The propagation of a cultural or biological trait by neutral genetic drift in a subdivided population

0.33 MB·English

by  R A Blythe

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0 Thepropagationof acultural or biologicaltraitby neutralgeneticdriftina
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2 subdividedpopulation

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a R.A.Blythe
J
School of Physics, Universityof Edinburgh, Mayfield Road, Edinburgh EH93JZ, UK
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EAbstract
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.Westudyfixationprobabilitiesandtimesasaconsequenceofneutralgeneticdriftinsubdividedpopulations,motivatedbyamodel
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of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We
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bfocus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with
-initialdistributionsofmutants.Ageneralformulaforthefixationprobabilityforarbitraryinitialconditionisderivedbyextending
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a duality relation between forwards- and backwards-time properties of the model from a panmictic to a subdivided population.
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From this we obtain new formulæ formally exact in the limit of extremely weak migration, for the mean fixation time from an

2arbitrary initial condition for Wright’s island model, presenting two cases as examples. For more general models of population
vsubdivision, formulæ are introduced for an arbitrary number of mutants that are randomly located, and a single mutant whose
7positionisknown.Theseformulæcontainparametersthattypicallyhavetobeobtainednumerically,aprocedurewefollowfortwo
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contrasting clustered models. These data suggest that variation of fixation time with the initial condition is slight, but depends
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strongly on thenatureof subdivision. Inparticular, we demonstrateconditions underwhich thefixation timeremains finiteeven
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in the limit of an infinite number of demes. In many cases—except this last where fixation in a finite time is seen—the time to
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fixation isshown tobein preciseagreement with predictions from formulæ for theasymptoticeffectivepopulation size.
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oKeywords: Random genetic drift,Population subdivision,Migration,Effective population size, Coalescent, Cultural evolution
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b
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1. Introduction fixationtimes inlargepopulations.
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This very same question has recently arisen in the con-
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X Genetic drift is a generic term for fluctuations in allele text of the cultural evolutionary phenomenon of language
rfrequencies that arise by sampling a finite population to change, in which the unit of variation is some aspect of
aproduceoffspringinthenextgeneration.Intheabsenceof spoken language, such as a vowel sound, through which
mutation,thesefluctuationscanleadtoextinctionofsome onecandistinguishdifferentdialectsofthesamelanguage,
alleles,ultimately causing one allele to fix. One canthere- butwhichareotherwisefunctionallyequivalent.Asimple,
fore ask whether it is feasible for some trait to propagate agent-basedmodeloflanguagereceptionandreproduction
across an entire population by neutral genetic drift alone. (Baxteretal., 2006), has a mathematical description that
Inthesimplestmathematicalmodels,suchasthosedueto coincideswiththatofneutralgeneticdriftinasubdivided
Fisher (1930), Wright (1931) and Moran (1958), individ- population,althoughanumberofdetailsoftheunderlying
uals from the entire population mate randomly and it is evolutionary processes are rather different—in particular,
knownthatfixationtime(measuredinunitsoftheexpected thelanguagedoesnotevolveduetogeneticchangesinthe
lifetime of one individual in the population) increases lin- speakers,butratherthefrequenciesoflinguisticvariantsin
early with the population size (KimuraandOhta, 1969; thepopulationofutteranceschangeovertimeinamanner
CrowandKimura,1970). This growthlaw callsinto ques- akintogeneticdrift.Thekeypointshere,however,arethat
tion the viability of neutral genetic drift as a mechanism fixation of an allele corresponds to a linguistic innovation
for population-level change, on the grounds that changes beingadoptedasacommunity’sconvention,andthatinter-
inlargepopulationsaresimplyfartooslow.Animportant actionsbetweenspeakersmaptomigrationsbetweenlarge
question then arising is whether non-random mating—for subpopulations.Thushereevenalineargrowthinfixation
example,thatseeninsubdividedpopulations—canreduce time with the number of speakers (each one correspond-
ing to a single subpopulation) is untenable: a change that
Preprintsubmitted toElsevier 8February 2008
becomes established in a small clique in a few days would same as those for an ideal panmictic population with an
then require several hundred years to propagate across a effective numberofgeneinstances N .
e
modestly-sized society. To see why this is a problem, one Oneprominentdefinitionofeffectivepopulationsizere-
should note that this long timescale roughly corresponds lates to the change per generation in the variance of a
with the age of language itself (ChristiansenandKirby, mutant allele frequency (CrowandKimura, 1970). Let x
2003), whilst society-wide language change has been seen be the frequency of mutants present in a population at
tooccurwithinoneortwohumangenerations,forexample sometime,andx′ theirfrequencyafterasinglegeneration.
in the case of new dialect formation (Gordonetal., 2004; Then, the variance effective size can be defined as N =
e
Trudgill,2004). x(1−x)/Var(x′). If this effective population size is large,
It is therefore tempting to suggest that some form of one can then use a diffusion approximation to model the
selection, i.e., intrinsic fitness of some linguistic variants evolution of the distribution of mutant frequencies. One
overothers,isrequiredforaninnovationtospreadrapidly canthencalculate the meantime to fixationto be
across an entire society. To be able to state this categori-
(1−x)ln(1−x)
cally,however,wemustunderstandquantitativelyhowfix- τ ≈−2Ne (1)
x
ationtimesandprobabilitiesdependonthenetworkofin-
teractions between speakers in the society (which, in the as was shown in the classic work by KimuraandOhta
biologicalinterpretationofourmodel,correspondstoaset (1969). Thus if one has the variance effective population
of migration rates between geographically separated sub- size for the kind of large, subdivided population that is
populations)andtheoverallsizeofthatnetwork(thenum- of interest in the present work, Eq. (1) can be used to
berofsubpopulations).We alsoneedtoexplorehowthese estimate the fixation time. In principle such an effective
fixationstatisticsvarywithchangesintheinitialcondition population size could be extracted from the diffusion ap-
(distributionofmutantsacrossthe totalpopulation)since proximation that applies to genetic drift in a subdivided
this isthe informationprovidedinthe historicaldatathat population. However, one typically finds that the effec-
wehopetousetoinferthepropagationmechanismsoflin- tive size depends on the mutant allele frequency x (see
guistic innovations. In this work, we aim to develop this e.g. Roze andRousset, 2003; Rousset, 2003) which makes
requiredunderstanding. the resulting expressions difficult to handle, unless ad-
A recurring theme in the considerable literature on ge- ditional approximations are made such as that used by
neticdriftinsubdividedpopulations—thisbeginningwith CherryandWakeley (2003). Instead, we exploit here the
Wright (1931) and continuing through to the present day fact that equilibrium values of genetic variables, such
(see e.g Charlesworthetal., 2003; Wakeley, 2005, for re- as the frequency of mutants, are all approached expo-
centreviews)—isthatpropertiesoffixationarenotstrongly nentially with a common time constant asymptotically
affected by such subdivision. For example, when migra- (Whitlock andBarton, 1997; Rousset, 2004). This time
tion is conservative, the probability a mutant allele fixes, constantcanthenbeusedtodefineaneffectivepopulation
whetherselectivelyadvantageousornot,isthesameasthat size.
as in an ideal panmictic population with the same over- One way to do this is to consider the change per gener-
all size (Maruyama, 1970b; Slatkin, 1981). In particular ationintheprobabilitythattwoindividualssampledfrom
this means that no matter where mutants are initially po- the population are identical by descent, i.e., share a com-
sitioned, they fix with equal probability. For this state of monancestor(Whitlock andBarton,1997;Rousset,2004).
affairstochange,oneeitherhaseitherhastointroducead- Let us recapitulate here the derivation of the formula for
ditional processes—such as extinction and recolonisation asymptotic effective size given by Rousset (2004) for the
(Barton, 1993)—or relax the assumption of conservative modelofpopulationsubdivisionthatwillbeusedthrough-
migration (Liebermanetal., 2005) in which case certain outthiswork.ThismodelcomprisesLdemes,eachhaving
geographicalstructures can result in an allele with even a afixedsubpopulationsizeNi.After agenerationofrepro-
smallselectiveadvantagefixingwithnearcertainty.Whilst duction and migration, the probability that an individual
thedependenceoffixationprobabilityontheinitialmutant indemeiisanoffspringofaparentindemej isµij,hence
distribution can be established rather easily (see below), jµij =1 foralli.Giventhe setofprobabilitiesFij that
the corresponding variation of fixation time seems harder two individuals, one sampled fromdeme i, one from deme
P
to obtain and progress in this direction forms the bulk of j,shareacommonancestor,onefindsonegenerationlater
the presentwork. thatthe correspondingsetofprobabilitiesaregivenby
Acontinuedstudyoftheliteraturerevealsthatthecon-
1−F
cept of effective population size—which again goes back F′ = µ µ F + kkδ . (2)
ij ik jℓ kℓ N k,ℓ
to Wright (1931)—has proved useful in characterising the k,ℓ (cid:18) k (cid:19)
X
overallfixationtimescale.Essentially,theideaisthatmany
properties of neutral genetic drift in a subdivided popu-
termsofanumberofdiploidorganisms,inwhichcasethenumberof
lation comprising in total N instances of a gene1 are the
genesNistwicethenumberoforganisms.Sinceploidyisirrelevantin
theculturalevolutionaryapplication,weshallsuppressthesefactors
1 It is traditional in the population genetics literature to talk in oftwo.
2
This formarisesbecauseifthe parentsofthe twosampled proach infinity, but one has lim µ N < ∞. There
Ni→∞ ij i
individuals came from different demes k and ℓ, they are are several reasons for this. Firstly, this is the limit that
identical by descent with probability F , whereas if the naturallyaroseinthemodeloflanguagechangepreviously
kℓ
parents arefromthe same deme k they havea probability described (Baxteretal., 2006). In a population genetics
1/N of being the same individual (and hence genetically context this limit is also relevant when the subpopulation
k
identicalintheabsenceofmutation)whereasotherwisethe sizesarelargeandoneis interestedintheevolutionovera
probabilitythey areidenticalis (1−1/N )F . numberofgenerationsthatisoftheorderofthesesubpop-
k kk
To simplify this expression,we introduce the important ulation sizes. Finally, it is a limit in which any deviation
quantityQ∗whichisdefinedasfollows.Considersomesub- from panmictic behaviour—such as sensitivity to an ini-
i
set of the present-day population, and trace its ancestry tialdistributionofmutants—wouldbemostlikelytoarise,
backwardsintimeuntilasinglecommonancestorisfound. andthus worthexploringfroma more theoreticalpoint of
Now,ifonekeepstracingbackwardsintime,this ancestor view (see,e.g.,Slatkin, 1981,for adiscussionofthe differ-
will hop from deme to deme and eventually the probabil- entphenomenologyin the fastandslow migrationlimits).
itythatitistobe foundindemeiapproachesthestation- We remark that this is not the only slow migration limit
ary value Q∗. Since a hop from deme i to j occurs with a that can be taken: one can also take µ ∼ µ → 0 at fixed
i ij
probability µ per generation,these stationary probabili- populationsizesandmeasuretimeinunitsof1/µ(see,e.g.,
ij
ties satisfythe equation Wakeley, 2004). For clarity, we reiterate that throughout
most of this work, we deal with a slow migration limit in
Q∗µ =Q∗ . (3)
i ij j whichdemepopulationsizesN areinfinite,andmigration
i
i
X rates are inversely proportional to this size. Furthermore,
Multiplying both sides of (2) by Q∗Q∗ and summing over
i j forreasonstobediscussedinthenextsection,wewillmost
alliandj,onethen finds that
often be dealing with the extreme case where the coeffi-
L cientsarevanishinglysmall.
1−F
∆F˜ ≡F˜′−F˜ = (Q∗i)2 N ii , (4) Inthislimit,itisunclearwhetheronecansimplyusethe
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i=1 expression(5)forthe effective sizethatappearsinEq.(1)
X
where F˜ = Q∗Q∗F . As t → ∞, the probability that forthefixationtime.Tracingtheancestryofastateoffixa-
ij i j ij
anypairofindividualsshareacommonancestorapproaches tionbackwardsintime,theextremeslowmigrationregime
unityunderPneutralevolution.Asthislimitofinfinitetime couldleadto asituationinwhichthe majorityofthe rele-
isapproached,onecanunambiguouslydefineanasymptotic vantcoalesenceeventsoccurlongbeforethetimeatwhich
effective sizeofthe subdivided populationthrough themeancoalescencerategivenbytheright-handsizeof(5)
isreached.Wealsoremarkthatithasalsobeenarguedthat
1 = lim ∆F˜ = lim Li=1 N1i(Q∗i)2(1−Fii) (5) only when lineagesare able to equilibrate througha rapid
Ne t→∞1−F˜ t→∞ PLi=1 Lj=1Q∗iQ∗j(1−Fij) migrationprocessbetweencoalescentevents,isitappropri-
atetocharacterisethecoalescenceratesbyasingleeffective
wheretisthenumberofgenPeratioPnsthathaveelapsedsince
population size (NordborgandKrone, 2002; Sjo¨dinetal.,
imposing someinitial condition.One wayto interpretthis
2005).Thisfactisparticularlyrelevantwhenconsideringa
equation is as an average of coalescence rates 1/N of two
i
finalstate offixation,since one musttrace the ancestryof
lineagespresentinthesamedemeiinthelimitt→∞given
theentirepopulation.Finally,whetherornotthe effective
thatthey havenotpreviouslycoalesced(Rousset,2004).
populationsizeturnsouttoprovidegoodestimatesoffixa-
If migration is a fast process, one can envisage that the
tiontimesintheslowmigrationlimit,itdoesnotprovidea
identity probabilityF is roughlythe sameforallpairsof
ij
frameworkforunderstandinghowfixationtimesvarywith
demesiandj.Undersuchconditions,theformulasimplifies
the initialdistributionofmutants.
to
1 L (Q∗)2 To assess the utility of the asymptotic effective popula-
= i . (6) tionsizeincharacterisingfixationtimescales,andtoinves-
N N
e i=1 i tigatetheeffectofvaryingtheinitialcondition,wedevelop
X
This formula was obtained by an independent means by a formalism within which the spatial distribution of mu-
Nagylaki(1980),andwasshowntobe validinthe limitof tantsasafunctionoftimecan(inprinciple,atleast)becal-
infinitedemesizeswhenmigrationratespergenerationsat- culatedexactlygivenanyinitialdistributionandmodelof
isfy lim µ N =∞,anexpressionthatdefinesafast a subdivided population undergoing neutral genetic drift.
Ni→∞ ij i
migration limit (see Wakeley, 2005, for a fuller discussion Specialisingtothecaseofafinalstateoffixation,wecanex-
ofthe variouslimitsthatareofinterest).Forintermediate tractviaEq.(1)ameasureofeffectivepopulationsizeand
migration rates, Whitlock andBarton (1997) presented a comparewithitsasymptoticcounterpartusingEq.(5).As
formulainwhichthedenominatorof(5)isreplacedby1−F¯ hasbeenhintedabove,theapproachweuseisbasedonthe
whereF¯istheidentityprobabilityaverageduniformlyover backward time coalescent process (Kingman, 1982) that
allpairsofindividuals inthe population. hasfoundmanyapplicationsinthecontextofdriftinsub-
In this work we are almost exclusively interested in the divided populations (see, e.g., Notohara, 1990; Takahata,
slow migration limit, where deme sizes again again ap- 1991;Nei andTakahata,1993;Donnelly andTavar´e,1995;
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Wakeley, 1998; Wilkinson-Herbots, 1998; Notohara, 2001; tant fixing with near certainty. The distinction is that in
Wakeley,2001;Charlesworthetal.,2003;WakeleyandLessard,that work, the mutant was considered to have a selective
2006). The main benefit of this approach is that lineages advantage and thus that the initial location played only a
that do not contribute to the final state of fixation are minorroleintheprobabilityoffixation.Underneutralge-
explicitly excluded from the mathematical description. neticdrift,thesituationisreversed:thesamemutationcan
However,a fixed initial condition does not enter naturally fix or go extinct with near certainty depending on where
in this description, and so some work is needed to match itoccurs.ThiswillbedemonstratedexplicitlyinSection5
thisupwiththedistributionofancestorsofthepresentday below.
stateoffixation.Aswehavealreadyremarked,population Before this, in Section 3 we derive from the exact for-
subdivisionishardertohandleindiffusion-equation-based mulæ presented in Section 2 an expression that holds for
approaches, although the initial condition is more easily arbitrary population subdivision and gives the mean fixa-
enforced and extensions to alleles with a selective ad- tion time from an initial condition in which mutants are
vantage can be treated more straightforwardly (see, e.g., randomly distributed with an overall frequency x. This
Maruyama, 1970b; Slatkin, 1981; Barton, 1993; Whitlock, would be the appropriate initial condition to use, for ex-
2002;CherryandWakeley,2003;Roze andRousset,2003; ample,whenmodellingahistoricalsituationforwhichini-
WakeleyandTakahashi,2004). tialmutantfrequencies,butnottheirlocation,areknown.
In the next section we will present the details of the We learn that the resulting expression involves the mean
derivation of the exact formula relating forward- and times to particular coalescence events from the final state
backward-timepropertiestooneanother,anddemonstrate of fixation, which need then to be obtained either analyt-
thesimplificationsthatoccurinthelimitofextemelyslow ically or by Monte Carlo sampling. One case in which the
migration. The rest of this work is then devoted to conse- former is possible is Wright’s island model (Wright, 1931;
quences of this formula under different models of subdivi- Maruyama,1970a;Latter,1973);furthermore,theformula
sion. It is worth at this early stage to fix the basic ideas can be extended to an arbitrary initial condition. There-
underlying the approachby considering the simple caseof fore,inSection4weareabletoperformanumberofexact
fixationprobabilityasafunctionofinitialcondition.Con- calculations for this very well-studied model, apparently
sider therefore the probability that a mutant allele fixes for the first time. For example, we can consider in addi-
when initially a fraction χ of the individuals in deme i tiontotheinitialconditionthathasmutantsinitiallyran-
i
are mutants. We denote this probability P∗(A), where at domlyscatteredacrossallislandswithsomeoverallmutant
this stage A is a shorthand for the initial condition (we frequency x the contrasting case where the same number
willdefineitmoreformallybelow).Ifwelookinfinitelyfar of mutants are all initially confined to the smallest possi-
forwardintimefromthisinitialstate,fixationofeitherthe ble number of islands. In genetics terms, these two differ-
mutant or the wild type is guaranteed to have occurred. entinitialconditionscorrespondtothetwoextremevalues
Then rewinding infinitely far from the final state back to of the inbreeding coefficient F = 0 and 1 respectively.
ST
the initial state, we find a single ancestor in deme i with For the island model, we show that the difference in fix-
probability Q∗ as previously discussed. Since the initial ation time for these two initial conditions is short on the
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assignment of mutants to individuals in the population is timescale of fixation which grows linearly with the num-
completelyindependentoftheensuingpopulationdynam- ber of demes. We further provide evidence that fixation
ics, this ancestor is a mutant with probability χ . Hence, fromanynon-randominitialconditionis slowerthanfrom
i
the fixationprobabilityis simply a random initial condition. For more general models, ex-
act calculations are much more difficult. Nevertheless, we
L
introduce a formula for fixation time from a single muta-
P∗(A)= χ Q∗ . (7)
i i tion with known location (a case of practical interest) to
i=1
X the statistics of the most recent common ancestor of the
Thustofindthefixationprobabilityfromanyinitialcondi- whole population. This will be explained in Section 5. By
tion,itis enoughto knowthe distributionQ∗ whichisthe combininganalyticalresultswithnumericaldatafromsim-
i
stationarydistributionoftheMarkovianmigrationprocess ulationsweexplorefixationtimes,andtheirvariationwith
running backwards in time. Since finding this stationary initial condition, in two concrete and contrasting models
distribution is a fairly standard problem in the theory of inwhichtheredemesfallintoclusterswithinwhichmigra-
stochasticprocesses,weshallnotconsideritinfurtherde- tionoccursatdifferentratesthanbetweendemesfromdif-
tail here, other than to remark that through variation in ferent clusters. Our interest here is in how fixation times
the migration rates, it is possible to construct models in grow with increasing number of demes, either by adding
whichQ∗forsomedemeiisclosetoone.Ifselectivelyneu- more demes to each cluster, or by adding clusters of fixed
i
tral mutants initially completely occupy that deme they size.Whilsttheformercasehasbeenconsideredinanum-
arethenalmostguaranteedtospreadtothewholepopula- berofworks(suchasWakeley,2001;WakeleyandAliacar,
tion.This is similarinspiritto the phenomenondiscussed 2001; WakeleyandLessard, 2006), we obtain in this work
by Liebermanetal. (2005), in which it was seen that the someevidencethatinthelattercase,themeantimetofix-
spatialstructureofthepopulationcouldgiverisetoamu- ation can remain finite in the limit of an infinite number
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of demes. This effect—and the general growth law for the N1 N2 NL bi
··· di P(B|A;t)=
fithxeateiffoencttiivmeesiizneafosrumbudliav(id5e)dinptohpeusllaotwionm—igirsatpiroendliicmteidt.bIny bX1=d1b2X=d2 bLX=dLYi (cid:0)Ndii(cid:1)
all cases other than that in which fixation in a finite time a1 a2 (cid:0) (cid:1)aL ai
··· ci Q(C|D;t). (8)
ipsresedeicnt,etdhebyco(e5ffi).ciWenetdoifstchuessgproowsstihblleaworiisgianlssofoarcctuhreatdeilsy- cX1=0cX2=0 cXL=0Yi (cid:0)Ncii(cid:1)
crepancythatremains,alongwithimplicationsofourfind- Inthiswork,wewillfindQ(C|D;t),(cid:0)or(cid:1)quantitiesderived
ingsformoreplausiblemodelsofsocialinteractionandmi- fromit,eitheranalyticallyornumerically,andusethisdu-
gration between biological populations in the conclusion, ality relation to derive new formulæ relating to fixation.
Section6. Thisrequiresustotorearrangetheimplicitexpression(8)
sothattheforward-timeprobabilityP (thatrelatestofix-
ation statistics) is given explicitly in terms of Q. This is
achievedby makinguse ofthe identity
N N−i k k
2. Fixationprobabilityas afunctionoftime k k−i
(−1)i+j i j−i j =(−1)i (−1)j
N i j−i
j (cid:0) (cid:1)(cid:0) j (cid:1)(cid:0) (cid:1) (cid:18) (cid:19)j=i (cid:18) (cid:19)
X X
We begin by deriving an expressionfor the distribution
(cid:0) (cid:1) =δi,k , (9)
ofmutantsthatisvalidforanymodelofpopulationsubdi-
visionandarbitraryinitialcondition.Wedohoweverstip- inwhichδ isthe Kroneckerdelta symbol.Here,the first
i,k
ulatethatunderthepopulationdynamicsthatanindivid- step is achieved by expanding the binomial coefficients in
ual’schancesofleavingoffspringinthenextgenerationare terms of factorials,making some cancellationsand recom-
unaffected by whether it is a mutant or not (i.e., there is bining remaining factorials as binomial coefficients. Then,
no selection), and that subpopulation sizes and migration if k < i, the sum is zero by definition; if k > i, the alter-
ratesareconstantintime. nating binomial coefficients sum to zero (see e.g., formula
As stated in the introduction, we consider a population (0.15.4)inGradshteynandRyzhik,2000);ifk =ionecan
thatissubdividedintoLdemeswithN individualsindeme easilyverifythatthattheresultingexpressionequalsunity,
i
i.Theinitialcondition,denotedA,isspecifiedbythenum- asrequired.Multiplying bothsides of(8)by
ber of mutants a in each deme, i = 1,2,...,L. Two dis-
tinctdistributionisarecentraltothiswork.Thefirstisthe (−1)b′i+di Ni N −b′i
b′ d −b′
probabilityP(B|A;t)thataftertgenerationsofreproduc- Yi Xdi (cid:18) i(cid:19)(cid:18) i i(cid:19)
tionwithin andmigrationbetweendemes,alldescendants andsumming overallb andd one finds
i i
of A form the set B, i.e., a distribution with precisely b
i
mutantsindemei.ThesecondistheprobabilityQ(C|D;t) a1 a2 aL N1 N2 NL
P(B|A;t)= ··· ···
thatallancestorsofapresent-daysampleD formedtgen-
erationspreviouslythe setC.The quantitiesdi andci are cX1=0cX2=0 cXL=0dX1=b1dX2=b2 dLX=bL
definedanalogouslytoai andbi forthese sets. (−1)bi+di Nbii Ndii−−bbii acii Q(C|D;t) (10)
tioTnhtehsaettwwaosdgiisvternibbuytiMono¨shalere(2c0o0n1n)efcotredthbeypaandmuaiclittiyccraeslae-, Yi (cid:0) (cid:1)(cid:0) Ncii (cid:1)(cid:0) (cid:1)
L = 1. The generalisation to L > 1 is found by recalling after dropping the prime on the(cid:0)va(cid:1)riables bi that remain.
For a final state of fixation, b = N and this expression
that—in the absence of selection—the assignment of mu- i i
simplifies to
tant alleles to individuals within the population is a pro-
cess independent of the populationdynamics. We proceed a1 a2 aL ai
P(A;t)= ··· ci Q(C;t), (11)
fboyrcQo—nsftorurctthiengpreoqbuaivbaillietnytoefxtphreeesvsieonntst—haotnaefsoertPofainnddivoinde- cX1=0cX2=0 cXL=0Yi (cid:0)Ncii(cid:1)
uals A contains all ancestors of some set of individuals D in which we have introduced the no(cid:0)tat(cid:1)ion P(A;t) for the
present in the population t generations later. To be clear, probability of fixation from an initial distribution of mu-
the set A must contain at least the ancestors of individu- tantsA,andQ(C;t)forthedistributionofancestorsofthe
als in D, and may additionally contain ancestors of indi- entirepopulationatimetpreviously.Althoughwearecon-
vidualsoutsideD,orindividualsthathavenodescendants cernedonlywithfixationinthiswork,weremarkthat(10)
at the later time; likewise, descendants of A may form a haswiderapplicabilityandcouldformthe basisoffurther
superset of the individuals in D. Therefore,we obtain the studiesofgeneticdriftinasubdividedpopulation.Wealso
desired probability by summing P(B|A;t) over all sets of note—althoughwehavenouseforithere—thatitispossi-
individuals B that contain D; meanwhile Q(C|D;t) must bleto writedownasimilarexplicitformulaforQ interms
be summedoverallsetsC thatarecontainedwithinA.In ofP by makinguse ofthe identity (9).
both cases, the sum must be weighted by the probability In the remainder of this work we specialise to the slow
thatonerandomlychosensetiscontainedwithintheother. migration limit for the reasons outlined in the introduc-
The expressionthatresults is tion. In particular we know (Nagylaki, 1980) that if µ is
ij
5
theprobabilitythatanindividualsampledatrandomfrom 1/m, and it is to be understood that any times T quoted
demeihadaparentfromthe previousgenerationindeme inthe followingarebelievedto havethe propertythat the
j, the overallpopulation behaves like a panmictic popula- combinationmT isexactinthelimitofextremelyslowmi-
tionwithaneffective sizegivenby Eq.(6)ifone has gration,m→0.
lim N¯µ =∞ (12) Weremarkthattheformulafortheultimateprobability
ij
N¯→∞ offixation,Eq.(7),thenfollowsbytakingthelimitt→∞
whereN¯ isthemeandemesize.Sincewearemostinterested (which is indicated by the asterisks on P and Q). In this
limit Q(c ,...,c ;t) is zero for any state with more than
inthosecaseswheredeviationfrompanmixiaismostlikely, 1 L
oneancestor;andconvergestoQ∗forthestatewithasingle
andgenetic diversityis maintainedforthe longestperiods i
lineagein deme i.Since lineageshopfromdeme i to deme
oftime before the onsetoffixation,weshallinsistthat
j goingbackwardintime, this distribution is givenby the
lim N¯µij =mij <∞ (13) solutionofthebalanceequations(3)recastintermsofthe
N¯→∞
parameterm ,viz,
ij
holds for all i and j. In this limit it is convenient to work
in rescaledtime units, so that that one unit of time corre- Q∗m = Q∗m ,i=1,2,...,L. (15)
i ij j ji
sponds to N¯ generationsof the underlying population dy-
j6=i j6=i
X X
namics.Theserescaledtime unitswillbeinforceuntilthe
Itisassumed—asiscustomary(Nagylaki,1980;Whitlock andBarton,
endofthis work.
1997)—that the stationary solution is unique. One way
Intheserescaledtimeunits,thebackwards-timehopping
thatthiscanbeassuredisifitispossibleforeverydemeto
oflineagesbecomesinthelimitN¯ →∞acontinuous-time
be reached from any other through some sequence of mi-
process in which a hop from deme i to j occurs with rate
grationevents,asiswellknownfromthetheoryofMarkov
m . Meanwhile, pairs of lineages collocated in deme i co-
ij chains(see e.g.,GrimmettandStirzaker,2001).
alesceatarateN¯/N¯ whichisassumedtoremainfinite in
i
thelimitN¯ →∞.Thefactthatbothoftheseprocessesoc-
3. Fixationfrom arandom initialcondition
cur on the same timescale makes calculation of Q(C;t) in
generaldifficult,sincecoalescenceandmigrationeventsare
intermingled.Tosimplifythecalculations,weshallfurther Ourfirstapplicationofthe formula(14)concernsaran-
assumethatallthemigrationratesmij areproportionalto dominitialconditionAxinwhichafractionxofallindivid-
some vanishingly small parameter m. This defines an ex- uals are mutants, but are spatially distributed uniformly
tremeslowmigrationlimitthatweshallhenceforthassume atrandom.Theprobabilityofhavinga ,a ,...mutantsin
1 2
is in operation. Then, we expect that the probability that demes1,2,...is thengivenby
a migration event occurs in the time it takes all lineages
L
1 N
thatarepresentinasingledeme tocoalescevanisheswith i .
m, even in the limit of an infinite system. This expecta- xLLN¯N¯ Yi=1(cid:18)ai(cid:19)
tion is based on the fact that the fixation time in an ideal
Theprobabilityoffix(cid:0)ation(cid:1)bytimetfromthisrandomcon-
populationisoforderone(inthe rescaledtimeunits),one
dition is obtained by summing the right-hand side of (14)
can choose m sufficiently small that the probability that overallinitialconditionsAthathave|A|≡ a =xLN¯,
all lineages coalesce before any of them migrate elsewhere i i
weighted by the previous expression. In this sum one en-
with arbitrarily high probability. We note that for large, P
countersthe combination
butfinite,populationsthisassumptionhasbeenusedprevi-
ouslybyTakahata(1991)andformallyprovedbyNotohara L ai ci Ni
(2001); furthermore, Griffiths (1984) gives grounds to be- N a δ|A|,xLN¯ .
A i=1(cid:18) i(cid:19) (cid:18) i(cid:19)
lievethatinaninfinitepopulation,onlyafinitenumberof XY
Thiscanbeevaluatedbynotingthatitisthecoefficientof
lineagesremainatanynonzero(rescaled)time.Giventhat
zxLN¯ inthe powerseriesexpansionof
this is the case, the probability for there to be more than
oneancestorinanydeme atatime oforder1/mvanishes, L 1 ∂ ci L
andhence allthe sums overc in (11)canbe truncatedat (1+zy )Ni = zci(1+zy )Ni−ci (16)
i i i
N ∂y
ci =1.Thisyieldsforthefixationprobabilityasafunction Yi=1(cid:20) i i(cid:21) Yi=1
oftime the expression evaluated at y = y = ··· = y = 1. This reveals the
1 2 L
L 1 fixationprobabilityPr(x;t)fromarandominitialcondition
P(A;t)= χciQ(c ,...,c ;t) (14) tobe
i 1 L
iY=1cXi=0 LN¯− jcj
whichisthefundamentalequationthatwillbeusedrepeat- P(A ;t)= L 1 xLN¯−Pjcj Q(c ,...,c ;t). (17)
edly in this work. Here, χi = ai/Ni is the initial fraction x (cid:0) LN¯ (cid:1) 1 L
of the individuals in deme i that are mutants and by con- iY=1cXi=0 xLPN¯
vention we take χ0 = 1 whenever χ = 0. Any timescales Note now that the combi(cid:0)nator(cid:1)ial factor appearing in the
i i
calculatedfromthisexpressionwillthusbeproportionalto sumdependsonlyonthetotalnumbern= c ofances-
j j
P
6
torsoftheentirepopulationthatremainaftergoingback- conditioncanbefullyexplored,andcomparisonmadewith
wardsatimetfromthepresentday,andnottheirlocation. the result for a random initial condition via the result of
Since,forany finiten,wehave the previous Section. These new results provide more de-
LN¯−n tailed information about the island model in the slow mi-
lim xLN¯−n =xn , (18) gration limit than the formulæ for the mean and variance
N¯→∞(cid:0) xLLN¯N¯ (cid:1) ofcoalescencetimes providedbyTakahata(1991).
Inthismodelofpopulationsubdivision,migrationoccurs
wefindinthelimitofinfini(cid:0)tesub(cid:1)populationsizes(butfixed,
at a uniform rate between every pair of demes, and all
finite deme numberL)the simple expression
demeshavethesamesizeN =N.Inorderthatresultsfor
i
L
differentnumbersofdemescanbecompared,weinsistthat
P(A ;t)= xnQ(n;t) (19)
x meannumber ofindividuals replacedineachgenerationis
nX=1 independent of the number of demes L. We therefore set
forthefixationprobability,inwhichQ(n;t)istheprobabil- m = m/(L−1) in which m defines an overall timescale
ij
ity thatthe entire populationhad preciselyn ancestorsat asobservedinSection2.
atimetpriortothepresentday.ComparisonwithEq.(14) Analysis of the model is relatively straightforward be-
reveals that this random initial condition gives the same cause the statistics of the genealogies are invariant under
fixation probability as a homogeneous initial condition, in relabellingofthe demes.Therefore,the distributionofan-
which every deme contains a fractionχi = ai/Ni = x mu- cestors Q(C;t) depends only on the number of lineages n
tants.Wealsoseethatsincelimt→∞Q(n;t)=δn,1,theul- that are contained within C and not their location. The
timate fixationprobabilityP∗(Ax)=x:thatis,nomatter fixationprobability(14)canthus be writtenas
what the migration rates are, the fixation probability for
L
a randomly distributed set of mutants is always equal to
P(A;t)= Γ (χ ,...,χ )Q(n;t) (23)
n 1 L
their initialoverallfrequency.
n=1
Since the probability that fixation occurs in the time X
interval[t,t+dt]is dP(A;t)dt,themeantimetofixation, where Q(n;t) is as defined in the previous sectionand the
dt coefficientsΓ (χ ,...,χ )areproportionaltotheelemen-
averagedoverthoserealisationswherefixationdoesoccur, n 1 L
tarysymmetricpolynomialsinχ:
is
1 ∞ d 1
τ(A)= t P(A;t)dt. (20) Γ (χ ,...,χ )= χ χ ···χ .
P∗(A) dt n 1 L L i1 i2 in
Forthe caseofthe randomZin0itialcondition, n 1≤i1<i2X<···<in≤L
(24)
(cid:0) (cid:1)
τ(Ax)= Ln=1xn 0∞tddtQ(n;t)dt . (21) aTshceaunltbimesaeteenfibxyatcioomnpparoribnagbwiliittyhPE∗q(.A(7))=.UΓs1in(χg1t,h.i.s.i,nχtLh)e,
x
P R denominatorof(20),wefind the meantime tofixationis
Noting that in any realisation of the dynamics, the n-
ancestorstateisenteredorexitedexactlyonce(exceptfor L Γ (χ ,...,χ )
n 1 L
the n=1 state whichis never exited), this expressioncan τ(A)=T1+ (Tn−Tn−1). (25)
Γ (χ ,...,χ )
1 1 L
alsobe writtenas n=2
X
L NotethesimilaritywithEq.(22)giveninthepreviousSec-
τ(A )=T + xn−1(T −T ) (22) tionforthecaseofarandominitialcondition.Here,anex-
x 1 n n−1
nX=2 pressionintermsofentrytimesTnispossibleonlybecause
in which T is the mean time at which the state with n allstateswithnancestorsareequiprobableatagiventime
n
ancestorsis entered,goingbackwardsintime froma state inthe historyofthe dynamics.
with one lineage per deme (i.e., the whole population in The entry times themselves canbe calculatedeasily be-
the extreme slow migration limit). We reiterate that this cause,asinanidealpopulation,therateofcoalescencefrom
expression, believed not to have appeared before, is valid astatewithnlineagestoonewithn−1lineagesisaPoisson
foranysetofmigrationrates(atleast,intheextremeslow processthatoccurswitharateproportionalton(n−1).In
migrationlimit) andfurther remarkthatwhenthe coales- theslowmigrationlimit,theconstantofproportionalityis
cencetimesT cannotbeobtainedanalytically,theycanbe m/(L−1)(see, e.g., Takahata, 1991), whereas in anideal
n
easily estimated by Monte Carlo sampling of the genealo- population with n ≪ N, it is 1. The mean time spent in
2
gies. the n ancestorstate isthen simply
L−1
T −T = . (26)
4. FixationinWright’sislandmodel n−1 n mn(n−1)
As the initialconditioncomprisesn=L lineages,wehave
Amodelforwhichfixationtimescanbe calculatedana- thatT =0,andtherefore
L
lytically for any initial condition is Wright’s island model
L
L−1 1
(Wright,1931;Maruyama,1970a;Latter,1973).Thispro-
T = (T −T )= 1− . (27)
1 n−1 n
vides one casewhere variationoffixationtime with initial m L
n=2 (cid:18) (cid:19)
X
7
The mean time to fixation from a random initial condi- deme, since this remains finite in the limit N¯ → ∞. This
tionA isthenobtainedbyusingtheseexpressionsin(22). correspondence between fixation times can be understood
x
We find by the fact that, as described above, the coalescence pro-
cess at the level of demes in the slow migration limit is
L
τ(A )= L−1 1− 1 + xn−1 1 − 1 (28) precisely the same as that for an ideal population, albeit
x
m "(cid:18) L(cid:19) n=2 (cid:18)n n−1(cid:19)# on a longer timescale. It is perhaps interesting to observe
X that whilst (1) was obtained using forward-time diffusion
L−1 L−1xn xL−x
equations,thesenewresultshavebeendeterminedentirely
= (1−x) + . (29)
mx " n L # within the backward-time coalescent formalism thus pro-
n=1
X
vidinganexplicitdemonstrationoftheequivalenceofthese
Whenthe numberofislandsLis large,wecanexpandthe
twocomplementaryapproaches.
terminsquarebracketsasaseriesin1/Ltoobtain
For large, but finite L, the difference between the fixa-
L−1 (1−x)ln(1−x) 1 tion time for the inhomogeneous and homogeneous initial
τ(Ax)∼ m − x − L +O(L−2) , conditions(A˜x andAx respectively)is
(cid:20) (cid:21)
(30) 1
τ(A˜ )−τ(A )∼ +O(L−1). (35)
whichcanbe comparedto (1),takinginto accountthatin x x
2m
thisexpressiontimeismeasuredinunitsofN¯ generations,
Asthisdifferenceissmallonthetimescaleoffixation,which
whereas(1)measurestime ingenerationsalone.
grows linearly with the number of demes L, we suggest
Since, as previously described, the random initial con-
thataninhomogeneousinitialconditionrelaxesquicklyto
dition is equivalent to a homogeneous initial condition
a homogeneous state. This state, which has a frequency
where a fraction x of the individuals in each deme are
x of mutants in each deme, would then persist for a time
mutants, the most interesting comparisonis with a highly
proportionaltoL.This accordswithwhatoneobservesin
inhomogeneous initial condition A˜ that has the same
x a Monte Carlo simulation of the forward-time population
number of mutants. This is attained by having a fraction
dynamics(Baxteretal.,2006).
x of the demes containing only mutants, and the rest
We conclude this Section by arguing that the homoge-
containing only the wild type. For such a distribution,
neous(orrandom)initialconditiongivestheshortestfixa-
Γ (χ ,...,χ ) = Lx / L . We then find that the mean
n 1 L n n tion time of all possible distributions that have an overall
time to fixationis
(cid:0) (cid:1) (cid:0) (cid:1) fractionx ofmutantsin the population.To showthis,one
τ(A˜x)= Lm−1"(cid:18)1− L1(cid:19)+ x1 nL=x2(cid:0)LLnnx(cid:1)(cid:18)n1 − n−1 1(cid:19)# . ntienegdsχLfir=stxtoLi−mposiL=e−11thχei.cTonhsetnr,aionntePfinidχsi t=heLexxtbreymsuemt-
X of τ(A) by setting all the derivatives of the right-hand
(31)
(cid:0) (cid:1) side of (25) withPrespect to the independent parameters
Tosimplifythissum,itisconvenienttoreplacethequantity
inthefinalsetofroundbracketswith(1− 1)−( 1 − 1). χ1,χ2,...,χL−1 to zero. It is a straightforward, but te-
n L n−1 L dious, exercise to show that ∂ Γ (χ ,...,χ ) is propor-
Collectingterms,oneeventuallyfindsaftersomerearrange- ∂χi n 1 L
tional to χ −χ with a non-negative constant of propor-
mentthat L i
tionality, except for the case n = 1 where the derivative
L−1 Lx Lx Lx 1 1 vanishesdue tothe constraint.Allderivativesof(25)thus
τ(A˜ )= n − n+1 − (32)
x mx nX=1"(cid:0)Ln(cid:1) (cid:0)nL+1(cid:1)#(cid:18)n L(cid:19) svpaonnisdhstwohtehnehalolmtohgeenfreaocutisoinnsitiχailcaornedeitqiuoanl.,Awlthhiochugchotrhreis-
(L−1)(1−(cid:0)x)(cid:1)Lx (cid:0)Lx (cid:1)1 demonstrates only that this is an extremal fixation time,
= n (33)
mx L n thefactthatthefixationoccursmoreslowlyfromaninho-
n=1(cid:0)n(cid:1)
X mogeneousinitialcondition(atleastforlargeL)issugges-
in which the second line wa(cid:0)s (cid:1)obtained from the first by tive that the extremum is a minimum, particularly since
expanding out the binomial coefficients and further rear- theanalysisjustoutlinedalsoshowsthatthepointχ =x
i
rangement.ForlargeL,this expressioncanbe writtenas (foralli)istheonlyextremumintheinteriorofthe(L−1)-
dimensionalspaceofindependent parametersχ .
L−1 (1−x)ln(1−x) 1 i
τ(A˜ )∼ − − +O(L−2) .
x
m x 2L
(cid:20) (cid:21)(34) 5. Fixationfrom the firstmutationevent
WeseethatbothinitialconditionsyieldinthelimitL→
∞thesamefunctionalformasthe classicresult(1)foran For models of migration that have more structure than
ideal population, up to a change of timescale. Taking into Wright’sislandmodel,calculationofthefixationtimefrom
accountthat(1)givesthefixationtimeintermsofanum- an arbitrary initial condition is much more difficult. We
ber of generations, we find the effective population size of thusspecialisetothecomparativelysimplecaseofaninitial
asinglepanmicticpopulationtobeN =N¯(L−1)/2m;in conditionthathasasinglemutantintheentirepopulation.
e
whatfollowsa usefulmeasurewillbe the effective popula- Our approachis to assume that certainstatistical proper-
tion size α = N /N¯ relative to the mean size of a single ties of the most recent common ancestor (MRCA) of the
e e
8
whole population are known, e.g., from an explicit calcu- ToattacktheremainingintegralwediagonalisetheL×L
lation or Monte Carlo sampling. Then, starting from (20) matrixM thathaselements
wederiveanewformulaforthefixationtimefromasingle
mutationasafunctionofitslocation.Wefirstpresentthis mij i6=j
derivation,andthenillustrateitsimplicationsthroughtwo [M]ij =− m i=j (42)
ij
explicitandcontrastingmodels ofpopulationsubdivision. 
i6=j
X
5.1. Relation between MRCAstatistics andfixation time and is the generator ofthe Markov process the describes
the backward-time hopping in the single-ancestor state.
This matrix has L eigenvalues, one of which λ is zero
If we have an initial condition A in which a fraction χ 1
i and correspondsto the stationary state: the left and right
of the individuals in deme i are mutants, andall others in
eigenvectors have elements [u ] = Q∗ and [v ] = 1 re-
the populationarethewildtype,wehave,forarbitraryχ, 1 i i 1 i
spectively.Theremainingeigenvaluesλ ,...,λ allhavea
the meanfixationtime 2 L
strictly negative real part, since the stationary state is by
τ ≡τ(A )= 0∞tddtQi(t)dt , (36) assumptionunique. The left andrighteigenvectorssatisfy
i i Q∗ the biorthogonalityrelationu ·v =δ .
R i n m n,m
The probabilitydensityQ (t) canthenbe writtenas
inwhichQ (t)istheprobabilitythat,atimetpriortothe ji
i
presentday,theentirepopulationhasasingleancestorthat
L
islocatedindemei.TomakeaconnectionwiththeMRCA, Q (t)=[eMt] =Q∗+ eλnt[u ] [v ] (43)
ji ji i n i n j
weintroducethreequantities:first,theprobabilitydensity
n=2
X
r (t)forthesingleancestorstatetobeenteredindemejat
j
timet;second,theintegralofthisquantityR∗ = ∞r (t)dt inwhichthestationarysolutionhasbeenseparatedoutfor
j 0 j clarity.We cannowevaluate the integralin(41),
whichgivesthetotalprobabilitythattheMRCAisindeme
R
j; and finally Q (∆t) for the single ancestor of the whole
ji ∞ d L [u ] [v ]
population to be in deme i a time ∆t after it was in deme t Q (t)dt= n i n j , (44)
ji
dt λ
j goingbackwardsintime.Withthesedefinitions,wethen Z0 n=2 n
X
havethat
whichallowsusfinallytowritedownanexpressionforthe
L t
fixationtime τ originallygivenby (36).It reads
Q (t)= dt′r (t′)Q (t−t′). (37) i
i j ji
Xj=1Z0 1 L [u ]
τ =T + n i (R∗·v ) (45)
The numeratorof(36)isthen i 1 Q∗ λ n
i n=2 n
X
∞ d L ∞ whereR∗isthe(row)vectorofprobabilitiesforthelocation
t Q (t)dt= t r (t)Q (0)+
i j ji
Z0 dt j=1Z0 (cid:20) of the MRCA. Thus the problem of calculating the mean
X time to fixation from the first mutation event in deme i
t d
dt′r (t′) Q (t−t′) dt. (38) is reduced to that of determining the mean time to the
j ji
dt
Z0 (cid:21) MRCA,T1,its spatialdistributionRi∗ andthe eigenvalues
The double integralinthis expressioncanbe writtenas andeigenvectorsoftheL×LmatrixofmigrationratesM.
Before applying this formula to concrete models, we re-
∞ ∞ d
dt′ dt(t′+t)r (t′) Q (j;t)= mark that shouldthe MRCA distribution R∗ and station-
j dt ji i
ZZ00∞dtZ′t0′rj(t′) Q∗i −Qji(0) +Rj∗Z0∞tddtQji(t)dt. scacarayulsaderistphtrreionbduRutci∗otniRsQt∗h∗i·evczoneirnvocailndeifest,heτeiisge=bnyvTet1chtefoorbrioaoflrlMthi,.ouTg0oh,niasanliidstytbhoee-f
(cid:2) (cid:3) (39) the eigenvectors. One situation where this occurs is when
Two simplifications now occur: first, the term containing any pair of demes can be exchanged without affecting the
Qji(0)cancelsthatin(38);second,we havethat dynamics, as in Wright’s island model: then both Ri∗ and
Q∗ areuniform.
i
L ∞ Ingeneral,the differencebetweenτ andT willbenon-
i 1
tr (t)dt=T , (40)
j 1 zero. Furthermore, by summing (45) weighted by Q∗, or
j=1Z0 i
X takingthelimitx→0inEq.(22),themeantimetofixation
since the total probability density for the MRCA to be fromarandomlylocatedmutationisequaltoT .Hence,τ
1 i
found at time t is the sum jrj(t). The expression (38) canbelargerorsmallerthanT1,aresultwhichatfirstsight
consequentlyreducesto seemscounterintuitive,butcanbeunderstoodfromthefact
P
that one averages over all realisations of the dynamics to
∞ d L ∞ d
t Q (t)dt=Q∗T + R∗ t Q (t)dt. (41) findthemeantimetotheMRCA,butonlyoverarestricted
dt i i 1 j dt ji
Z0 j=1 Z0 subsetinthe caseoffixation.
X
9
5.2. Two exampleapplications forthe effectivesize ofthe entire populationα relativeto
e
the meansubpopulationsizeN¯. Thus we needonly deter-
Wenowdeterminefixationtimesintwoconcretemodels minetheleadingcoeffecientsintheseriesexpansions.From
of population subdivision undergoing extremely slow mi- (47)we find linear equations satisfied by these coefficients
gration. The first is similar to Wright’s island model, in by againtaking the limit N¯ →∞ followedby m→0. For
that migration is permitted between every pair of demes. the casei6=j,we have
However,migration occurs at one of two rates, depending 1+ ν Y(0)+ ν Y(0)
onwhetherapairofdemesareconsideredtobelongtothe Y(0) = k6=i,j ik kj ℓ6=i,j jℓ iℓ (51)
ij ν + ν
same,ordifferent,clustersofdemes.Thesecondmodelhas P k6=i ik Pℓ6=j jℓ
afurtherrestriction,inthatmigrationbetweenclusterscan whicharepreciselythPeexpressioPnsobtainedifoneapprox-
onlyoccurifoneofthoseclustersisaspecialcentralcluster. imatesthecoalescenceprocessasonethatoccursatanin-
We will compare the data obtained with predictions for finite rate.Forthe casei=j,however,onefindsthe finite
the fixation time from (1) in the limit x → 0 and using result
the asymptoticeffectivesizegivenbyEq.(5).Ashasbeen
established (Slatkin, 1991) the limiting ratios of identity Yi(i0) =αi1+2 νijYi(j0) . (52)
probabilities appearing in (5) can be replaced by ratios of j6=i
X
meancoalescencetimesYijfortwoindividuals,onesampled Substituting this expressioninto (50)we findthe formula
fromclusteri andone fromj.Specifically,oneobtains
Q∗Q∗Y(0)
Ne = PLi=Li1=P1 NLj1=i(1QQ∗i∗i)Q2Y∗jiYiij . (46) fortherelaαtieve=effPecit(iQve∗i)Pp2o(cid:16)ip6=1uj+lat2iioPnjjs6=iizjieν.ijNYoi(tj0e)(cid:17)thatthis(5e3x)-
In the foregoing we haPve assumed that, on the timescale pression,validintheslowmigrationlimit,dependsonlyon
of migration, the rate of coalescence between pairs of lin-
fixationtimesbetweenpairsoflineagesindifferentdemes,
eages located in a single deme is infinitely fast. However,
andthedemesizesα donotenter.Finally,using(1)inthe
i
weclearlycannotsimplysetY =0inthepreviousexpres-
ii limitx→0,andreturningtorescaledtimeunits(i.e.,those
siontoobtainanestimateofthefixationtime;instead,we thathaveoneunitoftimecorrespondingtoN¯ generations
must take into account that coalescence occurs at a fast,
of the population dynamics), we arrive at an estimate of
but finite rate.
the fixationtime τ that behavesas
e
To this end, let use return to the original time units
2α
where there is a finite mean subpopulation size N¯, and τ ∼ e (54)
e
m
the population evolves in discrete time. We will take the
size of subpopulation i to be Ni = αiN¯, and migration in the limit m → 0, with αe given by (53) in terms of
probabilities µ = (mν )/N¯. We recall that the extreme the solutionsto (51).It is with this estimate that we shall
ij ij
slow migration limit (and a continuous-time dynamics) is comparedata forthe twodifferentconcretemodels.
reprisedby first taking N¯ →∞ and subsequently m → 0.
Themeancoalescencetimesarethengivenbythesolution 5.2.1. Two-level model
ofthe setoflinearequations Ourfirstconcretemodelhasℓequal-sizedclusters,n =
i
n. Every deme receives a fraction y of its migrants from
δ
k,ℓ
Yij =1+ µikµjℓ 1− Ykℓ . (47) demes within its own cluster, and the remaining fraction
N
k,ℓ (cid:18) k(cid:19) 1−y from demes in other clusters. So that this fraction
X
is well defined, we will insist that n ≥ 2 in all cases. We
We anticipate that the leading contribution to the co-
willalsohavethetotalrateofmigrationintodemei,m =
alescence times Y between pairs of lineages in the same i
ii
deme grows as N¯ in the limit N¯ → ∞ and m → 0, whilst jmijequaltothesmallparametermineachdeme.These
Y growsasN¯/minthislimit.Wecanthuswritetheexact considerationsspecifythe parametersνij appearinginthe
ij P
migrationratesm =mν as
solutions to (47)as powerseriesinthe (small)parameters ij ij
1/N¯ andm: y
ν = i,j fromsame cluster
s
n−1
Y =N¯ Y(0)+O(m)+O(1/N¯) (48) νij = 1−y . (55)
ii ii νd = n(ℓ−1) i,j fromdifferentclusters
N¯(cid:16) (cid:17)
Yij= m Yi(j0)+O(m)+O(1/N¯) fori6=j . (49) Weremarkthatthishierarchicalversionoftheislandmodel
has previously been considered by SlatkinandVoelm
(cid:16) (cid:17)
Substituting these expansions into (46), and taking the (1991). It has the special property that the dynamics are
limit N¯ →∞followedby m→0yields unaffected by exchanging any pair of demes, and so the
mN Q∗Q∗Y(0) distributionRi∗oftheMRCAisuniform;since j6=imij =
αe =mli→m0N¯l→im∞ N¯e = Pii6=(Qj ∗i)i2α1ijYii(ij0) (50) Paryj6=diimstjriibwuetioanlsoQh∗i.avTehferroemforEeqτ.i(d1o5)esanuontifdoPerpmensdtaotinoni-,
P
10