The Mid-Depth Method and HIV-1: A Practical Approach for Testing Hypotheses of Viral Epidemic History

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1 The Mid-Depth Method nd HIV-1: A Prcticl Approch for Testing Hypotheses of Virl Epidemic History Oliver G. Pybus, Edwrd C. Holmes, nd Pul H. Hrvey Deprtment of Zoology, University of Oxford We introduce the mid-depth method, prcticl pproch for testing hypotheses of demogrphic history using genelogies reconstructed from sequence dt. The reltive positions of internl nodes within genelogy contin informtion bout pst popultion dynmics. We explin how this informtion cn be used to (1) test the null hypothesis of constnt popultion size nd (2) estimte the growth rte nd current popultion size of n exponentilly growing popultion. Simultion tests indicte tht, s expected, estimtes of exponentil growth rtes re sometimes bised. The mid-depth method is computtionlly rpid nd does not require knowledge of the smple s muttion rte. However, it does ssume tht the reconstructed genelogy is correct nd is therefore best suited to the nlysis of vrition-rich virl dt sets. When pplied to HIV-1 sequence dt, the mid-depth method provides phylogenetic evidence of different exponentil growth rtes for subtypes A nd B. We posit tht this difference in growth rte reflects the different trnsmission routes nd epidemiologicl histories of the two subtypes. Introduction Homologous DNA sequences smpled from popultion crry informtion bout the demogrphic history of tht popultion. Such informtion hs wide-rnging pplictions, from virology (Holmes et l. 1995) to nthropology (Hrpending et l. 1993). Here, we present simple nd computtionlly rpid method for the inference of popultion dynmic history using genelogies reconstructed from sequence dt. Our method, clled the mid-depth method, is bsed on colescent theory nd tests hypotheses by compring reconstructed genelogies ginst genelogies generted using Monte Crlo simultion. Previous ttempts to infer demogrphic history from gene sequence dt hve followed two generl pproches. The nonphylogenetic pproch involves the clcultion of summry sttistics directly from sequence dt, such s the number of segregting sites in smple (Tjim 1989) or the pirwise differences between sequences (Sltkin nd Hudson 1991; Rogers nd Hrpending 1992). In contrst, the phylogenetic pproch tkes into ccount the genelogicl reltionships of the smpled sequences. This hs been done in two wys: (1) The genelogy is ssumed to be unknown. Mximum-likelihood estimtes of demogrphic prmeters re clculted by summing or smpling over ll possible genelogies, given explicit models of sequence evolution nd popultion chnge (Griffiths nd Tvré 1994; Kuhner, Ymto, nd Felsenstein 1995, 1998). (2) The genelogy is reconstructed using stndrd phylogenetic methods nd ssumed to be correct. Demogrphic informtion is then inferred from the reconstructed tree (Nee et l. 1995). The method presented here belongs to this ltter ctegory. Felsenstein (1992) demonstrted tht estimtes of popultion size bsed on phylogenetic pproches mke Key words: colescent theory, Monte Crlo simultion, HIV-1, mximum likelihood, mid-depth method. Address for correspondence nd reprints: Oliver Pybus, Deprtment of Zoology, University of Oxford, South Prks Rod, Oxford, OX1 3PS, United Kingdom. E-mil: oliver.pybus@zoo.ox.c.uk. Mol. Biol. Evol. 16(7): by the Society for Moleculr Biology nd Evolution. ISSN: more efficient use of the dt thn do summry sttistics. Furthermore, phylogenetic methods for the estimtion of popultion growth rtes re expected to outperform methods bsed on summry sttistics when the growth rte is low or negtive (Kuhner, Ymto, nd Felsenstein 1998). Unfortuntely, phylogenetic pproches which integrte over ll possible genelogies tend to be computtionlly intensive nd complex. The mid-depth method voids these problems by seprting the tsk of demogrphic prmeter estimtion from tht of phylogenetic reconstruction (sensu Nee et l. 1995). The drwbck of our pproch is tht uncertinty bout the genelogy is not incorported into the process of prmeter estimtion (cf. Kuhner, Ymto, nd Felsenstein 1998). Insted, the reconstructed genelogy is ssumed to be correct, requiring tht the sequences in question contin enough phylogenetic signl to ccurtely infer divergence times. The effect of phylogenetic error on prmeter estimtion hs yet to be quntified; for the time being, we dvise cution nd recommend tht our method should be pplied only to rpidly evolving virl genes. In this pper, we pply the mid-depth method to humn immunodeficiency virus type 1 (HIV-1) gene sequence dt nd find evidence of different growth rtes for HIV-1 subtypes A nd B. Methods A colescent tree is hypotheticl phylogeny which describes the genelogicl reltionships between smll number of individuls smpled from lrge popultion (Kingmn 1982). Typiclly, its tips re contemporneous, nd its internl nodes (colescence events) re ordered in time (Felsenstein 1992). Here, colescent trees will be considered solely in terms of the time intervls between their colescence events. Algorithms re vilble to generte the time intervls of colescent trees smpled from constnt-sized (Hudson 1990) nd exponentilly growing (Sltkin nd Hudson 1991) popultions. Both of these lgorithms re derived from probbility models with ll of the following ssumptions: (1) no recombintion; (2) phylogeneticlly-rn- 953

2 954 Pybus et l. dom smpled sequences; (3) n K N, where n is the number of different sequences nd N is the popultion size; nd (4) Wright-Fisher reproduction (genertions re nonoverlpping, with ech offspring choosing prent t rndom nd independent of the previous genertion). The constnt-sized (endemic) model hs only one prmeter, popultion size (N) (Hudson 1990). Most colescence events in n endemic colescent tree tend to occur ner the tips (Nee et l. 1995). The exponentilly growing (epidemic) model hs two prmeters, current popultion size (N 0 ) nd exponentil growth rte (r). When both N 0 nd r re lrge, the simulted tree resembles str phylogeny, with most colescence events close to the root (Sltkin nd Hudson 1991). It is therefore resonble to predict tht s r decreses, colescence events will tend to move wy from the root towrd the tips, such tht simulted epidemic tree will come to resemble simulted endemic tree s r pproches zero. The Mid-Depth Method The bove observtions suggest tht the reltive positions of colescence events within reconstructed genelogy contin informtion bout popultion dynmic history. To extrct this informtion, the mid-depth method uses new tree sttistic, clled. We define s the number of colescence events between the root nd the mid-depth point of genelogy. The mid-depth point is the time exctly hlfwy between the tree s tips nd its root. Importntly, is independent of ny fctor which is constnt multiple cross ll intervls, such s constnt muttion rte or the popultion size of constnt-sized popultion. The internl nodes of the reconstructed genelogy must represent divergence times condition which usully requires tht the tree reconstruction method used ssumes constnt rte of moleculr evolution. However, this ssumption is not binding, s it is sometimes possible to estimte divergence times in the bsence of constnt-rte moleculr clock (see, for exmple, Snderson 1997). If the moleculr clock ssumption is incorrectly pplied, then the resulting inccurtely reconstructed genelogy my give rise to misleding demogrphic inferences. The mid-depth method lso requires tht the genelogy stisfies ll of the colescent model ssumptions listed bove. Testing the Hypothesis of Constnt Popultion Size If genelogy represents smple from constntsized popultion, then, following from the observtions bove, we expect its vlue to be close to zero. Monte Crlo simultion shows this expecttion to be correct; more thn 95% of trees simulted using Hudson s (1990) endemic colescent model hve 3, provided 2 n 200 (see fig. 1). This result cn be used to test the null hypothesis, H 0, tht smple of gene sequences hs been tken from constnt-sized popultion. Let * be the vlue of genelogy reconstructed from the smple. If * 3, then we cn reject H 0 t the 95% confidence limit. FIG. 1. Testing the hypothesis of constnt popultion size. For ech vlue of n, 10,000 endemic colescent trees were simulted using the lgorithm of Hudson (1990). The ordinte shows the proportion of simulted trees with n tips which hve 3. More thn 95% of ll simulted trees with n 200 tips hve 3. Estimting Exponentil Growth Prmeters If the hypothesis of constnt popultion size is rejected, the next step is to discover the type of popultion chnge tht hs occurred. A useful tool for this purpose is the lineges-through-time (LTT) plot, which cn be used to infer the generl mode of popultion chnge from genelogy (Nee et l. 1995; Rmbut, Hrvey, nd Nee 1997). Informtion from n LTT plot, or perhps some other externl source, my provide priori evidence tht genelogy hs been smpled from popultion growing t constnt exponentil rte. If (nd only if) such evidence exists, then the mid-depth method cn be used to estimte N 0 r, the product of the current popultion size N 0, nd exponentil growth rte r of tht popultion. The reltionship between nd ws investigted using the simultion lgorithm presented by Sltkin nd Hudson (1991). Their lgorithm implements renormlized epidemic colescent model which mesures time in units of 1/r genertions nd which hs s its only prmeter. The renormliztion is equivlent to multiplying ll time intervls by the fctor 1/r nd therefore does not ffect the vlue of. Figure 2 shows the reltionship between log( ) nd for simulted epidemic colescent trees with 30 nd 100 tips. The plot demonstrtes tht colescence events move from the tipwrd hlf of tree to the rootwrd hlf s increses. For wide rnge of vlues, there is n pproximtely liner reltionship between nd log( ), suggesting tht cn be used to estimte. When 1 or 10 6, the sttistic is poor predictor of ; ll 10 6 or 1 generte simulted trees with (n 2) or 1, respectively. As trees with 3 re consistent with the hypothesis of constnt popultion size, it is irrelevnt tht is poor estimtor of when 1. Comprison of the 30- tip nd 100-tip curves indictes tht using more smples nrrows the rnge of vlues which generte uninformtive trees with (n 2) (see fig. 2).

3 Testing Virl Epidemic Hypotheses 955 Tble 1 Expected Estimtes of *, Given 30-Tip or 100-Tip Epidemic Colescent Trees Simulted with * Log( *) E[ ˆ *] n 100 n Clculted using C progrm vilble from the uthors on request. FIG. 2. The reltionship between the tree sttistic nd the demogrphic prmeter. The ordintes represent the men of 20,000 epidemic colescent trees simulted using Sltkin nd Hudson s (1991) lgorithm for ech vlue of. The curve with circulr mrkers corresponds to the left-hnd ordinte (trees with 30 tips). Confidence intervls for ech men were clculted using 1.96 SE( ) but re not displyed, s they were too smll to be noticeble. Given reconstructed genelogy, Monte Crlo simultion cn be used to clculte mximum-likelihood estimte (MLE) of *, the vlue of the popultion from which the genelogy ws smpled. Let ˆ denote our MLE of *, nd let * be the vlue of the reconstructed genelogy. We define x s the vlue of n epidemic colescent tree simulted using Sltkin nd Hudson s (1991) lgorithm with prmeter x. P( x *), the probbility tht x *, cn be estimted by simulting lrge number of epidemic colescent trees nd then clculting the proportion of those trees which stisfy x *. P( x *) is the probbility of obtining result * given the hypothesis * x nd is therefore proportionl to the likelihood of hypothesis * x, given the observed * (Edwrds 1972). Accordingly, ˆ is equl to the vlue of x which mximizes P( x *). Likelihood curves for ˆ, which re constructed by clculting P( x *) for mny vlues of x, become smoother s the number of simulted trees used to clculte ech point increses. The likelihood curves re usully unimodl, mking loction of the MLE strightforwrd. Conditions under which the likelihood curves become flt re discussed below. Upper nd lower limits of the MLE re clculted using the likelihood rtio test (Wilks 1938; Edwrds 1972). Given dt D, the difference in the support of hypotheses H 0 nd H 1 is mx[ln L(H 0 D)] mx[ln L(H 1 D)]. If H 0 is subset of H 1, then 2 is pproximtely 2 distributed, with degrees of freedom equl to the difference in the number of free prmeters between H 0 nd H 1 (Wilks 1938). Here, H 0 is the hypothesis tht ˆ tkes specific vlue v nd is subset of the hypothesis H 1 tht ˆ is free to vry. If , then v lies outside the pproximte 95% confidence limits. The vlidity of using the likelihood rtio test in our method ws confirmed by simultion; simulted vlues of 2 were found to be pproximtely 2 distributed (results not shown). Estimtes of * Are Sometimes Bised All estimtes of exponentil growth rte bsed on phylogenetic methods re expected to be bised (Kuhner, Ymto, nd Felsenstein 1998). We therefore performed extensive simultions to test the ccurcy of the mid-depth method. These simultions involved the clcultion of E[ ˆ *], the expected vlue of ˆ given epidemic colescent trees simulted with *. If the mid-depth method is bised upwrd, then E[ ˆ *] will be lrger thn *. For trees with n tips, E[ ˆ *] (1/[n 2]) n 2 i 0 P( * i) ˆ (i). As before, P( * i) is the prob- bility tht the vlue of n epidemic colescent tree (simulted with *) is equl to i. ˆ (i) is the middepth method MLE of * given tree with i. Tble 1 shows the simultion results. As expected, the middepth method shows significnt positive bis, but only when log( *) exceeds certin vlue. This vlue is higher, nd the rnge of unbised estimtion wider, when more smples re used. Comprison of tble 1 with figure 2 shows tht the rnge of unbised estimtion pproximtely coincides with the rnge of the liner reltionship between nd. There is lso negtive bis when log( *) 1. Anlysis of the simultion results reveled tht the positive bis is lmost entirely due to * being overestimted when (n 2). If * is sufficiently lrge, then ll simulted trees hve (n 2), thus cusing the likelihood curve for ˆ to be flt on one side (see fig. 2). In other words, ll sufficiently lrge vlues of * re eqully likely explntions of the observed ; hence, the MLE of * is bised upwrd. The negtive bis found when * is smll is cused by the sme effect; if 1, the prt of the likelihood curve representing low * vlues becomes flt, bising the MLE of * downwrd. Therefore, ˆ is unlikely to be bised unless one of the following circumstnces occurs: (1) the popultion hs incresed in size very rpidly (hence, [n 2] is more likely), (2) the popultion hs been decresing in size ( 1 is more likely), (3) n is smll (there re fewer possible vlues; hence, both [n 2] nd 1 re more likely).

4 956 Pybus et l. Tble 2 Detils of the HIV-1 Dt Sets Subtype DATA SET Gene LENGTH OF SEQUENCES (bp) NO. OF SEQUENCES ˆ CONFIDENCE LIMITS OF A env gg 1, B env 2, , , ,795 gg 1, , , ,182,730 pol 1, ,279 5, ,880,000 Clculted using C progrm vilble from the uthors on request. Lower ˆ Upper Exmple: The Epidemic Growth of HIV-1 To illustrte the use of the mid-depth method, we pplied it to HIV-1 sequences smpled from round the world. Phylogenetic nlysis hs been used to clssify the virus into series of subtypes, A J, which hve differing geogrphicl distributions (see McCutchn et l. 1996). Two of the most common subtypes (nd those for which the most sequences re vilble) re subtype A, which is most commonly found in sub-shrn Afric nd in migrnts from tht region, nd subtype B, which minly circultes in the developed world. Two previous nlyses of these subtypes using LTT plots indicted tht both hve spred t roughly constnt exponentil rte, with no evidence of difference in growth rte between them (Holmes et l. 1995; Holmes, Pybus, nd Hrvey 1999). To exmine whether the mid-depth method provides different picture of popultion dynmics, we collected ll vilble env, gg, nd pol gene sequences from the Los Almos HIV dtbse (Myers et l. 1996). Six dt sets were constructed, one for ech gene from ech subtype. Complete (or nerly complete) sequences were used if more thn 20 were vilble; otherwise, the lrgest region of the gene found in pproximtely 90% of the sequences ws used. Insufficient pol sequences from subtype A were vilble for nlysis. The FIG. 3. Vlues of ˆ for ech of the HIV-1 dt sets, estimted using the mid-depth method. The error brs represent confidence limits clculted using the likelihood rtio test (see text). sequence lignments vilble in the dtbse were used for ll dt sets. Bises in smple composition ffect the reconstruction of popultion dynmics; n overrepresenttion of sequences from very closely relted individuls will bis the smple towrd recent colescence events. Therefore, sequences were removed from ech dt set if they cme from the sme ptient or were relted by direct trnsmission to nother sequence in the lignment. This informtion ws obtined from the primry literture. Furthermore, ll sequences previously identified s intersubtype recombinnts were lso removed. Detils of the finl dt sets used re given in tble 2. Genelogies were estimted for ech dt set using mximum- likelihood pproch. Tree likelihoods were computed using the Hsegw-Kishino-Yno (HKY-85) substitution model nd codon- position-specific model of rte heterogeneity, under the ssumption of constnt-rte moleculr clock. Due to the lrge dt sets used, simultneous coestimtion of the tree topology nd model prmeters ws not possible. Therefore, the model prmeters were estimted on n initil neighbor-joining tree. Tree topologies were then evluted using heuristic serch pproch which implemented both tree bisection-reconnection nd nerest-neighbor interchnge perturbtions. Finlly, the brnch lengths of the mximum-likelihood topology were re-estimted using the generl reversible (REV) substitution model. The finl phylogenies re shown in figure 4. All nlyses were performed with bet test version of PAUP4 kindly provided by Dvid Swofford. Substitution model prmeter vlues nd sequence ccession numbers re vilble from the uthors on request. The End-Epi computer pckge (Rmbut, Hrvey, nd Nee 1997) ws used to nlyze the LTT plots of the reconstructed genelogies. In greement with previous LTT nlyses of HIV-1 (Holmes et l. 1995; Holmes, Pybus, nd Hrvey 1999), ll of the plots indicted tht the genelogies hd been smpled from popultions growing t constnt exponentil rte (results not shown). All five reconstructed trees hd 3, nd therefore the hypothesis of constnt popultion size ws rejected for ll dt sets (see tble 2). The mid-depth method ws used to clculte ˆ nd its confidence limits for ech dt set (see fig. 3 nd tble 2). It is cler from figure 3 tht HIV-1 subtypes A nd B hve very different vlues of ˆ ; the ˆ vlues of subtype B re two to three orders of mgnitude lrger thn re those of sub-

5 Testing Virl Epidemic Hypotheses 957 FIG. 4. Estimted mximum-likelihood phylogenies for ech of the five HIV-1 dt sets. The scle brs represent expected number of substitutions per nucleotide. type A. This difference is seen in both the env nd the gg genes nd is consistent for ll genes within subtype. Interprettion of the HIV-1 Results The higher ˆ vlues for subtype B suggest tht this subtype hs either lrger popultion size (N 0 )or higher exponentil growth rte (r) thn subtype A. The former explntion seems unlikely it is estimted tht 7.5% of individuls in sub-shrn Afric ged yers re HIV infected, compred with only 0.3% in Western Europe nd 0.6% in North Americ (UNAIDS nd WHO 1997). Sub-Shrn Afric is home to two thirds of the world s HIV-infected people (Mnn nd Trntol 1998), nd it is resonble to ssume tht mny of these infections re due to subtype A viruses. So, do contrsting exponentil growth rtes explin the different ˆ vlues of subtypes A nd B? Two importnt points must be considered here. First, the growth rte prmeter r represents n intrinsic rte of increse the number of new infections per infected individul per unit time. Hence, the huge number of new infections currently reported in sub-shrn Afric could be the consequence of high prevlence rther thn high growth rte. Second, our estimted ˆ vlues reflect the growth rtes of subtypes A nd B during the strt of their spred through humn popultions; s pointed out by Sltkin nd Hudson (1991), rndom smples of se-

6 958 Pybus et l. quences re unlikely to contin informtion bout recent decreses in growth rte. Moreover, mny of our smples come from individuls who were infected erly during the HIV-1 epidemic. It is cler tht the growth rte of the virus when it first emerged in North Americ nd Europe round 20 yers go ws much higher thn it is tody. At tht time, before the dvent of widespred behviorl intervention, trnsmission ws predominntly loclized to high-risk groups, notbly homosexul men nd injecting drug users. Within these groups, rtes of contct or prtner exchnge were often so high tht fully connected chin of trnsmission stnding network ws formed, llowing the virus to spred quickly through lrge number of individuls (Jcquez et l. 1994; Wtts nd My 1992). This process ws ided by the fct tht mny individuls within the network were t their most infectious, hving only recently been infected. Such trnsmission pttern would generte very high growth rtes. In contrst, most trnsmissions in Afric occur through unprotected heterosexul intercourse, so tht the verge witing times between trnsmission events re longer thn those experienced in stnding networks (Trntol nd Schwärtlnder 1997). Thus, subtype A s initil growth rte ws probbly considerbly slower thn tht of subtype B. If we ccept tht subtype A hs greter popultion size, then the erly growth rte of subtype B must hve been t lest two orders of mgnitude fster thn tht of subtype A. Hve the Assumptions of the Method Been Met? We will now consider the robustness of our results to possible violtions of the mid-depth method ssumptions. Assuming tht the genelogies re correct, we cn identify four mjor fctors which will ffect the interprettion of our results: (1) the smple is not phylogeneticlly rndom, (2) the sequences hve recombined, (3) the sequences hve been subject to selection, nd (4) the sequences hve not evolved t constnt rte. Smples collected from epidemiologiclly linked ptients will be bised towrd recently diverged sequences, leding to underestimtes of popultion growth rtes. We minimized this problem by removing sequences from our nlysis tht re clerly from the sme source. Unless sequence dt re collected in phylogeneticlly rndom mnner, this post hoc pproch is likely to be the most efficient wy to control for smpling bis. Recombintion nd nturl selection, which re both common occurrences in HIV-1 (Leigh Brown nd Holmes 1994; Robertson et l. 1995), significntly complicte the interprettion of our results. However, of singulr importnce here is whether these two processes occur t different rtes in subtypes A nd B. Although this is difficult question to nswer, there is no reson to believe tht the observed intersubtype differences re due to recombintion or nturl selection. At present, there is no strong evidence of subtype-level nturl selection (see, e.g., Pope et l. 1997) or of high levels of intrsubtype recombintion for either subtype, lthough this surely occurs. Furthermore, ll sequences known to be intersubtype recombinnts were removed from our nlysis, nd our results were consistent cross ll genes, suggesting tht they represent popultionwide phenomen. Considertion of the constncy, or inconstncy, of the HIV-1 moleculr clock hs recently incresed following ttempts to dte the origin of the virus using strin isolted in 1959 (Korber, Theiler, nd Wolinsky 1998; Zhu et l. 1998). Although there is rte vrition mong different HIV strins (Fuci 1996), estimtes of HIV origins bsed wholly on contemporry sequences re usully close to those mde using the fossil HIV- 1 strin (see Ksper et l. 1995). This suggests tht short-term rte fluctutions re not necessrily trnslted into long-term rte heterogeneity between subtypes. More pertinently, there is no evidence tht subtypes A nd B differ in substitution rte so s to cuse the differences observed here. Discussion The mid-depth method provides strightforwrd pproch to testing hypotheses of popultion dynmic history. It differs from most other methods in using reconstructed genelogies, which re ssumed to be correct, s its rw dt. This ssumption hs benefits nd costs. Computtionl complexity is gretly reduced by seprting the tsk of phylogenetic reconstruction from tht of demogrphic hypothesis testing. This seprtion lso increses the method s flexibility for ech dt set, the most pt tree reconstruction procedure cn be chosen without modifiction of the mid-depth method. In ddition, our method nlyses reltive (rther thn bsolute) intervl sizes, llowing demogrphic prmeters to be estimted without knowledge of the smple s muttion rte (sensu Nee et l. 1995). However, prmeter estimtes nd their confidence limits do not contin informtion bout the uncertinty of the genelogy. Further work is needed to incorporte this uncertinty into the mid-depth method. Acknowledgments Mny thnks to Andrew Rmbut, Nick Grssly, Mike Chrleston, John Huelsenbeck, nd Joe Felsenstein for technicl ssistnce nd inspirtionl discussion. This work ws supported by the Wellcome Trust (grnt ) nd The Royl Society. LITERATURE CITED EDWARDS, A. W. F Likelihood. Cmbridge University Press, Cmbridge, Englnd FAUCI, A. S Host fctors nd the pthogenesis of HIVinduced disese. Nture 384: FELSENSTEIN, J Estimting effective popultion size from smples of sequences: inefficiency of pirwise nd segregting sites s compred to phylogenetic estimtes. Genet. Res. Cmb. 59: GRIFFITHS, R. C., nd S. TAVARÉ Smpling theory for neutrl lleles in vrying environment. Philos. Trns. R. Soc. Lond. B Biol. Sci. 344:

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