrst US-Sok and BR-Pal

resulted in the best ﬁtted graph (worst ﬁtted f-statistics associated Z-score equal to -1.16). As before, three

alternative but equivalent graph, since they simply diﬀer according to the positioning of the Chinese related

source of BR-Pal, its distance from the CN node being always null (Figure 11).

addSokPal=graph.builder(scaf.tree.fit,leaves.to.add = c("US-Sok","BR-Pal"),dsu.fstats,

verbose=FALSE,drift.scaling=T,compute.ci=T)

#Difference of BIC wrt the best fitting graph

addSokPal.dbic=addSokPal$bic-min(addSokPal$bic)

format(addSokPal.dbic,digits = 3)

[1] "0.388" "0.388" "0.388" "0.000"

We thus simpliﬁed the graph by deﬁning CN (the ancestor of the two Chinese populations CN-Bei and

CN-Shi) as the Chinese source of BR-Pal (thereby removing one branch length parameter) and redeﬁned

some node names to clarify the corresponding inferred scenario. Indeed all graphs suggested that American

invasive populations originated from two independent admixture events among the populations from the

native area (including US-Haw):

•

A ﬁrst (older) admixture event between a Japanese and an Hawaiian related sources (called hereafter

J-Am1 and H-Am1 respectively) that lead to the internal node population called Am1 which is directly

ancestral to US-Sok

•

A second (and more recent) admixture event between a source related to Am1 (called S-Am2 hereafter)

and a Chinese closely related to the ancestor of CN-Shi and CN-Bei. We hereafter called this second

admixed population, directly related to BP-Pal, Am2

The corresponding graph, equivalent to the ones on Figure 11, was deﬁned (and ﬁtted) as follows and plotted

on Figure 12 (graph framed in red):

am.scaf.graph<-rbind(c('Am2','S-Am2','B'),c('Am2','CN','(1-B)'),c('BR-Pal','Am2',''),

c('S-Am2','Am1',''),c('US-Sok','S-Am2',''),c('CN-Shi','CN',''),

c('Am1','J-Am1','A'),c('Am1','H-Am1','(1-A)'),c('J-Am1','JP',''),

c('JP-Sap','J-Am1',''),c('US-Haw','H-Am1',''),c('H-Am1','JP',''),

c('CN','R',''),c('JP','R',''),c('CN-Bei','CN',''))

am.scaf.par=generate.graph.params(am.scaf.graph,dsu.fstats)

Total Number of Parameters: 12 (10 edges lengths + 2 adm. coeff.)

Total Number of Statistics: 15 (5 F2 and 10 F3)

am.scaf.fit=fit.graph(am.scaf.par,drift.scaling = T,verbose=FALSE)

am.scaf.fit@bic

[1] 269.2973

tmp.comp=compare.fitted.fstats(dsu.fstats,am.scaf.fit,n.worst.stats = 1)

1 Worst fit for:

Estimated Fitted Z–score

BR-Pal,US-Haw;CN-Bei,CN-Shi -5.09979e-05 0 1.187469

6.3.3 Positioning US-Col, US-Nca, US-Sdi, US-Wat and US-Wis onto the scaﬀold graph

extended with US-Sok and BR-Pal

We then explored the position of the ﬁve remaining American populations (US-Col, US-Nca, US-Sdi, US-Wat

and US-Wis) on the scaﬀold graph am.scaf that included BR-Pal and US-Sok using the add.leaf function:

As shown on Figure 12, only US-Wat could be placed without ambiguity on the graph (∆

BI C >

8and worst

ﬁtted f-statistic absolute Z-score

2) and could be interpreted as a recently admixed populations between

a source related to a (presumably) Northern American population Am1 and the admixed (presumably)

A )

worst fit: CN−Bei,CN−Shi;US−Haw,US−Sok (Z=−1.16)

B )

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Shi (Z=1.19)

∆BIC=0.388

C )

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Shi (Z=1.19)

∆BIC=0.388

D )

worst fit: BR−Pal,US−Haw;CN−Bei,CN−Shi (Z=1.19)

∆BIC=0.388

Figure 11: Admixture graphs resulting from the joint positioning of US-Sok and BR-Pal onto the scaﬀold

tree of native populations (Figure 7a) with BIC less than 6 units higher than the BIC with the best ﬁtting

graph (within red box and represented in Figure 9B). For each population, the graph (as obtained with the

function add.leaf) is displayed together with i) the worst ﬁtted f-statistics and its associated Z-score; and ii)

the diﬀerence of BIC of the graph with the graphs displaying the best ﬁtting graph (∆

BI C

) as a measure of

support. For all the graphs, the ﬁtted edge lengths are in drift units (x1,000) since drift.scaling argument

was set to TRUE.29

Southern Am2 population. The positioning of US-Sdi also resulted in a best ﬁtting graph with a high support

(∆

BI C >

33) and a rather good ﬁt, the worst ﬁtted f-statistic Z-score being equal -2.41. The resulting scenario

suggested US-Sdi derived from a population admixed between a source deriving from Am2 (and actually

closely related to BR-Pal) and a source related to an (another?) Hawaiian population distantly related to

H-Am1 the Hawaiian source of Am1 (that also contributed to Am2).

This suggested that US-Sdi resulted from multiple admixture events between populations closely related

the Am2 and Hawaii that are rather diﬃcult to ﬁt with simple admixture graph (worst ﬁtted f-statistics

associated Z-score slightly higher than 2). Accordingly, when jointly adding US-Wat and US-Sdi (whatever

the order of inclusion) to the am.scaf graph, the best ﬁtted graph was consistent with the ones obtained

above when adding the US-Sdi and US-Wat, the Am2 related source being common for the two populations

(the inferred distance between S1-US-Wat and S1-US-Sdi being null) and the Hawaiian related source of

US-Sdi being closely related to H-Am1 (Figure 13A). The inferred distance were also more consistent with

geographical and historical origins of the populations and the high contribution of the S1-US-Sdi/S1-Wat

source was also consistent with the observed

values (see section 5). Nevertheless, the ﬁt remained poor as

the worst ﬁtted f-statistics were clearly oﬀ (absolute Z-score

6) and a likely alternative scenario would be

that the second US-Sdi source is a ghost population deriving from H-Am1 according to the Figure plotted in

Figure 13B. However, as indicated by the error message generated by the ﬁt.graph function, such a scenario

is not ﬁttable because the resulting system of equations is singular.

#Joint positioning of US-Wat and US-Sdi on the am.scaf graph

am.scafWatSdi=graph.builder(am.scaf.fit,leaves.to.add = c("US-Wat","US-Sdi"),dsu.fstats,

drift.scaling=TRUE,verbose = FALSE)

#An alternative (not fittable) admixture graph scenario

am.scafWatSdi.sc<-rbind(c('Am2','S-Am2','B'),c('Am2','CN','(1-B)'),

c('BR-Pal','Cal',''),c("Cal","Am2",""),

c("CN","R",""),c('S-Am2','Am1',''),

c('US-Sok','W',''),c('CN-Shi','CN',''),

c("W","S-Am2",""),c('Am1','J-Am1','A'),

c('Am1','H-Am1','(1-A)'),c('J-Am1','JP',''),

c('JP-Sap','J-Am1',''),c('US-Haw','H-Am1',''),

c('H-Am1','JP',''),c('JP','R',''),c('CN-Bei','CN',''),

c("US-Wat","SW",""),c("SW","W","D"),c("SW","Cal","(1-D)"),

c("US-Sdi","SSdi",""),c("SSdi","Cal","E"),

c("SSdi","S-SSdi","(1-E)"),c("S-SSdi","H-Am1",""))

am.scafWatSdi.sc.par=generate.graph.params(am.scafWatSdi.sc,dsu.fstats)

Total Number of Parameters: 19 (15 edges lengths + 4 adm. coeff.)

Total Number of Statistics: 28 (7 F2 and 21 F3)

#The graph can be plotted using plot(am.scafWatSdi.sc.par) but

#This scenario is not fittable as indicated by the error message from the fit.graph function

am.scafWatSdi.sc.fit=fit.graph(am.scafWatSdi.sc.par,drift.scaling = T)

The system is singular: the rank of the incidence matrix is lower than n.edges-1 (e.g., some branches are not identifiable) and/or the number of parameters is larger than the number of basis f-statistics

It is not possible to properly fit this admixture graph.

In agreement with the analysis of

-based admixture tests results that suggested complex admixture histories

among these closely related populations, no proper admixture graph could be found when trying to position

US-Col, US-Wis or US-Nca onto the am.scaf scaﬀold graph (Figure 12). Nevertheless, for each of these

populations, all the resulting best ﬁtted graph suggested a high contribution of the Am2 admixed source

(BR-Pal being its best proxy) that may be closely related to the common source of US-Sdi and US-Wat

mentioned above, a second contributing source being related to Japanese populations.

Finally, no proper admixture graph could be obtained when trying to jointly include each possible pairs of

the ﬁve populations:

Scaffold graph

US−Col

worst fit: BR−Pal,US−Col;CN−Shi,US−Haw (Z=6.15)

∆BIC=0

US−Nca

worst fit: BR−Pal,US−Nca;US−Haw,US−Sok (Z=−5.86)

∆BIC=0

US−Sdi

worst fit: CN−Bei,CN−Shi;US−Haw,US−Sdi (Z=−2.41)

∆BIC=33.841

US−Wat

worst fit: CN−Bei,CN−Shi;US−Haw,US−Wat (Z=−1.83)

∆BIC=8.365

US−Wis

worst fit: BR−Pal,US−Wis;CN−Shi,US−Haw (Z=4.37)

∆BIC=0

Figure 12: Admixture graphs resulting from the positioning of US-Col, US-Nca, US-Sdi, US-Wat and US-Wis

onto the am.scaf scaﬀold graph that relates the scaﬀold tree of native populations and BR-Pal and US-Sok

(red frame). The best ﬁtting graphs (as obtained with the function add.leaf) are displayed together with i)

the worst ﬁtted f-statistics and their associated Z-score; and ii) the diﬀerence of their BIC with respect to

the graphs displaying the second lowest BIC (∆

BI C

) as a measure of support. The target populations are

highlighted in yellow. For all the graphs, the ﬁtted edge lengths are in drift units (x1,000) since drift.scaling

argument was set to TRUE.

worst fit: BR−Pal,US−Sok;CN−Bei,US−Haw (Z=−5.87)

Figure 13: Connecting US-Wat and US-Sdi to the am.scaf scaﬀold graph. A) Best ﬁtted admixture graph

obtained with the graph.builder function (the ﬁtted edge lengths are in drift units (x1000) since drift.scaling

argument was set to TRUE). B) Possible (but not ﬁttable) scenario connecting US-Wat and US-Sdi to the

am.scaf scaﬀold graph.

cleangrid<-function(x){n=ncol(x);x[apply(x,1,f<-function(y){length(unique(y))==n}),n:1]}

pops.to.include<-expand.grid(c("US-Col","US-Nca","US-Sdi","US-Wat","US-Wis"),

c("US-Col","US-Nca","US-Sdi","US-Wat","US-Wis")) %>%

as.matrix() %>% cleangrid()

#exploring and evaluating the graphs

for(i in 1:nrow(pops.to.include)){

tmp.graph<-graph.builder(am.scaf.fit,leaves.to.add = pops.to.include[i,],

fstats=dsu.fstats,verbose=F)

tmp.comp.fit=compare.fitted.fstats(dsu.fstats,tmp.graph$best.fitted.graph,

n.worst.stats = 0)

if(max(abs(tmp.comp.fit$`Z--score`))<3){

cat("##########################################\n")

cat("Inclusion Order:",pops.to.include[i,],"\n")

#number of graphs within Dbic<6 (default)

cat("\tNumber Of Graphs:",tmp.graph$n.graphs,"\n")

#min BIC

cat("\tMin BIC:",min(tmp.graph$bic),"\n")

#worst fitted fstats (best fitting graph)

cat("\tWorst fstats fit:\n")

print(tmp.comp.fit[which.max(abs(tmp.comp.fit$`Z--score`)),])

}

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Nov 2024

Large scale genomic resources can place genetic variation into an ecologically informed context. To advance our understanding of the population genetics of the fruit fly Drosophila melanogaster , we present an expanded release of the community-generated population genomics resource Drosophila Evolution over Space and Time (DEST 2.0; https://dest.bio/). This release includes 530 high-quality pooled libraries from flies collected across six continents over more than a decade (2009-2021), most at multiple time points per year; 211 of these libraries are sequenced and shared here for the first time. We used this enhanced resource to elucidate several aspects of the species' demographic history and identify novel signs of adaptation across spatial and temporal dimensions. We showed that patterns of secondary contact, originally characterized in North America, are replicated in South America and Australia. We also found that the spatial genetic structure of populations is stable over time, but that drift due to seasonal contractions of population size causes populations to diverge over time. We identified signals of adaptation that vary between continents in genomic regions associated with xenobiotic resistance, consistent with independent adaptation to common pesticides. Moreover, by analyzing samples collected during spring and fall across Europe, we provide new evidence for seasonal adaptation related to loci associated with pathogen response. Furthermore, we have also released an updated version of the DEST genome browser. This is a useful tool for studying spatio-temporal patterns of genetic variation in this classic model system.

Extending Approximate Bayesian Computation with Supervised Machine Learning to infer demographic history from genetic polymorphisms using DIYABC Random Forest

Article

Full-text available

May 2021

Simulation‐based methods such as Approximate Bayesian Computation (ABC) are well‐adapted to the analysis of complex scenarios of populations and species genetic history. In this context, supervised machine learning (SML) methods provide attractive statistical solutions to conduct efficient inferences about scenario choice and parameter estimation. The Random Forest methodology (RF) is a powerful ensemble of SML algorithms used for classification or regression problems. RF allows conducting inferences at a low computational cost, without preliminary selection of the relevant components of the ABC summary statistics, and bypassing the derivation of ABC tolerance levels. We have implemented a set of RF algorithms to process inferences using simulated datasets generated from an extended version of the population genetic simulator implemented in DIYABC v2.1.0. The resulting computer package, named DIYABC Random Forest v1.0, integrates two functionalities into a user‐friendly interface: the simulation under custom evolutionary scenarios of different types of molecular data (microsatellites, DNA sequences or SNPs) and RF treatments including statistical tools to evaluate the power and accuracy of inferences. We illustrate the functionalities of DIYABC Random Forest v1.0 for both scenario choice and parameter estimation through the analysis of pseudo‐observed and real datasets corresponding to pool‐sequencing and individual‐sequencing SNP datasets. Because of the properties inherent to the implemented RF methods and the large feature vector (including various summary statistics and their linear combinations) available for SNP data, DIYABC Random Forest v1.0 can efficiently contribute to the analysis of large SNP datasets to make inferences about complex population genetic histories.

Near-chromosome level genome assembly of the fruit pest Drosophila suzukii using long-read sequencing

Article

Full-text available

Jul 2020

Over the past decade, the spotted wing Drosophila, Drosophila suzukii, has invaded Europe and America and has become a major agricultural pest in these areas, thereby prompting intense research activities to better understand its biology. Two draft genome assemblies already exist for this species but contain pervasive assembly errors and are highly fragmented, which limits their values. Our purpose here was to improve the assembly of the D. suzukii genome and to annotate it in a way that facilitates comparisons with D. melanogaster. For this, we generated PacBio long-read sequencing data and assembled a novel, high-quality D. suzukii genome assembly. It is one of the largest Drosophila genomes, notably because of the expansion of its repeatome. We found that despite 16 rounds of full-sib crossings the D. suzukii strain that we sequenced has maintained high levels of polymorphism in some regions of its genome. As a consequence, the quality of the assembly of these regions was reduced. We explored possible origins of this high residual diversity, including the presence of structural variants and a possible heterogeneous admixture pattern of North American and Asian ancestry. Overall, our assembly and annotation constitute a high-quality genomic resource that can be used for both high-throughput sequencing approaches, as well as manipulative genetic technologies to study D. suzukii.

A Whole-Genome Scan for Association with Invasion Success in the Fruit Fly Drosophila suzukii Using Contrasts of Allele Frequencies Corrected for Population Structure

Article

Full-text available

Apr 2020
MOL BIOL EVOL

Evidence is accumulating that evolutionary changes are not only common during biological invasions but may also contribute directly to invasion success. The genomic basis of such changes is still largely unexplored. Yet, understanding the genomic response to invasion may help to predict the conditions under which invasiveness can be enhanced or suppressed. Here we characterized the genome response of the spotted wing drosophila Drosophila suzukii during the worldwide invasion of this pest insect species, by conducting a genome-wide association study to identify genes involved in adaptive processes during invasion. Genomic data from 22 population samples were analyzed to detect genetic variants associated with the status (invasive versus native) of the sampled populations based on a newly developed statistic, we called C2, that contrasts allele frequencies corrected for population structure. We evaluated this new statistical framework using simulated data sets and implemented it in an upgraded version of the program BayPass. We identified a relatively small set of single nucleotide polymorphisms (SNPs) that show a highly significant association with the invasive status of D. suzukii populations. In particular, two genes, RhoGEF64C and cpo, contained SNPs significantly associated with the invasive status in the two separate main invasion routes of D. suzukii. Our methodological approaches can be applied to any other invasive species, and more generally to any evolutionary model for species characterized by non-equilibrium demographic conditions for which binary covariables of interest can be defined at the population level.

Ryacas: A computer algebra system in R

Article

Full-text available

Oct 2019

Ryacas is an R (R Core Team, 2018) package that enables a computer algebra system (CAS) within R via the open source CAS yacas (A. Z. Pinkus & Winitzki, 2002; A. Pinkus, Winnitzky, & Mazur, 2016), which is short for “yet another computer algebra system”. Ryacas includes both a high-level (symbol) interface using R objects like matrices and vectors as well as direct access to the underlying yacas such that the user can use the full yacas system, including for example defining new summation rules.

admixr-R package for reproducible analyses using ADMIXTOOLS

Article

Full-text available

Sep 2019
BIOINFORMATICS

We present a new R package admixr, which provides a convenient interface for performing reproducible population genetic analyses (f3, D, f4, f4-ratio, qpWave and qpAdm), as implemented by command-line programs in the ADMIXTOOLS software suite. In a traditional ADMIXTOOLS workflow, the user must first generate a set of text configuration files tailored to each individual analysis, often using a combination of shell scripting and manual text editing. The non-tabular output files then need to be parsed to extract values of interest prior to further analyses. Our package simplifies this process by automating all low-level configuration and parsing steps, making analyses as simple as running a single R command. Furthermore, we provide a set of R functions for processing, filtering and manipulating datasets in the EIGENSTRAT format. By unifying all steps of the workflow under a single R framework, this package enables the automation of analytic pipelines, significantly improving the reproducibility of population genetic studies. Availability and implementation: The source code of the R package is available under the MIT license. Installation instructions, reference manual and a tutorial can be found on the package website at https://bioinf.eva.mpg.de/admixr. Supplementary information: Supplementary data are available at Bioinformatics online.

Measuring Genetic Differentiation from Pool-seq Data

Article

Full-text available

Jul 2018

The advent of high throughput sequencing and genotyping technologies enables the comparison of patterns of polymorphisms at a very large number of markers. While the characterization of genetic structure from individual sequencing data remains expensive for many non-model species, it has been shown that sequencing pools of individual DNAs (Pool-seq) represents an attractive and cost-effective alternative. However, analyzing sequence read counts from a DNA pool 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 FST 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 FST estimator may question the interpretation of population structure inferred from previous analyses.

Deciphering the demographic history of allochronic differentiation in the pine processionary moth Thaumetopoea pityocampa

Article

Full-text available

Nov 2017
MOL ECOL

Understanding the processes of adaptive divergence, which may ultimately lead to speciation, is a major question in evolutionary biology. Allochronic differentiation refers to a particular situation where gene flow is primarily impeded by temporal isolation between early and late reproducers. This process has been suggested to occur in a large array of organisms, even though it is still overlooked in the literature. We here focused on a well-documented case of incipient allochronic speciation in the winter pine processionary moth Thaumetopoea pityocampa. This species typically reproduces in summer and larval development occurs throughout autumn and winter. A unique, phenologically shifted population (SP) was discovered in 1997 in Portugal. It was proved to be strongly differentiated from the sympatric “winter population” (WP), but its evolutionary history could only now be explored. We took advantage of the recent assembly of a draft genome and of the development of pan-genomic RAD-seq markers to decipher the demographic history of the differentiating populations and develop genome scans of adaptive differentiation. We showed that the SP diverged relatively recently, i.e. few hundred years ago, and went through two successive bottlenecks followed by population size expansions, while the sympatric WP is currently experiencing a population decline. We identified outlier SNPs that were mapped onto the genome, but none were associated with the phenological shift or with subsequent adaptations. The strong genetic drift that occurred along the SP lineage certainly challenged our capacity to reveal functionally important loci. This article is protected by copyright. All rights reserved.

A Unified Characterization of Population Structure and Relatedness

Article

Full-text available

May 2017
GENETICS

Many population genetic activities, ranging from evolutionary studies to association mapping to forensic identification, rely on appropriate estimates of population structure or relatedness. All applications require recognition that quantities with an underlying meaning of allelic dependence are not defined in an absolute sense, but instead are made "relative to" some set of alleles other than the target set. The 1984 Weir and Cockerham FST estimate made explicit that the reference set of alleles was across populations, whereas standard kinship estimates do not make the reference explicit. Weir and Cockerham stated that their FST estimates were for independent populations, and standard kinship estimates have an implicit assumption that pairs of individuals in a study sample, other than the target pair, are unrelated or are not inbred. However, populations lose independence when there is migration between them, and dependencies between pairs of individuals in a population exist for more than one target pair. We have therefore re-cast our treatments of population structure, relatedness and inbreeding to make explicit that the parameters of interest involve the differences in degrees of allelic dependence between the target and the reference sets of alleles and so can be negative. We take the reference set to be the population from which study individuals have been sampled. We provide simple moment estimates of these parameters, phrased in terms of allelic matching within and between individuals for relatedness and inbreeding, or within and between populations for population structure. A multi-level hierarchy of alleles within individuals, alleles between individuals within populations, and alleles between populations allows a unified treatment of relatedness and population structure. We expect our new measures to have a wide range of applications, but we note that their estimates are sensitive to rare or private variants: some population-characterization applications suggest exploiting those sensitivities whereas estimation of relatedness may best use all genetic markers without filtering on minor allele frequency.

Applying f4 -statistics and admixture graphs: Theory and examples

Article

Jul 2020

Mark Lipson

A popular approach to learning about admixture from population genetic data is by computing the allele‐sharing summary statistics known as f‐statistics. Compared to some methods in population genetics, f‐statistics are relatively simple, but interpreting them can still be complicated at times. In addition, f‐statistics can be used to build admixture graphs (multi‐population trees allowing for admixture events), which provide more explicit and thorough modeling capabilities but are correspondingly more complex to work with. Here, I discuss some of these issues to provide users of these tools with a basic guide for protocols and procedures. My focus is on the kinds of conclusions that can or cannot be drawn from the results of f4‐statistics and admixture graphs, illustrated with real‐world examples involving human populations.

Genetic Data Analysis: Methods for Discrete Population Genetic Data.

Article

Jun 1991
Syst Zool

f ‐Statistics estimation and admixture graph construction with Pool‐Seq or allele count data using the R package poolfstat

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