STAT 536: Genetic Drift

Transcription

1 STT 536: Genetic Drift Krin S. Dormn Deprtment of Sttistics Iow Stte University October 05, 2006

2 Finite Popultion Size In finite popultions, rndom chnges in llele frequency result becuse of Vrition in the number of offspring per prent. Lw of segregtion in diploid species. The rndom chnges in llele frequency re clled genetic drift. Genetic drift is nother force cting on genes in popultions nd it hs two min consequences: It removes genetic vrition t rte inversely proportionl to the popultion size. It ffects the probbility of survivl of new muttions, in mnner pproximtely independent of the popultion size.

3 Neutrl Theory Just like there is selection/muttion blnce, there lso exists drift/muttion blnce, with genetic drift removing vrition nd muttion restoring vrition. The ide tht much of the genetic vrition present in popultions is the consequence of the drift/muttion blnce is clled the Neutrl Theory, very powerful concept in popultion genetics. The theory hs remined controversil since its inception becuse it is difficult to test it seems to negte the importnce of nturl selection, the core of Drwin s theory of evolution.

4 Simulting Popultions For diploid popultion of size N = 20 nd strting llele frequency of p(0) = 0.2, follow these steps to simulte genertion t + 1, Set K = 0. Choose to copy n llele with probbility p(t) nd increment K, otherwise generte B (non-) llele. Repet 2 N times. Compute the genertion t + 1 llele frequency p(t + 1) = K /2/N. Repet for 100 genertions. llele Frequency Genertion The result for 10 independent runs is shown in the plot.

5 Observtions bout simultions There re few things to observe bout these llele frequency fluctutions There re rndom chnges in llele frequency. ll 10 popultions disply different llele frequencies over time, thus evolution cn never be repeted. lleles re lost from the popultion, 3 times B ws lost, 6 times ws lost. Only one time did both lleles persist for 100 genertions. llele Frequency Genertion The direction of rndom chnges is neutrl, i.e. not preferentilly up or down. We will probbly need to prove this mthemticlly to truly convince you.

6 Inbred Individuls nd Finite Popultion Sizes n individul is inbred if she/he contins lleles t locus tht re IBD (identicl by descent). n individul becomes inbred becuse his/her ncestors re relted. The only wy to not be inbred is to hve completely unrelted ncestors. But, if there were no reltionships mong your ncestors, then if you re genertion t, you hve 2 distinct ncestors in genertion t 1 (your prents), 4 = 2 2 distinct prents in genertion t 2 (your grndprents), 8 = 2 3 distinct prents in genertion t 3 nd so on. It doesn t tke long before we re tlking bout terbytes of ncestors, more ncestors thn the size of the popultion N. Therefore, finite popultions led to inbred individuls nd everyone is t lest little inbred.

7 Inbred Popultions popultion is inbred if the probbility tht two lleles selected without replcement re IBD is positive. s popultion becomes inbred, so too the individuls. In rndomly segregting, rndomly mting popultions, the level of popultion inbreeding is equl to the level of individul inbreeding, i.e. the probbility tht two rndom lleles re IBD is the sme s the probbility tht the two lleles in rndom individul re IBD. We will often refer genericlly to inbreeding without distinguishing whether we men the individul or the popultion becuse of this equivlency.

8 Mintining Diversity in Finite Popultions in Finite Popultions diploid bse popultion of size N will hve 2N distinct lleles t ech locus. Wht is the probbility tht none of these lleles is lost when pssing lleles to the next genertion? ssume tht ech prent hs precisely two offspring. Then the prent must pss one llele to one offspring (hppens with probbility 1) nd the other llele to the other offspring (hppens with probbility 1 2 ). So, the probbility tht ll prents pss on both lleles is ( ) N 1 2 very smll number! Therefore, it is very likely to lose one or few lleles during ech genertion. Plese contrst this conclusion with the sme sitution under HWE. n llele tht strts out t proportion 1 2N (i.e. it is present s one copy in size N popultion), will persist in HWE forever t the sme llele frequency 1 2N.

9 Genetic Drift In finite popultion, the persistence of n llele is not gurnteed. We lredy know there is very good chnce tht not ll lleles will mke it to the next genertion. So, some will be lost. The lost lleles re replced with IBD copies of other lleles. These other lleles gin in frequency nd the overll frction of ibd lleles in the next genertion will hve incresed. So, smll popultion size nd inbreeding go hnd-in-hnd. Genetic drift is synonymous with incresing levels of inbreeding. Genetic drift is dominting force in smll popultions.

10 llele Fixtion If we crry the bove process of llele loss t ech genertion forwrd in time, it leds to the conclusion tht ultimte there will remin only one llele in the popultion. While t first it is very esy to remove lleles, the numbers of the remining lleles increse nd it is less likely tht they will be removed t ech genertion. However, there is lwys positive, though smll, chnce tht n llele goes extinct in ech genertion. Combine these probbilities over enough genertions, nd eventully ll lleles but one will go extinct. In the end, genetic drift leds to the fixtion of one llele t every position in smll popultion. The popultion will become completely inbred. Clerly, something in our rgument does not pply to rel life since ll popultions re finite, but they re not entirely inbred. There re other forces (muttion, migrtion, selection) tht cn counterct the effects of genetic drift. We will discuss them nd their reltionship to genetic drift lter.

11 Hploid Finite Popultion Consider hploid popultion of size N. If every individul copies itself to produce the next genertion, there is no chnge due to genetic drift. However, if some individuls fil to copy themselves, while other copy themselves more thn once, there will be rndom chnges in llele frequencies nd genetic drift will pply to the popultion. Hence, genetic drift in hploid popultions is determined by the vrince in number of offspring. If there is no vribility, there is no genetic drift. There re mny wys tht the number of offspring could vry mong individuls. The simplest nd most mthemticlly elegnt is to think bckwrds gin. ssume tht ech offspring rndomly selects his/her prent.

12 Biologicl Plusibility If (1) ech prent produces huge number of offspring initilly, but (2) ech prent produces exctly the sme huge number of offspring, nd (3) these offspring re killed off t rndom (without regrd to genotype) until there re only N left, then the bove model will pply quite well. Killing off t rndom cn lso be thought of s selecting few lucky survivors (N to be precise) t rndom. The key observtion is tht ech time one of the lucky survivors is selected, the pool of offspring from tht prent is not substntilly decresed.

13 Popultion Inbreeding Recurrence Reltion Let f t be the mount of inbreeding t genertion t. Here, we men popultion inbreeding, since there is no concept of inbred individuls. We will develop recurrence reltion for f t with time. f t = P(two individuls hve ibd lleles) = P(ibd two indivs. select sme prent)p(two indivs. select sme prent) + P(ibd two indivs. select diff. prents)p(two indivs. selected diff. prents) = 1 1 N + f t «. N The solution is f t = 1 N 1 N ( ) 1 1 t 1 N N = 1 ( 1 1 ) t. 1 N N

14 Interprettion of Solution Looking t the solution ( f t = ) t, N we see tht in the limit f t 1. nd ech genertion, the probbility of non-ibd lleles h t = 1 f t declines by nother frction 1 N. Note lso tht the bove is the verge behvior of mny popultions. Now rndom chnce is very importnt force in these smll popultions. So the ctul level of inbreeding in popultion t genertion t will be something round f t but my show substntil vrition from popultion to popultion.

15 Diploid Equivlent Now, consider diploid popultion where ll the prents produce huge pools of gmetes tht re thrown into common gmetic pool. ssume rndom union of gmetes nd ssume hermphroditic dults tht cn self-fertilize. Then, the offspring cn be thought of s rndomly selecting two individuls from the previous genertion to be its prents. In fct, this model is equivlent to the hploid model in the sense tht ech offspring cn be thought of s rndomly selecting two lleles from the preceding genertion. s before, the probbility tht these two lleles re IBD is given by the recursion eqution: f t = 1 2N + f t 1 since now there re 2N gmetes. ( 1 1 2N The solution s before is f t = 1 ( 1 1 2N ) t with 2N replcing N. ),

16 Rte of Loss of Heterozygosity Inbreeding results in the increse of ibd lleles nd obligte loss of heterozygosity (ech new inbred individul replces either homozygote or heterozygote, the ltter replcement resulting in grdul loss of heterozygotes from the popultion). How fst does this occur? Let h t = 1 f t be the probbility tht the two lleles t locus re not ibd. Then, ( h t = 1 1 ) t. 2N Clerly, the smller N, the fster the increse in heterozygosity. nd exmining the updte eqution f t = 1 2N + ( 1 1 2N ) f t 1, we see in ech genertion frction 1 lleles (i.e. new inbreeding). 2N of ll ibd lleles re new ibd

17 Rte of New Inbreeding Genertions 1 to 100 Genertions 100 to 1000

18 nother View of the Rte If we write f = 1 2N for the new inbreeding introduced ech genertion, then the updte eqution looks like (Note: f f t f t 1!) Rerrnge this eqution to find f t = f + (1 f ) f t 1. f = f t f t 1 1 f t 1. This is the rte of chnge of the inbreeding coefficient f t genertion t reltive to the totl mount it hs left to go before chieving complete inbreeding f = 1.

19 Recovery from Bottleneck Sometimes popultions re subject to bottlenecks (brief periods of time when their numbers drop to very low numbers where genetic drift domintes). n interesting question is whether restoring their numbers will reverse the dmge done to during the bottleneck. To be specific, is inbreeding reduced fter the popultion numbers re restored? To nswer this question, we will ssume the popultion ends the bottleneck with inbreeding t f 0 > 0 nd then subsequently the popultion immeditely rises to infinite size N =. Wht will hppen to inbreeding? We exmine the updte eqution f t = 1 2N + ( 1 1 2N ) f t 1 = f t 1 = = f 0, so the level of inbreeding remins fixed t f 0 > 0 despite the huge increse in popultion size.

20 Time to Trget Heterozygosity Loss We cn use the time-dependent equtions to lso find how much time is needed to lose pre-defined mount of heterozygosity. For exmple, the hlf-life t 0.5 for heterozygosity, the time it tkes to remove 1 2 of the current heterozygosity h 0 is ( h t0.5 = 1 1 ) t0.5 = h 0 2N 2 t 0.5 = ln h 0 ln 2 ln ( ) 1 1 2N For smll x, ln(1 x) x by Tylor s series so for lrge N, t 0.5 ln h 0 ln N Or when the current heterozygosity h 0 = 1, mening no inbreeding, t N. (Note: For hploids, the hlf-life is t N.)

21 Hlf-Life of Heterozygosity t 0.5 cn be thought of the hlf-life of heterozygosity. The diploid result t 0.5 = 2N ln(2) shows tht the hlf-life is proportionl to the popultion size. It tkes longer to lose heterozygosity if the popultion is lrger. For exmple, popultion size of 1 million tht reproduces every 20 yers would require 28 million yers to lose hlf of its heterozygosity. Lrge browsing mmmls nd the first monkey-like primtes were first ppering long with the lps nd Himlys here on plnet erth. For resonbly lrge popultions, genetic drift is slow process.

22 Wright-Fisher - Hploid In popultion of size N, the frequency of llele t genertion t is { 0 p t N = 0, 1 N, 2 N,..., N } N = 1. Suppose tht currently p t = i N, so there re i copies of llele in the popultion. The next genertion offspring will ech independently select lleles t rndom from the current genertion, nd with probbility p t, ech will select n llele. If X t is the number of llele copies in the popultion t genertion t, then X t+1 Binomil(N, p t ). We cn define trnsition probbilities P ij (t) = P (X t+1 = j X t = i), where ( ) N P ij (t) = p j t j (1 p t) N j, with p t = i N.

23 Wright-Fisher - Diploid For the simple diploid model, the offspring select their lleles rndomly from the previous genertion (without worrying to mke sure tht they obtined one from fther nd the other from femle). Then, the trnsition probbilities re ( ) 2N P ij (t) = p j t j (1 p t) 2N j, with p t = i 2N. With the simple diploid model we need not chnge the equtions except to multiply ll N s by 2. We hve defined simple Mrkov chin.

24 Some Results from Mrkov Chin Theory The stte of this Mrkov chin is the number of lleles in the popultion. t ech genertion, this stte is updted ccording to the trnsition probbilities defined bove. How the chin s stte is updted depends only on the current stte of the system (the current number of lleles). How it will be updted does not depend on the whole history of this popultion. For this reson, the stochstic process is clled Mrkovin. In ddition, we observe tht there re two bsorbing sttes (0 nd 1) from which the chin cnnot escpe, mening once tht stte is entered, the chin will never leve tht stte. This is of course true only in the bsence of muttion nd migrtion. ll other sttes re wht re clled trnsient sttes mening tht they will only be visited finite number of times. Eventully the chin will bump into one of the bsorbing sttes, preventing ny more visits to the trnsient sttes.

25 Properties of this Process While the Mrkov chin cn provide us with mny useful results bout this process we re modeling it fils to give us closed form solution for the distribution of future llele frequencies. In fct, there is no known solution to this complete specifiction of the process. Insted, we must rely on summry descriptions (e.g. men nd vrince) to describe the rndom process of genetic drift.

26 Men of Genetic Drift Recll X t is the number of lleles in the popultion t time t, nd X t+1 is binomilly-distributed rndom vrible with success probbility Xt 2N. Then, by the properties of the binomil distribution E (X t+1 X t ) = 2N X t 2N = X t. In generl, becuse E[E(X Y )] = E[X], we hve E (X t+1 ) = E (X t ) = E (X t 1 ) = = X 0, where X 0 is the initil number of lleles in the popultion. Divide through by 2N, to obtin E (p t+1 ) = E (p t ) = E (p t 1 ) = = p 0.

27 Fixtion Probbilities ll popultions ultimtely become fixed. Exmine collection of popultions tht ll strt with the sme llele frequency p 0. t time t, let Y it indicte whether the ith popultion hs fixed llele. We wnt to know wht proportion of popultions fixed llele s t, i.e. lim t P(Y it = 1) = lim t E(Y it ). But becuse the expecttion does not chnge in time, the overll frequency of llele cross ll these popultions must be p 0, therefore p 0 = P(Y i0 = 1) = P(Y i1 = 1) = = P(Y it = 1) = = lim t P(Y it = 1) nd we conclude tht the probbility tht fixes in the popultion is equl the initil strting frequency of llele. The more llele present t the beginning, the more replicte popultions it is likely to persist in or the more likely it is to persist in single popultion.

28 Fixtion in Replicte Popultions

29 Vrince in llele Frequency Since X t+1 Binomil(2N, p t ), the properties of the binomil distribution provide Vr(X t+1 p t ) = 2Np t (1 p t ). Converting to vrince in llele proportions, we hve Vr(p t+1 p t ) = p t(1 p t ). 2N We cn lso think of the next genertion count s X t+1 = X t + e x, or the next llele frequency s p t+1 = p t + e, where e is some rndom devition from the previous llele frequency. The rndom devition e hs men E(e) = 0, since E(p t+1 ) = E(p t ).

30 Vrince in llele Frequency In ddition, ll the vrince of the updte to p t+1 = p t + e is contined in the devition e Vr(p t+1 p t ) = Vr(e) = E(e 2 ), since p t is non-rndom by the conditioning nd E(e) = 0. E(pt+1 2 p t ) = E [ (p t + e) 2] = E ( pt 2 ) + 2E (pt e) + E ( e 2) = p 2 t + 2p t E(e) + E(e 2 ) = p 2 t + p t(1 p t ) 2N = p 2 t ( 1 1 ) + p t 2N 2N.

31 Vrince of llele Frequencies However, we seek the unconditionl probbility E(p 2 t+1) = E [ E(p 2 t+1 p t ) ], which is obtined by tking second expecttion over ll possible p t. ( E(pt+1) 2 = E(pt 2 ) 1 1 ) + E(p t) 2N 2N but we know llele frequency is unchnging in expecttion E(p t ) = p 0 nd of course, by definition of vrince E(p 2 t ) = Vr(p t ) + p 2 0. Thus, Vr(p t+1 ) + p0 2 = ( Vr(p t ) + p0 2 ) ( 1 1 ) + p 0 2N 2N.

32 Vrince of llele Frequencies The eqution we hve just derived cn be rerrnged into recurrence reltion for the llele frequency vrince ( Vr(p t+1 ) = Vr(p t ) 1 1 ) + p 0(1 p 0 ), 2N 2N with initil condition Vr(p 0 ) = 0. This is stndrd recurrence reltion you know how to solve. The solution is [ ( Vr(p t+1 ) = p 0 (1 p 0 ) ) ] t. 2N s t, the vrince in llele frequencies pproches p 0 (1 p 0 ).

33 Predicted llele Frequency nd Vrince Predicted llele Frequency N=1000 N= Future Genertions N=200 N=100

34 Genetic Drift nd Inbreeding Notice tht ( Vr(p t+1 ) p 0 (1 p 0 ) = ) t = f t. 2N This quntittes the reltionship between genetic drift nd inbreeding. The mount of inbreeding t genertion t is direct proportionl to the degree of vrince in llele frequencies between replicte popultions of the sme process.

35 Hndling ssumptions - effective popultion size Of course we hve mde some rther strong ssumptions in deriving these results. How cn we model more relistic popultions? One wy is to compre rel popultions to the idel Wright-Fisher popultion we hve been ssuming. Specificlly, we cn find the size of the idel popultion tht would hving the sme level of inbreeding s we observe in rel popultion. The (inbreeding) effective popultion size of rel popultion is the size of the idel popultion (stisfying our hermphroditic rndom mting model describe previously) tht would hve the sme level of inbreeding s our rel popultion. The symbol used to represent (inbreeding) effective popultion size is N e.

36 Multiple Definitions of N e To mtch levels of inbreeding, we cn choose N e such tht the frction of new inbreeding produced ech genertion (recll this is 1 2N e for the idel Wright-Fisher model) mtches the frction of new inbreeding produced in the rel popultion. Most often this is how the mtching between idel nd rel popultions is done, but there cn be other wys to compre popultions nd mke them mtch. The concept of effective popultion size cn then be difficult to interpret, especilly if it is not clrified how the equivlency is mde.

37 Disposing of Selfing ssumption The ssumption tht might hve you most worried is the ssumption of hermphroditic species. Very few species ctully stisfy this ssumption nd it seems tht it might cuse some difficulties. To quntitte how big n effect distinct sexes might hve on the results, we will explicitly model it in two phses. First we remove the ssumption tht prents cn self (fertilize themselves). Next, we remove the ssumption tht nyone cn mte with nyone, nd force mle plus femle mtings. s before, let h t be the probbility tht two lleles t locus re non-ibd. Let k t be the probbility tht two lleles selected t rndom from two rndomly selected individuls re not-ibd. First note h t+1 = k t. We ll derive k t+1 = 1 ( 2N h t ) k t. N

38 Second Eqution k t+1 is the probbility tht two rndomly selected lleles re not ibd in two rndom individuls. They cn only be non-ibd if the two individuls selected these genes from different prents or they selected different lleles from the sme prent. They select the sme prent with probbility 1 N, then with probbility 1 2 they select seprte lleles. With probbility h t, those two lleles will be non-ibd. They select different prents with probbility 1 1 N. The probbility tht they select non-ibd from these two prents is k t. ll together, the eqution resulting is k t+1 = 1 2N h t + ( 1 1 N ) k t.

39 Solving Recurrence Reltions Using h t = k t 1, we hve k t+1 = 1 2N k t 1 + ( 1 1 ) k t. N If we ssume logrithmic liner grow, k t+1 = λk t = λ 2 k t 1 or λ 2 k t 1 = 1 ( 2N k t ) λk t 1. N The relevnt solution is λ = 1 1 N N 2 which is just slightly different from 1 1 2N, the fctor tht pplies in the Wright-Fisher model. 2,

40 N e for Non-Selfers To compute the effective popultion size, we need the popultion size N e tht if evolving like the idel popultion would hve the sme single-genertion chnge in inbreeding s observed in this non-idel popultion. In other words, we need 1 1 N N 2 2 = 1 1 2N e. It turns out tht N e N + 1 2, so this popultion, where individuls must select two distinct prents, behves like n idel popultion where there is 1 2 dditionl individul in the popultion. Tht 1 2 extr individul will slow the effects of inbreeding very slightly. Essentilly, there is very little impct of forcing selection of distinct prents.

41 dding Distinct Sexes Suppose there re N f femles nd N m mles nd ech offspring must select one rndom fther nd one rndom mother from these pools. If the llele of interest is not sex-linked, then sex is ssigned independent of the genotype. We use the sme quntities h t nd k t. s before, h t+1 = k t. Two genes from different individuls cn only be non-ibd if they selected these lleles s two distinct lleles from the sme femle prent the sme mle prent, or entirely different prents.

42 Distinct Sexes (cont.) Two individuls select the sme femle prent with probbility 1 N f or the sme mle prent with probbility 1 N m. When we select to lleles t rndom from two individuls there is chnce tht they re both mternl or both pternl. 1 4 k t+1 = 1 [ ) ] 1 h t + (1 1Nf k t + 1 [ ( 1 h t ) ] k t N f 4 2N m N m 2 k t ( 1 = + 1 ) ( h t ) k t 8N f 8N m 4N f 4N m Compre this eqution to the one for distinct prents: k t+1 = 1 ( 2N h t ) k t. N

43 N e for Distinct Sexes The size N of the distinct-prent popultion tht would cquire inbreeding s fst s this distinct-sex popultion is given by 1 N = 1 + 1, 4N f 4N m which yields but then N = 4N f N m N f + N m, N e N = 4N f N m N f + N m Notice, tht when N f = N m = N 2, N = N, nd there is no effect of distinct sexes beyond tht of distinct prents when the sexes re blnced.

44 N e for Distinct Sexes Nevertheless, there cn be substntil impct when the sex rtio is not equl. For popultions with N = 100 but unequl sex rtios, the effective popultion size is N f N m N e

45 N e for Monogomous Species Of course the humn species fvors monogmy for the reproduction process. How does monogmy effect N e? To model this, hve the N dults split into N 2 pirs tht lst for life (we ssume N f = N m = N 2 so tht the sexes mtch up). Then, when offspring pick their prents, they first pick pir (fmily) nd then strt picking genes. h t+1 = k t k t+1 = 2 N ( = 1 2N h t + 2 h t k t ( 1 1 2N ) ) k t, ( ) k t N nd we conclude tht monogmy when there re equl frctions of ech sex does not chnge the rte of inbreeding fixtion.

46 Vrying Popultion Size Wht if the popultion size N t depends on time t? When the popultion size vries, the updte equtions revel t 1 ( h t = 1 1 ). 2N i i=0 The equivlent idel popultion would hve eh t = ( 1 1 ) t. 2N e Equting these such tht the sme level of inbreeding (or non-inbreeding ctully) is chieved by genertion t, we hve ( 1 1 ) = 2N e t 1 i=0 ( 1 1 ) 1/t. 2N i

47 N e for Vrying Popultion Size When ll N t re lrge despite the vrition from genertion to genertion, the pproximtion ( 1 1 ) 1/t 1 1 2N i 2N i t, so we hve ( 1 1 ) = 2N e t 1 i=0 ( 1 1 ). 2N i t Solving for N e, we hve N e = t t 1 i=0 1 N i, i.e. N e is the hrmonic men of the popultion sizes cross ll those genertions. Hrmonic mens tend to strongly weight low vlues over high vlues, so few dips in the popultion size will strongly dip the corresponding effective popultion size N e.

48 Oscillting Popultion Size Popultion Size N e N t Genertions

49 Rndom Vrition in Offspring Numbers We will now consider the effect of vribility in the number of offspring. ll individuls in the idel popultion produce n infinitely lrge number of offspring which re then selected rndomly to proceed into the next genertion. This is clerly n unresonble ssumption becuse obviously, no one cn produce infinitely mny offspring, nd the number of offspring produced per dult my vry (perhps becuse of rndom locl environments good or bd for rising offspring). In relxing this ssumption, we will continue to ssume infinitely mny offspring re produced, but different levels of infinity pply to different prents.

50 Probbility of Selecting Two Gmetes from Sme Prent Suppose the fitness of the ith individul is w i. These fitness vry from individul to individul not for genetic resons but becuse of rndom environmentl effects. In the infinite gmete pool from ll dults, there will be proportion gmetes produced by individul i. The probbility tht two gmetes come from prent i is ( ) 2 wi i w. i P w i i w i The probbility tht ny two gmetes come from the sme prent is the sum of the bove over ll possible prents i w 2 i ( j w j) 2. of

51 New Inbreeding per Genertion The mount of new inbreeding introduced per round of repliction is just the probbility tht the sme two gmetes re selected from the sme prent i w 2 i 1 ( 2 2. j j) w The frction 1 2 is the diploid fctor, the probbility tht the sme llele is selected both times from tht prent. The mount of new inbreeding introduced by ech ech round of repliction of n idel popultion is 1 2N e, so we conclude the effective popultion size for this rel popultion with vrying offspring numbers is ( j w j) 2 N e = i w 2 i.

52 N e for Offspring Vrition We re now going to write this in terms of the vrince in offspring numbers (or equivlently the vrince in the individul fitnesses w i ). where w = 1 N Therefore, Vr(w i ) = V w = 1 N i w 2 i w 2, i w i is the men fitness cross ll individuls. N e = = N 2 w 2 ( NVw + N w 2) N 1 + V w / w 2 = N 1 + Cw 2, where C 2 w = Vw w 2 is the squred coefficient of vrition of fitness.

53 Interprettion Notice N e < N, so vrition in offspring numbers reduces the effective popultion size below the census size nd increses the rte of ppernce of inbreeding. This should mke intuitive sense. Some individuls will contribute more offspring to the next genertion nd others will fil to contribute t ll. The results is more ibd lleles in the next genertion. The derivtion requires tht the fitnesses w i re constnt throughout the genertions, well t lest tht the men nd vrince re constnt. So, since these fitness differences re cused by the environment, the environment must be effectively constnt from genertion to genertion. There cn be locl vrition, but the overll verge environment nd vribility in the environment must remin constnt.

54 Vrying Surviving Gmetes It is lso possible to write the effective popultion size in terms of the vrition in the number of gmetes n i tht survive into the next genertion from individul i. Now, we re mesuring bsolute contribution to the next genertion, rther thn reltive contribution (it might be esier to mesure). Then (detils not shown), N e = 4N V n, where V n is the vrince in n i cross the popultion. When ll prents contribute exctly 2 gmetes to the next genertion, then V n = 0 nd N e = 2N 1, so the rel popultion cts like much lrger idel popultion; inbreeding is slowed.

55 Wright-Fisher Vrince V n Notice, the Wright-Fisher model lso hs n intrinsic vrince in offspring number. In Wright-Fisher, n i is the number of successes in 2N drws of Bernoulli rndom vrible with probbility of success 1 N. The vrince is ( ) ( 1 V n = 2N 1 1 ) = 2 2 N N N. This vrince is the smllest chievble vrince when offspring rndomly select gmetes from prents. So, ny other form of selecting gmetes tht is still rndom will decrese the effective popultion size. On the other hnd, if offspring do not select gmetes t rndom the effective popultion size cn be incresed. If gmetes re not selected t rndom, the bove formul is useful for computing V n nd N e. The derivtion of N e vi V w ssumes rndom selection of gmetes nd cnnot be used.

56 Summry The effective popultion size is n efficient mens of compring popultions by relting them ll to stndrd, idel popultion. It llows us to quickly evlute popultion nd mting scheme to determine how much inbreeding it will produce. If N e > N t, where N t is the census size of the rel popultion t time t, then the rel popultion hs ccumulted less inbreeding thn we would hve expected under idel conditions. If N e < N t, the the rel popultion hs been ccumulting more inbreeding thn the sme size idel popultion.