ESTIMATION of population parameters in classical

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1 Copyrght Ó 005 by the Genetcs Socety of Amerca DOI: /genetcs Lkelhoods From Summary Statstcs: Recent Dvergence Between Speces Scotland C. Leman,* Yuguo Chen,*,1 Jason E. Stach, Mohamed A. F. Noor and Marcy K. Uyenoyama, *Insttute of Statstcs and Decson Scences, Department of Molecular Genetcs and Mcrobology and Department of Bology, Duke Unversty, Durham, North Carolna 7708 Manuscrpt receved December 9, 004 Accepted for publcaton August 5, 005 ABSTRACT We descrbe an mportance-samplng method for approxmatng lkelhoods of populaton parameters based on multple summary statstcs. In ths frst applcaton, we address the demographc hstory of closely related members of the Drosophla pseudoobscura group. We base the maxmum-lkelhood estmaton of the tme snce specaton and the effectve populaton szes of the extant and ancestral populatons on the pattern of nucleotde varaton at DPS00, a noncodng regon tghtly lnked to a paracentrc nverson that strongly contrbutes to reproductve solaton. Consderaton of summary statstcs rather than entre nucleotde sequences permts a compact descrpton of the genealogy of the sample. We use mportance samplng frst to propose a genealogcal and mutatonal hstory consstent wth the observed array of summary statstcs and then to correct the lkelhood wth the exact probablty of the hstory determned from a system of recursons. Analyss of a subset of the data, for whch recursve computaton of the exact lkelhood was feasble, ndcated close agreement between the approxmate and exact lkelhoods. Our results for the complete data set also compare well wth those obtaned through Metropols-Hastngs samplng of fully resolved genealoges of entre nucleotde sequences. ESTIMATION of populaton parameters n classcal populaton genetcs has tradtonally proceeded through a moments approach, based on closed-form expressons for the expectaton and varance of summary statstcs (revewed by Hey and Machado 003; Beaumont 004). For the purposes of ths dscusson, we regard as a summary statstc any measure of genetc varaton that can be determned drectly from numbers of derved mutatons observed n a sample wthout explct reference to the genealogy of the sampled genes. Felsensten (199) proposed a maxmum-lkelhood (ML) approach to the estmaton of populaton parameters, treatng the ntervals separatng the nodes of the sample genealogy as mssng data. He showed that ML methods that use knowledge of the node ntervals scaled to the mutaton rate have greater statstcal power than classcal moments-based methods that use summary statstcs. Metropols-Hastngs (MH) samplng to smulate the posteror dstrbuton of the gene genealogy based on entre nucleotde sequences now forms the bass of the domnant ML and Bayesan approaches to both populaton parameter estmaton and phylogeny reconstructon (Kuhner et al. 1995; Huelsenbeck and Ronqust 001). 1 Present address: Department of Statstcs, Unversty of Illnos, Champagn, IL Correspondng author: Department of Bology, Duke Unversty, Box 90338, Durham, NC E-mal: marcy@duke.edu Among the appealng propertes of estmaton procedures based on summary statstcs are greater smplcty and potental for customzaton to specfc bologcal systems. Whle computaton of summary statstcs by defnton requres no knowledge of the genealogy of the sample, the ntmate relatonshp between the pattern of segregatng varaton and the sample genealogy les at the core of coalescence theory and molecular populaton genetcs (Ewens 197; Watterson 1975). In dervng not only the frst two moments but also entre probablty dstrbutons of summary statstcs, those works lad the bass for parameter estmaton through lkelhoods as well as for the parametrc tests for whch they are well known. Our obectve here s to develop a lkelhood-based method for nferrng demographc hstory from multple summary statstcs. Classes of segregatng mutatons: Wakeley and Hey (1997) ntroduced a set of summary statstcs sutable for explorng the demographc hstory of two groups: numbers of stes polymorphc n both groups, polymorphc n only one group, or showng fxed dfferences. Our classfcaton expands thers by usng an outgroup to dstngush between ancestral and derved bases (see materals and methods). Wth respect to the genes sampled from a gven group, we descrbe each segregatng derved mutaton as absent (a), segregatng (s; present n at least one but not all), or fxed (f; present n all). A ont classfcaton wth respect to two groups comprses only seven types because segregatng mutatons are nether absent nor fxed n both groups. Genetcs 171: (November 005)

2 140 S. C. Leman et al. TABLE 1 Segregatng mutatons Type Group 1 Group per a /pse b per/bog c bog/pse 1 Segregatng Absent Fxed Absent Absent Segregatng Absent Fxed Segregatng Segregatng Fxed Segregatng Segregatng Fxed a D. persmls, 13 sequences. b D. p. pseudoobscura, 19 sequences. c D. p. bogotana, 13 sequences. Dstngushng between the effects of recent dvergence and ntrogresson after dvergence on the sharng of polymorphsms between closely related speces can be dffcult, partcularly n sngle-locus analyses (Wang et al. 1997). To address the tme snce dvergence alone, we chose to study varaton at DPS00, a noncodng regon for whch tght lnkage to a paracentrc nverson strongly assocated wth multple reproductve solatng mechansms precludes ts ntrogresson between the closely related Drosophla speces Drosophla pseudoobscura and D. persmls (Noor and Smth 000; Noor et al. 001a,b) (see materals and methods). Table 1 shows the array of mutatons observed n DPS00 haplotypes sampled from D. persmls and two subspeces of D. pseudoobscura, D. p. pseudoobscura and D. p. bogotana (Machado et al. 00). Consstent wth the absence of ntrogresson at DPS00, the D. persmls sample shares no mutatons wth the D. p. bogotana sample and only one ( f/s) wth the D. p. pseudoobscura sample. In contrast, the two subspeces of D. pseudoobscura share 10 mutatons, suggestng recent dvergence or ongong gene flow. In the nterspecfc comparson nvolvng D. p. bogotana (per/bog), the observaton of both recprocal arrangements of fxed dfferences ( f/a and a/f ) ndcates that the most recent common ancestor (MRCA) of genes sampled from each speces postdates any between-speces coalescence events: the DPS00 gene genealogy shows recprocal monophyly. Whle the recprocal numbers of DPS00 polymorphsms restrcted to one speces (s/a and a/s) appear smlar between D. p. bogotana and D. persmls, D. p. pseudoobscura shows fourfold more polymorphsms than D. persmls, perhaps reflectng larger effectve populaton sze. These observatons suggest that the numbers of the varous knds of mutatons segregatng n the sample contan nformaton about genealogcal and demographc hstory. Models of specaton: Under the Dobzhansky-Muller scenaro for the orgn of speces (revewed by Turell et al. 001), genetc factors contrbutng to reproductve solaton arse durng an ntal phase of allopatry between the ncpent speces. We dentfy speces dvergence wth the onset of the allopatrc phase and not n partcular wth the orgn of genetc solatng mechansms. Specaton corresponds to a change n coalescence structure, the most recent pont at whch ancestral lneages wth descendants n dfferent speces can have coalesced. We assume that the paracentrc nverson that precludes ntrogresson at DPS00 between D. pseudoobscura and D. persmls arose as a neutral mutaton and became assocated wth solatng barrers durng the allopatrc phase. That ths second chromosome regon s homosequental n D. pseudoobscura and outgroup D. mranda (Dobzhansky and Tan 1936) ndcates that t s the D. persmls arrangement that s derved. Ths nverson appears to correspond to a fxed dfference, not only n our sample but also between the speces (Dobzhansky and Powell 1975). For smplcty, we assume nstantaneous fxaton of the alternatve gene orders n the two descendant speces, wth the nverson havng arsen mmedately before the MRCA of the sampled nverted chromosomes. Lkelhoods from summary statstcs: Maroram et al. (003) ntroduced a Markov chan Monte Carlo (MCMC) method that drectly approxmates the posteror dstrbuton of the model, obvatng the need to determne the probablty of the data. Numercally generatng genealogcal and mutatonal hstores that match the data may requre extensve smulaton. We address methods that base nferences about demographc hstory and populaton parameters (M) on a set of summary statstcs (D). Implementaton of lkelhood-based approaches entals the development of a practcal means of determnng or approxmatng probablty dstrbutons that depend on the unknown genealogcal hstory (G) of the sampled genes and the number and locaton (U ) of mutatons that occurred on t, LðMÞ ¼P M ðdþ ¼ X X P M ðd; U ; GÞ; ð1þ G U V G for V G, the set of all possble placements of mutatons on a gven G. We adopt the nfnte-stes model, under whch neutral base substtutons occur ndependently, wth the number n a gven tme nterval followng a Posson dstrbuton. Many lkelhood-based methods proceed from the pror dstrbuton of the genealogy, dependent on the populaton parameters alone, whle others, ncludng fully Bayesan approaches (Wlson and Baldng 1998), ncorporate the posteror dstrbuton of hstores gven D (revewed by Stephens and Donnelly 000). Wth respect to tme snce dvergence of populatons, exact analytcal expressons are avalable for certan smple demographc hstores (Watterson 1985; Takahata et al. 1995); however, the dervaton of closed-form solutons s ntractable for most systems of bologcal

3 Dvergence-Tme Estmaton 141 nterest. Uyenoyama and Takebayash (004) descrbed a method for the recursve determnaton of exact lkelhoods from ont probablty-generatng functons of correlated summary statstcs (compare Grffths and Tavaré 1996). Takebayash et al. (004) used ths approach to obtan a maxmum-lkelhood estmate (MLE) of the rate of recombnaton between a determnant of matng type n a flowerng plant and a closely lnked marker locus from the numbers of segregatng stes at the two loc. Whle ths method can n prncple accommodate multple summary statstcs and general evolutonary models, the computaton tme makes ts applcaton mpractcal. Lkelhoods have been approxmated by smulatng complete genealogcal and mutatonal hstores under a coalescent model and computng summary statstcs for each realzaton (for example, Fu and L 1997; Wess and von Haeseler 1998; Prtchard et al. 1999). Whle full smulaton can accommodate a wde range of evolutonary questons, analyss of even small data sets under smple models may mpose a consderable computatonal burden (Wall et al. 00; Wall 003), reflectng the exceedngly low probablty of any partcular realzaton (Stephens and Donnelly 000; Maroram et al. 003). Tavaré et al. (1997) ntroduced an acceptancereecton algorthm for the estmaton of the age of the MRCA of a sample of genes that entals smulatng a gene genealogy and acceptng the value for ths random varable on the bass of the probablty for the tree of the observed total number of segregatng mutatons. Beaumont et al. (00) used a local regresson method to approxmate the lkelhood of the actual data from realzatons wthn a certan tolerance (Estoup et al. 004; Tallmon et al. 004; Hamlton et al. 005). Alternatvely, the fxed-s approach (e.g., Hudson 1993; Depauls and Veulle 1998) entals generatng a genealogy under a standard coalescent model, randomly placng the observed total number of segregatng stes (S) on the genealogy, and determnng the fracton of outcomes consstent wth the remanng observed summary statstcs (H, for D ¼fS, Hg). Markovtsova et al. (001) showed that the dstrbuton generated by ths procedure does not n fact approxmate P M (HS), the condtonal dstrbuton of H gven S. In partcular, the genealogy should be sampled, not from a pror dstrbuton determned by the standard coalescent, but from a condtonal dstrbuton, gven S and the model parameters M. Ths dscrepancy can become problematc under large departures of the observed value of S from ts expectaton (Depauls et al. 001; Markovtsova et al. 001; Wall and Hudson 001). Approach through mportance samplng: We develop an mportance-samplng (IS) approxmaton (see Lu 001) to the lkelhood (1). We use a proposal dstrbuton to sample a genealogcal and mutatonal hstory consstent wth the observed array of seven types of segregatng stes and then correct the bas by determnng the exact probablty of the hstory. Here, a genealogcal path (G) corresponds to an ordered lst of the states assumed by the process at the nodes of the full genealogy, wthout specfcaton of the mutatons. Observaton of the segregatng stes present n a sample (D) provdes multple knds of nformaton, D ¼fD 1 ; D g; for D 1 the types and D the numbers of base substtutons observed. We rewrte (1) as LðMÞ¼P M ðdþ ¼ X X P M ðd;u ;GÞ Q G U V M ðd;u ;GÞ Q M ðd ;U D 1 ;GÞQ M ðd 1 ;GÞ; G ðþ for Q M (D 1, G), the statonary dstrbuton of genealoges compatble wth D 1, and Q M (D, UD 1, G), a heurstc dstrbuton of placements of mutatons on G compatble wth D. We approxmate ths average () by P M ðdþ 1 m X m ¼1 P M ðd;u ;G Þ Q M ðd;u ;G Þ ; ð3þ for (U, G ) ndependent and dentcally dstrbuted (..d.) samples from the proposal densty Q M (D, UD 1, G)Q M (D 1, G). Lkelhoods approxmated through ths procedure may serve as the bass of ether Bayesan or maxmumlkelhood analyses. Here, we use IS to determne MLEs of the tme snce dvergence between closely related speces of Drosophla and the effectve populaton szes of the extant and ancestral speces. MATERIALS AND METHODS Sequence nformaton: We studed the pattern of nucleotde varaton segregatng among DPS00 sequences obtaned by Machado et al. (00) from the D. pseudoobscura speces group, ncludng 19 D. p. pseudoobscura, 13 D. p. bogotana, and 13 D. persmls sequences (GenBank nos. AF AF450734). Ths regon, 940 bp n length, shows numerous sngle-nucleotde polymorphsms and varable olgonucleotde repettve motf tracts, but no detectable open-readng frames. For each ste segregatng wthn the ngroup, we assumed that the base present n the sngle D. mranda sequence represented the ancestral base. Although Machado et al. (00) localzed DPS00 wthn a fxed paracentrc nverson that dstngushes D. persmls from D. pseudoobscura, the Noor group has recently determned that t les 1.5 Mb outsde ths nverson, on the telomerc sde (M. A. F. Noor, unpublshed data). In general, recombnaton n nverson heterozygotes appears to be suppressed beyond the bounds of nverson breakponts, perhaps reflectng dsrupton of chromosome parng or producton of unbalanced gametes upon crossng over (see Navarro et al and references theren). In partcular, DPS00 has been shown to be tghtly lnked to the nverted regon n nverson heterozygotes (0/357 recombnants) (Noor and Smth 000).

4 14 S. C. Leman et al. TABLE Incompatbltes Type Group 1/group Incompatble Compatble 1 s/a None All f/a s/s, s/f a/f or f/s 3 a/s None All 4 a/f s/s, f/s f/a or s/f 5 s/s f/a, a/f f/s or s/f 6 f/s a/f, s/f f/a or s/s 7 s/f f/a, f/s a/f or s/s In the present study, we assume complete lnkage of stes wthn DPS00 and of DPS00 to the nverson. Assocaton of the nverson wth multple reproductve solatng mechansms, ncludng hybrd male sterlty, hybrd nvablty, hybrd male courtshp dysfuncton, and behavoral solaton, prevents ntrogresson of lnked regons, ncludng DPS00 (Noor and Smth 000; Noor et al. 001a,b). Classfcaton of mutatons: Our assumpton of the absence of ntragenc recombnaton entals that all stes wthn a locus share a sngle genealogcal hstory. Ths common hstory constrans the observed array of neutral, ndependent mutatons, n ¼ðn 1 ; n ;... ; n 7 Þ; for n the number of mutatons of type (Table ). For example, the observaton of a mutaton fxed n group 1 and absent from group mples a topology n whch the MRCA of the haplotypes sampled from group 1 postdates all between-group coalescence events. Ths topology excludes the presence n group of mutatons segregatng n group 1(s/s and s/f ). Wth the excepton of types 1 and 3, the presence of each mutatonal type excludes the presence of two other types, mplyng observaton of a maxmum of four dstnct types of mutatons. Fgure 1 llustrates the four possble dstngushable topologes (modulo recprocals) assumed by the sample genealogy: ff/a, a/fg, ff/s, f/ag, f f/s, s/sg, and fs/sg. Summary statstcs: We used CLUSTAL-W (Thompson et al. 1994), wth mnor manual modfcatons, to generate a multple algnment of the 46 DPS00 sequences. After elmnaton of stes wth gaps n any of the sequences, 89 homologous stes remaned. For each of three trads (outgroup D. mranda and a par of ngroup taxa), we restrcted attenton to stes at whch more than one base segregated wthn the Fgure 1. Possble topologes of gene genealoges of a nonrecombnng regon. Brackets below the trees ndcate sequences sampled from the same group. Branch labels ndcate branch type (Table ). ngroup taxa under consderaton. In accordance wth an nfnte-stes model of mutaton, we desgnated the base n the outgroup sequence as ancestral and any other base as a mutaton. After removng stes at whch the ancestral base was absent from both ngroup taxa, we counted the number of mutatons n each category shown n Table 1. For stes at whch more than two bases segregated, both derved bases contrbuted to our mutaton counts: for example, at a ste at whch C was desgnated ancestral, C and G segregated n the sample from one ngroup taxon, and C and T n the other, we counted each mutaton (G and T) as segregatng n one group and absent n the other. A Boperl (Stach et al. 00) scrpt for countng the varous types of mutatons s avalable from ts author on request ( J.E.S.: ason.stach@duke.edu). METHODS OF ESTIMATION We present an evolutonary model for the demographc hstory of a sample of haplotypes from two speces. We then construct a recurson n the exact probablty of the observed array of the seven knds of segregatng stes under ths model and descrbe ts mportance-samplng approxmaton (3). Evolutonary model: We assume that the tme snce dvergence of speces 1 and from ancestral speces 0 follows an exponental dstrbuton wth parameter l, treatng l as the rate of speces fuson. Durng the nterval spanned by the gene genealogy, speces ( ¼ 0, 1, ) mantans a constant effectve sze of N genes (for autosomal regons, twce the effectve number of ndvduals). At any pont wthn the gene genealogy of a sample, we classfy each lneage accordng to ts speces membershp and the dstrbuton of ts descendants between the two groups of the ntal sample. A type 1 lneage has descendants among the genes sampled from speces 1 but not from speces, and type has those from speces but not from speces 1. Type 3 lneages have descendants n both groups. On level l of the gene genealogy (the nterval n whch a total of l ancestral lneages reman), the state or confguraton of the process corresponds to (l 01, l 0, l 03, l 11, l ), for l, the number of ancestral lneages of type ( ¼ 1,, 3) n speces ( ¼ 0, 1, ). Specaton corresponds to a change n populaton structure: transton from two solated groups to a sngle panmctc group. Our assumpton of the nstantaneous fxaton of the alternatve chromosomal types upon specaton entals that postspecaton coalescence events occur at rate l =N ( ¼ 1, ), n whch l ¼ 0 forl,: Our specaton scenaro stpulates the orgn of the nverson mmedately before the MRCA of the D. persmls

5 Dvergence-Tme Estmaton 143 subsample. For cases n whch ts orgn predated the specaton event, we assume that t segregated n the ancestral speces at frequency p. Durng the nterval between the specaton event and the MRCA of the nverted chromosomes, coalescence among type 1 lneages occurs at rate l 01 =pn 0 and among type lneages at rate l 0 =ð1 ÿ pþn 0 ; wth no between-type coalescence events. Upon the orgn of the nverson, all pars of lneages coalesce at the same rate (1/N 0 ), rrespectve of type. A genealogcal path G corresponds to an ordered lst of descrptons of the nodes of the full genealogy, wthout specfcaton of mutatons or wthn-level transtons, G ¼ðS L ; S Lÿ1 ;...;S ; S 1 Þ; ð4þ for S l ; the entry (most recent) state on level l, and L total sample sze. For a sample comprsng L 1 haplotypes from speces 1 and L from speces (L ¼ L 1 1 L ), the ntal state S L corresponds to (0, 0, 0, L 1, L ) and the MRCA of the entre sample S 1 to (0, 0, 1, 0, 0). Characterzaton of the statonary dstrbuton of genealogcal paths requres only determnaton of Markov matrces of wthn- and between-level transton rates (appendx a). We extend the procedure ntroduced by Wuf and Donnelly (1999) to condton the gene genealoges proposed n the IS procedure to have a topology compatble wth the observed combnaton of mutatonal types (appendx b). Restrctng samplng to compatble genealoges affords a consderable ncrease n effcency for a modest computatonal cost. Exact recurson n lkelhoods: We derve a recurson n the ont probablty-generatng functon (PGF) for the array of summary statstcs. Array of mutatons n the sample: For each confguraton on level l, we determne a PGF for the array of mutatons accumulated n the subtree extendng from level l to the MRCA. Let g l ðaþ denote the vector of these PGFs, for a ¼ða 1 ; a, a 3, a 4, a 5, a 6, a 7 ) comprsng PGF parameters correspondng to the seven types of segregatng mutatons (Table 1). Fgure llustrates that the total number of mutatons accumulated wthn the subtree begnnng on level l corresponds to the sum of numbers accumulated on level l and n the subtree begnnng on level l ÿ 1: g l ðaþ¼r l ðaþg lÿ1 ðaþ: ð5þ appendx a presents the dervaton of R l ðaþ; the PGF of mutatons accumulated wthn level l. Recurson (5) has ntal condton Fgure. Recurson n probablty-generatng functons. Mutatons n the subtree extendng from the MRCA to level l comprse those occurrng wthn level l and those n the subtree extendng to level l ÿ 1. Independence of the mutatonal process n these dsunct tme perods mples that the PGFs of mutaton numbers to level l correspond to the product of PGFs of mutatons to level l ÿ 1 and wthn level l (5). g 1 ðaþ[1; reflectng that mutatons n the MRCA do not segregate n the sample. The PGF of the numbers of the seven types of mutatons observed n a sample of sze L corresponds to g L ðaþ¼ YL l¼ R l ðaþ; n whch the matrx product begns on the left wth the largest ndex value. Recursve computaton of exact lkelhoods: We determne lkelhoods from a recurson n probabltes rather than n the PGFs themselves. Takng dervatves of (5), we obtan an expresson for the probablty of observng the array p ¼ðp 1 ; p,..., p 7 ) of mutatons n the subtree extendng from level l to the MRCA, g ðpþ l ð0þ Q ¼ X p! q R ðqþ l ð0þ lÿ1 ð0þ Q Q q! ðp ÿ q Þ! ; g ðpÿqþ ð6þ n whch q denotes the array of mutatons that arose wthn level l and p ÿ q the remanng mutatons; the summaton n q runs over all possble subsets of the total array p; and superscrpts of the form (p) ndcate the order of dervatves wth respect to the parameters representng the mutatonal types (the p th dervatve wth respect to a, ¼ 1,,..., 7). We ntalze the recurson n probabltes (6) by consderng all possble assgnments of mutatons to level, g ðqþ ð0þ= Y q! ¼ R ðqþ ð0þ= Y q!; for q, a subset of the observed mutatons n. We then determne g ðpþ 3 ð0þ= Q p! for all possble mutatonal arrays p that can occur n the subtree comprsng levels and 3. Ths recurson ends wth g ðnþ L ð0þ= Q n!; the probablty of the array of mutatons observed n the ntal sample.

6 144 S. C. Leman et al. Importance-samplng approxmaton: To approxmate (), we frst sample from an analytcal statonary dstrbuton a genealogcal path G (4) consstent wth the types of mutatons observed n the sample (D 1 )as well as the specaton scenaro, place the observed numbers of mutatons n along G accordng to a heurstc dstrbuton, and then correct the bas ntroduced by the proposal, usng the exact probablty of the genealogcal and mutatonal hstory. Genealogcal path: Gven entry state S l on level l, the statonary dstrbuton of the level l ÿ 1 entry state S lÿ1 s gven by the correspondng row of Ṽ l 1Ũ l Ṽ l 1Ũ l Ṽl 1 ¼½I ÿ Ũ l Š ÿ1 Ṽ l ; ð7þ for Ũ l and Ṽ l matrces of rates of wthn- and betweenlevel transton probabltes (appendx a), condtonal on the observed types of mutatons (D 1 ; appendx b). Begnnng wth the state of the ntal sample (0, 0, 0, L 1, L ), we construct a genealogcal path G by samplng, for each successve level untl termnaton n the MRCA, a path segment from the row of (7) that corresponds to the entry state for the level. Proposal dstrbuton: Gven a genealogcal path G,we propose placements of the observed mutatons accordng to a multnomal dstrbuton, Q M ðd ;U D 1 ;G Þ¼ Y7 ¼1 n! YL l¼ ðr l; Þ n ;l ; n ;l! n whch n,l represents the number of type mutatons on level l ( P n ;l ¼ n ) and r l, the probablty that a lneage on level l receves a type mutaton. For e l, the number of lneages on level l that are elgble to receve mutatons of type, e l;w l r l; ¼ P Lk¼ ; e k; w k ð8þ n whch w l represents the relatve weght assgned to level l. Weght w l reflects the expected duraton of level l, whch has an exponental dstrbuton wth parameter correspondng to the rate of coalescence wthn the level (appendx c). For the path segment correspondng to level l, we obtan the true probablty P M (D, U,l, G,l ) from the element of R ðqþ l ð0þ= Q q! (A8) n the row and column assocated wth the entry (most recent) states on levels l and l ÿ 1, respectvely, for q, the array of mutatons assgned to level l. Lkelhood functon: Grffths and Tavaré (1994) descrbed an mportance-samplng procedure for generatng lkelhoods of arbtrary models (M) from those obtaned under a partcular drvng model (M 0 ). Ths approach entals frst ntensvely samplng genealogcal paths and placements of mutatons under M 0 and then characterzng the entre lkelhood functon by rescalng the probabltes (see Grffths and Tavaré 1994; Kuhner et al. 1995; Felsensten et al. 1999; Stephens and Donnelly 000). For (U, G )..d. samples from Q M0 ðd ; U D 1 ; G ÞQ M0 ðd 1 ; G Þ; we approxmate the lkelhood by LðMÞ¼P M ðdþ¼ X G ff 1 m TABLE 3 D. p. bogotana/d. persmls dvergence X m ¼1 Parameter Unconstraned N 0 ¼ N 1 ¼ N l/u un un un Lkelhood ÿ ÿ6 P-value 0.83 X P M ðd;u ;GÞ Q M0 ðd;u ;GÞ Q ðd M0 ;U D 1 ;GÞQ M0 ðd 1 ;GÞ U VG P M ðd;u ;G Þ Q M0 ðd;u ;G Þ : Because the choce of M 0 affects the relablty of the approxmaton (Kuhner et al. 1995, 1998; Stephens and Donnelly 000), we frst estmate the MLE through a two-ter search (appendx d) and then sample ntensvely under ths drvng model. APPLICATION We began our exploraton wth a comparson between D. p. bogotana and D. persmls. For ths smaller data set (Table 1, per/bog), determnaton of both the exact lkelhoods by recursve computaton (6) and ther mportance-samplng approxmatons was n fact feasble. Havng establshed a bass for confdence n our IS mplementaton, we then addressed the estmaton of populaton parameters from the comparson between D. p. pseudoobscura and D. persmls (Table 1, per/pse). That the nverson dfference appears to be fxed between the speces as well as n our sample (Dobzhansky and Powell 1975) suggests the absence or segregaton n very low frequency (p) of the derved gene order n the ancestral speces. In accordance wth ths expectaton, prelmnary results (not shown) ndcated hgher lkelhoods of lower values of p. As the data set contans lttle nformaton about ths aspect, we arbtrarly assgned p as n estmatng the remanng parameters. Comparson of D. p. bogotana and D. persmls: We used our two-phase search procedure (appendx d) to locate the maxmum-lkelhood values of the four parameters of the system (l/u, un 0, un 1, un ). Our IS approxmatons compare well to the exact lkelhood computed usng (6). Maxmum-lkelhood estmates: Table 3 provdes the MLE array ( Unconstraned column), wth probabltes estmated usng (3), based on a sample of

7 Dvergence-Tme Estmaton 145 Fgure 3. Exact and IS-approxmated condtonallkelhood functon of the nstantaneous rate of specaton (l/u), wth un 0, un 1, and un assgned to ther MLE values. Usng the unconstraned MLEs (Table 3) as the drvng values, we generated 18 IS curves, each based on 500,000 sampled genealoges. Red ndcates the exact lkelhood surface of l/u, computed usng (6) under the same assgnments of the other parameters, and blue shows the average of the IS curves ( total samples). genealogcal paths. A lkelhood-rato test detected no sgnfcant dfferences n effectve populaton sze among the speces (comparson to N 0 -N 1 -N ). Comparson of exact and IS lkelhood curves: To assess the accuracy of our IS approxmaton, we constructed condtonal-lkelhood curves for the specaton rate (l/u) wth the remanng parameters assgned to ther MLE values (Table 3) under both the exact recurson (6) and our IS method. Fgure 3 presents the exactlkelhood functon and 18 IS curves, each based on 500,000 sampled genealoges generated usng the MLEs as the drvng values. The average of the IS curves (blue lne), based on a total of samples, corresponds well to the exact condtonal lkelhood (red lne), although t somewhat underestmates the absolute value of the lkelhood. Computaton usng (6) of a sngle pont of the exactlkelhood functon requred 4 hr on a Macntosh PowerPC G5 (.5-GHz processor, 3.5 GB DDR SDRAM). Constructon of the 00 ponts consttutng the entre condtonal-lkelhood curve n Fgure 3 requred 800 hr usng the exact recurson, compared to 30 mn usng our IS method based on 500,000 sampled genealogcal hstores. Whle computaton tme under the exact recurson ncreases roughly wth the product of the numbers of mutatons observed (elements of n), observaton of more mutatons has vrtually undetectable effect on computaton tme under IS. Fgure 4 compares the exact log-lkelhood functon for the specaton rate (l/u) computed under (6) to the Fgure 4. Exact and IS-approxmated log-lkelhood functons of specaton rate (l/u) under the same condtons as ndcated for Fgure 3. The red lne represents the log of the exact-lkelhood functon, scaled to ts maxmum value, and the blue lne shows our IS approxmaton, based on sampled genealoges. IS approxmaton. Our IS curve provdes an excellent ndcaton of both the MLE and the breadth of the lkelhood functon. As noted by Stephens and Donnelly (000), error of the IS approxmaton generally ncreases farther from the drvng value. In the case studed here, however, detectable dscrepances arse only n regons qute dstant from the MLE, well beyond the 95% confdence range spanned by log-lkelhood unts. Comparson of D. p. pseudoobscura and D. persmls: The nterspecfc comparson nvolvng D. p. pseudoobscura (Table 1) comprses more sequences (longer genealoges) and many more mutatons (87 compared to 4). Both factors, but especally the ncrease n mutaton number, render computaton of the exact-lkelhood functon mpractcal. Maxmum-lkelhood estmates: Table 4 provdes maxmumlkelhood estmates of the populaton parameters, determned by our IS method usng sampled genealoges. Lkelhood-rato tests suggest that the effectve sze of D. p. pseudoobscura (un ) sgnfcantly exceeds those of both D. persmls (un 1 ) and the ancestor (un 0 ). Profle-lkelhood curves: Varous approaches exst for conveyng a sense of the level of confdence n the estmate of a parameter n a multple-parameter model (e.g., Berger et al. 1999). Wthn the maxmum-lkelhood framework adopted here, we have chosen to present our results n terms of the profle lkelhood, under whch the lkelhood of a gven value of a parameter corresponds to the maxmum acheved over all assgnments of the other parameters. appendx e descrbes our procedure for approxmatng the full four-dmensonal lkelhood surface usng nterpolatng splnes.

8 146 S. C. Leman et al. TABLE 4 D. p. pseudoobscura/d. persmls dvergence Parameter Unconstraned N 0 ¼ N 1 N 0 ¼ N N 1 ¼ N l/u un un un Lkelhood ÿ ÿ ÿ ÿ9 P-value ÿ ÿ4 Fgure 5 shows the profle-lkelhood curve for the relatve rate of specaton (l/u), and Fgure 6 shows the relatve effectve sze of D. persmls (un 1 ). Table 5 provdes 90% confdence ntervals for our MLEs, determned under the assumpton of a x -dstrbuton for the log-lkelhood. DISCUSSION We have descrbed a lkelhood-based method for the estmaton of populaton parameters. Ths frst applcaton addresses the tme snce dvergence of closely related speces n the D. pseudoobscura group. More mportantly, t serves as a proof-of-concept demonstraton of the speed and relablty of our approach, based on summary statstcs rather than on entre nucleotde sequences. Dvergence between D. pseudoobscura and D. persmls: Our maxmum-lkelhood analyss ndcates a sgnfcantly larger effectve populaton sze for D. pseudoobscura than for D. persmls (Table 4) and a specaton event somewhat more ancent than some prevous estmates. Table 6 presents our estmates (Table 5) scaled to the rate of mutaton per klobase (ũ ¼ u1000=89; for our 89-bp regon). Numbers for effectve populaton sze correspond to ũn ( ¼ 0, 1, ) and those for dvergence tme to ũ=l; the expectaton of the exponentally dstrbuted specaton-tme varable. Calbraton of mutaton rate: Rescalng of our estmates nto unts of numbers of years or ndvduals requres determnaton of the rate of substtuton of neutral mutatons n noncodng regons. For the same Drosophla speces studed here, Hey and Nelsen (004) (HN) estmated ÿ6 mutatons per klobase per year. Ths number, an average across 14 regons for whch ther analyss ndcated substantal heterogenety, reflects dvergence n both codng and noncodng regons and both synonymous and nonsynonymous substtutons. For mammals (humans and rodents), Bustamante et al. (00) estmated a 70% reducton n the rate of substtuton at slent stes n expressed genes compared to ther homologous pseudogenes. Ths accelerated substtuton n pseudogenes, partcularly marked n genes wth hgh GC content, reflects n part the release from selectve constrants on hypermutable CpG dnucleotdes (Sved and Brd 1990). In prmates, Subramanan and Kumar (003) found a sgnfcant overall excess n neutral Fgure 5. Profle-lkelhood functon for the scaled nstantaneous rate of specaton (l/u). Fgure 6. Profle-lkelhood functon for the scaled effectve populaton sze of D. persmls (un 1 ).

9 Dvergence-Tme Estmaton 147 TABLE 5 MLEs and confdence ntervals for D. p. pseudoobscura/ D. persmls dvergence Parameter MLE a 90% confdence ntervals b l/u 0.1 (0.0, 0.46) un (0.1, 5.0) un 1.71 (1.5, 5.75) un 18.1 (8, ) c a Based on sampled genealoges. b From profle lkelhood. c Beyond bounds of approxmated surface. substtuton n exons compared to noncodng regons (pseudogenes, ntrons, and ntergenc tracts), whch dsappeared upon excluson of CpG stes. Whle Drosophla genomes do not show sgnfcant CpG defcency (Gentles and Karln 001), selectve constrants on codon usage do nfluence the pattern of substtuton at synonymous stes (Akash 1995). Usng the skew n base composton as an ndex of codon usage bas, Tamura et al. (004) (TSK) nferred a neutral substtuton rate of ÿ5 mutatons per klobase per year for recently dverged (5.1 mllon years) Drosophla speces. Effectve populaton sze: We dentfy Wrght s nbreedng effectve number of ndvduals (Crow and Dennston 1988) wth half the nverse of the rate of coalescence (half the expected number of generatons to coalescence between a random par of genes; see Slatkn 1991). Our estmates of scaled populaton sze (un ), a rato of rates, have unts of mutatons per coalescence. Dvson of these estmates by twce the rate of mutaton per generaton converts the scale to generatons per coalescence, whch we dentfy wth effectve numbers of ndvduals. Table 6 reports absolute effectve populaton szes (mllons of ndvduals), obtaned under the assumpton of four generatons per year (Schaeffer 1995) and HN and TSK mutaton rates. Under the TSK rate, our analyss suggests an effectve populaton sze for D. p. pseudoobscura of , wth TABLE 6 Maxmum-lkelhood estmates Absolute (310 6 ) Parameter Scaled a HN b TSK c Dvergence tme Ancestral effectve sze D. persmls effectve sze D. p. pseudoobscura effectve sze a Mutatons per klobase. b ÿ6 mutatons/kb/year. c ÿ5 mutatons/kb/year the lower bound of the 90% confdence nterval, comparable to Schaeffer s (1995) estmates ( , ), based on the Adh regon. Hey and Nelsen s (004) estmate obtaned from DPS00 alone corresponds to and under the HN and TSK rates, respectvely. Credble ntervals, presumably obtanable from ther method, were not reported. For the effectve populaton sze of D. persmls, we obtaned an MLE of , wth 90% confdence nterval (0.5, 1.17). Hey and Nelsen s (004) value based on DPS00 alone ( u 3 u S n ther Table 1) corresponds to and under the HN and TSK rates, respectvely (credble ntervals not provded). Dvergence tme: Under the TSK mutaton rate, our MLE of the tme snce specaton between D. pseudoobscura and D. persmls corresponds to 850 thousand years (KY), dentcal to the number obtaned by Tamura et al. (004) from ther moments-based analyss of nuclear proten-codng genes located throughout the genome. Our estmate shows an nsgnfcant excess over that of Aquadro et al. (1991) (500 KY), based on restrcton ste dfferences at the Amy regon calbrated by DNA-DNA hybrdzaton data. Hey and Nelsen (004) based ther ML analyss on MCMC reconstructon of the gene genealogy of entre nucleotde sequences from 14 genomc regons. It permts varaton among regons n rates of substtuton and ntrogresson, but does not accommodate the populaton substructurng nduced by lnkage of DPS00 to the D. persmls nverson, whch prevents ts ntrogresson. Ther estmates of dvergence tme scaled to substtuton rate (t 3 u S, analogous to our ũ=l) vary over a 13-fold range across the 14 regons, and those of absolute dvergence tme vary 38-fold. Ther overall estmate of absolute dvergence tme (589 KY) corresponds to the average scaled dvergence tme dvded by the average substtuton rate. Hey and Nelsen s (004) estmate of dvergence tme based on DPS00 alone exceeds the average across the 14 regons by.86%. Ths number corresponds to 1113 KY under the average substtuton (HN) rate and to 536 KY under the TSK rate. Expressed on the same scale, our MLEs (l/u ¼ 0.1, un 0 ¼ 0.81, un 1 ¼.71, un ¼ 18.1) show nonsgnfcant dfferences from thers for DPS00 alone (0.19, 0.9, 1.9, 17.), n whch we assocate ther number for dvergence tme wth the nverse of the parameter of the exponentally dstrbuted specaton-tme varable. Approaches to estmaton: Basng the estmaton on summary statstcs rather than on entre nucleotde sequences permts consderable smplfcaton of the descrpton of genealogcal hstory. Streamlnng of the evdentary and computatonal bass affords greater computatonal and analytcal freedom to address more realstc demographc scenaros and bologcal processes.

10 148 S. C. Leman et al. Genealoges as nusance parameters: Our prmary nterests le n the characterzaton of the evolutonary process of specaton and genetc dvergence. In ths context, the genealogy of the sampled genes represents a nusance parameter, an unknown aspect that nfluences the estmaton of populaton parameters but that holds lttle nterest n tself. More precsely, the gene genealogy s not a parameter but another manfestaton of the evolutonary process under study (Donnelly and Tavaré 1995; Stephens 001). To accommodate some of the dversty of evolutonary processes, a number of methods ental estmaton of a great many parameters. For example, full genealogcal reconstructon n structured populatons requres estmaton of the ages of all nodes, mutatons, mgraton events, changes n populaton sze, and populaton dvergences (Beerl and Felsensten 001; Nelsen and Wakeley 001; Wlson et al. 003; Hey and Nelsen 004). A gven bologcal system may correspond to a set of parameter assgnments wthn the full model (for example, Yang 1996; Hey and Nelsen 004). However, a model that accommodates the unque orgn of the second chromosome nverson that prevents ntrogresson at the DPS00 regon s not nested wthn avalable data analyss packages. Grffths and Tavaré (1996) showed that the varance of estmates of populaton parameters obtaned from full genealogcal reconstructon of entre nucleotde sequences tends to be smaller than that based on summary statstcs. Even so, one may well prefer smpler methods based on summary statstcs n cases for whch they are nearly suffcent for the estmaton of the populaton parameters of nterest (Maroram et al. 003). The development and mplementaton of analyses capable of accommodatng addtonal evolutonary processes can demand lterally years of effort (Felsensten et al. 1999), and even entre sequences may contan nsuffcent nformaton to support the estmaton of fully resolved gene genealoges as well as all parameters of a heavly parameterzed model (Wuf 003). A related observaton s that less detaled models can sometmes generate more accurate estmates (Takahash and Ne 000; Pontkvska 004; Kosakovsky Pond and Frost 005). Reflectng our nterest n populaton parameters rather than gene genealoges themselves, our method adopts a much-condensed genealogcal descrpton. A genealogcal path n our analyss corresponds to an ordered lst of lneage types assocated wth the nodes of the gene genealogy (4). It dffers, n partcular, from the genealogcal hstory of Grffths and Tavaré (1994), for whch the state space ncludes the mutatons. Reducton of the computatonal burden nvested n estmatng genealogy may permt analyss of more realstc bologcal processes or demographc hstores for whch full genealogcal reconstructon of entre nucleotde sequences may be altogether nfeasble. For the applcaton at hand, our model explctly condtons the genealogcal hstores on the unque orgn of the nverson that prevents ntrogresson n the genomc regon studed (appendx b). Incorporaton of ths bologcal nformaton nto the estmaton procedure entaled only modfcaton of Markov matrces of rates of wthn- and between-level transtons (appendx a). Ths structural flexblty may permt customzaton of the analyss to a wde varety of bologcal systems. Estmaton of dvergence tmes: Speces dvergence corresponds to a change n coalescence structure: the most recent pont at whch ancestral lneages wth descendants n dfferent speces can have coalesced. Usng the age of the most recent node wth descendants sampled from dfferent populatons as a surrogate for dvergence tme generates neglgble error only for ancent dvergence events nvolvng small ancestral populaton szes and lttle nterpopulaton gene flow (see Nchols 001). A number of recent revews of moment- and lkelhood-based approaches have addressed the estmaton of populaton dvergence apart from node age (Arbogast et al. 00; Rosenberg and Feldman 00; Takahata and Satta 00). Many lkelhood-based methods related to ours treat tme snce specaton as a parameter. Takahata et al. (1995), Rannala and Yang (003), and Wall (003) based the estmaton of dvergence tme and ancestral populaton sze on numbers of segregatng stes n a present-day sample comprsng one sequence from each of two or more speces. Nelsen and Wakeley (001) and Hey and Nelsen (004) used MH samplng to approxmate the posteror dstrbuton of fully resolved gene genealoges, ncludng all node ages and tme snce specaton. To ncorporate tme nto the method of Grffths and Tavaré (1994), for whch the genealogcal hstores record only the relatve order of events, Nelsen (1998) determned the probablty dstrbuton of the numbers of ancestral lneages remanng n each group at the dvergence event. Our method characterzes the tme snce specaton as an exponentally dstrbuted random varable and estmates the nstantaneous rate of specaton (l/u). Ths constructon obvates the need to ncorporate tme nto the backward constructon of genealogcal hstores. Unque evolutonary events: Slatkn and Rannala (1997, 000) based the estmaton of the age of an advantageous or deleterous mutaton on the relatve magntudes of neutral varaton segregatng wthn the affected and unaffected subsamples. The sample genealogy reflects coalescence of affected lneages only among themselves, wth the number of ancestors from whch they could have descended determned from a branchng-process model. In contrast, Wuf and Donnelly (1999) addressed the age of a neutral marker restrcted to a subset $ of genes sampled from a populaton. Determnaton of the lkelhood entals

11 Dvergence-Tme Estmaton 149 condtonng a random gene genealogy to contan a node from whch all members of $ and none of the other sampled genes descend and requrng the occurrence of exactly one neutral mutaton on the branch mmedately ancestral to that node. These approaches dffer wth respect to more than statstcal phlosophy. In the latter case, the parttonng of the sample arses only after observaton of the mutaton of nterest, whle n the former, the dstncton between affected and unaffected genes exsts before observaton of the sample. We chose to study the DPS00 regon precsely because tght lnkage to a second chromosome nverson precludes ts ntrogresson. We have mposed the smplfyng assumpton that the orgn of the nverson occurred mmedately before the MRCA of the nverted lneages. Further, for cases n whch the MRCA of the nverted lneages predates the specaton event, our model assumes a constant frequency (p) of the nverson n the ancestral speces and permts coalescence only wthn and not between gene orders. A more detaled analyss would ncorporate a descrpton of evolutonary change n p (e.g., Slatkn and Rannala 1997; De Ioro and Grffths 004). Importance samplng: The hgh complexty of vrtually all bologcal systems of nterest ensures that any partcular realzaton of the evolutonary process occurs wth extremely low probablty, makng approxmaton of lkelhoods by nave Monte Carlo smulaton mpractcal (Stephens and Donnelly 000). Importance samplng offers a means of compensatng for dscrepances generated by samplng from convenent but ncorrect proposal dstrbutons. As dscussed n the ntroductory secton, sprnklng the observed number of mutatons over a random genealogy under the fxed- S procedure very rapdly generates genealogcal and mutatonal hstores consstent wth the data, but approxmates an ncorrect dstrbuton (Markovtsova et al. 001). Our method () proposes genealogcal paths (4) by samplng, not from a posteror dstrbuton gven the full data (D ¼ fd 1, D g), but from an analytcally determned statonary dstrbuton (7) of paths consstent wth the types of segregatng mutatons observed (Q M (D 1, G)). It then sprnkles the observed numbers of mutatons on the genealogcal path (Q M (D, UD 1, G)) accordng to a heurstc weghtng scheme (8). We then correct the bas ntroduced by the proposal dstrbuton usng the exact probablty of the proposed genealogcal and mutatonal hstory (appendx a). Expanson of the evdentary bass to nclude addtonal summary statstcs would extract more nformaton from the sampled sequences. Through straghtforward redefnton of the state space, our method can ncorporate counts of mutatons of varous knds, ncludng numbers of mutatons classfed accordng to ther dstrbuton among groups (Wakeley and Hey 1997; Wakeley et al. 001), the number of haplotypes and ther frequency spectrum (Ewens 197), and the frequency spectrum of mutaton numbers (Fu 1995). Our approach s less well suted to summary statstcs defned as varous moments, ncludng average parwse dfferences, varances, regressons, and correlatons. However, the dstrbuton of mutatons among groups can replace parwse F ST values as the bass for the characterzaton of gene flow (Wakeley and Hey 1997), and the relatve numbers of segregatng stes at lnked loc can replace parwse lnkage dsequlbrum as the bass for the estmaton of recombnaton rate (Takebayash et al. 004). The smplcty of our approach (appendx a) facltates structural modfcaton both to ncorporate more nformaton contaned n the sampled sequences and to broaden the scope of evolutonary processes amenable to analyss. We thank Beatrx Jones for percevng the relevance of mportance samplng to ths problem, Mchael Lavne for comments and suggestons that mproved the analyss, Sudhr Kumar for nsghts nto rate calbraton and CpG avodance, John Wlls for remndng us of the unque evolutonary orgn of nversons, and two anonymous revewers and John Wakeley for comments. Ths research was supported n part by fundng from Natonal Scence Foundaton (NSF) [DMS (Y.C.), NSF DEB (M.A.F.N.), and an NSF predoctoral fellowshp (J.E.S.)] and by the Natonal Insttutes of Health [GM (M.K.U.)]. LITERATURE CITED Akash, H., 1995 Inferrng weak selecton from patterns of polymorphsm and dvergence at slent stes n Drosophla DNA. Genetcs 139: Aquadro, C. F., A. L. Weaver, S.W. Schaeffer and W. W. 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