Genetic estimates of contemporary effective population size: to what time periods do the estimates apply?

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1 Universiy of Nebraska - Lincoln DigialCommons@Universiy of Nebraska - Lincoln Publicaions, Agencies and Saff of he US Deparmen of Commerce US Deparmen of Commerce 005 Geneic esimaes of conemporary effecive populaion size: o wha ime periods do he esimaes apply? Robin Waples NOAA, robinwaples@noaagov Follow his and addiional works a: hp://digialcommonsunledu/usdepcommercepub Waples, Robin, "Geneic esimaes of conemporary effecive populaion size: o wha ime periods do he esimaes apply?" (005) Publicaions, Agencies and Saff of he US Deparmen of Commerce 46 hp://digialcommonsunledu/usdepcommercepub/46 This Aricle is brough o you for free and open access by he US Deparmen of Commerce a DigialCommons@Universiy of Nebraska - Lincoln I has been acceped for inclusion in Publicaions, Agencies and Saff of he US Deparmen of Commerce by an auhorized adminisraor of DigialCommons@Universiy of Nebraska - Lincoln

2 Molecular Ecology (005) 14, doi: /j X x Geneic esimaes of conemporary effecive populaion size: Blackwell Publishing, Ld o wha ime periods do he esimaes apply? ROBIN S WAPLES Norhwes Fisheries Science Cener, 75 Monlake Blvd Eas, Seale, WA 9811 USA and Laboraoire d Ecologie Alpine (LECA), Génomique des Populaions e Biodiversié, Universié Joseph Fourier, Grenoble, France Absrac Alhough mos geneic esimaes of conemporary effecive populaion size ( ) are based on models ha assume is consan, in real populaions changes (ofen dramaically) over ime, and esimaes (ˆe) will be influenced by in specific generaions In such cases, i is imporan o properly mach ˆe o he appropriae ime periods (for example, in compuing /N raios) Here I consider his problem for semelparous species wih wo life hisories (discree generaions and variable age a mauriy he salmon model), for wo differen sampling plans, and for esimaors based on single samples (linkage disequilibrium, heerozygoe excess) and wo samples (emporal mehod) Resuls include he following Discree generaions: (i) Temporal samples from generaions 0 and esimae he harmonic mean in generaions 0 hrough 1 bu do no provide informaion abou in generaion ; (ii) Single samples provide an esimae of in he parenal generaion, no he generaion sampled; (iii) single-sample and emporal esimaes never provide informaion abou in exacly he same generaions; (iv) Recen bolenecks can downwardly bias esimaes based on linkage disequilibrium for several generaions Salmon model: (i) A pair of single-cohor (ypically juvenile) samples from years 0 and provide a emporal esimae of he harmonic mean of he effecive numbers of breeders in he wo parenal years (N b(0) and N b() ), bu adul samples are more difficul o inerpre because hey are influenced by N b in a number of previous years; (ii) For single-cohor samples, boh onesample and emporal mehods provide esimaes of N b in he same years (conras wih resuls for discree generaion model); (iii) Residual linkage disequilibrium associaed wih pas populaion size will no affec single-sample esimaes of N b as much as in he discree generaion model because he disequilibrium diffuses among differen years of breeders These resuls lead o some general conclusions abou geneic esimaes of in ieroparous species wih overlapping generaions and idenify areas in need of furher research Keywords: discree generaions, heerozygoe excess, linkage disequilibrium, /N raio, Pacific salmon, emporal mehod Received January 005; revision acceped 15 June 005 Inroducion Effecive populaion size ( ) is a criical parameer in evoluionary biology, bu i is difficul o collec enough demographic daa from mos naural populaions o calculae direcly As a consequence, geneic mehods for esimaing have been used widely by boh evoluionary biologiss and conservaion biologiss For some Correspondence: R S Waples, Fax: ; robinwaples@noaagov esimaors, performance measures such as bias and precision have been evaluaed (Nei & Tajima 1981; Waples 1989; Pudovkin e al 1996; Wang 001) However, one fundamenal problem has received lile aenion: mos esimaors of conemporary are based on models in which is assumed (eiher implicily or explicily) o be consan, bu his is highly unlikely in naural populaions These esimaes apply o a specific generaion(s) or ime period(s), and if varies i is imporan (for wo reasons) o deermine o which ime period(s) he esimae applies Firs, several differen geneic mehods for esimaing 005 Blackwell Publishing Ld

3 3336 R S WAPLES are available (see below), and in comparing resuls he ime periods relevan o each esimae should be congruen Second, if one wans o esimae he raio /N, failure o properly mach he esimae wih he appropriae esimae of census populaion size (N) can resul in biologically misleading conclusions A consensus has no emerged in he scienific lieraure regarding he ypical range of /N values in naural populaions (Nunney 1993; Hedgecock 1994; Frankham 1995; Scribner e al 1997; Hauser e al 00) This remains an acive area of research, and geneically based esimaes of /N are regularly used o draw conclusions on opics such as maing sysems and reproducive success (Kaeuffer e al 004; Maocq 004); /N raios in differen axa (Turner e al 00; Hoffman e al 004); paerns of change in /N over ime (Ardren & Kapuscinski 003; Shrimpon & Heah 003); and appropriae conservaion and managemen acions (Hoffman & Blouin 004; Johnson e al 004) To illusrae some of he implicaions of properly maching ime periods for and N, consider he hypoheical siuaion depiced in Fig 1: geneic samples are aken in wo consecuive generaions, and he emporal mehod is used o esimae, yielding (in his example) = 100 Does his esimae apply o he effecive size in he firs generaion, he second generaion, or some combinaion (harmonic or arihmeic mean) of he wo? In his example, he esimae of /N varies almos sevenfold depending on he generaion or combinaion of generaions o which is applied Noe also ha boh samples could be used individually o esimae based on eiher of he onesample mehods discussed below To wha ime period(s) would hose esimaes of apply, and how would hey compare wih he ime period(s) covered by he emporal esimae? The issue is complex because resuls depend on he life hisory of he species in quesion, he mehod used o esimae, and he sampling regime Here I consider his opic for esimaes of conemporary based on single samples (linkage disequilibrium, heerozygoe excess) and wo samples (emporal changes in allele frequency); for ypical sampling regimes (before vs afer reproducion; single- vs muliple-cohor); and for semelparous species wih wo differen life hisories (discree generaions and variable age a mauriy) Pacific salmon (Oncorhynchus spp) are perhaps he bes-known species ha exhibi semelpariy wih variable age a mauriy, bu his ype of life hisory is also found in a variey of oher axa Alhough discree generaions occur in relaively few species, discree generaion models have been widely used o esimae in species wih overlapping generaions Finally, I will commen on he relevance of he resuls of he semelparous models o ieroparous, age-srucured species Mehods Effecive populaion size Fig 1 Schemaic diagram of esimaion of effecive populaion size based on samples (S 0, S 1 ) aken in consecuive generaions In his hypoheical example, census size (N) in he wo generaions is 800 and 10, and he emporal mehod yields an esimae = 100 The esimae of /N varies grealy depending on which generaion (or combinaion of generaions) is applied o Random geneic processes occur a a rae inversely relaed o populaion size However, i is no he census size ha deermines he rae of hese random processes, bu raher he populaion s effecive size is he size of an ideal populaion (Wrigh 1931) ha would have he same rae of geneic change as he populaion under consideraion In an ideal populaion (one in which maing is random, sex raio is equal, generaions are discree, and variaion in reproducive success is random), N and are he same, bu ha is rarely he case in naure In addiion o direcly affecing raes of geneic change, effecive size also deermines he relaive imporance of migraion and selecion; hese forces are deerminisic in large populaions bu can be overwhelmed by random processes in small ones The models discussed here all assume ha populaions are closed o migraion and he markers considered are selecively neural Selecion or migraion could bias esimaes of and, under some circumsances, could also affec he ime periods o which esimaes of effecive size apply Several effecive sizes have been idenified in he lieraure (Ewens 1979; Crow & Dennison 1988; Caballero 1994);

4 GENETIC ESTIMATES OF CONTEMPORARY 3337 Table 1 Noaion used A j Fracion of breeders ha maure a age j; ΣA j = 1 g Generaion lengh in years; g = ΣjA j Elapsed ime beween wo samples, scaled in years in he salmon model and in generaions in he discree generaion model N i Number of individuals in he populaion in ime period i (i) Effecive populaion size in generaion i Effecive number of breeders in ime period i N b(i), N b e ( b) P i S i X (Y ) V(X Y) F r R An esimae of or N b Harmonic mean effecive populaion size (effecive number of breeders) over a period of ime Frequency of an allele in (i) genes represening effecive populaion size in generaion i or in N b(i) genes represening effecive breeders in ime period i Number of individuals sampled a ime period i Frequency of an allele in S 0 (S ) genes in a sample aken in ime period 0 () Variance of he difference in allele frequency beween wo samples Sandardized variance of allele frequency change; F = V(X Y)/[P(1 P)] Squared correlaion of allele frequencies a differen gene loci Asympoic r reached in a populaion of consan size here we will be concerned wih he wo mos commonly used: variance effecive size (relaed o he rae of allele frequency change) and inbreeding effecive size (relaed o he rae of increase in inbreeding) Effecive size can be hough of as applying o eiher a single generaion or o muliple generaions A single-generaion is associaed wih ransmission of geneic maerial from one generaion o he nex Kimura & Crow (1963) showed ha inbreeding perains o he number in he parenal generaion, whereas variance perains o he number in he progeny generaion However, as discussed below, his conclusion applies o populaion parameers; geneic esimaes of depend on samples, and he sampling process can affec he ime period(s) o which he esimaes apply This paper will focus on esimaes of conemporary ha is, one or more generaions ha span he ime frame represened by he samples I do no consider esimaes of long-erm effecive size (which generally depend on equilibrium relaionships beween and measures of geneic diversiy) or mehods specifically designed o deec bolenecks; however, I do evaluae effecs of recen changes in ha can affec conemporary esimaes of effecive size Here we will be concerned wih geneic daa for wo samples aken in differen ime periods (samples S 0 and S ; see Table 1 for noaion) In he discree generaion model, he ime periods are generaions; in he variable age a mauriy model (henceforh known as he salmon model ), he ime periods are referred o as years Two samples provide a basis for using he emporal mehod o esimae ; depending on he species life hisory, he esimae applies eiher o he effecive populaion size per generaion ( ) or he effecive number of breeders per year (N b ) In addiion, each of he wo samples can be analysed separaely for informaion abou effecive size The analyses below assume he species is diploid, bu simple modificaions could be made for haploid or polyploid species Populaion geneic models Discree generaions The model used here (Fig ) is similar o ha of Waples (1989), he only imporan difference being ha here we wan o disinguish effecive sizes ha may differ among generaions In each generaion i, here are N i adul breeders having effecive size (i), which (o be compleely general) can be hough of as represening (i) genes (having frequency P i ) raher han a paricular number of ideal individuals The (i) effecive breeders produce a large pool of gamees from which individuals in he nex generaion are drawn The gamee pool is assumed o be large enough ha is allele frequency is also P i Drif occurs in choosing he nex generaion s oal populaion (N i+1 ) and effecive populaion ((i+1) ) from he gamee pool Salmon model Species wih his life hisory reproduce in only one ime period bu can maure a a variey of ages j wih specified probabiliy A j (Σ A j = 1); as a consequence, he populaion allele frequency in ime period i depends no only on he frequency in ime period i 1, bu also on frequencies in one or more previous ime periods For salmon, he relevan ime period is 1 year, so represens he elapsed number of years beween samples A model of geneic change ha incorporaes hese life hisory feaures is shown in Fig 3 Each large circle represens he N i aduls making up he pool of poenial breeders in 1 year Smaller circles each year are he effecive number of breeders, which can be hough of as represening N b(i) genes ha are he equivalen of N b(i) ideal individuals In his example, aduls can maure a ages j = 3, 4, or 5 Thus (for example), aduls in year 0 are derived from breeders in years 3, 4, and 5, and aduls in year are derived from breeders in years 1,, and 3 Breeders in years 0 and share one common source breeders in year 3 bu he degree of overlap could

5 3338 R S WAPLES Fig Model of geneic change over ime and sampling from a populaion wih discree generaions Poin of reference for all geneic changes is he allele frequency (P 1 ) in he gamee pool preceding generaion 0 N i represens he census populaion size in generaion i, and he (i) genes represening he effecive populaion size in generaion i produce he gamee pool from which he nex generaion is formed Samples (of S i individuals) for geneic analysis are aken a wo or more poins in ime according o wo sampling plans In Plan I, individuals (ypically aduls) are sampled afer reproducion or nonlehally before reproducion, so ha some individuals in he sample may also conribue o fuure generaions (poenial overlap indicaed by shaded circle) In Plan II, individuals (ypically juveniles) are sampled wihou replacemen before reproducion, so here is no overlap beween individuals comprising he sample and he census or effecive populaion size in ha generaion By convenion (Nei & Tajima 1981; Waples 1989), boh Plan I and Plan II samples are associaed wih he generaion in which he sampled individuals would maure be more or less depending on he iming of he samples and he age srucure Waples (1990a, b) showed ha in his model, effecive size per generaion ( ) is approximaely equal o gn b, where g is he generaion lengh (average age of parens = average age a mauriy weighed by age-specific fecundiy) When N b varies, g b, where b is he harmonic mean N b across he years making up he generaion (Waples 1990b, 00) The model used here is similar, excep ha care will be aken o idenify which specific years mos srongly influence he esimae of N b esimaors Single-sample esimaes Single-sample esimaors of depend on deparures from one- or wo-locus geneic equilibrium ha arise when progeny are derived from a finie number of parens These samples provide an esimae of he effecive number of parens ha produced he progeny from which he sample is drawn; hence, hey can be associaed wih he inbreeding effecive size (Laurie-Ahlberg & Weir 1979) In he discree generaion model, he sample can be hough of as being produced by he effecive populaion size of he parenal generaion In he salmon model, he single-sample esimaes apply o a paricular number of breeders ha generally do no represen a generaion The heerozygoe excess mehod (Pudovkin e al 1996) measures increases in he observed proporion of heerozygoes (in comparison wih expeced Hardy Weinberg proporions) due o random differences in allele frequency beween males and females Because a single generaion of random maing is sufficien o resore HW proporions, he heerozygoe excess mehod provides an esimae of he effecive number of parens ha produced he sample ha is, (or N b ) in he parenal generaion (or year) The linkage disequilibrium mehod, which measures deparures of digenic gameic frequencies from hose

6 GENETIC ESTIMATES OF CONTEMPORARY 3339 is biased unless S (R Waples & P England, unpublished) Alhough a correcion can be applied o remove mos of he bias (R Waples, unpublished), he bias is no subsanial enough o affec conclusions of his paper, so equaions 1 and are used in he examples below The above formula all assume ha linkage disequilibrium is esimaed from genoypic daa and maing is random wihou permanen pair bonds; see Weir & Hill (1980) for adjusmens for oher maing sysems and for haploype daa Fig 3 Model of geneic change over ime and sampling from a semelparous populaion wih variable age a mauriy (he salmon model) Large circles (labelled N i ) represen he breeding populaion in successive ime periods (in his example, years); smaller nesed circles (labelled N b(i) ) represen he effecive number of breeders in year i No individual is a breeder in more han 1 year Because age a mauriy varies, he breeders in year i represen progeny of parens from more han one previous year In his example, he breeders in years 0 and are boh derived from parens ha reproduced 3, 4, and 5 years before (as indicaed by curved, dashed lines wih arrows) As in Figure, samples are aken according o wo differen plans Plan I samples (ypically aduls) include individuals from more han one cohor Plan II samples (ypically juveniles) for year i are from a single cohor ha represens progeny of aduls reproducing in year i expeced based on random maing and binomial sampling, depends on chance associaions of alleles a differen gene loci Unlike single-locus HW deparures, linkage disequilibria do no disappear afer a single generaion of random maing; insead, hey decay a an exponenial rae deermined by he recombinaion fracion, c (0 c 05) In a populaion of consan size, an asympoic level of disequilibria due o drif is evenually reached, in which decay of exising disequilibria is balanced by random creaion of new disequilibria each generaion For neural gene loci ha are unlinked (c = 05), he asympoic linkage disequilibrium is approximaed by E( r 1 1 ) + 3N S e (Hill 1981), (eqn 1) where s is an esimae (from a sample of S individuals) of he mean squared correlaion of allele frequencies a differen gene loci The firs erm in equaion 1 is disequilibrium due o drif; he second is due o sampling a finie number of progeny for geneic analysis This equaion can be rearranged o provide an esimaor of (Weir & Hill 1980; Waples 1991): = 1 3( r 1/ S) (eqn ) Equaion 1 assumes ha second order erms in S and can be ignored (Weir & Hill 1980), and as a resul equaion Temporal esimaes Temporal esimaes depend on he premise ha he rae of geneic change over ime is a funcion of ; his mehod hus provides an esimae of he variance effecive populaion size If X and Y are allele frequencies in samples S 0 and S, he quaniy V(X Y) can be expressed as V(X) + V(Y) cov(x, Y), where V denoes a variance and COV a covariance (Nei & Tajima 1981) V(X Y) is of ineres because he sandardized variance, F = V(X Y)/P(1 P), can be expressed as a funcion of and he ime beween samples: E ( ) 1 f is he harmonic mean of S 0 and S (Waples 1989) and is he number of generaions beween samples This leads o an esimaor of as: N e (eqn 3) Equaion 3 assumes Plan II sampling (see nex secion); a sligh modificaion is needed if sampling is according o Plan I For species wih salmon life hisory, he analogue o equaion 3 is (Waples 1990a) N b (eqn 4) where b is a consan ha varies according o age srucure and he number of years beween samples Waples (1990a) obained values of b via simulaion, bu Tajima (199) showed ha hey can be derived analyically For boh he discree generaion and salmon models, i will be necessary o revisi previous derivaions of f o deermine which generaions or years of geneic drif conribue o f and hence o he esimae of (or N b ) Sampling = ( f 1 /@) = b ( f 1 /@), Two sampling plans have been idenified in he lieraure for each life hisory model (Figs and 3; Table ) Alhough he names of he plans (Plans I and II) are he same in boh models, some imporan differences will be noed briefly here; more deails appear below

7 3340 R S WAPLES Table Feaures of wo sampling plans discussed in he ex Model Plan I Plan II Discree generaions Feaures Sampling before reproducion, wih replacemen; or sampling afer reproducion Sampling before reproducion, wihou replacemen Examples Aduls sampled afer maing Lehal sample of juveniles Pre-maing aduls sampled nonlehally Salmon model Feaures Sample conains > 1 cohor Sample consiss of a single cohor Examples Adul spawners Same-age juveniles Mixed-age juveniles Aduls grouped by cohor Discree generaions (Nei & Tajima 1981; Waples 1989; Fig ) In sampling Plan I, individuals are sampled afer reproducion or nonlehally (wih replacemen) before reproducion This ype of sample hus could include individuals ha also conribue offspring o he nex generaion In Plan II, samples are aken wihou replacemen before hey can reproduce and do no conribue o any fuure generaions The disincion is imporan because in Plan I he allele frequencies in he wo emporal samples are posiively correlaed o he exen ha some genes appear boh in S 0 and (0) (Nei & Tajima 1981) Plan II samples are generally juveniles and Plan I sampling ypically involves aduls; however, wih he increasing populariy of nonlehal biopsy sampling for DNA analyses, Plan I would apply o many juvenile samples as well Salmon model (Fig 3) Waples (1990a) used his same before / afer reproducion framework o define Plan I and Plan II samples in Pacific salmon However, as discussed in more deail below, wih variable age a mauriy he disincion beween sampling before or afer reproducion is less imporan han i is wih discree generaions In he salmon model, a more imporan consideraion is wheher he sample represens a single cohor (progeny of a single year s breeding populaion) or muliple cohors In he following herefore Plan I will refer o samples ha include muliple cohors, and Plan II will refer o samples from a single cohor In Pacific salmon, juvenile samples are ypically Plan II because differen cohors are spaially and emporally isolaed, whereas aduls samples are ypically Plan I because of variable age a mauriy However, excepions occur in boh direcions Mixed-cohor samples of juveniles are common in species such as Oncorhynchus mykiss, and hese should be analysed according o Plan I Conversely, i may be possible o pariion an adul sample ino individual cohors, in which case each of hese reconsruced cohors can be analysed as a Plan II sample Reference generaion For analyical purposes i is essenial o clarify which generaion or ime period a paricular sample is associaed wih, and in his I have followed convenions esablished by previous auhors In he discree generaion model, Nei & Tajima (1981) idenified boh Plan I and Plan II samples wih he progeny generaion ha is, he generaion a which he sampled individuals would become maure (Fig ) This is logical when generaions are discree, and i is also logical for samples of adul salmon, which can readily be associaed wih a specific year of spawning Bu his framework is problemaical when applied o a sample of juvenile salmon, which in general canno reliably be associaed wih any single year of breeders in he fuure However, a juvenile sample generally can be idenified wih a single parenal populaion For his reason, Waples (1990a) considered Plan II salmon samples o be associaed wih he parenal generaion ha is, hey are progeny of he same year s breeders ha an adul sample would be associaed wih (Fig 3) I should be noed ha alhough he reference generaion is imporan o clarify for bookkeeping purposes, choice of he reference generaion does no affec he underlying processes of geneic drif ha influence F and esimaes of effecive size The more imporan quesion biologically is, Wha ime period(s) affec an esimae of (or N b ) developed using a paricular sampling regime? This quesion is he focus of he res of his paper Simulaed daa Linkage disequilibrium Because equaions 1 and assume ha s has reached is asympoic value, can be biased if he populaion has recenly changed in size, and he biases will differ depending on wheher he populaion has recenly increased or decreased Some limied heoreical evaluaion has been carried ou of he effecs of populaion hisory on r, bu none has considered he consequences for esimaes of effecive populaion size Therefore, compuer

8 GENETIC ESTIMATES OF CONTEMPORARY 3341 simulaions were used o model he rae of increase of disequilibrium due o drif in a declining populaion I used a Wrigh Fisher model (wo sexes, random maing, random variaion in reproducive success) o rack mulilocus genoypes for eigh independen, diallelic gene loci over a number of discree generaions Iniial genoypes were drawn binomially a each locus wih parameric allele frequency 05 (equivalen o assuming a populaion of infinie size in generaion 0); each subsequen generaion was produced by maing a fixed number of parens ( = eiher 40 or 00) Samples of S = progeny were aken each generaion o compue he mean s across all pairs of loci (using Weir s 1979 unbiased esimaor of he Burrows mehod) Because he ineres here is he general behaviour of s as a funcion of changing populaion size (raher han on variaion in s among replicae samples), mean s was averaged across a large number of replicaes o generae an empirical approximaion o E(s ), which was hen used o calculae using equaion Temporal esimaes in he salmon model To pinpoin specific years o which an esimae of N b applies in he emporal salmon model, I conduced simulaions similar o hose described by Waples (1990a, b), wih an iniial populaion consising of 5 years wih P = 05 a 0 diallelic loci Reproducion was Wrigh Fisher, so N b(i) = N i every year, and he age of parens of each progeny produced was deermined by random variaion around a fixed probabiliy of age a mauriy (A j, j = 1, 5) In each replicae, a warm-up period of 40 years was run a a consan (Baseline) N b (in mos cases, Baseline N b was 100) In year 41, and for 1 year only, he populaion size was changed o N b = N 1, and a year 41 + j, j = 1, 5, he size was changed o N b = N In all oher years, effecive size was he Baseline N b Plan II samples (generally of S = 50) were aken annually for 15 years beginning in year 35, and N b was compued for all pairs of samples 1 5 years apar, using equaion 4 This forma allowed evaluaion of he following combinaions of rue N b values in he wo years sampled: N 1 N, N 1 Baseline, N Baseline, and Baseline Baseline The laer group was spli ino comparisons in which boh sampled years were prior o he firs year of populaion change ( before ), and hose for which a leas one of he samples was aken afer year 41 ( afer ) Wihin each group of comparisons, a disribuion of N b values was generaed For each parameer se, a large number of replicaes (10 000) was run o ensure ha resuls refleced real age-srucure effecs and no random noise Resuls Discree generaions Single-sample esimaes As discussed above, in general single-sample esimaes provide informaion abou effecive size in he parenal generaion ha is, he populaion of breeders ha produced he sample From Fig, i is apparen ha boh Plan I and Plan II samples in generaion i are produced by he effecive breeders in generaion i 1 Thus, juvenile and adul samples from generaion i boh provide informaion abou in generaion i 1 When effecive size is no consan, he heerozygoe excess mehod will sill provide an esimae of (i 1), bu in he linkage disequilibrium mehod s and will also be affeced by in generaions prior o (i 1) Sved (1971; see also Sved & Feldman 1973) provided a formula for evaluaing he rae a which drif disequilibria accumulae in a populaion ha iniially begins wih r = 0 (ie as would occur in a populaion of infinie size) and hen sabilizes a a consan, finie This scenario hus represens an exreme example of populaion decline If c = 05, hen in generaion he expeced disequilibria in a sample can be approximaed by: E( r ) [ (/ )]( ) drif + R S sample (eqn 5) where R is he asympoic r due o drif for he new effecive size (firs erm in equaion 1) and 1/S is he conribuion o E(s ) from sampling (second erm in equaion 1) Since he erm [1 (1/4) ] 1 very rapidly as increases (being 094 for = and 098 for = 3), he drif erm rapidly approaches is asympoic value, meaning ha if populaion size has recenly decreased, he disequilibrium mehod will primarily reflec he new (smaller) raher han in previous generaions Resuls for simulaed populaions under he populaion decline scenario modelled by Sved are shown in Fig 4A Wih new = 40, equaion 1 slighly underesimaes E(s ), primarily because he erm 1/S does no compleely accoun for disequilibria due o sampling (R Waples, unpublished daa) However, for boh new effecive sizes he empirical rae of increase in s closely maches he expecaion from equaion 5 More imporanly, he esimae is srongly upwardly biased only in he firs generaion afer he populaion change and quickly sabilizes a approximaely he new effecive size Even under his exreme scenario herefore he effecs of he hisorically larger populaion size on do no persis beyond 3 generaions For populaions ha increase in size, he opposie effec mus be considered decay of exising disequilibria ha are higher han he asympoic value for he new Assuming again ha he loci are unlinked, exising disequilibria decay a a rae of 75% per generaion, leading o a rapid convergence rae similar o (bu from he opposie direcion as) ha described by equaion 5 A he same ime, new drif disequilibria are generaed each generaion ha will converge on he asympoic value for he new, larger If we ignore sampling for he momen, he overall effecs on drif disequilibria can be approximaed using

9 334 R S WAPLES Fig 4 Rae of approach of linkage disequilibrium o is new asympoic value afer a change in populaion size, and resuling effecs on Discree generaions are assumed (A) (Populaion decline; comparison of simulaed daa wih heoreical expecaion) The populaion begins wih r = 0 in generaion 0 and subsequenly has consan = 40 (riangles) or 00 (circles) Open symbols are empirical s from simulaions; doed lines are expeced values based on equaions and 5 Filled symbols show approach of o he new effecive size as disequilibria accumulae in he populaion (B) (Populaion expansion; heoreical dependence of on rae of decrease in r ) Resuls shown are parameric expecaions for he populaion (equivalen o assuming a sample of an infinie number of progeny) The populaion experiences a boleneck of = 10 ending in generaion 0 and increases o consan = 00 (riangles) or = 1000 (circles) in generaions 1 9 Open symbols show expeced rae of decrease of r as residual linkage disequilibria break down; filled symbols show convergence of (calculaed using equaion assuming 1/S = 0) o he new, larger a recursive equaion o generae Er ( ) for a given number of generaions () following a populaion increase: Er ( ) ( 14 / )( r ) + [ 1 (/ 14)]( R ), r 0 0 (eqn 6) where is disequilibrium a he ime of he boleneck and R again is he asympoic r for he new (larger) effecive size If r 0 = 0, equaion 6 reduces o he drif erm in equaion 5 As an example, assume ha a populaion experienced a boleneck of = 10 and subsequenly grew in one generaion o a new, consan (00 or 1000; Fig 4B) Assume also ha he boleneck was long enough (a leas 3 generaions; see above) ha he drif disequilibrium approached is asympoic value [R = 1/(3 ) = 0033 for = 10; equaion 1] One generaion afer he boleneck, he residual disequilibrium from he small populaion size is sill much higher han he asympoic values for he new effecive sizes (open symbols in Fig 4B), leading o downward bias in (closed symbols) If he new is 00, he esimae will be wihin 0% of he rue value by posboleneck generaion 3 If he new is 1000, however, a generaion 3, sill will be downwardly biased by over 50%, and no unil generaion 5 will he bias due o he boleneck drop below 10% Thus, even wih rapid decay he residual disequilibrium from a small boleneck can affec r and for a number of generaions Bias from less severe bolenecks would no be as subsanial for as many generaions (Table 3) We can conclude ha populaion declines are no likely o seriously affec single-sample esimaes of conemporary ; if he effecive size in he generaion producing he

10 GENETIC ESTIMATES OF CONTEMPORARY 3343 Table 3 Rae of approach of o he new (consan) following bolenecks of = 10, 40, and 100, wih pos boleneck effecive sizes (new ) = 1000, 00, 100, and 40 Values in body of able are he raio /new was calculaed from equaion based on E(r ) calculaed from equaion 6 See Fig 4B for a plo of some of hese daa Boleneck size = 10 = 40 = 100 Gen New sample is small, he mehods will (on average) deec ha small size, regardless wheher he populaion was larger in previous generaions For populaions ha have recenly increased, can be biased downwards for several generaions, wih duraion and magniude of bias proporional o severiy of he boleneck and he relaive increase in These conclusions apply o unlinked loci Linked loci have greaer precision for esimaing provided he recombinaion fracion c is known (Hill 1981), bu inerpreaion of he esimae is more complicaed because hisorical populaion size has a sronger effec If he loci used o calculae s are ighly linked, can be more srongly affeced by in he disan pas han by conemporary effecive size Temporal esimaes A general formula for V(X Y) for boh Plans I and II is (Nei & Tajima 1981; Waples 1989): VX ( Y) = P ( P ) ( XY, ) S N e S cov 0 P ( P ) ( S S 1 N cov X, Y), e 0 (eqn 7) where P 1 is he allele frequency in he pool of gamees preceding generaion 0 The covariance erm differs in he wo sampling plans In Plan I, cov(x, Y) = P 1 (1 P 1 )/(N 0 ) (because boh samples are derived from he same N 0 individuals in generaion 0), whereas in Plan II he samples a ime 0 and ime are independenly derived from he iniial gamee pool, so cov(x, Y) = 0 (Waples 1989) If he samples a imes 0 and are aken a he same life sage, hey are separaed by exacly generaions of geneic drif On he oher hand, he ime period from generaion 0 hrough generaion, inclusive, encompasses + 1 generaions of effecive sizes, and i is no so obvious exacly which of hese ime periods affec he esimae of To pinpoin he generaion(s) o which he esimaes of apply when effecive size varies over ime, i is necessary o decompose he erm [1 1/( )] ino a series of erms of magniude 1 1/((i) ) I is sraighforward bu edious (see deails in Appendix) o obain he following resuls: 1 If one considers he populaion as a whole, he magniude of allele frequency change depends on effecive populaion sizes in generaions 1 hrough bu does no depend on (0) This illusraes a poin made by Kimura & Crow (1963) he variance effecive size is deermined by he number in he progeny generaion(s) When sampling is considered, however, he generaions ha affec V(X Y) are no 1 hrough bu raher 0 hrough 1 This resul can be undersood from examinaion of Fig Wih Plan II sampling, he difference in allele frequencies beween he samples a imes 0 and canno depend on (), since he sample in generaion is aken independenly of () and before any breeding occurs However, he magniude of V(X Y) will depend on (0) and all subsequen generaions hrough 1 Plan I sampling appears more complicaed bu leads o he same conclusion: he sample in generaion does no provide informaion abou (), even hough some individuals in he sample may also be represened in () For boh sampling plans herefore, VX ( Y) P ( P ) cov( X, Y) S S i N ei 0 = 0 (eqn 8) Subsiuing he cov(x, Y) erms from above, and noing ha E(f) V(X Y)/P 1 (1 P 1 ) = harmonic mean of S 0 and S, leads o he following:

11 3344 R S WAPLES Plan I: E( f) + ; Ne N N ( f + 1/ N ) 1 1 Plan II: E( f) + ; Ne N ( f Noe ha i= 1 0[/ 1Nei ()] = ()/( Ñe), demonsraing ha E(f) and are a funcion of he harmonic mean from generaions 0 hrough 1, inclusive In Plan I sampling, he esimae is also affeced by he populaion size in generaion 0 (N 0 ) Populaion size in subsequen generaions does no affec he esimaes under eiher sampling plan Salmon model 1 i= 0 1 i= 0 ei ei 0 0 Single-sample esimaes Plan II samples associaed wih year i are progeny of parens reproducing in year i and hus provide informaion direcly abou N b(i) As in he discree generaion model, esimaes based on linkage disequilibrium can also be affeced by N b in prior years However, since breeding populaions in he salmon model are no conneced by a simple Markov chain, he effec is less one of residual disequilibrium persising hrough ime han of mixure disequilibrium creaed each year by breeding among individuals of differen ages derived from parens wih slighly differen allele frequencies Waples & Smouse (1990) evaluaed his ineryear effec and found ha i was generally small compared o disequilibrium due o a mixure of differen salmon populaions Unless recen flucuaions in N b(i) have been very exensive, herefore, a Plan II sample in year i should primarily be affeced by N b(i) Plan I samples are more difficul o inerpre because hey are derived from parens in wo or more previous years Two differen facors, opposie in sign and of uncerain magniude, will end o bias Plan I esimaes of N b On he one hand, he oal number of parens producing a ypical Plan I sample will be larger perhaps wo o hree or more imes larger han he number ha produces a Plan II (single-cohor) sample This will end o upwardly bias a Plan I esimae compared o he effecive number of breeders in any single year, oward an esimae ha is inermediae beween N b and On he oher hand, a Plan I sample includes individuals from wo or more cohors produced by parens wih allele and genoypic frequencies ha vary randomly among years This will produce a kind of Wahlund effec (heerozygoe deficiency a single loci and mixure disequilibrium a pairs of loci; Sinnock 1975) ha will end o downwardly bias esimaes of N b The relaive imporance of hese wo effecs is likely o vary considerably depending on a number of facors, and unil his issue is evaluaed quaniaively i is difficul o assess heir ne effec on N b based on Plan I sampling Temporal esimaes Previous resuls for he salmon model (Waples 1990a, b) were obained by compuer simulaions because analyical soluions for V(X Y) and E(f) are difficul owing o he complexiy of he sysem Here, I have exended previous resuls o obain approximae analyical soluions, which were hen esed wih simulaed daa To beer undersand populaion-level processes, for he momen we will ignore sampling In he salmon model, he variance in parameric populaion allele frequency beween years 0 and is given by k 1 VP ( P P P C P P k 0 ) ( 1 ) 1 1 ( 1 ) N + + N C e e (eqn 9) (Waples 1990b), where k is he elapsed number of generaions (k = /g) and C is a consan Thus, he rae of allele frequency change in he populaion is a funcion of he effecive size per generaion, jus as in he discree generaion model; however, over any given ime period he observed variance is elevaed by an addiional, consan magniude C (Fig 5A) This laer effec occurs because a single year of spawners represens only a porion of a generaion, and comparing wo years of spawners is equivalen o aking wo samples from he overall populaion Because he value of C depends on sampling N b genes from he populaion as a whole o form he breeders in years 0 and, is value is approximaely given by C P(1 P)/N b(0) + P(1 P)/N b() = P(1 P)/ b(0,), where b(0,) is he harmonic mean N b in years 0 and Thus, k 1 VP ( 0 P ) P( 1 P) + N Ñ k 1 EF ( ) + N Ñ e b( 0, ) e b( 0, ) (eqn 10) If we noe ha = g b, where b is he harmonic mean N b over he ime period under consideraion, equaion 10 can be expressed as: k 1 k 1 EF ( ) + = + N Ñ gñ Ñ (eqn 11) Therefore, E(F) in he salmon model is deermined by wo drif componens, one relaed o a general background effecive size and one relaed o N b in he wo years under consideraion I is useful o examine he relaive imporance of hese wo erms If k = g, he firs erm becomes 1/ b and wo erms will be approximaely equal as long as he populaion is flucuaing around a long-erm mean value, so ha b(0,) b If g = 4 years, a common generaion lengh for salmon, he conribuion o F from overall populaion change will be as large as he conribuion from N b in he wo sampled years only afer approximaely k = 8 generaions (over e b( 0, ) b b( 0, ) and

12 GENETIC ESTIMATES OF CONTEMPORARY 3345 Fig 5 Temporal variance of allele frequency V(P 0 P ) in a simulaed Pacific salmon populaion wih N b = 50 each year, g = 4 years (A j = 0, 0, 05, 05, 05), and iniial allele frequency (P 0 ) = 05 (A) Long-erm observed rae of increase of V(P 0 P ) (filled circles) is as prediced assuming = 4N b (doed line), bu absolue magniude a any poin in ime is inflaed by he approximae amoun C = P 0 (1 P 0 )/N b = 05/50 = 0005 (B) On shorer ime frames (a few generaions), yearly flucuaions in V(P 0 P ) due o age srucure are relaively more imporan Modified from Waples (1990a, 004) 30 years) elapse beween he samples Mos emporal comparisons of salmon populaions span only 1 generaions or less, and wih k = 1 he overall F is srongly dominaed by N b in he wo sampled years (1/ b(0,) vs 1/(8 b) for he background drif effec) This suggess ha emporal esimaes spanning up o abou wo generaions in he salmon model primarily reflec he harmonic mean of he effecive numbers of breeders in he wo years sampled: N b < g Ñ b ( 0, ) ( ) (eqn 1) We expec a priori ha his approximaion should be bes when effecive size in he sampled years is small relaive o he background N b, in which case he 1/ b(0,) erm will be relaively large To evaluae he accuracy of equaion 1 for ypical (shor-erm) salmon comparisons, I conduced simulaions (see Mehods) ha compared N b wih b(0,) Resuls (Fig 6, Fig 6 Temporal esimaes of N b in he salmon model compared wih he harmonic mean N b in he wo years sampled (indicaed wih an *) Simulaed populaions had a consan populaion size (N b = Baseline N = 100) excep for years wih higher or lower N b [N 1, N = 30/30 (Panel A) or 50/00 (Panel B)] Plan II samples were aken each year for 15 years spanning he period of populaion change Esimaes for which boh samples were from years wih N b = 100 are spli ino hose for which boh samples were before he firs year of populaion change and hose for which a leas one sample was aken afer he populaion change Verical bars show he disribuion of N b esimaes based on samples aken 1 5 years apar, wih he rue N b in he years of sampling as shown Sample size was 50 and he fracion mauring a ages j = 1 5 was A j = 0, 0, 05, 05, 05 Table 4) show ha under mos circumsances N b is close o b(0,) For example, wih Baseline N b = 100 and N 1 = N = 30, median N b for he N 1 N (30 30) comparisons was 38 (Table 4), close o he expeced value of 30 Similarly, median N b for N 1 Baseline = N Baseline (30 100) comparisons was 468, almos idenical o he harmonic mean of 30 and 100 (46) For he Baseline Baseline comparisons, he differen disribuions of he before and afer esimaes illusrae he effecs of age srucure The before esimaes are very ighly clusered, wih 80% falling beween 974 and 995, reflecing he abiliy of equaion 4 o accuraely accoun for age srucure in esimaing he

13 3346 R S WAPLES Table 4 Temporal esimaes of N b in he salmon model Simulaed populaions had a consan populaion size (N b = Baseline) excep for wo years wih higher or lower N b (N 1, N ) Plan II samples were aken each year for 15 years spanning he period of populaion change Each line shows he median and perceniles of N b esimaes based on samples aken 1 5 years apar, wih rue N b in he years of sampling as shown (see Fig 6 for a graphical presenaion of daa for wo of he parameer ses) A (B) indicaes ha sampling occurred afer (before) change in N b Expeced N b is he harmonic mean of N b in he wo years of sampling In he sandard parameer se, Baseline N was 100, N 1 and N were 30, sample size was 50, and he fracion mauring a ages j = 1 5 was A j = 0, 0, 05, 05, 05 In oher simulaions, changes from his sandard se are shown in bold Esimaed N b Baseline (N 1, N ) N b in years sampled Median Perceniles 10% 90% Expeced N b 100 (30, 30) 30/ / /100 B /100 A A j = 0, 0, 033, 034, / / /100 B /100 A A j = 0, 01, 0, 03, 04 30/ / /100 B /100 A A j = 0, 0, 0, 06, 0 30/ / /100 B /100 A A j = 0, 0, 01, 08, 01 30/ / /100 B /100 A Sample size = / / /100 B /100 A (30, 70) 30/ / / /100 B /100 A (00, 00) 00/ / /100 B /100 A (500, 500) 500/ / /100 B /100 A (50, 00) 50/ / / /100 B /100 A (100, 100) 100/ / /00 B /00 A

14 GENETIC ESTIMATES OF CONTEMPORARY 3347 rue N b (100) in populaions of consan size The afer esimaes are more dispersed, alhough no biased on average This modes dispersion (80% of esimaes wihin 8% of he rue N b ) reflecs he uneven effecs of changing size in age-srucured populaions All afer esimaes were poenially affeced by low N b (30) in a leas one prior year, bu hese effecs can ake many generaions o even ou in he populaion; on shorer ime frames, such as considered here, he effecs can be eiher negligible or no, depending on he age srucure and elapsed years beween samples In general, when N 1 and N were boh < Baseline N b, N b agreed well wih b(0,) (Table 4) In conras, when N 1 and/ or N > Baseline N b, esimaes involving samples in years wih higher N b ended o be lower han b(0,) For example, median N b for he N b = comparisons shown in Fig 6B was 164, abou 5% less han he harmonic mean of 100 and 00 (1333) If boh sampled years had much higher N b han he Baseline, he difference beween N b and he rue N b was larger (median N b abou 15% oo low for N 1 = N = 00, Baseline = 100 and abou 35% oo low for N 1 = N = 500, Baseline N b = 100) This resul is due o he relaively weak drif signal when effecive size is large in he sampled years, compared o background flucuaions in allele frequency associaed wih he smaller Baseline N b The above resuls were obained using an age srucure wih 50% of aduls mauring a age 4 and 5% each a ages 3 and 5 (so A j = 0, 0, 05, 05, 05), which has been used in previous salmon models (Waples 1990a, b, 00) I also examined he scenario shown in Fig 6A (Baseline N = 100; N 1 = N = 30) wih a variey of differen age srucures (Table 4) Performance was comparable o ha shown in Fig 6A as long as no more han 50% of he aduls maured a a single age, even if he age disribuion was highly asymmerical (eg A j = 0, 01, 0, 03, 04) When 60% of he aduls maured a one age, small biases were apparen in median N b and dispersion increased; wih a srongly unimodal age srucure (A j = 0, 0, 01, 08, 01), behaviour of N b became more erraic This paern esimaes of N b srongly affeced by demographic flucuaions in prior years when a single age a mauriy srongly dominaes has been noed previously (Waples 00) Resuls discussed above apply o Plan II; Plan I sampling is more complex and resuls are quie differen In his case, each (generally adul) sample can be represened as a composie of individuals produced by wo or more previous breeding populaions (Fig 3) Allele frequencies in adul samples hus will be direcly affeced by N b in a series of previous years Two general approaches are possible for dealing wih his complicaion If each individual in he sample can be assigned o a parenal breeding populaion (eg by marking, paerniy analysis, or ageing), a series of new samples can be consruced ha can be reaed under Plan II Alernaively, i could be recognized ha he Plan I sample is affeced by N b in muliple prior years, in which case N b can bes be inerpreed as esimaing he harmonic mean N b over he previous few generaions In he salmon model, correlaions in allele frequencies beween emporal samples can arise in wo ways: eiher direcly (if he breeders in year 0 make a direc conribuion of progeny o he breeders in year ) or indirecly (hrough cascading effecs of prior years breeders on years 0 and ) Covariances associaed wih he laer effec are capured in he coefficien b ha depends on age srucure and years beween samples Direc effecs differ according o he sampling plan Under Plan II, boh he sample in year 0 and he breeders in year are derived independenly by binomial sampling from he breeders in year 0; herefore, cov(x, Y) = 0, jus as i is in he discree generaion model Plan I samples could include some individuals ha reproduce and conribue o subsequen generaions; however, because he salmon model is no a simple Markov chain, his facor is less imporan han i is for species wih discree generaions For example, in Fig 3 breeders in year 0 make no geneic conribuion o he populaion in year, so wheher aduls sampled in year 0 include any individuals ha also reproduce has no effec on he difference in allele frequencies in samples S 0 and S If he second sample were aken in year 3 insead of year, here would be some direc conribuion from aduls in year 0 o he second sample, bu ha would apply o only a fracion of he breeding populaion Waples (1990a) did no quaniaively evaluae adul sampling in he salmon model bu suggesed ha a correcion of magniude 1/N 0 (comparable o he adjusmen for Plan I sampling in he discree generaion model) migh be appropriae The above argumens indicae ha, a a maximum, he effec will be smaller han his and in many cases will be zero So, cov(x, Y) can probably be safely ignored even for Plan I samples in he salmon model Discussion Resuls for boh models are summarized in a generalized form in Table 5 In he discree generaion model, geneic drif is a simple Markov process, and i is possible o specify precisely he generaions ha conribue o a emporal esimae of The heerozygoe excess mehod also provides informaion abou a specific ime period (he parenal generaion) In conras, Plan I samples in he salmon model are derived from a mixure of cohors and are affeced by N b in a variey of previous brood years Inermediae o hese exremes are esimaes based on Plan II samples in he salmon model and esimaes based on linkage disequilibrium in he discree generaion model In hese cases, he esimaes can be affeced by effecive size in previous ime periods, bu hese effecs are funnelled hrough (and modulaed by) specific years or generaions As a resul, in hese cases i is possible o idenify he ime