The Structured Ancestral Selection Graph and the Many-Demes Limit

16 Pages·2005·0.17 MB·English

Copyright©2005bytheGeneticsSocietyofAmerica
DOI:10.1534/genetics.104.032276
The Structured Ancestral Selection Graph and the Many-Demes Limit
Paul F. Slade*,1 and John Wakeley†
*School of Mathematics and Statistics, School of Biological Sciences, SUBIT, University of Sydney, New South Wales 2006, Australia and
†DepartmentofOrganismicandEvolutionaryBiology,HarvardUniversity,Cambridge,Massachusetts02138
ManuscriptreceivedJune21,2004
AcceptedforpublicationOctober28,2004
ABSTRACT
Weshowthattheunstructuredancestralselectiongraphappliestopartofthehistoryofasamplefrom
apopulationstructuredbyrestrictedmigrationamongsubpopulations,ordemes.Theresultholdsinthe
limitasthenumberofdemestendstoinfinitywithproportionatelyweakselection,andwehavealsomade
the assumptions of island-type migration and that demes are equivalent in size. After an instantaneous
sample-size adjustment, this structured ancestral selection graph converges to an unstructured ancestral
selectiongraphwithamutationparameterthatdependsinverselyonthemigrationrate.Incontrast,the
selectionparameterforthepopulationisindependentofthemigrationrateandisidenticaltotheselection
parameterinanunstructuredpopulation.Weshowanalyticallythatestimatorsofthemigrationrate,based
onpairwisesequencedifferences,derivedundertheassumptionofneutralityshouldperformequallywell
in the presence of weak selection. We also modify an algorithm for simulating genealogies conditional
onthefrequenciesoftwoselectedallelesinasample.Thispermitsefficientsimulationofstrongerselection
than was previously possible. Using this new algorithm, we simulate gene genealogies under the many-
demes ancestral selection graph and identify some situations in which migration has a strong effect on
thetimetothemostrecentcommonancestorofthesample.Wefindthatasimilareffectalsoincreases
thesensitivityofthegenealogytoselection.
KIMURA(1983)stronglypromotedtheideathatthe Hudson (1983), and Tajima (1983). The reason for
abundantgeneticvariationseeninnearlyeveryspe- thisisclear:itismuchsimplertopredictthefrequencies
ciesstudiedmustbeneutral.OhtaandKimura(1971) ofselectedallelesusingdiffusiontheorythanusingthe
suggestedaless strictversionofthis inwhichpolymor- backward-timeapproachofthecoalescent.Forneutral
phisms are explained by the constant input of nearly loci,thecoalescentprovidesasimpleandusefuldescrip-
neutralmutations,i.e.,variantswithselectiveadvantages tionofthegeneticancestryofasampleofgeneticdata—
ordisadvantagessmallerthanthereciprocalofthepop- thegenealogyforshort—fromalargewell-mixed,orpan-
ulationsize.Afteragreatdealofdebateandmanyanaly- mictic,populationofconstantsizethroughtime.Dueto
ses,summarizedinGolding(1994),therearenowfew the close connection between genealogies and genetic
adherentstothestrictneutralmutationhypothesis.Mo- data,andtotheeasewithwhichgenealogiescanbesim-
lecular techniques currently allow huge numbers of ulated,agrowingsetofcomputationaltoolsusethecoales-
polymorphismstobeassayedwithrelativeease,andthe cent to make inferences about population history from
resulting genomic data can be used to estimate the
DNAsequences;seeStephens(2001)andTavare´(2004)
strengthofselectionassociatedwithgeneticdifferences
for reviews.
(Sawyer and Hartl 1992; Bustamante et al. 2002).
The coalescentis robust to manykinds of deviations
Recent statistical inferences made from such data em-
fromitsunderlyingassumptions(Kingman1982b;Mo¨hle
phasizetheimportanceofselectionandshowthatselec-
1998c),butitdoesnotholdwhenmembersofthesam-
tive advantages or disadvantages on the order of the
ple, or the ancestral lineages of the sample, are not
reciprocalofthepopulationsizemaybefairlycommon
exchangeable.Infact,therelativesimplicityoftheneu-
(Sawyer et al. 2003).
tralcoalescentprocessflowsdirectlyfromtheexchange-
Themethodsusedintheseworkstoestimateselection
ability of the lineages. Exchangeability, which results
parametersfromintraspecificgenomicdatarelyonpre-
in the model from the assumptions of neutrality and
dictions of sample allele frequencies derived from for-
panmixia,meansthatthestatisticalpropertiesofasam-
ward-time diffusiontheory. They do notuse the ances-
pledonotdependonhowthesampleditemsarelabeled
tralgeneticprocessknownasthecoalescent,whichwas
(Kingman 1982b; Aldous 1985). Possible “labels” in-
introduced in the early 1980s by Kingman (1982a,b),
cludethegeographiclocationswherethesampleswere
collectedortheallelicstatesofthesamplesiftheseare
known. Whensamples arenot exchangeable,a related
1Correspondingauthor:SchoolofMathematicalSciences,University
ofAdelaide,SA5005,Australia. E-mail:[email protected] but more complicated model called the stuctured coa-
Genetics169:1117–1131(February2005)
1118 P.F.SladeandJ.Wakeley
lescent exists (Slatkin 1987; Strobeck 1987; Noto- able to identify cases in which simple models may be
hara 1990; Wilkinson-Herbots1998), and computa- applied even though the underlying dynamics appear
tional methods of inference are being developed for complicated. We present a rigorous derivation of the
this model as well (Nath and Griffiths 1996; Beerli ASG with very many demes (subpopulations) in an is-
and Felsenstein 1999, 2001; Bahlo and Griffiths landmodelofmigration(Wright1931)andshowthat
2000; De Iorio and Griffiths 2004). itpossessesasimplestructureakintothatunderneutral-
Genealogies in the presence of strong selection, i.e., ity (Wakeley 1998). Five parameters collapse into just
withselectiveadvantagesordisadvantagesmuchgreater three(oneeachformigration,selection,andmutation)
thanthereciprocalofthepopulationsize,canbemod- in the limit as the number of demes tends to infinity.
eled using the structured coalescent with the subdivi- Interestingly,thepopulationratesofselectionandmu-
sions being the alternative alleles (Kaplan et al. 1988, tation scale differently (see below) in the presence of
1989). For weak selection, Krone and Neuhauser population subdivision. This was found previously in
(1997)andNeuhauserandKrone(1997)showedthat forward-time analyses (Wakeley 2003; Wakeley and
thegenealogyofasampletakenatrandomwithrespect Takahashi2004),buttheresultismoreintuitiveinthe
to allelic variation is described by an exchangeable an- presentcase.Wealsodescribeanewlyenhancedprocess
cestral process. The ancestral selection graph (ASG) is for the simulation of genealogies conditional on the
abranching-coalescingrandomgraphwithinwhichthe frequenciesofallelesinthesample.Weshowusingcon-
simple,bifurcatinggenealogyofasampleisembedded. ditionalsimulationsthatpopulationsubdivisionandmi-
TheelegantexchangeabilityoftheASGcomeswithacost: grationamongverymanydemesaccentuatetheaction
itlendsitselfonlytoverycomputationallyintensivetech- of selection on the genealogy.
niques. In addition, there appears to be very little about
genealogiesintheASGthatdiffersgreatlyfromgenealo-
THEORY
gies under neutrality (Krone and Neuhauser 1997).
For these reasons, and also to be able to control for Thestructuredancestralselectiongraphistheances-
sampling, recent extensions to the ASG have modeled tralprocessforaclassofforward-timemodels.Following
genetic ancestry conditional on the frequencies of se- Kroneand Neuhauser(1997)webegin bydescribing
lected alleles in the sample (Slade 2000a,b; Stephens acontinuous-timemodelinwhichreproductionoccurs
and Donnelly 2003; Barton et al. 2004). This leads accordingtotheMoran(1958,1962)model,butwhere
to dramatic improvements in computation time while thepopulationissubdividedintodemes(subpopulations)
preserving some degree of exchangeability among the connected by migration. There are D demes, and in a
sampledlineages(Slade2000a).Theconditionalsimu- general model these might have sizes N, i (cid:1) 1, ..., D.
i
lation of selected genealogies by Slade (2000a,b) en- Weassumethateachofthe(cid:1)Di(cid:1)1Ni lineagesinthepopu-
ables the timing structure of common ancestry to be lation can be in one of two possible allelic states, A
1
quantified.IncontrasttotheunconditionalASG,gene- and A , with relative rates of reproduction (cid:2) and (cid:2),
2 1 2
alogiesconditionaluponallelefrequenciesinthesam- respectively. However, as in Krone and Neuhauser
plemaybeeitherlargerorsmalleronaveragethangene- (1997),webeginwithadescriptioninwhichthestatesof
alogiesunderneutrality(Slade2000a);seealsothereview lineagesarenotspecifiedandsoeachlineageundergoes
of Slade and Neuhauser (2003). The extent of these two kinds of reproduction (birth/death) events. Type
differencesdependsstronglyontheinitialsamplecon- 2birth/deatheventsoccurwithrate(cid:2) (cid:3)(cid:2) perlineage.
2 1
figuration of alleles, particularly for small mutation These represent the action of selection since they are
rates. Under neutrality, samples in which there are ap- realized only if the lineage that reproduces is of the
proximately equal numbers of each allele have longer favored allelic type at the time of the event. Type (cid:4)
timestocommonancestrythanmoreunbalancedsam- birth/death events occur with rate (cid:2) per lineage. The
1
ples.However,theselectiveeffectongenealogiesforsam- majority of birth/death events will be of this type, and
plesin whichthere areapproximately equalnumbers of theycanbethoughtofasneutralinthesensethatthey
each allele is greater than that of more unbalanced arerealizedregardlessoftheallelictypeoftheparental
samples. We show that restricted migration can cause lineage.Note thatat thispoint, withoutspecifiying the
unbalanced samples to be converted to balanced sam- allelic types, lineages are exchangeable within demes
ples,viatheinstantaneoussample-sizeadjustmentmen- but are not exchangeable between demes. Note also
tioned above, and thus the time to common ancestry that, while this is a model of directional selection, it is
under selection is reduced. possibletoincludefrequencydependence(Neuhauser
Our purpose is to consider the ASG in the presence 1999) or general diploid selection (Slade 2000a).
of population subdivision and migration. Population Subdivision is mediated by a collection of D (cid:5) D
structureisevidentinmanygeneticdatasetsbutmodel- migration probabilities, m , for i, j (cid:1) 1, ..., D. When
ij
ing it presents difficulties (Slatkin 1985; Avise 2000; a lineage in deme j reproduces, either by a type (cid:4) or
Hey and Machado 2003), creating impediments both by a type 2 event, the individual its offspring will re-
toinferenceandtounderstanding.Therefore,itisvalu- place (i.e., the individual picked to die) is chosen uni-
Many-DemesAncestralSelectionGraph 1119
formlyatrandomfromamongtheN lineagesindeme Neuhauser (1997) refer to as the percolation diagram.
i
i with probability m . With probability 1 (cid:3) m, where This is shown in Figure 1 and is the structured analog
ij
m (cid:1) (cid:1)Di(cid:1)1mij, the individual to die is chosen uniformly ofFigure1inKroneandNeuhauser(1997).Following
atrandomfromwithinthesamedemeasthereproduc- a single lineage, either forward or backward in time,
ingindividual.AsusualinMoran-typemodels,theindi- 2-arrows are encountered at rate (cid:2)s and (cid:4)-arrows are
1 D
vidualchosentoreproducemightalsobetheonecho- encountered at rate (cid:2). Each arrow has a probability m
1
sentodie.Althoughthemodeldoesallow“migration” ofconnectingtoalineagesampleduniformlyatrandom
backtothesamedeme(withprobabilitym )theeffectof from the entire population and probability 1 (cid:3) m of
ii
thisbecomesnegligibleinthelarge-Dlimitweconsider connecting to a lineage sampled uniformly at random
below. from the same deme. Thus, there are four kinds of
The general model described above forms the basis branchesandtheseappearrepeatedlyinthehistoryof
forastructuredancestralselectiongraph.Thiswouldbe every lineage with rates (cid:2)(1 (cid:3) m), (cid:2)m, (cid:2)s (1 (cid:3) m),
1 1 1 D
obtainedviatheusuallarge-Nlimit,withDfinite,inthe and (cid:2)s m. Note that (cid:2) is simply a scaling factor that
1 D 1
same way that the structured coalescent is obtained in specifiestheunitsinwhichtimewillbemeasured;(cid:2) (cid:1)
1
theneutralcase(Notohara1990;Wilkinson-Herbots 1 corresponds to the usual notion of measuring time
1998).Thatis,wewouldletN (cid:1) (cid:1)Di(cid:1)1Ni andassumethat in generations.
c (cid:1)N/N,M (cid:1)Nm ,(cid:6)(cid:1)(cid:2) (cid:3)(cid:2),and(cid:7)(cid:1)Nu,where Thedualorancestralprocessisobtainedbyfollowing
i i ij i ij 2 1
u is the neutral mutation rate, are all finite as N → ∞, lineages back in time, i.e., up the graph in Figure 1.
and time is measured in units of N generations, where Because2-arrowsarerealizedonlyiftheparentalallele
a generation is defined to be (cid:2)(cid:3)1 time units. Note that is of the favored type, both lineages must be followed
1
acrucialassumptionofthestructuredcoalescentisthat intheancestralprocess.Asinglegenealogyisobtained,
migrationoccursonlybetweentheDdemesenumerated in the unconditional ancestral process, by tracing lin-
intheinitialsample,andtheancestryisthereforealways eages back to the ultimate ancestor, assigning its type
restricted to those particular demes. from the equilibrium distribution of allele frequencies
Wedonotpursuethislarge-Nlimithere,butinstead (see below), and then following lineages forward in
consideralimitingancestralprocessthatapproximates time,withmutation,andtrimmingoffbranchesappro-
the behavior of a population divided into very many priately(KroneandNeuhauser1997;Neuhauserand
demes and in which selective differences are small. Krone1997).Ratherthanthisunconditionalancestral
Thus,weassumethatthesamplesizeissmallrelativeto process,weconsiderthegenealogyofasamplecontain-
thenumberofdemesinthepopulation,ratherthanto ing known numbers of different alleles. This leads to
thedemesize.Historically,migrationcantakeancestors moreefficientsimulationofselectedgenealogies.Using
toanydemewithinthepopulation.Thisfollowsrecent theconditionalapproach,itispossibletolabelancestral
coalescent work under the assumption of neutrality lineages as either virtual or real, and it is necessary to
(Wakeley 1998) and work on the forward-time diffu- trace lineagesback only tothe most recentcommon an-
sion approximation for allele frequencies in the pres- cestorofthesample,i.e.,amongthereallineages,rather
enceofselection(Wakeley2003;WakeleyandTaka- than all the way back to the ultimate ancestor (Slade
hashi 2004). We set (cid:2) (cid:1) (1 (cid:8) s )(cid:2), and we use (cid:2)s 2000a). There is also a minimal representation of the
2 D 1 1 D
in place of (cid:2) (cid:3) (cid:2) for the rate of type 2 birth/death ancestral process that reduces the number of possible
2 1
events below. This corresponds to a model in which eventsrequiredateachancestraltransition(Sladeand
therearetwoalleles,A andA ,andalleleA hasfitness Neuhauser2003).Wedescribehowthisisdoneinthe
1 2 2
equal to 1 (cid:8) s while A has fitness equal to 1. We put present, structured model in methodology.
D 1
a subscript on the selection coefficient in recognition Theessenceofthelarge-Dresulthere,asintheneu-
ofthefactthatthelimitweseekhasDs(andDu)finite tral case, is that the rates at which events occur that
asD→∞.FollowingLessardandWakeley(2004),the affect the history of a sample from the population are
large-N, large-D limit can also be studied. Similar limit verydifferentwheneverylineageisinaseparatedeme
resultshererequiremorestringentconditions:roughly, than when at least one deme contains more than one
that D goes to infinity faster than N (see appendix a). lineage.ThedependenceonDissuchthataseparation-
For simplicity, we assume that migration occurs ac- of-timescales result applies, as, for example, in Mo¨hle
cordingtoWright’s(1931)islandmodel,butadapted (1998a,b). Because of this, the history of a sample can
forMoran-typereproductionasinWakeleyandTaka- be divided into two phases (Wakeley 1999). The first
hashi(2004).Thatis,weassumethatm (cid:1)m/Dforall is a short scattering phase in which coalescent events
ij
iandj,andthatN (cid:1)Nforalli.Thus,wehaveamodel occurbetweensamplesfromthesamedemeandmigra-
i
in which there are D demes, each of size N, and each tioneventsinwhichlineagesmovetounoccupieddemes
of which receives a migrant from the total population (i.e.,demesthatdonotcontainanyancestrallineages).
withprobability mat eachbirth/death event.As inthe Thescatteringphaseoccursonafasttimescaleandbe-
unstructured ancestral selection graph, it is helpful to comes an instantaneous sample size adjustment in the
consider a graphical representation, which Krone and limit as D tends to infinity. The scattering phase ends
1120 P.F.SladeandJ.Wakeley
Figure1.—Graphicaldepictionofthestruc-
tured selection process for the case of D (cid:1) 3
demes.
when allremaining lineages are indifferent demes. At usedtoshow thatthecollectionofdemes isalwayssuf-
this point, the collecting phase begins in which pairs of ficiently close to (cid:9) that changes in X depend only on
lineagescometogetherintoasingledemeandcoalesce, the ensemble properties E(cid:9)[K] (cid:1) (cid:1)Nk(cid:1)0k(cid:9)k (cid:1) Nx and
eventuallyfindingacommonancestoroftheentiresam- Var(cid:9)[K](cid:1)x(1(cid:3)x)N2/(Nm (cid:8)1(cid:3)m)andthatthedif-
ple.Themigrationeventsthatcouldbeignoredinthe fusion of X isidentical to the usual unstructured diffu-
scatteringphase,i.e.,thoseinwhichlineagesmovetooc- sionexceptthatthetimescaleisincreasedbythefactor
cupieddemes,arenowessentialsinceacoalescentevent 1 (cid:8) (1 (cid:3) m)/(Nm) (Wakeley and Takahashi 2004).
canoccuronlybetweenapairoflineagesiftheyarein By considering overall limiting rates of type (cid:4) and
thesamedeme.Weusethescattering/collectingtermi- type2birth/deathevents—whichare(cid:2)NDand(cid:2)s ND
1 1 D
nology even though we have incorporated selection. (1(cid:3)x),respectively—andconditioningonthenumber
Forward-timeanalysis:Tosimulategenealogiesinboth of copies of allele A in the deme where the reproduc-
1
theconditionalandtheunconditionalASG,itisneces- tion event occurs, it can be shown that
sarytoknowtheequilibriumdistributionofthefrequen-
⎧ 1 Nm
ciesof A1and A2in thepopulation.In fact,the ASGis ⎪X(cid:8)ND withrate(cid:2)1NDx(1(cid:3)x)Nm(cid:8)1(cid:3)m
ariummodpeolpsupleactiifionca.lTlyhfeorreawsiallmbpelenofreoqmuisluibcrhiuamnweqituhioliubt- ⎪⎪ (cid:8)(cid:2)ND(1(cid:3)x)u (cid:2)1(cid:8)2x Nm (cid:3)
⎪ 1 D Nm(cid:8)1(cid:3)m
mutation, and here we follow Krone and Neuhauser ⎪ (cid:8)O(s u )
(1997) in assuming that there is a probability of muta- X→⎨ 1 D D Nm
⎪X(cid:3) withrate(cid:2)NDx(1(cid:3)x)(1(cid:8)s )
tionuDperbirth/deathevent.Wenotethatasymmetric ⎪ ND 1 D Nm(cid:8)1(cid:3)m
mutation can also be accommodated (Slade 2000a). ⎪ (cid:2) Nm (cid:3)
The forward-time dynamics of allele frequencies in a ⎪ (cid:8)(cid:2)1NDxuD1(cid:8)2(1(cid:3)x)Nm(cid:8)1(cid:3)m
⎪
subdivided population may be complicated, but in the ⎩ (cid:8)O(sDuD). (2)
caseofalargenumberofdemestheyarenearlyassimple
We let
as in an unstructured population (Wakeley 2003). In
amodelverysimilartotheoneweconsiderhere,Wake- ND(cid:4) 1 (cid:3) m(cid:5)
(cid:2) (cid:1) 1 (cid:8) (3)
leyandTakahashi(2004)showedthatthefrequencies 1 2 Nm
of alleles in the total population change according to
adiffusionprocessthatisidenticaltothediffusionpro- and we further assume that
cessinanunstructuredpopulation,onlywithadifferent NDs → (cid:6) and NDu [1 (cid:8) (1 (cid:3) m)/(Nm)] → (cid:7)
timescale.Further,asthesefrequencieschange(slowly) D D
the collection of demes closely tracks an equilibrium as D → ∞. (4)
distribution of allele frequencies.
Then, the diffusion process for the frequency of allele
Let the random variable X(t) be the frequency of A has drift parameter a(x) (cid:1) (cid:3)((cid:6)/2)x(1 (cid:3) x) (cid:8)
alleleA1attimet,andletxrepresentaparticularvalue ((cid:7)1/2)(1 (cid:3) 2x) and diffusion parameter b(x) (cid:1) x(1 (cid:3)
ofX(t).BecauseweconsiderthelimitD→∞whileDs
D x). This process has a unique stationary distribution
and Du remain finite, mutation and selection do not
D
appear in the equilibrium h(x) (cid:1) Bx(cid:7)(cid:3)1(1 (cid:3) x)(cid:7)(cid:3)1e(cid:3)(cid:6)x, (5)
(cid:6)
(cid:4)N(cid:5)(cid:4) Nmx (cid:5) (cid:4)Nm(1 (cid:3) x)(cid:5) (cid:4) Nm (cid:5) whereBisanormalizingconstant;thatis,h(x)satisfies
(cid:9) (cid:1) (1)
k k 1 (cid:3) m (k) 1 (cid:3) m (N(cid:3)k) 1 (cid:3) m (N) 0 (cid:1) 1 d2 [b(x)h(x)] (cid:3) d [a(x)h(x)] (6)
2dx2 dx
(Wakeley and Takahashi 2004), which here is the
fractionofdemesinthepopulationthatcontainkcopies (Wright 1949; Kimura 1955). With reference to Fig-
ofalleleA andinwhicha (cid:1)a(a(cid:8)1)···(a(cid:8)k(cid:3)1) ure1,thetwo-partequilibrium(1)and(5),whichpre-
1 (k)
isanascendingfactorial.Clearly,theequilibriumvector dicts the distribution of A among demes and across
1
(cid:9)doesdependonthefrequencyofalleleA inthetotal the total population, is what would be observed by as-
1
population, and this will change over time. However, signingA ’s andA ’sto thelineagesand thenrunning
1 2
Theorem 3.3 of Ethier and Nagylaki (1980) can be the process forward a very long time.
Many-DemesAncestralSelectionGraph 1121
TABLE1
Theninepossibleresultsoftype(cid:1)andtype2birth/deatheventsinthehistoryofnlineagescurrentlyinndifferentdemes
Rate Event Outcome Magnitude
U (cid:1)(cid:2)n(1(cid:3)m) (cid:4),nomigration Nochange O(1)
1 1
D(cid:3)(n(cid:3)1)
U (cid:1)(cid:2)nm (cid:4),migrationtou-deme Nochange
2 1 D
n(cid:3)1 1
U (cid:1)(cid:2)nm (cid:4),migrationtoo-deme,coalescent Coalescentevent O(1/D)
3 1 D N
n(cid:3)1N(cid:3)1
U (cid:1)(cid:2)nm (cid:4),migrationtoo-deme,nocoalescence Possiblecoalescent
4 1 D N
1
U (cid:1)(cid:2)s n(1(cid:3)m) 2,nomigration,coalescent Nochange(self-collision)
5 1 D N
N(cid:3)1
U (cid:1)(cid:2)s n(1(cid:3)m) 2,nomigration,nocoalescence Possiblebranch
6 1 D N
D(cid:3)n
U (cid:1)(cid:2)s nm 2,migrationtou-deme Branchevent
7 1 D D
n 1
U (cid:1)(cid:2)s nm 2,migrationtoo-deme,coalescent Collision O(1/D2)
8 1 D DN
nN(cid:3)1 2,migrationtoo-deme,nocoalescence Possiblebranch
U (cid:1)(cid:2)s nm
9 1 D D N
U-demeando-dememeanunoccuppieddemeandoccupieddeme,respectively.
The collecting phase: Consider the case in which n coalescenceaswefollowthehistoryofthesampleback
lineagesarein,oraresampledfrom,ndifferentdemes. in time.
By tracing the lineages back in time without knowing It is appropriate that we have ignored mutations
theirallelictypes,wefindasimpleancestralprocessin aboveandinTable1becauseherewearedealingwith
thelimitD→∞andgenerateanintuitiveunderstanding the unconditional ancestral process. Mutations will oc-
forthedifferentscalingofmutationvs.selectionin(4) cur with probability u , 0 (cid:10) u (cid:10) 1, at both types of
D D
andinWakeley(2003)andWakeleyandTakahashi arrows or reproduction events. Along a single lineage,
(2004).Thefourkindsofarrows,orreproductionevents, theyoccurwithrate(cid:2)u (cid:1)ND(1(cid:8)s )u [1(cid:8)(1(cid:3)m)/
2 D D D
discussedabovewillbeencounteredbythesenlineages (Nm)]/2,whichbecomes(cid:7)/2inthelimitD→∞.Asin
and will sometimes move them to occupied demes so Krone and Neuhauser (1997) and Neuhauser and
thattheymightcoalesce.Wecanrecognizeninepossible Krone (1997), we superimpose this mutation process
eventsforsuchasample,andthesearelistedinTable1. ontheunconditionalancestralprocess.Intheconditional
For example,the fourthkind of eventis that a(cid:4)-arrow ancestralprocessweconsiderbelow,itisnecessarytotreat
takes one of the n lineages to one of the other n (cid:3) 1 mutation, coalescence, and branching simultaneously.
occupied demes but does not connect to the resident When D is large, the first two events in Table 1 will
ancestrallineage.Theresultisthatnowthenlineages, account for the vast majority of events. These events
while still distinct, reside in n (cid:3) 1 demes. Thus, at the have rates of O(1), which here means that they have a
nexteventthetwolineagesthatresidetogetherinone finite and nonzero limit as D → ∞ for a given (cid:2). It is
1
deme can coalesce. The interpretations of the other easy to see that this is true because the rates of these
eventsinTable1areequallystraightforward.Notethat events depend on D only through (cid:2). These events are
1
the events are categorized according to their effect on lineage switches within and between demes, but ones
theancestral lineagesand alsoby theirprobabilities of that preserve the basic sample structure of n lineages
occurring with reference to the limit D → ∞ (while in n different demes. They do not change the rates at
Ds remains finite). As in the unstructured ancestral which events occur and Table 1 continues to apply.
D
selection graph, when a 2-arrow is encountered, both Note that with other kinds of (non-island) population
paths are followed and the lineage splits into two lin- structure, these events would change the state of the
eages. We are interested in the rates of branching and sample by moving lineages among different types of
1122 P.F.SladeandJ.Wakeley
demes (Wakeley and Aliacar 2001). Here, they can where(cid:4)n(cid:5) (cid:1) n(n (cid:3) 1)/2and(cid:6)(cid:1)NDs .Again,thefirst
2 D
be ignored due to the symmetry of the island model. twotypesofeventsdonotchangethestateofthesample,
Thenextmostnumerouseventswillbeevents3–7,and soitisnotproblematicthattheirratesbecomeinfinite.
these have rates of O(1/D), which means that these Itsimplymeansthattheancestrallineageswillencoun-
rates will approach zero as D → ∞ for a given (cid:2)1, but ter many birth/death events before anything of rele-
that their rate of approach to zero will be inversely vance happens to the sample. Event 3 is a coalescent
proportional to D. Again, the limiting process we seek event in which the number of lineages decreases from
is for DsD finite as D → ∞, so that sD is of O(1/D). Four nton(cid:3)1andthesen(cid:3)1lineagesareallindifferent
oftheseevents—3,4,6,and7—dofundamentallyalter demes. Event7 is a branchingevent, in whichcase the
the state of the sample. These are migration events to samplegoesfromnton(cid:8)1lineages,andamigration
occupied demes, both with and without coalescence, event leaves all n (cid:8) 1 lineages in different demes. In
and branching events, both with and without a migra- event4thenumberoflineagesremainsn,butnowtwo
tion event to an unoccupied deme.
of them are in the same deme, while in event 6 the
KroneandNeuhauser(1997)andNeuhauserand number of lineages increases to n (cid:8) 1, and again two
Krone(1997)usethetermcollisiontodenotetheevent
areinthesamedeme.Tosummarize,thefirststepthat
that an ancestral lineage splits and immediately co- matters in the limiting (D → ∞) process for n lineages
alesces with another ancestral lineage. In the unstruc-
in n different demes can be a coalescent event or a
tured ancestral selection graph, all collisions become
branching event, but in either case the remaining lin-
negligible in the limit (N → ∞ and D (cid:1) 1), and it is
eages are all in different demes; alternatively it can be
not necessaryto distinguishbetween differentkinds of
a migration event or a branching event that results in
collisions.Here,becauseNisfinite,somecollisionswill
two lineages residing together in the same deme while
occurwithratescomparabletoregularcoalescentevents.
the rest are in separate demes.
Event5inTable1isofthissortandhasarateofO(1/
We note that Equations 13 and 14, correspond to
D).However,event5isacollisioninwhichthelineage
events that would disrupt the dual process or greatly
splits and then immediately coalesces with itself. Both
complicate the branching structure, respectively. The
forward and backward in time, this has no effect. The
zero rates with which we have described them above
descendant lineage has the same parent regardless of
refertotheirinstantaneousratesinthelimitingprocess,
whethertheparentalalleleisA andthe2-arrowisnot
1 butdonotfullyrevealthatevenovertheentiregraph,
followed or the parental allele is A and the 2-arrow is
2 untiltheultimateancestorisreached,theseeventswill
followed.Werefertothiseventasaself-collision.Only occurwithprobabilityzerointhelimitasD→∞.Proof
self-collisionshavethepotentialtooccurwithratescom-
of this that allows us to omit these events without loss
parable to regular coalescent events. Other kinds of
of generality is deferred to appendix a.
collisions, for example, event 8 in Table 1, which in-
WiththesamelevelofdetailasinTable1,27different
cludes some self-collisions, have rates of O(1/D2). The
events can be distinguished for a sample of n lineages
number of occurrences of all events with rates of this in n (cid:3) 1 demes, i.e., where a single pair resides in one
magnitudeisshownlater,inappendixa,tobenegligi-
deme. However, it is unnecessary to distinguish all of
ble in the limiting (D → ∞) process.
these, and Table 2 shows them grouped into just four
Now,considerwhathappensinthelimitingprocess.
kindsofevents.TheimportantdifferencefromTable1
Bysubstitutingthevalueof(cid:2) fromEquation3intothe
1 isthateventswithratesofO(1)nowhavethepotential
entries of Table 1, the limiting rates become
toaffectthestateofthesample.TheO(1)eventsinTa-
U* (cid:1) limU (cid:1) ∞ (7) ble 2 are a coalescent event between the two lineages
1 1
D→∞ thatshareademeandamigrationeventthatsendsone
U* (cid:1) limU (cid:1) ∞ (8) ofthesetwolineagestoanunoccupieddeme.Theother
2 2
D→∞ events that can change the sample are of O(1/D) or
(cid:4)n(cid:5)Nm (cid:8) 1 (cid:3) m smaller,andthesearebranchingeventsandmigration
U* (cid:1) limU (cid:1) (9)
3 D→∞ 3 2 N events to occupied demes. These fast events would be
problematiciftheyweretoactuallyoccurinthelimiting
(cid:4)n(cid:5)(N (cid:3) 1)(Nm (cid:8) 1 (cid:3) m)
U* (cid:1) limU (cid:1) (10) process, but whenever a pair of lineages resides in the
4 D→∞ 4 2 N same deme events with rates of O(1) will dominate.
(cid:6)(1 (cid:3) m)(N (cid:3) 1)(Nm (cid:8) 1 (cid:3) m) Theseevents,1and2inTable2,endwithallremaining
U*6 (cid:1) lDi→m∞ U6 (cid:1) n2 N2m lineagesindifferentdemes.Thisguaranteesthatthere
(11) will never be more than one multiply occupied deme
U* (cid:1) limU (cid:1) n(cid:6)Nm (cid:8) 1 (cid:3) m (12) and,thatwhenthereis,itwillcontainjusttwolineages
7 D→∞ 7 2 N (appendix a). That is, in the limiting process, either a
U* (cid:1) limU (cid:1) 0 (13) V or a V event always preempts a V event during the
8 8 1 2 4
D→∞ state where precisely one deme contains two lineages.
U* (cid:1) limU (cid:1) 0, (14)
9 D→∞ 9 Onceintheresultingstate,Table1appliesandthehighest
Many-DemesAncestralSelectionGraph 1123
TABLE2
The27distinguishabletype(cid:1)andtype2birth/deatheventsinthehistoryofnlineages
currentlyinn(cid:3)1differentdemes,groupedintojustfourkindsofevents
Rate Outcome
V (cid:1)(cid:2)2m(cid:8)O(1/D) Migration(nlineagesinndemes)
1 1
V (cid:1)(cid:2)2(1(cid:3)m)(1/N)(cid:8)O(1/D) Coalescence(n(cid:3)1lineagesinn(cid:3)1demes)
2 1
V (cid:1)(cid:2)[(n(cid:3)2)(cid:8)2(1(cid:3)m)(N(cid:3)1)/N](cid:8)O(1/D) Nochange(nlineagesinn(cid:3)1demes)
3 1
V (cid:1)(cid:2)2s (cid:8)(cid:2)(n(cid:3)2)m(n(cid:3)2)/D(cid:8)O(s /D) Twoormorelineagesinatleast1deme
4 1 D 1 D
(cid:6)
rate events that affect the sample configuration are of U* (cid:8) P(mig)U* (cid:1) n , (18)
7 2 6
O(1/D). 2
Thus,inarelativelyshorttime,asampleofnlineages
in n (cid:3) 1 demes will be converted to a sample of k respectively. Therefore, the rates of coalescence and
branching in the limiting (D → ∞) ancestral process
lineagesinkdemes.Thevalueofkdependsonwhether
become identical to the rates of coalescence and
a coalescent event or a migration event occurs. If it is
branchinginapanmicticancestralselectiongraph,but
acoalescentevent,thenk(cid:1)n(cid:3)1,andifitisamigration
wheretimeismeasuredinunitsof(cid:2) generations.From
event, then k (cid:1) n. On the timescale above, with (cid:2) (cid:1) 1
1 Equations 3 and 15 we can write (cid:2) (cid:1) ND/(2P(mig)).
ND[1(cid:8)(1(cid:3)m)/(Nm)]/2,theseeventsbecomeinstan- 1 2
Since we assume that 0 (cid:11) m (cid:10) 1 and N (cid:12) 1, we have
taneousbecausetheirratesapproachinfinityinthelimit
0(cid:11)P(mig)(cid:10)1and(cid:2) (cid:12)ND/2.Withamigrationrate
D → ∞. The result is an instantaneous adjustment that of m (cid:1)2 1, the present1model reduces to a panmictic
has two possible outcomes, with probabilities modelwith timescale(cid:2) (cid:1)ND/2.Otherwise, theeffect
1
V Nm ofisland-modelsubdivisionistolengthenthetimescale
P(mig) (cid:1) lim 1 (cid:1) (15)
2 D→∞ V1 (cid:8) V2 Nm (cid:8) 1 (cid:3) m of the ancestral (and the forward-time) process.
Note that the processes of selection and mutation
and responddifferentlytosubdivision.Mutationscaleswith
(cid:2), while the selection parameter (cid:6) scales simply with
V 1 (cid:3) m 1
P(coal) (cid:1) lim 2 (cid:1) . (16) ND/2.Inotherwords,therearedifferenteffectivepopu-
2 D→∞ V1 (cid:8) V2 Nm (cid:8) 1 (cid:3) m lationsizesforselectionandformutation.Theeffective
size for selection is smaller and is equal to P(mig) (cid:1)
Thiswill actas afilteron thefour-rate Poissonprocess 2
Nm/(Nm(cid:8)1(cid:3)m)timestheeffectivesizeformutation.
described by Equations 9–12. Starting with the sample
The reason for this is that, when a branching event
ofnlineagesinndifferentdemes,wheneveramigration
creates a new lineage in the ancestral process, it has a
event to an occupied deme occurs without immediate
P(coal) (cid:1) (1 (cid:3) m)/(Nm (cid:8) 1 (cid:3) m) chance of being
coalescence (event 4), there is a chance P(mig) that 2
2 erased by a coalescent event, so only P(mig) of them
the sample will be returned to its original state. The 2
are observable. In the present model, this cancels, ex-
rest of the time, i.e., with probability P(coal), event 4 is
2 actly, the factor 1/P(mig) by which the number of
converted to event3, which is acoalescent event. When- 2
branching events that occur in the limiting process is
ever a branching event occurs where both parents stay
increased relative to the number under panmixia. For
inthesamedeme(event6), thereisachanceP(coal)
2 thesamereason,thereisalsoadifferencebetweenthe
thatitisconvertedintoself-collision,sothatthesample
scalings for recombination, which splits lineages, and
returns to its original state, and a chance P(mig) that it
2 mutation in similarly structured populations (Nord-
isconvertedintoanobservablebranchingevent(event7).
borg 2000; Lessard and Wakeley 2004).
It is straightforward to show that a Poisson process
Thescatteringphase:Theresultsobtainedaboveap-
filtered in this way is equivalent to a Poisson process
plytothehistoryofasampleofnlineagesinndifferent
withanadjustedrate;forexample,seeWakeley(1999).
demes.Theyfollowfromthefactthatcoalescentevents
Thus,theprobabilitiesP(mig)andP(coal)narrowthe
2 2 within multiply occupied demes and migration events
relevant number of O(1/D) events to just two. Using
from multiply occupied demes to unoccupied demes
Equations 9–12, and simplifying, the limiting rates of
occur with rates that are D times larger than the rates
coalescence and branching become
of branching events and migration events to occupied
(cid:4)n(cid:5) demes.Becauseofthis,whenevermultiplesamplesare
U* (cid:8) P(coal)U* (cid:1) (17)
3 2 4 2 taken from one or more demes, there will be a brief
scattering phase and this will leave the remaining lin-
and eagesindifferentdemes(cf.Table2)sothatthecollect-
1124 P.F.SladeandJ.Wakeley
ing-phase results above apply. The number of lineages witheachremaininglineageinaseparatedeme,E[Y|n(cid:13)]
thatremainattheendofthescatteringphase,andthat istheexpectedvalueofY forasampleofsize|n(cid:13)|inour
then enter the collecting phase, will depend on the rescaled,unstructured,collecting-phaseASG.Below,we
numberofcoalescenteventsthatoccurwithinthemulti- usetheframeworkofEquation21bothanalyticallyand
plysampleddemeordemes.InthelimitasD→∞,the insimulations,whereourresultsarealsoconsistentwith
durationofthescatteringphasebecomesnegligibleso keeping track of allelic configurations during the scat-
that it can be treated as instantaneous when time is tering phase.
measured in proportion to D generations. A compression of the Markov process that describes
Becausewehaveassumedthats isO(1/D),whereas the conditional evolution of the unstructured ASG is
D
the rates of within-deme coalescence and migration to found. This continuous-time Markov jump chain is an
occupieddemesdependonlyonN,m,andthesample extension of the conditional ancestral selection graph
size(s), thescattering phasehere is thesame as thatin of Slade (2000a) and is derived with new clarity. The
apopulationinwhichallgeneticvariationisselectively result is a maximal compression of the ancestral selec-
equivalent.AsinWakeley(1998),thescatteringphase tiongraphinwhichthetimingpropertiesofthegraph
forasampleofsizenallfromasingledemeisequivalent areretained.Thiscontinues,andimprovesupon,asim-
to a series of n Bernoulli trials with probabilities of ilar enhancement of the conditional graph by Slade
success (2000b)thatreducesitsbranchingrate.Thisisanalter-
nativeversionoftheminimalrepresentationofthecon-
(cid:1)
P(mig) (cid:1) , (19) ditional graph as discussed in Slade and Neuhauser
k (cid:1) (cid:8) k (cid:3) 1
(2003). Without the timing structure of the graph in
for k (cid:1) 1, 2,..., n, and in which (cid:1) (cid:1) Nm/(1 (cid:3) m). place,resultsthatdescribesomeprobabilisticproperties
Note that k (cid:1) 2 yields P(mig) given in Equation 15. ofasingleancestrallineagewithintheASGareobtained
2
Success is defined to be the migration of one of the byFearnhead(2002).Thedetailsofourgeneralderiva-
lineages to an unoccupied deme, and each of these tion are deferred to appendix b.
increasesbyonethenumberoflineagesthatwillenter Thesimulationspresentedinresultsareperformed
thecollectingphase.Ifweusen(cid:13)todenotethenumber usingthecorrespondingadaptationofacomputational
of lineages that will enter the collecting phase, then Monte Carlo method as in Slade (2000a,b). The pur-
pose of such a simulation scheme is the calculation of
|S(n(cid:13))|(cid:1)n(cid:13)
P[n(cid:13)|n] (cid:1) n (20) anapproximateprobabilitydistributionovertherealiza-
(cid:1)
(n) tionsofthegenealogy.Foraparticulargenealogythen,
a weight is attributed to its associated time to the most
is its probability function (Wakeley 1998), in which
the|S(n(cid:13))|areunsignedStirlingnumbersofthefirstkind recentcommonancestor(TMRCA),andtheweightedaver-
n
and (cid:1) (cid:1) (cid:1)((cid:1) (cid:8) 1)···((cid:1) (cid:8) n (cid:3) 1). Finally, the ageisevaluateduponcompletionofalargenumberof
(n)
repetitions.Theimprovedefficiencyinhavingattained
scatteringphaseoccursindependentlyamongdemes,so
the maximal compression of the genealogical Markov
thatwecanmultiplytheprobabilities(20)oversamples
chain is substantial; running time until convergence
from different demes.
thatwaspreviouslyrequiredisnowatleastfullyhalved.
The feasibility of at least doubling the branching and
METHODOLOGY mutation rates is also achieved. The performance gets
betterstillwhencalculatingstatisticsoftheT distri-
We have shown that the ancestry of a sample from a MRCA
bution,asitwouldalsodoforpropertiesofsubtreesof
subdivided population, with equal deme sizes and is-
thegenealogy.Ourcompressionreducesthenumberof
land-typemigration,andinwhichtwoallelesaresubject
possibletransitionsrequiredtodescribetheconditional
to selection and mutation, includes an instantaneous
ASG at any event, from 10 to only 6. Thus, the size of
scattering phase and then enters a collecting phase,
thestatespaceofarealizationoftheconditionalgraph
whichisequivalenttotheunstructuredASGwithmuta-
decreases exponentially. Hence, variance of any corre-
tionandselectionparametersscaledappropriately.This
spondingsimulation techniqueis reduced.Depending
meansthatifYisanymeasurementonthesample,such
on the parameter combinations, to produce the com-
as the time to common ancestry or the total length
posite PDFsin resultswith a Pentium3.06GHz Xeon
of the genealogy, we can compute properties of Y by
processor,computingtimerangedfromafewhoursto
conditioning on the outcome of the scattering phase.
several days.
For example, if E[Y|n] is the expected value of Y for
thesamplen(cid:1)(n1,...,nd)ofsize|n| (cid:1) (cid:1)di(cid:1)1ni fromd
different demes, then RESULTS
E[Y|n] (cid:1) (cid:1)E[Y|n(cid:13)]P[n(cid:13)|n], (21) From Equation 21 we can see that simple estimators
n(cid:13) ofthemigrationparameter(cid:1)areasusefullwhenweak
where P[n(cid:13)|n] is the product of probabilities like (20) selection is present as they are when all variation is
over demes. Because the scattering phase always ends neutral. For example, let q and q be the probabilities
0 1
Many-DemesAncestralSelectionGraph 1125
ofidentityinstateforsamplesofsizetwotakenfromthe collectingphase.Lowmigrationratesyieldsamplecon-
samedemeandfromtwodifferentdemes,respectively. figurationsatthestartofthecollectingphasethatcon-
Taking the scattering phase into account, we have sistofclosetodA allelesanddA alleles,providedall
1 2
of thed demes initiallysampled contained atleast one
(cid:1)
1 (cid:3) q (cid:1) (1 (cid:3) q), (22) ofeachallele.Thehighestsensitivitytoselectionthere-
0 (cid:1) (cid:8) 1 1 fore is achieved when the population is structured be-
tween very many demes connected by an exceedingly
and if qˆ and qˆ are estimates of q and q from some
0 1 0 1 lowmigrationrate.Notethatthiseffectisindependent
data, then
ofwhethertheinitialsamplecontainsmostlythefavored
(cid:7)(cid:1)(cid:1) (cid:2)11(cid:3)(cid:3)qˆqˆ1 (cid:3) 1(cid:3)(cid:3)1 (23) orTthoecounnvfearvtoarecdoaalleleslcee.nttimeunitintoyears,itisnec-
0
essarytocalculatetheproductofthe(estimated)effec-
is an estimate of (cid:1). Note that (23) has the same form tive population size and number of years per repro-
asthevariousestimatorsofgeneflowbasedonFSTorpair- ductivegeneration.Whenthemigrationrate(cid:1)(cid:14)1we
wisesequencedifferencesundertheassumptionofselec- compare the structured model only since according to
tive neutrality; see, for example, Hudson et al. (1992). thefactorcalculatedintheory,1(cid:8)(cid:1)(cid:3)1,acoalescent
Neither theunstructured ASGnor ourslightly more timeunitintheunstructuredASGrepresentsincompa-
complicated structured ASG lends itself to analytical rablyfewernumbersofgenerations.Ontheotherhand,
computations much more involved than the above. when(cid:1)(cid:15)1thereisameaningfulcomparisonbetween
From Equation 21 it is clear how analysis of our struc- results with and without very many demes. When (cid:1) (cid:1)
tured ASG is related to analysis of the unstructured 0.01 in the structured genealogies (both neutral and
graph. So, for example, it is possible to average such ex- selected)everygenerationthatelapsesaccordingtocoa-
pressionsasKroneandNeuhauser’s(1997)Equation lescent time translates as 100 additional generations
3.5 for the expected time to the ultimate ancestor of in an unstructured genealogy. Population subdivision
the sample over n (cid:1) |n(cid:13)| at the start of the collecting magnifies the extent of changes to coalescence times
phase. Instead of this, we have used simulations that andthusimmediatelyaccentuatestheeffectofselection
includethescatteringphaseandthemaximalcompres- on genealogical time depth.
sion of the ASG described in appendix b to compute The interaction between migration and selection as
theprobabilitydensityfunctionofthetimetothemost it affects the distribution of the conditional T is
MRCA
recent common ancestor of a sample (TMRCA). shown in Figures 2 and 3. The mutation rate is set at
In unstructured nonneutral genealogies with small three levels, (cid:7) (cid:1) 0.01, 0.1, and 1. Low mutation rates
mutationratescertainsampleconfigurations,amonga areappropriatesinceweconsiderallelescorresponding
sample of size n, have reduced mean coalescent times toaminoacidreplacementsatahypotheticalnucleotide
whencomparedtotheirneutralcounterparts,andother sitewithinasinglelocus.(Wenotethat(cid:7)(cid:1)0.001yields
sample configurations have enlarged coalescent times a distribution of the T under neutrality that is ex-
MRCA
(Slade 2000a; see also Slade and Neuhauser 2003). tremely similar to that obtained for (cid:7) (cid:1) 0.01.) The
The mixtures of reduced sample configurations at the selectionrateinourthreecasesischoseninaccordance
start of the collecting phase in the structured model withtheweightedaveragepredictedtobeoperatingat
resultinnoveldistributionsoftheTMRCAunderselection. nucleotide sites by Sawyer et. al(2003), namely (cid:6) (cid:1) 5
This mixture is the same under neutrality as it is with and 7.5. The migration rate is set to two levels, (cid:1) (cid:1)
selection.Theeffectofthescatteringphaseunderneu- 0.01 in Figure 2 and (cid:1) (cid:1) 10 in Figure 3.
tralityisgovernedbythelevelofmutationinthecollect- The probability density function (PDF) for the low
ing phase. Although the sample size is reduced, it can migrationrateandlowmutationratecaseispresented
easilycorrespondtoanincreasedTMRCAbecausethead- in Figure 2A. A substantial change in the shape of the
justed sample configuration requires longer to assimi- PDF is clear under selection. Increasing the mutation
late its ancestors (particularly for low mutation rates). rate, the next lowest migration rate case is shown Fig-
IntheunstructuredASGasthemutationratedecreases ure2B.Theextentoftheeffectofselectionappearsto
the conditional genealogy becomes more sensitive to diminish at our highest level of the mutation rate in
thepresenceofselection(Slade2000a).However,the Figure 2C. The parameters (mean and standard devia-
sample configurations that yield the greatest selective tion) of these distributions are compared in Table 3.
effect are also the samples least likely to be obtained, In the last case just mentioned, we have a particularly
havingextremelysmallsamplingprobabilities.Thesam- novel result as the mean starts to increase again, i.e.,
pling distribution is increasingly U-shaped as (cid:7) de- toward neutrality, with the selection strength above
creases,andunderselection,samplesdominatedbythe some threshold. In both Figure 2 and Table 3 it is as-
favoredallelearemorelikelythanthosedominatedby sumedthatsamplesofsize12weretakenfromeachof
the unfavored allele. Introducing population subdivi- fourdemes,andthateachofthefoursamplesincluded
sionmakesthemostsensitivesamplesaccessiblebecause onecopyofalleleA and11copiesofalleleA ,or(1,11)
1 2
the scattering phase delivers adjusted samples to the for short.
1126 P.F.SladeandJ.Wakeley
Figure3.—PDFsfortheT ofasampleofsize48spread
MRCA
evenlyamongfourdemes,butcomparingtwosampledistribu-
tionsofallelesA andA :(1,11)asinFigure2and(11,1).
1 2
In all cases, (cid:1) (cid:1) 10 and (cid:7) (cid:1) 0.1. As in Figure 2, the solid
curve shows the PDF for (cid:6) (cid:1) 0; the long-dashed curves, for
(cid:6)(cid:1)5.0;andtheshort-dashedcurves,for(cid:6)(cid:1)7.5.
(11, 1), respectively. The effect on the shape of the
PDFscanbediscerned;however,inTable4comparisons
of the parameters of the distributions under panmixia
are also shown. With a migration rate as high as (cid:1) (cid:1)
10, the effects of many-demes population structure in
Table4,forbothneutralandselectedgenealogies,can
still be distinguished. In the neutral case (Table 4, left
column),evenwiththesamplesizeadjustmentthemean
Figure2.—PDFsfortheTMRCAofasampleofsize48spread TMRCA is considerably higher than it is for panmixia.
evenlyamongfourdemesandwithsampleallelecounts(1A ,
1 ComparingthetworowsofTable4,wecanseethatthis
11A ). A–Cdepict resultsfor threedifferent mutationrates
asind2icated.Themigrationparameterwas(cid:1)(cid:1)0.01,andthe also reflects the 10% lengthening of the genealogies
solidcurveshowsthePDFfor(cid:6)(cid:1)0;thelong-dashedcurves, expectedwhen(cid:1)(cid:1)10.Thedifferenceintheselective
for (cid:6) (cid:1) 5.0; and the short-dashed curves, for (cid:6) (cid:1) 7.5. The effectbetweenthetwocompletelydifferentinitialsam-
caseof(cid:7)(cid:1)0.01and(cid:6)(cid:1)7.5couldnotbecompleteddueto ples is only minor.
prohibitivecomputationalrequirements.
DISCUSSION
Incontrast,whenthemigrationrateishighthesam-
ple sizeadjustment is moderate, andthe initial sample Wehaveobtainedasimplestructuredancestralselec-
configurationsshowverysimilarresponsestoselection. tion graph in the limit as the number of demes tends
The PDFs shown in Figure 3, A and B, correspond to toinfinityandwithisland-typemigrationamongdemes
different initial samples within each deme of being ofequalsize.Theresultisunderstoodasanapproxima-
either all (1, 11) of A and A alleles or vice versa all tiontotheancestralprocessforsamplesinthepresence
1 2