Admixture as a tool for finding linked genes and detecting that

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1 Proc. Ntl. Acd. Sci. USA Vol. 85, pp , December 1988 Genetics Admixture s tool for finding linked genes nd detecting tht difference from llelic ssocition between loci (linkge disequilibrium/genetic epidemiology) RANAJIT CHAKRABORTY* AND KENNETH M. WEISSt *Center for Demogrphic nd Popultion Genetics, University of Texs Grdute School of Biomedicl Sciences t Houston, P.O. Box 2334, Houston, TX 77225; nd tdeprtment of Anthropology nd Grdute Progrm in Genetics, The Pennsylvni Stte University, University Prk, PA 1682 Communicted by Willim T. Snders, My 25, 1988 ABSTRACT Admixture between geneticlly different popultions my produce gmetic ssocition between gene loci s function of the genetic difference between prentl popultions nd the dmixture rte. This ssocition decys s function of time since dmixture nd the recombintion rte between the loci. Admixture between geneticlly long-seprted humn popultions hs been frequent in the centuries since the ge of explortion nd coloniztion, resulting in numerous hybrid descendnt popultions tody, s in the Americs. This represents nturl experiment for genetic epidemiology nd nthropology, in which to use polymorphic mrker loci (e.g., restriction frgment length polymorphisms) nd disequilibrium to infer genetic bsis for trits of interest. In this pper we show tht substntil disequilibrium remins tody under widely pplicble situtions, which cn be detected without requiring inordintely dose linkge between trit nd mrker loci. Very disprte prentl llele frequencies produce lrge disequilibrium, but the smple size needed to detect such levels of disequilibrium cn be lrge due to the skewed hplotype frequency distribution in the dmixed popultion. Such situtions, however, provide power to differentite between disequilibrium due just to popultion mixing from tht due to physicl linkge of loci-i.e., to help mp the genetic locus of the trit. A grdient of dmixture levels between the sme prentl popultions my be used to test genetic models by relting dmixture to disequilibrium levels. The publiction costs of this rticle were defryed in prt by pge chrge pyment. This rticle must therefore be hereby mrked "dvertisement" in ccordnce with 18 U.S.C solely to indicte this fct. Admixture between two popultions with different llele frequencies t two loci will produce gmetic ssocition between these loci in ny dmixed popultion (1). Here, we refer to such gmetic ssocition s "mixture disequilibrium" to distinguish it from gmetic ssocition between closely linked loci. Such mixture disequilibrium will decy over time, but if the two loci re not linked, or their linkge is loose, nontrivil levels of disequilibrium my persist for long time periods. Even if recombintion between loci occurs with constnt rte, mixture disequilibrium in n dmixed popultion my remin over substntil period of time, if the history of dmixture is not very old. Chkrborty nd Smouse (2) showed tht in the presence of recombintion, estimtes of dmixture from hplotype dt my be errorprone, if the genetic ssy of the dmixed popultion is not done immeditely fter the dmixture event. There re mny instnces where humn popultions hve been formed through dmixture of the sme two stocks of rcil groups, yet the degree of dmixture vries mong the dmixed groups (with pproximtely the sme historicl depth of dmixture). Such popultions my offer n opportunity to use dmixture s tool for nthropologicl reserch. Erlier, we (3) showed the utility of using dmixed popultions for fitting genetic models of inheritnce of complex diseses. The objective of the present pper is to show tht the observed levels of disequilibrium between ny two loci in such n rry of dmixed popultions my be used to detect their linkge reltionship nd to differentite the cse of mixture disequilibrium between loci from the disequilibrium tht cn be scribed to genetic linkge. MATHEMATICAL TREATMENT Mixture Disequilibrium in n Admixed Popultion. As in the cse of trditionl dmixture models, we consider two loci (A nd B) tht re not ffected by selection. Let A nd, B nd b be the two segregting lleles t these loci, respectively. Suppose tht n dmixed popultion (Z) obtins frction (m) of its genes from ncestrl popultion X, nd frction (1 - m) from ncestrl popultion Y. We ssume tht the dmixture event hs tken plce in single pulse t genertion, nd the popultions re surveyed t genertions fter this event. This theory is discussed in more detil elsewhere (ref. 2 nd unpublished work). Let r denote the recombintion rte between the A nd B loci, nd let PA(J), P(J)A pb(j), nd Pb(J) be the llele frequencies in popultion j (j = X, Y, or Z). Note tht for nyj, P(i) = 1 - PA(J) nd Pb(i) = 1 - PB(j), nd none of the llele frequencies chnge over genertions, in the bsence of selection nd genetic drift (whose effects re ignored for the present discussion). Under these ssumptions, it is known (2, 4) tht the mixture disequilibrium between the A nd B loci in the dmixed popultion Z, produced by dmixture, t genertion cn be written s -( )= md( ) + (1 -m)d(t) + m(l-m)a6b, [1] where 8A = [PA(X) -PA(Y)] nd 8B = [PB(X) - PB(Y)]. In ddition, for ny popultion ] (X, Y, or Z), the mixture disequilibrium decys with time s function D(-)= J (1 -r)td() [2] Therefore, even if the prentl popultions (X nd Y) re initilly t linkge disequilibrium, if their llele frequencies re different (SA #, BB + ), then the dmixed popultion (Z) will exhibit mixture disequilibrium t the outset, due to its dmixed origin. The disequilibrium will decy over time, due to recurrent recombintion over genertions, following the eqution since in this cse D~') = (1- r)'m(1 -m)sasb, D-() m(1 - M) 6AA6B. Thus, for given vlues of 8A nd 8B (llele frequency differences between the two prentl popultions) nd mixture Abbrevition: lod, log-likelihood rtio. [3] [4] 9119

2 912 Genetics: Chkrborty nd Weiss Proc. Nd Acd Sd USA 85 (1988) proportion (m), we cn determine wht level of mixture disequilibrium will remin in n dmixed popultion fter t genertions of the dmixture event. This quntity is function of the recombintion rte r, which is surrogte mesure of the physicl distnce between the loci A nd B. Eq. 3, therefore, shows tht in n dmixed popultion of known history of dmixture (i.e., with known AA, BB, t, nd m), the mount of mixture disequilibrium between two loci cn be used to drw inferences regrding the linkge reltionship between loci. Fig. 1 shows the result of such computtions, bsed on representtive vlues of the llele frequencies in prentl popultions for two loci, A nd B, by plotting the decy of mixture disequilibrium over time in the dmixed popultion s function of t for two vlues of r, for loci t different recombintion distnce from ech other. The figure compres popultions with very different llele frequencies (fixtion t both loci) with popultions with similr llele frequencies. The bsolute vlue of D is highly dependent on these frequencies, nd the rnge ofd vlues is much lrger if the gene frequencies re more disprte (note tht the four pnels hve very different bsolute rnges on the verticl scle). High dmixture levels induce higher bsolute D vlues becuse the rnge of possible D vlues is greter. The four pnels of this figure cover rnges of t tht encompss most situtions in which humn studies might be done (t up to 1 genertions) nd resonble distnces of mrker from second locus in 13 bse pirs (kb) if recombintion is roughly 1-3 per kb. It is cler from these computtions tht if in fct the two loci re linked (r < '/2), even if the prentl popultions re t linkge equilibrium, dmixed popultions rising from popultions of substntil llele frequency difference will exhibit mixture disequilibrium for long period of time. Note tht the bsolute mgnitude of disequilibrium in popultion my not be very lrge t ny point of time. This is expected becuse the mgnitude of disequilibrium is lso dependent on llele frequency (5), the dmissible vlue of D(') being given by -min[pa(z)pb(z), P(Z)Pb(Z)] < Dz < min[pa(z)pb(z), P(Z)PB(Z)I. [5] Power of Detection of Given Level of Disequilibrium. Brown (6) considered the problem of determining the sttisticl power of detecting given linkge disequilibrium, Dz, b oo.oo C d.1 t t FiG. 1. Mixture disequilibrium vlues (D,) s function of time since dmixture (t), for different levels of dmixture (m). The four pnels encompss some representtive vlues of llele frequency differences of prentl popultions nd recombintion distnce between loci A nd B. Curves represent m vlues from.1 (bottom curve) to.5 (top curve) in increments of.1. () PA(X) = PB(X) = 1., PA() = PB(Y) =., nd r =.1. (b) pa(x) = PB(X) = 1., PA(M = P(Y) =., nd r =.2. (c) pa(x) = pb(x) =.6, PA(Y) = P(Y) =.4, nd r =.1. (d) pa(x) = pb(x) =.6, PA(Y) = P8(Y) =.4, nd; =.2.

3 from survey of n gmetes. When the null hypothesis Ho: D(t) = is tested ginst the lterntive H1: DLjz, the test criterion for rejecting the null hypothesis Ho is given by {C: IDZ"I > 1.96[PA(Z)P(Z)PB(Z)Pb(Z)]½} [6] for 5% level of significnce test (6). For specific lterntive D(t) +, the power of this test procedure is given by where nd Genetics: Chkrborty nd Weiss,8= 1-[Q(r2)-Q(rl)], [7] -1.96[PA(Z)P(Z)PB(Z)Pb(Z)]½ - Dz(n)½ [PA(Z)P(Z)PB(Z)Pb(Z) + DZEA(Z)EB(Z) - Dz]½' r2 = l.96[pa(z)p(z)pb(z)pb(z)1½2- Dz(n)½ [PA(Z)P(Z)PB(Z)Pb(Z) + DZEA(Z)EB(Z) - Dz]½ Q(r) is the cumultive (lower til) probbility of stndrd norml vrite r, nd EA(Z) = PA(Z) p(z) nd similrly for EB(Z). This method of - power evlution, we my note, is more ccurte thn the pproximtion used by Brown (6), who used one-sided norml integrl to pproximte Eq. 7. The ccurcy of the norml devite test (Eq. 6) is known to be better thn tht of the X2 test of llelic ssocition (7). Fig. 2 shows the computtions of the power, following Eq. 7, for the prmeter vlues used in Fig. 1. Here the power is computed s functions of the stndrdized disequilibrium vlues, Di = D(z/D(z(mx), where D(z)(mx) is the mximum bsolute vlue, given by the bounds shown in expression 5 bove. Fig. 2 shows tht for very disprte prentl llele frequencies, even with reltively smll smple sizes, the power to detect linkge disequilibrium is quite dequte so long s there is resonble mount of dmixture. For intermedite prentl llele frequencies the rnge of D(z) is nrrow, but even very smll smples will be dequte to detect smll devition from linkge equilibrium. This is so becuse in such instnces the hplotype frequencies in the dmixed popultion re more evenly distributed, nd hence even in smll smple, ll of them will be observed, mking it esier to detect deprture from equilibrium for smll bsolute levels of mixture disequilibrium. There is, however, discontinuity t pa(x) = PA(Y), t which point D(z) = ; i.e., there is no mixture disequilibrium, even t the outset. It is interesting tht in the former cse (disprte prentl frequencies), the power of the test cn be smll for very low levels of dmixture, becuse of skewed hplotype frequencies in the dmixed group. Genetic Linkge Versus Mixture Disequilibrium Due to Mixture of Dissimilr Gene Pools. The mixture disequilibrium between two loci A nd B in popultion of dmixed origin my be merely due to the dmixture process lone nd my not signify ny linkge reltionship between these loci. However, if we hve n rry of dmixed popultions, ech of which rose from the sme two prentl popultions, but which hve undergone different levels of dmixture, there will be trend of disequilibrium vlues depending upon their dmixture history. An pproximte goodness-of-fit test for such trend cn be constructed bsed on the estimted linkge disequilibrium vlues in these popultions, contrsting the disequilibrium vlues with their expecttions bsed on Eq. 3. In single dmixed popultion of known history of dmixture (i.e., with known prentl llele frequencies nd known vlues of t), log-likelihood rtio test criterion my lso be constructed to exmine whether the observed vlue whose vlue cn be computed for ny r < '/2, given the other prmeters in prticulr survey. In nlogy with linkge tests from fmily dt, the vlue of the lod score itself cn be used to decide whether Ho or Hi is in conformity with the observed dt. Since the specific vlue of the lod score de) C n- 1.2 ) Proc. NtL Acd Sci USA 85 (1988) Stndrdized Disequilibrium FIG. 2. Power to detect given levels of mixture disequilibrium s function of dmixture levels in n dmixed popultion. The curves represent dmixture (m), in increments of.1, rnging from m =.1 (bottom curve) to m =.5 (top curve). The two pnels contrst the power s function of disprity of llele frequencies in the prentl popultions. () PA(X) = PB(X) = 1., PA(Y) = PB(Y =., nd n = 1. (b) pa(x) = PB(X) =.6, PA(Y) = PB(Y) =.4, n = 5. Note tht the disequilibrium is plotted s frction of its mximum nd minimum, which re relted to prentl llele frequencies nd dmixture levels (see text). of mixture disequilibrium is more likely to hve come from the linkge of loci or from the process of dmixture lone. For this, let us consider n dmixed popultion Z, s before, for which fter t genertions of the dmixture event the four gmetic frequencies in survey of n gmetes nd their expected probbilities under the two hypotheses HO: r = 1/2, nd H1: specific vlue of r, less thn 1/2, re shown in Tble 1. Denoting these multinomil probbilities by nri(r) for H1 nd wr('/2) for the hypothesis Ho, for i = 1,..., 4; we my write the log-likelihood rtio of these two hypotheses s 4 lod = Eni[log iri(r) - log 1ri('/2)], i=1 b [8]

4 9122 Genetics: Chkrborty nd Weiss Tble 1. Observed number of two locus gmetic types nd their probbilities under the hypotheses of linkge nd dmixture Probbility under the hypothesis Observed Gmete frequency H1: r < 1/2 Ho: r = 1/2 AB ni PAPB + DHI PAPB + DHO Ab n2 PAPb - DHl PAPb - DHO B n3 PPB - DHI PPB - DHO b n4 PPb + DHI PPb + DHO DH1 = (1 - r)'m(1 - m)8a8b; DHO = m(l - m)8a8b/2. pends upon the observed gmetic frequencies (ni vlues), it it is not possible to nswer in dvnce the strtegic question whether or not discrimintion between these two hypotheses is possible from prticulr survey design. However, conservtive decision my be reched if we wnt to evlute the expected lod score for ny given r in survey with smple size n. For this we replce the ni vlues of Eq. 8 by their respective expecttions, E(nilr) = niri(r), to obtin the expected lod score s function of r, given by 4 Er(lod) = nj i r(r)[log Ti (r) - log iri('/2)], [9] i=1 which cn be plotted ginst vlues of r (O < r s '/2) for ny given smple size n. Fig. 3 shows some representtive vlues of such computtions bsed on some prmeter vlues used in our erlier computtions (Figs. 1 nd 2). Fig. 3 shows tht with disprte prentl llele frequencies, the power to resolve the two hypotheses is very high under relistic smpling circumstnces, for resonbly closely linked loci, becuse in such n instnce the decy of initil mixture disequilibrium will be smll due to the lck of recombinnts due to close genetic linkge. With highly overlpping prentl llele frequencies, even with much lrger smples nd smll recombintion rte, genetic linkge cnnot be demonstrted sttisticlly, becuse even with linkge ctul recombinnt hplotypes will be indistinguishble from existing prentl hplotypes. When the expected lod score is lrge enough to suggest true linkge (i.e., mximum lod score 3, in the trdition of usul convention in genetic epidemiology), then it will be profitble to collect detiled fmily dt from the popultion. In tht cse such dt will be informtive for the segregtion nd mpping of the trit locus. n 2' - U - x.w Q).126 _ Q).8- bi Proc. NtL Acd ScL USA 85 (1988) Recombintion Rte DISCUSSION The bove theory suggests n interesting wy to mke use of the pst history of humn popultions s reserch strtegy in nthropologicl nd epidemiologicl studies. During the evolution of vrious rcil groups of humns, genetic isoltion over long time periods, documentble in historicl, linguistic, rcheologicl, nd other kinds of dt, hs llowed muttions nd gene substitutions to generte genetic diversity with highly ptterned geogrphic distribution. Some of these muttions hve cused complex phenotypes to evolve with their specific geogrphic distributions. While the mode of inheritnce of such complex phenotypes my be perplexing, their geogrphic distribution often suggests involvement of genetic fctors. Exmples of such complex phenotypes re disese susceptibilities: dibetes in Amerindins nd Polynesins, rheumtoid rthritis in Amerindins, skin cncer in Cucsins, nd hypertension in Blcks. In plnt nd niml genetics, when strins crrying unusul phenotypes re discovered, the inheritnce of such trits is often studied by creful crossing of these strins with those tht do not crry them. In humn genetics, this is not possible. However, during recent centuries lrge-scle movement of popultions over continentl distnces hs givb.4.5 FIG. 3. Expected lod scores to differentite between disequilibrium due to dmixture nd tht due to genetic linkge, s function of recombintion rte between loci, for different levels of dmixture. The curves represent dmixture (m) in increments of.1, rnging from m =.1 (bottom curve) to m =.5 (top curve). The two pnels show the pttern for different prentl llele frequencies nd smple size (n = number of hplotypes smpled). () PA(X) = PB(X) = 1., PA(Y) = PB() =., t = 1, nd n = 1. (b) PA(X) = pb(x) =.6, PA(Y) = PB() =.4, t = 1, nd n = 5. en rise to dmixed popultions, where very different gene pools hve mixed. Such dmixed popultions re reminiscent of genetic crosses. Before discussing the dvntges of genetic studies in dmixed groups, we must mention the consequences of the simplified ssumptions mde in our model. We ssumed tht the dmixed group is formed by single "pulse" of dmixture. In nture, however, dmixture is continul process tht occurs over mny genertions. We will del with this problem elsewhere (unpublished work); when dmixture continues for certin number of genertions, lrger mixture disequilibrium will be exhibited in the dmixed group, nd it will persist for longer period of time fter the dmixture process ceses. Though qulittively similr, continuous dmixture reduces the power of discrimintion between mixture disequilibrium nd physicl linkge. The dmixing of groups with very different gene frequencies, especilly if importnt lleles re nerly fixed in one nd nerly bsent in nother of the groups, will led to high

5 Genetics: Chkrborty nd Weiss level of segregting mtings. Such gene vrints my be present in gretly different (perhps completely different) hplotype bckgrounds in the two prentl popultions, mking linkge studies from rndomly smpled fmilies quite fesible in dmixed popultions. In mny of these circumstnces, not only is genetic vrition gret between the prentl popultions, but the time since dmixture is short ( few centuries, or on the order of t = 1 genertions). Mrker loci (restriction frgment length polymorphisms) within smll recombintion distnce (sy, r <.1) of the trit locus will hve good sttisticl properties in regrd to drwing genetic inference s well s in mpping genes. It is lso often possible to scertin smples from vriety of popultions with vrying levels of dmixture between the sme two prentl popultions. Elsewhere it hs been shown how dmixture my be used to infer the existence of genetic etiologicl fctors in complex phenotypes such s non-insulin-dependent dibetes mellitus in Amerindins (8-1), s well s how models of genetic custion my be tested with dt representing grdient of popultion dmixture levels (3). Proc. NtL Acd Sci USA 85 (1988) 9123 We thnk Drs. P. E. Smouse nd J. C. Long for their comments on the mnuscript. This work ws supported in prt by Grnts GM2293 nd CA19311 from the Ntionl Institutes of Helth nd grnt from the Wenner-Gren Foundtion for Anthropologicl Reserch. 1. Nei, M. & Li, W.-H. (1973) Genetics 75, Chkrborty, R. & Smouse, P. E. (1988) Proc. Ntl. Acd. Sci. USA 85, Chkrborty, R. & Weiss, K. M. (1986) Am. J. Phys. Anthropol. 7, Thomson, G. & Klitz, W. (1987) Genetics 116, Lewontin, R. C. (1964) Genetics 49, Brown, A. H. D. (1975) Theor. Popul. Biol. 8, Chkrborty, R. (1984) Genetics 18, Chkrborty, R. (1986) Ybk. Phys. Anthropol. 29, Chkrborty, R., Ferrell, R. E., Stern, M. P., Hffner, S. M., Hzud, H. P. & Rosenthl, M. (1986) Genet. Epidemiol. 3, Hnis, C. L., Chkrborty, R., Ferrell, R. E. & Schull, W. J. (1986) Am. J. Phys. Anthropol. 7,