3 Effective population size

Transcription

1 36 3 EFFECTIVE POPULATION SIZE 3 Effctiv population siz In th first two chaptrs w hav dalt with idalizd populations. Th two main assumptions wr that th population has a constant siz and th population mats panmictically. Ths idal populations ar good to start with bcaus thy allow us to driv som important rsults. Howvr, natural populations ar usually not panmictic and th population siz may not b constant ovr tim. Nvrthlss, w can oftn still us th thory that w hav dvlopd. Th trick is that w dscrib a natural population as if it is an idal on by adjusting som paramtrs, in this cas th population siz. This is th ida of th ffctiv population siz which is th topic of this sction. Human Population Siz Exampl As an xampl, w will analys a datast from Hammr t al. 2004). Th datast, which may b found in th fil TNFSF5.nx, contains data from diffrnt human populations: Africans, Europans, Asians and Nativ Amricans. It contains 4 squncs from 4 mals, from on locus, th TNFSF5 locus. TNFSF5 is a gn and th squncs ar from th introns of this gn, so thr should b no constraint on ths squncs, in othr words vry mutation should b nutral. Th gn lis on th X-chromosom in a rgion with high rcombination. What that mans for th data will bcom clarr latr in th cours. Exrcis 3.. Import th data in DNASP and dtrmin θ π pr sit and θ S pr sit using all 4 squncs. As you hav sn in Sction 2, both θ π and θ S ar stimators of th modl paramtr θ = 4N µ whr µ is th probability that a sit mutats in on gnration. Howvr, th TNFSF5 locus is on th X-chromosom and for th X-chromosom mals ar haploid. Thrfor th population of X-chromosoms can b sn as a population of.5n haploids instad of 2N haploids for autosoms) and thrfor in this cas θ π and θ S ar stimators of 3Nµ. Th rason that Hammr t al. 2004) lookd at X-chromosoms is mainly bcaus th squncing is rlativly asy. Mals hav only on X-chromosom, so you don t hav to worry about polymorphism within on individual mor about polymorphism within an individual in Sction 4). Th mutations in ths data ar singl nuclotid polymorphisms. SNPs ar frquntly usd to dtrmin θ π and θ S pr sit. Thir pr sit) mutation rat is stimatd to b µ = by comparing human and chimpanz squncs. Exrcis 3.2. Rcall Sction. Assum that th divrgnc tim of chimpanzs and humans is 0MY with a gnration tim of 20 yars and th mutation rat is pr nuclotid pr gnration... What is th xpctd prcntag of sits that ar diffrnt or what is th divrgnc) btwn human and chimp?

2 3. Th concpt Both θ π and θ S ar stimators for 3Nµ and both can b dirctly computd from th data. What stimat of N do you gt, whn using th stimatd θ valus from th last xrcis? 3. Thr ar about popl on arth. Dos th human population mat panmictically? Is th population constant ovr tim? Can you xplain why your stimat of N is so diffrnt from 6 0 9? Th numbr of popl on arth is rfrrd to as th cnsus population siz. This sction is about a diffrnt notion of population siz which is calld th ffctiv population siz. 3. Th concpt Bfor w start with calculations using ffctiv population sizs w introduc what thy ar. W us th following philosophy: Lt b som masurabl quantity that rlats to th strngth of gntic drift in a population. This can b.g. th rat of loss of htrozygosity or th probability of idntity by dscnt. Assum that this quantity has bn masurd in a natural population. Thn th ffctiv siz N of this population is th siz of an idal nutral panmictic constant-siz quilibrium) Wright-Fishr population that givs ris to th sam valu of th masurd quantity. To b spcific, w call N th -ffctiv population siz. In othr words, th ffctiv siz of a natural population is th siz of th idal population such that som ky masur of gntic drift is idntical. With an appropriat choic of this masur w can thn us a modl basd on th idal population to mak prdictions about th natural on. Although a larg numbr of diffrnt concpts for an ffctiv population siz xist, thr ar two that ar most widly usd. Th idntity-by-dscnt or inbrding) ffctiv population siz On of th most basic consquncs of a finit population siz - and thus of gntic drift - is that thr is a finit probability for two randomly pickd individuals in th offspring gnration to hav a common ancstor in th parnt gnration. This is th probability of idntity by dscnt, which translats into th singl-gnration coalscnc probability of two lins p c, in th contxt of th coalscnt. For th idal Wright-Fishr modl with 2N haploid) individuals, w hav p c, = /2N. Knowing p c, in a natural population, w can thus dfin th idntity-by-dscnt ffctiv population siz N = 2p c,. 3.)

3 38 3 EFFECTIVE POPULATION SIZE W will s in th nxt chaptr that th dgr of inbrding is on of th factors that influncs N. For historic rasons, N is thrfor usually rfrrd to as inbrding ffctiv population siz. Sinc all coalscnt tims ar proportional to th invrs coalscnt probability, thy ar dirctly proportional to th inbrding ffctiv siz. On also says that N fixs th coalscnt tim scal. Th varianc ffctiv population siz Anothr ky aspct about gntic drift is that it lads to random variations in th alll frquncis among gnrations. Assum that p is th frquncy of an alll A in an idal Wright-Fishr population of siz 2N. In Sction 2, w hav sn that th numbr of A allls in th nxt gnration, 2Np, is binomially distributd with paramtrs 2N and p, and thrfor Var W F [p ] = 2N) 2 Var[2Np ] = p p) 2N. For a natural population whr th varianc in alll frquncis among gnrations is known, w can thrfor dfin th varianc ffctiv population siz as follows N v) = p p) 2Var[p ]. 3.2) As w will s blow, th inbrding and varianc ffctiv sizs ar oftn idntical or at last vry similar. Howvr, thr ar xcptions and thn th corrct choic of an ffctiv siz dpnds on th contxt and th qustions askd. Finally, thr ar also scnarios.g. changs in population siz ovr larg tim scals) whr no typ of ffctiv siz is satisfactory. W thn nd to abandon th most simpl idal modls and tak ths complications xplicitly into account. Loss of htrozygosity As an application of th ffctiv-population-siz concpt, lt us study th loss of htrozygosity in a population. Htrozygosity H can b dfind as th probability that two allls, takn at random from a population ar diffrnt at a random sit or locus). Suppos that th htrozygosity in a natural population in gnration 0 is H 0. W can ask, what is th xpctd htrozygosity in gnration t =, 2, 3, if w assum no nw mutation i.. w only considr th variation that is alrady prsnt in gnration 0). In particular, for t =, w find H = 2N 0 + ) H 0 = 2N ) H ) Indd, if w tak two random allls from th population in gnration, th probability that thy hav th sam parnt in gnration 0 is. Whn this is th cas thy hav probability 0 to b diffrnt at any sit bcaus w ignor nw mutations. With probability

4 3.2 Exampls 39 thy hav diffrnt parnts in gnration 0 and ths parnts hav by dfinition) probability H 0 to b diffrnt. By itrating this argumnt, w obtain H t = ) t H0 for th htrozygosity at tim t. This mans that, in th absnc of mutation, htrozygosity is lost at rat vry gnration and dpnds only on th inbrding ffctiv population siz. Estimating th ffctiv population siz For th Wright-Fishr modl, w hav sn in Sction 2 that th xpctd numbr of sgrgating sits S in a sampl is proportional to th mutation rat and th total xpctd lngth of th coalscnt tr, E[S] = µ[l]. Th tr-lngth L, in turn, is a simpl function of th coalscnt tims, and thus of th inbrding ffctiv population siz N. Undr th assumption of ) th infinit sits modl no doubl hits), 2) a constant N ovr th gnrations constant coalscnt probability), and 3) a homognous population qual calscnt probability for all pairs) w can thrfor stimat th ffctiv population siz from polymorphism data if w hav indpndnt knowldg about th mutation rat.g. from divrgnc data). In particular, for a sampl of siz 2, w hav E[S 2 ] = 4N µ and thus N = E[S 2] 4µ. In a sampl of siz n, w can stimat th xpctd numbr of pairwis diffrncs to b Ê[S 2 ] = θ π s 2.6)) and obtain th stimator of N from polymorphism data as N = θ π 4µ. A similar stimat can b obtaind from Wattrson s stimator θ S, s Eq. 2.7). Whil th assumption of th infinit sits modl is oftn justifid as long as 4N µ n, with µ n th pr-nuclotid mutation rat), th assumption of constant and homognous coalscnt rats is mor problmatic. W will com back to this point in th nxt sction whn w discuss variabl population sizs and population structur. 3.2 Exampls Lt us now discuss th main factors that influnc th ffctiv population siz. For simplicity, w will focus on N. W will always assum that thr is only a singl dviation from th idal Wright-Fishr population.

5 40 3 EFFECTIVE POPULATION SIZE Offspring varianc On assumption of th idal modl is that th offspring distribution for ach individual is Binomial approximatly Poisson). In natural populations, this will usually not b th cas. Not that avrag numbr of offspring must always b, as long as w kp th cnsus) population siz constant. Th offspring varianc σ 2, howvr, can tak any valu in a wid rang. Lt X i b th numbr of offspring of individual i with i m i = 2N. Thn th probability that individual i is th parnt of two randomly drawn individuals from th offspring gnration is m i m i )/2N2N )) 2N i= m i m i ) 2N2N ) 3.4) is th probability for idntity by dscnt of two random offspring. Th singl-gnration coalscnt probability p c, is th xpctation of this quantity. With E[m i ] = and E[m 2 i ] = σ 2 + and th dfinition 3.) w arriv at N = N /2 σ 2 N σ ) By a slightly mor complicatd drivation not shown), w can stablish that th varianc ffctiv population siz N v) taks th sam valu in this cas. Sparat sxs A larg varianc in th offspring numbr lads to consqunc that in any singl gnration som individuals contribut much mor to th offspring gnration than othrs. So far, w hav assumd that th offspring distribution for all individuals is idntical. In particular, th xpctd contribution of ach individual to th offspring gnration was qual = ). Evn without slction, this is not ncssarily th cas. An important xampl ar populations with sprat sxs and unqual sx ratios in th brding population. Considr th following xampl: Imagin a zoo population of primats with 20 mals and 20 fmals. Du to dominanc hirarchy only on of th mals actually brds. What is th inbrding population siz that informs us, for xampl, about loss of htrozygosity in this population? 40? or 2?? Lt N f b th numbr of brding fmals 20 in our cas) and N m th numbr of brding mals in th xampl). Thn half of th gns in th offspring gnration will driv from th N f parnt fmals and half from th N m parnt mals. Now draw two gns at random from two individuals of th offspring gnration. Th chanc that thy ar both inhritd from mals is. In this cas, th probability that thy ar copis from th sam 4 patrnal gn is 2N m. Similarly, th probability that two random gns ar dscndnts from th sam matrnal gn is 4 2N f. W thus obtain th probability of finding a common ancstor on gnration ago p c, = + = + ) 4 2N m 4 2N f 8 N m N f

6 3.2 Exampls 4 and an ffctiv population siz of N = 2p c, = 4 N m + N f = 4N fn m N f + N m. In our xampl with 20 brding fmals and brding mal w obtain N = = Th idntity-by-dcnt or inbrding) ffctiv population siz is thus much smallr than th cnsus siz of 40 du to th fact that all offspring hav th sam fathr. Gntic variation will rapidly disappar from such a population. In contrast, for an qual sx ratio of N f = N m = N w find N 2 = N. Sx chromosoms and organlls Tak two random Y -chromosom allls from a population. What is th probability that thy hav th sam ancstor on gnration ago? This is th probability that thy hav th sam fathr, bcaus Y-chromosoms com only from th fathr. So this probability whr N m is th numbr of mals in th population, so N is N m mitochondrial gns N = N m. Similarly, for = N f whr N f is th numbr of fmals in th population. In birds th W-chromosom is th fmal dtrmining chromosom. WZ individuals ar fmal and ZZ individuals ar mal. So for th W-chromosom N = N f. For th X- chromosom in mammals and th Z-chromosom in birds it is a bit mor complicatd. Tak two random X-chromosom allls, what is th probability that thy ar from th sam ancstor? This is Nf + N ) m 2 N m N f 2N f + N m. Exrcis 3.3. Explain th last formula. Hint: You nd to trat mals and fmals in both th offspring and parnt gnrations spratly.) What is N for th X-chromosom if th sx ratio is :? Fluctuating Population Sizs Anothr assumption of th idal Wright-Fishr modl is a constant population siz. Of cours, th population siz of most natural populations will rathr fluctuat ovr tim. In this cas, th cnsus siz - and also th ffctiv siz - of a population changs from gnration to gnration. Howvr, if fluctuations in th population siz occur ovr cycls of only a fw gnrations, it maks sns to dfin a singl long-trm ffctiv population siz to captur th avrag ffct of drift ovr longr volutionary priods. Considr th volution of a population with varying siz ovr t gnrations and imagin that w hav alrady calculatd th ffctiv population siz for ach individual gnration

7 42 3 EFFECTIVE POPULATION SIZE N 0 to N t. Th N i tak brding structur tc. into account. Th xpctd rduction of htrozygosity ovr t gnrations thn is H t = ) ) H 0 2N 0 2N t = p c, ) t H 0 whr p c, is th rlvant avrag singl-gnration coalscnc probability that dscribs th loss of htrozygosity. W thn hav p c, = [ 2N 0 ) = xp 2t )] /t [ 2N t )) N 0 N t 2t xp and gt an avrag inbrding) ffctiv population siz of N = 2 p c, t N N t xp ) 2N ) N 0 N t 2N t )] /t So in this cas th avrag) inbrding-ffctiv population siz is givn by th harmonic man of th population sizs ovr tim. Othr than th usual arithmtic man, th harmonic man is most strongly influncd by singl small valus. E.g., if th N i ar givn by 00, 4, 00, 00, th arithmtic man is 76, but w obtain a harmonic man of just N = 4. W can summariz our findings by th rmark that many populations ar gntically smallr than thy appar from thir cnsus siz, incrasing th ffcts of drift. Exrcis 3.4. In Exrcis 3.2 you stimatd N for th X-chromosom in humans. Th human population was not of constant siz in th past. Assum that within th last yars i.. within th last 0000 gnrations) th human population grw gomtrically. That mans that N t+ = gn t. How larg must g b in ordr to xplain your ffctiv population siz? Two toy modls Lt us dal with two xampls of populations that ar unralistic but hlpful to undrstand th concpt of ffctiv sizs. Th first xampl that is givn rprsnts a zoo population. In ordr to kp polymorphism as high as possibl, car is takn that vry parnt has xactly on offspring. Th ancstry is givn by Figur 3.A). Th scond xampl is similar to th xampl of unqual sx ratios whr th population had 20 mals, but only on of thm had offspring. Howvr, th nxt xampl is vn mor xtrm. In this cas th individuals ar haploids, and only on ancstor is th parnt of th whol nxt gnration. A scnario lik this is shown in Figur 3.B).

8 3.3 Effcts of population siz on polymorphism 43 A) B) Figur 3.: A) Th ancstry of a population that is kpt as polymorphic as possibl. B) Th ancstry of a population whr ach gnration only has on parnt Exrcis 3.5. Figurs 3.A) and B) clarly do not com from idal Wright-Fishr populations. So N c N and w can try to calculat th ffctiv population sizs for thm. Givn th cnsus siz in th two cass is N c what ar th varianc ffctiv siz, inbrding ffctiv siz. 3.3 Effcts of population siz on polymorphism W hav alrady sn that gntic drift that rmovs variation from th population. So far, w hav nglctd mutation that crats nw variation. In a natural population that volvs only undr mutation and drift, an quilibrium btwn ths two procsss will vntually b rachd. This quilibrium is calld mutation-drift balanc. Mutation-drift balanc Th nutral thory of molcular volution tris to xplain obsrvd pattrns in nuclotid frquncis across populations with only two main volutionary forcs: mutation and drift. Mutation introducs nw variation, and drift causs thm to sprad, but also causs thm to b lost. An quilibrium is found whn ths two procsss balanc. W hav alrady drivd that gntic drift rducs th htrozygosity ach gnration by H = H. In ordr to driv th quilibrium frquncy, w still nd to know th chang in htrozygosity H undr mutation alon. In th infinit sits mutation modl, vry nw mutation hits a nw sit. Thrfor, mutation can nvr rduc htrozygosity by making allls idntical.

9 44 3 EFFECTIVE POPULATION SIZE Howvr, vry pair of idntical allls has th chanc to bcom htrozygot if ithr of th gns mutats. Thn, htrozygosity incrass du to mutation lik E[H H] = H + 2µ H). Summing ovr th ffcts of mutation and drift and ignoring trms of ordr µ 2 w obtain: E[H H] = H 2N H + 2µ H), E[ H H] = Th quilibrium is obtaind for E[ H H] = 0 and so 2µ H) H 2N and so th quilibrium htrozygosity is at = 0, H 2µ + ) 2N = 2µ H + 2µ H). H = 2µ ) = θ 2µ + θ ) As xpctd, th quilibrium htrozygosity incrass with incrasing mutation rat bcaus mor mutations ntr th population) and with incrasing ffctiv population siz bcaus of th rducd ffct of drift). Not that only th product θ of both quantitis ntrs th rsult. Thr is an altrnativ way of driving th sam formula using th coalscnt. What w ar looking for is th probability that two individuals ar diffrnt at som gn or at som nuclotid. If you follow thir history back in tim, two things can happn first: ) ithr on of th two mutats or 2) thy coalsc. If thy coalsc first thy ar idntical, if on of th two mutats thy ar not idntical. Sinc mutation in ithr linag) occurs at rat 2µ and coalscnc occurs at rat / ), it is intuitiv that th rlativ probabilitis of both vnts to occur first ar 2µ : [/ )] rsulting in th probability for mutation first as givn in Eq. 3.6). For a rigorous mathmatical tratmnt, w nd th following rsult about xponntial distributions: Maths 3.. Lt X and Y b xponntially distributd with rats µ and ν thn P[X < Y ] = 0 = µ µ + ν. P[X = x] P[Y x]dx = 0 µ µx νx dx = µ µ + ν xµ+ν) 0 Assum X and Y ar waiting tims for crtain vnts. Thn th probability that X occurs bfor Y is dtrmind by th fraction of th rat for X with rspct to th total rat for both vnts. Exrcis 3.6. Using Maths 3. can you rdriv 3.6)?

10 3.4 Fixation probability and tim Fixation probability and tim Th probability that a nw mutation that has occurd in a population is not quickly lost again, but rachs fixation, and th tim that it taks to do so, ar fundamntal quantitis of molcular population gntics. Blow, w will introduc ths quantitis for th nutral modl. Fixation probability of a nutral mutation What is th fixation probability of a singl nw mutation in a population of haploid) siz 2N undr nutrality? W can find th answr to this qustion by a simpl argumnt that is inspird by gnalogical thinking. Obviously, th mutation will vntually ithr fix in th population or gt lost. Assum now that w mov fast forward in tim to a gnration whr this fat has crtainly bn sortd out. Now imagin that w draw th gnalogical or coalscnt) tr for th ntir population for this latr gnration. This gnalogy will trac back to a singl ancstor in th gnration whr th mutation that w ar concrnd with happnd. Fixation of th mutation has occurd if and only if th mutant is that ancstor. Sinc undr nutrality ach individual has th sam chanc to b pickd as th ancstor, th nutral fixation probability must b p fix = 2N. You can hav a look again at th simulation of wf.modl) that you saw in chaptr 2, and chck how oftn th lowr lft individual is ancstor to all individuals in th last gnration of th simulation. Exrcis 3.7. Us a similar argumnt to driv th fixation probability of a mutation that initially sgrgats at frquncy p in th population. Exrcis 3.8. This xrcis is about th quilibrium lvl of htrozygosity or th maintnanc of htrozygosity in th fac of drift. It has always bn and still is) a major qustion in volutionary biology why thr is variation. Slction and drift ar usually thought to rmov variation, so whn looking at data, th obvious qustion is always: how is variation maintaind. You will b daling with th thory dvlopd abov. For this xrcis you will us th function maint) from th R-packag. If you typ maint), th function maint) is carrid out with standard valus N = 00, u = 0.00, stoptim=0, init.a=, init.c=0, init.g=0, init.t=0 is simulatd. You can.g. chang th popualtion siz by using maintn=000). If you want to plot th frquncis of th allls ovr tim, you can put th rsult of maint) dirctly into a plot command, for xampl by >plotmaintn=000, stoptim=0000, u=0.0000))

11 46 3 EFFECTIVE POPULATION SIZE If you just want to look at how th frquncy of G changs, you can us th option plotmaint...), what="g"). This modl simulats a population undrgoing drift and continual mutation at a nutral locus. Th locus is rally just on nuclotid and it can thrfor b in four stats: A, C, G or T. At gnration 0, all mmbrs of th population hav th sam homozygous gnotyp i.., th frquncy of th singl alll in th population is qual to on). In subsqunt gnrations, nw allls ntr th population through mutation.. St th modl paramtrs for a population of 300 individuals with a mutation rat at this locus of 0 4. Obsrv th population for 0000 gnrations. Viw th graph of th frquncis of th diffrnt allls in th population ovr tim. Run this simulation a fw tims. 2. What happns to most nw mutants that ntr th population? How many allls in this population attaind frquncis abov 0.? Do any nw mutant allls rach a frquncy abov 0.9? 3. Basd on this population siz and mutation rat, what is th rat at which nw mutants ntr th population? Not th appropriat formula as wll as th numrical answr.) What is th rat at which you would xpct nw mutants to bcom fixd? You can also viw th numbr of nw mutations that occurrd in th population using what="numofmut"). Dos it fit with your xpctation? Basd on this rat, how many nw mutants would you xpct to bcom fixd during th 0000 gnrations for which you obsrvd th population chck using what="fixd")? Explain what this valu mans. 4. How dos th numbr of fixd mutations dpnd on th population siz? What dos this man for divrgnc btwn two spcis with diffrnt population sizs? 5. Now viw th graph of htrozygosity ovr tim what="h"). What dos this graph suggst about th lvl of variation in th population i.. is it fairly constant through tim or dos it chang, is H abov zro most of th tim)? Giv a rough stimat for th avrag valu of H throughout ths 0000 gnrations. 6. Using 3.6), prdict th quilibrium valu of H. You can also plot th prdictd valu using what=c"h","h.prd"). 7. Would you xpct th htrozygosity to incras, dcras, or rmain th sam if you significantly incrasd th mutation rat? What would you xpct if you incrasd th population siz? What if you incras mutation rat but dcras population siz? 8. Incras th mutation rat for th modl to Viw th graph of htrozygosity ovr tim and th graph of alll frquncis. Dos this simulation confirm your xpctation givn in th last task)? What dos th formula prdict H to b in this situation?

12 3.4 Fixation probability and tim 47 Fixation tim undr nutrality In gnral, fixation tims ar not asily drivd using th tools of this cours. For th spcial cas of a singl nutral mutation in an idal Wright-Fishr population on can show that th xpctd fixation tim is qual to th xpctd tim to th MRCA th hight of th coalscnt tr) of th ntir population. W hav alrady calculatd this quantity in sction 2.3 s q. 2.4). If th sampl siz quals th population siz 2N, w nd to tak th larg-sampl limit and obtain T fix 4N. Exrcis 3.9. Us maint) from th R- packag. In this xrcis you will look at th fixation tim for a nutral mutation in a population. St th mutation rat so that you a hav only fw on or two) fixations in th graph. You nd to st stoptim to a larg valu, mayb and N to a low valu, mayb 00.. Us N = 00 and look at last at 0 fixation vnts. Rcord th tim it taks btwn apparanc of th nw muation and its fixation. 2. Tak a diffrnt valu for N and rpat th xrcis. Plot roughly) th rlationship btwn N and th man tim to fixation for a nutral mutation. Hint: First crat on instanc of th volution of th nutral locus using th command rs<-maint...). Aftrwards you can look at spcific tims during th volution,.g. by using plotrs, xlim=c000,300)). 3. For humans, how long in yars) would it tak for a nutral mutation to fix? Exrcis 3.0. W hav introducd th nutral fixation probability and fixation tim for an idal Wright-Fishr population. How do w nd to adjust th rsults if w think of a natural population and th various concpts of ffctiv population sizs?