The effects of admixture and population subdivision on cytonuclear disequilibria

THEORETICAL

POPULATION

BIOLOGY

39, 273-300 (1991)

The Effects of Admixture and Population on Cytonuclear Disequilibria

Subdivision

MARJORIE A. ASMUSSENAND JONATHAN ARNOLD Department of Genetics, University of Georgia, Athens, Georgia 30602 Received January 16, 1990

We examine the generation of cytonuclear disequilibria by admixture and continued gene flow. General formulas analogous to the nuclear case are first derived showing that the allelic and genotypic disequilibria from admixture or population subdivision equal their expected value across the contributing (sub) populations plus the covariance across these sources between the cytoplasmic gene frequency and the relevant nuclear frequency. A detailed study is then presented of the cytonuclear dynamics, in a random-mating population under two different migration scenarios. In both cases closed-form solutions are given for all variables as a function of the initial conditions and relevant migration parameters. The dynamics of the gene frequencies and allehc disequihbria, which dominate each system, are the same as those involving two unlinked nuclear loci, while the dynamics of the genotypic disequilibria and cytonuclear frequencies have no nuclear counterpart. The continent-island formulation focuses on a population receiving continued immigration from a large source of constant composition. A major discovery is that cytonuclear disequilibria can transiently build up on the “island” to levels far exceeding those found at equilibrium. In contrast, the admixture formulation focuses on the dynamics within two populations undergoing continued intermigration. Although in this case all cytonuclear associations must ultimately decay to zero, long-term transient disequilibria can develop which are many times their initial admixture values. For both migration scenarios it is shown that the time of population censusing relative to migration and reproduction dramatically affects both the amount and pattern of the nonrandom associations produced. The empirical relevance of these models is discussed in light of nuclear-mitochondrial data from a hybrid zone between European and North American eels and from a cri 1991 Academic Press, Inc. zone of racial admixture in humans.

Significant nonrandom associations have now been found between diploid biparentally inherited nuclear loci and haploid uniparentally inherited cytoplasmic loci (e.g., Lamb and Avise, 1986). Such cytonuclear disequilibria could in theory be generated by a variety of evolutionary forces, such as nonrandom mating, migration, selection, drift, and founder effects. Moreover, due to the contrasting modes of inheritance of the two 273 0040-5809/91 $3.00 CopyrIght 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved.

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ASMUSSEN

AND

ARNOLD

genomes, such associations can provide a qualitatively new kind of information about the genetic structure and evolutionary history of natural populations. In order to interpret better the growing body of nuclear-cytoplasmic data, we (Asmussen et al., 1987) recently introduced and determined the properties of four interrelated measures of cytonuclear disequilibria. These include the allelic disequilibrium (D) and three genotypic disequilibria (0,) D,, D3), which quantify the nonrandom associations between two cytotypes and the alleles and genotypes at a diallelic nuclear locus. As our first step toward a general evolutionary theory for these cytonuclear associations, we examined their dynamics and patterns under several mating systems relevant to hybrid zones (Asmussen et al., 1987; Arnold et al., 1988). We showed that although the allelic and genotypic associations eventually all decay to zero if mating is random or assortative based on nuclear genotype, permanent nonrandom associations may be generated by an assortative mating pattern in which female mating preference is determined by an epistatic interaction between individual nuclear and cytoplasmic loci. In contrast, all cytonuclear disequilibria again ultimately decay to zero if mate choice is based on an epistatic interaction between cytotype and the multilocus nuclear genotype characteristic of the parental species. More recently, we investigated the effects of nonrandom mating and continued immigration of the two parental species on the cytonuclear dynamics in a hybrid zone (Asmussen et al., 1989). Two key findings were that (i) continued migration of pure parentals can generate permanent cytonuclear disequilibria, whether hybrid zone matings are random or parental females preferentially mate with conspecific males; and (ii) joint, nuclear-cytoplasmic frequency data can provide particularly sensitive estimates of gene flow into hybrid zones. The results also revealed that the timing of population censusing (before or after mating and reproduction by immigrants) can have a dramatic effect on the magnitude of the cytonuclear disequilibria observed. Another recent study extended the basic cytonuclear framework (Asmussenet al., 1987) to encompassthe nonrandom associations within three-locus, nuclear-dicytoplasmic systems (Schnabel and Asmussen, 1989). Here we expand the theory of two-locus cytonuclear disequilibria by examining more general effects of admixture, population subdivision, and migration, whose relevance extends beyond the context of hybrid zones. Although many other migration schemes remain to be addressed (e.g., Feldman and Christiansen, 1975; Christiansen, 1989), we focus here on two simple models which have direct applicability to nuclear-cytoplasmic data currently being gathered. Specific issues include formulas for cytonuclear disequilibria due to admixture/population subdivision, and the cytonuclear

275

CYTONUCLEAR DISEQUILIBRIA

dynamics within two randomly mating populations experiencing continued intermigration, paralleling classical studies of the gametic disequilibrium between two nuclear loci (Nei and Li, 1973). We also examine the dynamical behavior within a single randomly mating population receiving continued migration from a large source of fixed genetic composition, analogous to the classical continent-island model (Wright, 1931). A general concern of both dynamical studies is the effect of census timing on the ability to detect and utilize nonrandom associations between nuclear and cytoplasmic loci. CYTONUCLEAR VARIABLES

Following our previous investigations (Asmussen et al., 1987, 1989; Arnold ef al., 1988), we consider a cytonuclear system determined by two cytotypes (44, m) at a haploid cytoplasmic locus and two alleles (A, a) at a diploid, autosomal nuclear locus. Within any single population the frequencies of the six possible cytonuclear genotypes are denoted as in Table I, with the row and column sums respectively providing the marginal frequencies of the two cytotypes and the three nuclear genotypes. From the latter, the nuclear allele frequencies are in turn readily calculated as q = freq.(a) = w + iv,

p = freq.(d) = u + iv,

(1)

where “freq.” denotes “frequency of.” The statistical association between alleles at the nuclear and cytoplasmic loci is quantified by the allelic disequilibrium, D = freq.(A/M) - freq.(A) freq. (M) = u1+ iv1 - px,

(2)

while the statistical associations between cytotypes and nuclear genotypes are measured by the three genotypic disequilibria, TABLE I Frequencies of Cytonuclear Genotypes Nuclear Genotype cytotype

AA

Aa

aa

Total

276

ASMUSSENAND ARhjOLD

D,=freq.(AA/M)-freq.(AA)freq.(M)=u,

-U.Y

D, = freq.(Aa/M) - freq.(Aa) freq.(M) = t’, - ux

(3)

D, = freq.(aa/M) - freq.(au) freq.(M) = M’,- LVX.

These four measures are interconnected by the relations and

D, +D,+D,=O

D=D,+$D,.

(4)

CYTONUCLEAR ADMIXTURE FORMULAS

We begin our investigation of population subdivision and migration by deriving formulas for the cytonuclear disequilibria generated by the admixture of two or more (sub)populations. Consider the general case where n populations or subpopulations are combined and viewed as a single entity. The mean value of each variable z across the underlying (sub)populations is then the weighted average E(z)= i

m,z(‘),

,=I

where z(j) is the value of z in (sub)population i and m, is the fraction of the combined population originating in (sub)population i, with Cy= r mi = 1. The overall cytonuclear disequilibria in the combined population are given by DT=E(D)+cov(p,x) D:=E(D,)+cov(u,

x)

D;=

E(D,) + cov(v, x)

D;=

E(D,) + cov(w, x)

(5)

and thus differ from their average values whenever there is a nonzero covariance across the n (sub)populations, between the cytoplasmic frequency and the relevant nuclear frequency. These admixture formulas which represent a two-locus Wahlund principle, parallel the classical formula for nuclear linkage disequilibrium derived by Nei and Li (1973) and Prout (1973), and follow directly from the basic properties of mathematical expectation and covariance. For instance, since the overall frequencies are simply their average values, E(z), across the

CYTONUCLEAR

DISEQUILIBRIA

277

(sub)populations, we see from definition (2) that the overall allelic disequilibrium in the combined population is o’=E(u,)+fE(u,)-E(p)E(x) =E(u,)+;E(u,)-E(px)+cov(p,x) = E(D) + cov( p, x) and similarly from (3) that the overall value of each genotypic disequilibrium Di is =E(g,)-E(gx)+cov(g,x) = E(Di) + COV( gyX), where g = U, u, or w for i = 1, 2, or 3, respectively. Note that the same basic method can be used to provide an alternative, and somewhat simpler, derivation of the classical nuclear formula. As in the latter case the cytonuclear formulas in (5) apply both to an actual population formed by the physical admixture of two or more populations, and to a “statistical” population created when a group of distinct subpopulations are aggregated into a single larger population during data analysis. The results have the most straightforward interpretation in the special case of n = 2 (sub)populations, where admixture disequilibria (unequal to their average values) arise if and only if the underlying populations differ in both their cytoplasmic and (relevant) nuclear frequencies. THE CONTINENT-ISLAND

MODEL

Consider a randomly mating population which consists each generation of a fixed fraction m of migrants from a large source population, with the remaining fraction, 1 -m, derived from existing residents. It is assumed that the cytonuclear frequencies and disequilibria in the migrant pool are constant over time, and are denoted by u,, p, X, B, etc. If migrants are derived from multiple sources, the frequencies in the overall migrant pool (Ui, etc.) are given by their average values across these sources, while the immigrant disequilibrium measures (6, etc.) can be calculated from the admixture formulas in (5). We are interested in the dynamical behavior under each of the two possible censusing times relative to mating and migration (Fig. 1). For each case it is assumed that (i) the cytonuclear frequencies are the same in both sexes; (ii) the cytotypes are uniparentally inherited; (iii) there are no selec-

278

ASMUSSENAND ARNOLD CENSUS BEFORE MATING

MATING AND REPRODUCTION

CENSUS AFTER MATING

FIG. 1. Alternative immigrants.

censusing times within

a generation

cycle of a population

receiving

tive differences among the six cytonuclear genotypes; (iv) the population is large so that the effects of drift can be ignored; and (v) the population has discrete, nonoverlapping generations. The basic model is like that used in our earlier hybrid zone study (Asmussen et al., 1989), in which explicit time-dependent solutions were given only for the special random-mating case where migrants are derived from two genetically distinct parental species. Census Method 1: After Migration

and Before Mating

Under this census method the censused frequencies in Table I and the derived variables in Eqs. (l)-(3) denote the values in the population after all migrants have entered and before mating has occurred. Since we have assumed that the cytonuclear genotypes are selectively neutral, censusing may be based on either adults or juveniles. Each generation cycle (or census interval) begins with random mating among the current residents of the population, and ends after all new migrants have entered (Fig. 1). At the following census, the cytonuclear frequencies are thus simply the weighted average of the migrant frequencies and the frequencies in the residents’ offspring, where the latter are obtained from the random mating recursions in Asmussen et al., (1987). This yields the basic transformations u;=mti,

+(I-m)(p2x+pD)

u;=mG,

+(l-m)[2pqx+(q-p)D]

w;=m@,+(l-m)(q2x-qD) u;=mii,

+(l-rn)(~~~-pD)

u;=rnti,

+(l-m)[2pqy-(q-pp)D]

w; = m%, + (1 - m)(q2y + qD),

(6)

CYTONUCLEAR

279

DISEQUILIBRIA

where primes (‘) denote the values at the next census. From these the new marginal, single locus frequencies are readily computed as u’=mti+(l-m)p2,

p’=W@+(l

-m)p

v’=mfi+2(1

x’=m?+(l

-m)x

-m)pq,

(7)

w’=mG+(l-m)q2. The new cytonuclear associations can now be calculated either directly, by substituting the relevant frequency recursions from (6)-(7) into the disequilibrium definitions in (2)-( 3), or by substituting the migrant associations (6, D,, D,, 6,) and the associations in the offspring of random matings among the residents (Asmussen et al., 1987) into the admixture formulas in (5). The resulting disequilibrium recursions are D’ =mD

+i(l

-m)D+m(l

-m)(p-p)(x-2)

D;=n?D,+(l-m)pD+m(l-m)(p2-ti)(x-2) D; = d,

+ (1 - m)(q - p) D + m( 1 - m)(2pq - 6)(x - X)

(8)

D;=m&(l-m)qD+m(l-m)(q2-@(x4).

The full model in (6t(8) shows that the cytonuclear dynamics are completely determined by the migration parameters in conjunction with the trajectories of the gene frequencies and the allelic disequilibrium. Moreover, the three dominant variables (p,, xr, D”‘) exhibit the same dynamics as for a system of two unlinked nuclear loci. The nuclear and cytoplasmic gene frequencies monotonically approach the corresponding migrant frequencies at the constant geometric rate of 1 -m per generation, P,=D+(Po-PN-m)‘+F

as t-+cc

x,=X+(x,-X)(1--m)‘-,.?

as t-co.

(9)

The allelic disequilibrium has a somewhat more complicated dynamic, D(t)

=

(D’0’-6-h)[$(l

-m)]‘+b(l

if m#$ if m=$,

-wz)*~+D

[D’“‘-~+t(po-~)(xo-.f)](~)‘+b

(10)

where for m # $, b = 2m(po - PNxo - 2) 1-2m

’

(11)

and for all O-cm< 1, DC*)

l+m

as

t-km.

(12)

280

ASMUSSEh

AhU

AKI\;OLIl

After substituting these three time-dependent solutions into the other recursions in (6))(8) we find that the genotypic associations and cytonuclear frequencies (Appendix A) have much more complex dynamics than the standard nuclear variables, with each equalling (for m # $) its equilibrium value plus a linear combination of five geometric terms: ( I - m I”, (1 -m)“, [i( 1 -m)]‘, and (1 -m)‘. Moreover, a com[i( 1 -m)*]‘, parison with the simpler allelic disequilibrium solution in (lo)-(12) shows that while the allelic association reaches equilibrium at the asymptotic rate of (1 - m)* per generation if m < i and f( 1 - m) per generation if m > !, the genotypic variables ordinarily reach equilibrium at the slower asymptotic rate of 1 -m per generation, corresponding to the geometric approach to equilibrium by the gene frequencies. Although the complexity of the genotypic solutions precludes a full analytic characterization of their qualitative behavior through time, considerable insight can be gained from their limiting values alone. For example, as t + CCthe genotypic associations approach 2m(l -m)pD l+m

B,=mmB,+(l-m)pB=mD,+ Bz=mB,+(l

2m(l -m)(cj-p)D l+m

-m)(g-p)fi=mD,+

2m(l -m)gD l+m

b,=mD,-(l-m)ijB=mB,-

(13)

’

while the frequencies of the cytonuclear genotypes approach C,=mti,

+(l-m)(~*X+~6)=mti,+(l-m)

ti,=mC,

+(l-m)[2@$%+(q-p)d] 2&Y+

=miT, +(1-m) [

2m(ij - p)B l+m

~,=m~,+(1-m)(~2~-~~)=m~,+(1-m)

I

2mpd l+m

L

2mijD , etc. l+m

p2X+-

1

1 q2X---

1

An immediate observation from the equilibrium values in (12))( 14) is that continued migration from a constant source can generate permanent nonzero cytonuclear associations in randomly mating populations, just as it can generate nuclear linkage disequilibrium. A closer examination reveals several other noteworthy points. For example, (12) shows that the equilibrium allelic association, fi (which is the same as that for two unlinked

CYTONUCLEAR

DISEQUILIBRIA

281

nuclear loci), depends only on the constant migration rate (m) and the migrants’ allelic disequilibrium (a). Furthermore, the ratio, B/d = 2m/( 1 + m), monotonically increases from 0 to 1 as the fraction of migrants, m, increases from 0 (no migrants) to the limiting value of 1 (only migrants). As a result, the equilibrium allelic association always has the same sign but a magnitude smaller than the migrants’ allelic disequilibrium. In contrast, the equilibrium genotypic associations in (13) are not simple reflections of the corresponding migrant disequilibria, but instead depend on the migrants’ nuclear gene frequencies and allelic association, in addition to the migrants’ genotypic disequilibria and migration rate; there is consequently no simple, general relationship between the sign and magnitude of the equilibrium genotypic associations and those of the corresponding genotypic disequilibria in the migrant pool. (Analogous statements apply to

the frequencies of the six cytonuclear genotypes.) Moreover, in contrast to the distinctive transient pattern in isolated random-mating populations (Asmussen et al., 1987), there can be virtually any transient and limiting disequilibrium sign pattern in random-mating populations which experience continued migration from an arbitrary (constant) source, if censusing is after migration. A distinctive pattern is found, however, if migrants are derived from two genetically distinct source populations, as is the case for many hybrid zones (Asmussen et al., 1989). Two final results can be deduced from an examination of the full timedependent solutions in (lo)-( 12) and Appendix A. First, the trajectories of the four cytonuclear disequilibria and six cytonuclear genotypes are not necessarily monotonic except in special circumstances, such as when the nuclear or cytoplasmic gene frequencies initially equal those in the migrants. In particular, as illustrated by Fig. 2, continued migration can generate transient cytonuclear associations of magnitude even greater than those found at equilibrium. Moreover, in light of their slower convergence,

the genotypic disequilibria will in such casespersist above their final values considerably longer than will the allelic disequilibrium. A second, more technical point is that at any census t the cytonuclear structure depends solely on the population’s initial allelic association (DC”) and initial deviations from the migrants’ gene frequencies (pa - p and x0 - X), in addition to the time t, the migration rate, and the cytonuclear constitution of the migrants. Census Method 2: After Mating and Before Migration

Under the alternative census scheme in Fig. 1 the cytonuclear structure is tracked following mating and reproduction. This results in a different set of dynamics, which is most easily obtained by applying the random mating recursions (Asmussen et al., 1987) to the solutions under the first census method. Only the gene frequency trajectories remain as in (9) since these

282

ASMUSSENAiW ARNOLD

a

0.102 -

----------------

5 aI m

3 3

o,ooo :

............. ... .. ... ..... .“‘.‘. ...... .............. .. ....... .. .. . .. .... ... ... ...... .... .......

., ” : .’ _ :; .:’ 7.’

: 5

O.l7Oj-

-D ---0, .... ... ..... 4

t

.o.1366

15

b

30

GENERATION

FIG. 2. Cytonuclear disequilibrium dynamics under the continent-island model with censusing before (a) or after (b) reproduction by migrants. In both cases the migration parameters are M = 0.25, U, = 0.4, 3, = 0.6, and the initial island composition was given by uIo) - , 1 -

CYTONUCLEAR

283

DISEQUILIBRIA

are the only variables unchanged by random mating. When censusedafter reproduction, the time-dependent frequencies of the joint, cytonuclear genotypes are uy

2mpii -+ p’x + $7 = p2x +l+m

= pfx, + pp

UT(‘) = 2p,q,x, + (q, - p,) DCr)+ 2&f + (4 - Is) 6

= 2pqx +

2m(q-- p)d

(15)

l+m

2m$ w;C(‘)= q:x, - qlD(‘) -+ q2.f - i@ = q2X - l+m,

etc.

while the cytonuclear disequilibria dynamics become D*(f) = -1 D(t) + -1 3 = - md

2

2

l+m 2mpD -

D;“’

= (qr - p,) DC’) + (4 - ppj

D:“’

= -q,D(‘)

+ -qfi

- 2mgD =~

l+m

(16)

= 2my;;)D

’

where stars (*) denote values after mating and p,, x~, D,, and B are as in (9)-(12). In applying these time-dependent formulas, it must be borne in mind that the expressions taken from the first census method represent the values immediately preceding the same round qf mating, so that the two sets of initial conditions are connected by the relations, x$ =x,,, p$ = pO, and D$ = $D,.

Although the full time-dependent solutions have similar functional forms under the two census methods (Appendix A), they have several important qualitative differences. One obvious, but nonetheless important effect of census timing is that the allelic association censused after mating will always be half that censused before mating, corresponding to the halving of the allelic disequilibrium between two unlinked loci by a round of random mating. Census timing also has a substantial effect on the qualitative behavior of the genotypic disequilibria and frequencies (see Fig. 2). First, unlike populations censused after migration, the trajectories of the genotypic associations and cytonuclear frequencies censused after 653!39/3-4

284

ASMUSSEN AND ARNOLD

muting are independent of the corresponding values in the migrants (D,, U,. etc.). The transient values at any census t 3 1 are entirely determined by those of pr, x,, and D*“) = iD(“, and thus depend only on the time t, the migration rate (m), the migrants’ gene frequencies and allelic disequilibrium (p, 2, D), and the population’s initial deviations from the migrants’ gene frequencies and initial allelic disequilibrium (p. - II, x0 - -U,D*“” = fD’“‘). The final cytonuclear structure is determined solely by the four migration parameters, m, p, Z, and D. A further contrast to the first census method is that the cytonucleur disequilihriu here always have the relative, transient sign pattern found in isolated, random-mating populations (Asmussen et al., 1987). Although the precise signs are fixed in the latter case by that of the initial allelic association (plus the initial gene frequency in the case of D2), here the actual signs may vary along a trajectory (although not the relative sign relationships) because of possible sign changes by D “) through time. The final genotypic associations are also quite different at the two census times. Unlike the complex and highly variable disequilibrium patterns found in (13) under censusing after migration, censusing after reproduction always produces a distinctive equilibrium sign pattern in which L?*, l?T, and -6: have the sign of the migrants’ allelic disequilibrium (D), while fi: is either zero or has the same or opposite sign as 4 depending on whether the migrants’ nuclear gene frequency (p) equals or is below or above i. Cytonucleur disequilibriu will thus be generated with censusing after muting tf and only tf the migrants have nonrandom allelic associations. The population’s allelic and genotypic associations all eventually decay to zero if nuclear alleles are randomly associated with the two cytotypes in the migrants (i.e., B = 0), whereas the population’s cytonuclear disequilibria all (except possibly 0;) approach nonzero values when the migrants have a nonrandom allelic association (i.e., d # 0). The final cytonuclear structure can thus be quite different from that found with censusing before mating, where the final genotypic disequilibria are composites of the migrants’ allelic and genotypic associations and thus may be nonzero when D = 0, provided the migrants have completely nonrandom genotypic associations. Several further observations follow by viewing the equilibria under the first census scheme as the (unobserved) equilibrium values midway through the census interval (i.e., after migration) under the second method. Formally, each cytonuclear variable z satisfies the relation, 2 = rn? + (1 - m)i*, where 2 and I;* denote the equilibrium values before and after reproduction, respectively, and Z denotes the corresponding value in migrants. Each equilibrium genotypic association/frequency censused before mating must therefore always lie between the corresponding association/frequency in the migrants and the equilibrium value censused after mating. This last key fact in turn implies two final general results, which were also found in our

CYTONUCLEAR DISEQUILIBRIA

285

earlier hybrid zone study (Asmussen et al., 1989). First, the two equilibrium values for a given genotypic disequilibrium or frequency will either both exceed, both equal, or both be less than the corresponding migrant value. Last, in contrast to the allelic association and allelic frequencies, the actual migrant values determine which of the censusmethods will generate a greater genotypic association/frequency at equilibrium. In particular, censusing before mating will result in a higher genotypic association (or genotypic frequency) if and only if the corresponding equilibrium value in (15)-(16) is less than the migrant value.

ADMIXTURE MODEL

We turn next to the analog of Nei and Li’s (1973) classical model of nuclear linkage disequilibrium in subdivided populations to delimit the cytonuclear dynamics under continued intermigration between two random-mating populations. Here, for simplicity, the frequencies and disequilibria in population 1 are denoted by all lower case letters (i.e., ui ,..., w,; u, v, w; p, x; d, .... d3) and those in population 2 by the corresponding upper case letters (Vi, .... Wz; U, V, W; P, X, D, .... D3). It is assumed that each generation population 1 consists of a fixed fraction m, of migrants from population 2, with the remaining fraction, 1 - m,, derived from existing residents of population 1. Similarly, population 2 consists each generation of a fixed fraction, m*, of migrants from population 1, with the remaining fraction, 1 - m,, stemming from existing residents of population 2. Except for the new migration scheme,the other general assumptions of the continent-island model are still assumed. We are again interested in the dynamics under each of the two possible census timings in Fig. 1. CensusMethod 1: After Intermigration and Before Mating The censused frequencies here reflect the individual values in the two populations after migrants have moved between the populations and before mating has occurred. Each census interval begins with random mating within each population and ends after the next round of intermigration. The new frequencies in the two populations are thus weighted averages of those in the offspring of random matings within the individual populations (Asmussen et al., 1987). In analogy to the first continent-island model above, the basic recursions for population 1 are u;=m,(P’X+PD)+(l-m,)(p2x+pd) v;=mI[2PQX+(Q-P)Dl+(l-m,)[2pqx+(q-p)dl w;=m,(Q2X-QD)+(l-m,)(q2x-qd),etc.

(17)

286

ASMUSSEN AND ARNOLD

while the new marginal, single locus frequencies are u’=m,P’+ u'=2m,PQ+2(1

(1 -m,)p’. -m,)pq,

r’=m,X+(l

-m,)x

p’=?n,P+(l

-m,)p

(18)

“‘=m,Qz+(l-V?ll)q~. The new cytonuclear disequilibria can either be computed directly from the recursions in (17-( 18) and the definitions in (2)-(3) or by applying the admixture formulas in (5) to the progeny of the two randomly mating components. In either case the transformations for population 1 are found to be d’=$z,D++(l-m,)d+m,(l-m,)(P-p)(X-x) d,‘=m,PD+(l-m,)pd+m,(l

-m,)(P’-p2)(X-x) (19)

d;=m,(Q-P)D+(l-m,)(q-p)d+2m,(l-m,)(PQ-pq)(X-x) d;= -m,QD-(l-m,)qd+m,(l-m,)(Q*-q*)(X-x).

The recursions for population 2 are analogous to those in (17)-( 19), with the left-hand variables converted to upper case and the right-hand sides modified by replacing m, by 1 - m, and 1 - m, by m,. The cytonuclear dynamics are here driven by the intermigration rates in conjunction with the gene frequencies and allelic disequilibria, where the latter have the same trajectories as for two unlinked nuclear loci (Nei and Li, 1973). The joint nuclear gene frequency dynamics are in this case

P,=B+(mllm)(po-po)(l

-m)‘+B

P, = li + (m21m)(Po- pd(l -m)’ + B

as t+m as t--+03,

(20)

where m = m, + m, is the total intermigration rate each census interval. The two populations reach the common equilibrium frequency d= (miPi+m2po)lm

(21)

in which the initial value in each population is weighted by the fraction of migrants originating from that population. The cytoplasmic frequencies, X, and X,, follow equivalent dynamics with p’s replaced by x’s and P’s by x’s Taking an alternative approach to the three-dimensional transformation originally employed by Nei and Li (1973), we see after substituting the gene frequency solutions into the recursions for the allelic disequilibria that

CYTONUCLEAR

287

DISEQUILIBRIA

the latter dynamics can be described by a system of two equations, which can be written in matrix form as

If the two populations initially have equal nuclear or cytoplasmic gene frequencies (i.e., PO= p0 and/or A’,,= x,), this reduces to a two-dimensional linear transformation that only differs from that for the gene frequencies by a factor of 4. By analogy to (20), the solution is d”‘=6’0’(~)‘+(m,/m)(d’O’-D’O’)[~(l

-m)]’

II(r) = DO’($) + (m2/m)(ls0’ - d’O’)[;( 1 -Hz)]‘,

Pa)

where D(O)= (m,D’“‘+m2d’o’)/m is a weighted average of the initial allelic associations in the two populations, paralleling the gene frequency equilibrium in (21). If the populations initially differ in both their nuclear and cytoplasmic gene frequencies, the allelic associations are given at any census t 2 1 by the homogeneous solutions in (23a) plus new, inhomogeneous terms due to the covariance factors from admixture: d”‘=~‘0’(~)‘+(m,/m)(d’O’-D(O’)[~(l-m)]’ + (mJm)(Po - PONXO-

xoHm2(2

-m) fi”

+ Cm,(l -m,)-m,(l -~dlf:“> D(r) = D’O’($ + (m2/m)(lY0’ - d’O’)[f( 1 - m)]’ + (m,lm)(Po - Po)Wo - xoHd2 +

Cm2(1

Wb)

-ml f!”

-~,nm

-m,)-m,(l

where, for m # i or 1 f &/2, flr) = (3’ - (1 - mJ2’ ;-(1-m)’ ’

fFl=[f(1-m)]‘-(1-m)2’ i(1-m)-(l-m)2

Pa)

while, for m = 4, fy’=4[(;)‘-

($J,

fy = t( $)‘- ’

(24b)

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and, for m = 1 * 4512, .f’:“=r(+)’

‘,

,fl’, = ( T :&I’, -t;$-;

($1’

(24~)

These results are formally derived in Appendix B. The time-dependent solutions for the genotypic disequilibria and genotypic frequencies can now be obtained by substituting those for the gene frequencies and allelic disequilibria into the remaining recursions in (17)-(19). An immediate observation is that unlike the continent-island model censused after migration, the initial gene frequencies and allelic disequilibria determine (in conjunction with the intermigration rates) the subsequent values of all variables in the two populations. Although the genotypic associations (and frequencies) are completely independent of their own initial values, the genotypic disequilibria still lack any obvious general sign pattern through time when censusedafter migration. The solutions also show that in contrast to the continent-island model, but paralleling nuclear linkage disequilibrium in subdivided populations (Nei and Li, 1973), all cytonuclear disequilibria eventually decay to zero in two random-mating populations undergoing constant intermigration. If the total intermigration rate, m = m , + m,, is between the critical values, 1 - d/2 ( ~0.293) and 1 + d/2 ( g 1.707) the disequilibria all reach zero at the asymptotic rate of 4 per generation, corresponding to the constant geometric decay rate in isolated, random-mating populations. If, however, the total intermigration is below 0.293 (or in the unlikely event it exceeds 1.707) the allelic and genotypic associations ultimately decay at the retarded rate of (1 -m)2 per generation. The genotypic and allelic associations thus all behave asymptotically like the linkage disequilibrium between two unlinked nuclear loci (Nei and Li, 1973), which was not the case in the continent-island formulation. Although continued intermigration inevitably results in completely random associations, it can serve to generate transient cytonuclear disequilibria in the two populations as well as retard their loss, in much the same way it does nuclear linkage disequilibrium (Nei and Li, 1973). This effect is most apparent when each population begins with random allelic associations (i.e., d (‘) = D(O)= 0). The trajectories in Eq. (23a) then show that whenever PO= p. or X0 = x0 the allelic associations will remain at zero for all subsequent censuses. This in turn implies from (19) that all genotypic disequilibria will also be fixed at 0 for all t B 1, since the genotypic admixture terms all vanish in the event PO= p. or X0 = x0. This means that if the nuclear and cytoplasmic alleles are initially randomly associated in each of the two populations, aileiic and genotypic cytonuclear disequilibria will be generated by intermigration only if the populations initially differ in both their nuclear and cytoplasmic gene frequencies.

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289

As found for nuclear systems by Nei and Li (1973), additional insight is possible in the special case of symmetric intermigration rates (i.e., m, = m2 = m/2) in conjunction with the initial conditions, d(O)= D”’ = 0, Po#po, and X,#x,. Under these conditions, the trajectories in (23b) show that the two populations have the same allelic associations at every census t 3 1, with d(')-D(')=ml(

1 - ml)(Po-p,)(X,-x,)f\'),

(25)

where the time-dependent factor, f’,“, is defined in (24), and its coefficient is the allelic disequilibrium generated in each population by the first round of intermigration. Several important points can be deduced from the common trajectories in (25). First, since all three possible formulas forf’,” are positive at every census t B 1, the transient allelic associations always have the sign of the original admixture disequilibrium, which is set by the product of the initial gene frequency differences, (PO- p,)(X,-x,). The same is true for two nuclear loci, although it was not explicitly pointed out by Nei and Li (1973). Second, inspection of the conditions under which ID(‘)1> (DC’- ‘)I reveals that the allelic associations always initially increase in magnitude, after which they monotonically decay to zero (see Fig. 3). The maximum allelic disequilibrium occurs at census t,,, = [t*] 2 1, where

t* = 1-lnC8ml(l -m,)l ln[2(1-2m,)*]

(26)

and [ .] is the greatest integer function. Note that this formula for t,,, gives the precise censusat which DC’)and d(l) change direction, and thereby allows a more detailed investigation of the transient increase in allelic associations than was possible via Nei and Li’s real-valued approach. The new results, described below, apply equally to two unlinked nuclear loci and are readily extended to nuclear systems with arbitrary linkage. The first observation from (25)-(26) is that both the number of generations the allelic associations increase, t,,,, and the corresponding maximum allelic associations, Id(‘maX)I = ID(fmaX)Iare symmetric about m, = 1. Moreover, t,,, increases from 1 to CCas m, decreasesfrom 1 to 0, with t ,,,>l only if m,i+fi/4 (~0.854). The allelic associations thus continue to increase above their original admixture values if and only if the intermigration rate is below roughly 0.15 (or above 0.85), with the increase lasting at most four censusesfor 0.01
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FIG. 3. Cytonuclear disequilibrium dynamics under the admixture model. The four panels show the effects within population 1 of the level of gene flow (ml =m,=O.Ol in the upper graphs; m, = m, = 0.1 in the lower graphs), and of the census timing (before reproduction in the left-hand graphs; after reproduction in the right-hand graphs). In all cases, the critical initial values were P, = sa = 1.0, pU = A’, = 0, and d’“’ = D”’ = 0.

m, = 4 + fi/4,

the allelic associations remain at the original admixture value at census t = 2 before monotonically decaying to zero.) Returning to the actual trajectories in (25), we see that the longer increase in allelic disequilibria with lower gene flow coincides with a steady decreasein the original allelic associations generated by the first admixture. The latter process in fact offsets the continued increase in allelic disequilibria for low m, values, so that even though the allelic associations increase to many times their initial admixture value under low intermigration rates, the maximum allelic association attained along a trajectory nonetheless monotonically increases with m,, increasing from 0 to

0.25 1(PO- p,)(X, - x0)1 as the intermigration rate increases from 0 to 4. These trends are illustrated numerically in Table II. The results suggest that the intermigration rate (m,) must be at least 0.01 in order to generate significant allelic associations, with the maximal disequilibrium generated

CYTONUCLEAR

TABLE Transient Cytonuclear

291

DISEQUILIBRIA II

Associations in the Admixture

Model

D”‘/(PO - Po)(~o -x”) ml 0.01 0.02 0.03 0.05 0.06 0.07 0.08 0.10 0.14 0.15 0.25 0.50

'm 4 4 3 3 2 2 2 2 2 1 1 1

D'hx' #I' 1.712 1.563 1.473 1.311 1.274 1.240 1.206 1.140 1.018 1.000 1.000 1.000

I= 'ma, 0.017 0.03 1 0.043 0.062 0.072 0.08 1 0.089 0.103 0.123 0.128 0.188 0.250

t=5

l=lO

t=25

r=50

0.017 0.030 0.039 0.049 0.05 1 0.052 0.05 1 0.049 0.041 0.038 0.023 0.016

0.014 0.021 0.022 0.019 0.016 0.013 0.011 0.007 0.003 0.002 0.001 0.001

0.008 0.006 0.003 0.001 * * * * * * * *

0.003 0.001 * * * * * * * * * *

Note. The effect of the intermigration rate m, is shown on the number of generations allelic associations increase in each population (t ,,J, the relative increase above the initial admixture value, D"maxl/D"), and the normalized allelic association, D"'/(P, - po)(Xo - x0). at various census times t. Values are based on the symmetric case where m, = m2 = fm and d”’ := D"' = 0, with censusing after migration. A ‘L*” entry denotes a value below 5 x 10e4. The corresponding values for censusing before migration differ only in that the normalized disequilibria are halved.

(for m, = m2 and 8’) = D”)=O) when m, = 0.5 and the two populations are fixed for different alleles at both the nuclear and cytoplasmic loci. (An investigation of the statistical power to detect cytonuclear disequilibria will be presented elsewhere (Fu and Arnold, 1991)) The intermigration rate has a more complex effect on the residual allelic disequilibrium at subsequent censuses(i.e., for t > r,,,), becausethe greater associations generated by higher migration rates are in turn more rapidly lost than those produced by lower migration rates. As a result, it is possible for populations with low intermigration rates to retain higher residual allelic associations than populations with greater gene flow (and greater initial admixture associations). This is illustrated graphically in Fig. 3 and more precisely in Table II, where it is seen, for example, that the normalized allelic disequilibrium, D”‘/[(P, - pO)(Xo- x0)], is 0.022 at census t = 10 when m, = 0.03, whereas under the higher intermigration rate, m, = 0.25, the normalized allelic association is already reduced to 0.023 at t = 5 and is only 0.001 at t = 10.

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Other numerical examples show that this is not a universal phenomenon, however, because the allelic association at a given census t appears to be first enhanced by increasing migration until a critical migration level mf(t) is reached, whose value depends on t. Thereafter, further increases in migration reduce the residual disequilibrium found at that census. Table II shows, for example, that the (normalized) allelic disequilibrium at census t = 5 increases from 0.017 to 0.052 as m, increases from 0.01 to m?(5) = 0.07, and then slowly, but steady declines with greater migration rates, reaching 0.023 when m, = 0.25 and 0.016 when m, = 0.5. In contrast, the maximum residual disequilibrium five censuses later (t = lo), occurs at a much lower intermigration rate (m, = 0.03). This, in fact, illustrates a final general feature from the numerical examples: the critical migration level, m:(t), which leads to the greatest residual allelic association at a given census t, appears to steadily decrease as t increases, until by census t = 25, when the residual disequilibrium has been uniformly reduced below reasonably detectable levels, the allelic associations are effectively decreasing functions of m, . Moving on to the other cytonuclear associations, we find (still assuming d(O)= D(O) = 0 and m, = mz) that despite their common allelic associations, the populations have different trajectories for their genotypic disequilibria. Substituting (20))(21) its cytoplasmic analog, and (25) into the last three recursions in (19) shows, for instance, that at any census t 3 1 the genotypic associations in population 1 equal

where m = 2m,, and d”‘=m,(l

-m,)(Po-po)(Xo-x0)

is the original allelic disequilibrium produced by the first round of intermigration. The genotypic disequilibria in population 2 differ from those in (27) in that PO- p. entry is replaced by the complementary term, p,, - PO. The resulting time-dependent solutions are quite complex, and, indeed, unlike those for the allelic associations, are not appreciably simplified by the symmetric case under consideration. In the usual (and most complex) situation where m # 1 f $12, each genotypic association solution is here, for example, a linear combination of the four geometric terms, (1 - m)3’, (1 - m)2’, [$( 1 - m)]‘, and (i)‘. Although it is therefore difficult to characterize their qualitative dynamics analytically, it is nonetheless clear from (27) that transient genotypic disequilibria can be generated by constant inter-

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293

DISEQUILIBRIA

migration, and these associations are also all strongly influenced by the allelic disequilibrium generated by the first admixture.

Numerical examples of the trajectories in (27), in fact, indicate that the genotypic associations exhibit the same basic behavior and effects of intermigration found above for the allelic associations. Under low intermigration rates the genotypic disequilibria generally continue to increase above their generally small, initial admixture values before slowly decaying to zero (Fig. 3a), while higher gene flow generates much greater genotypic disequilibria during the first round of intermigration which then quickly decay (Fig. 3~). Most importantly, these numerical examples show that constant intermigration can generate substantial transient genotypic associations which in some cases even exceed the maximum allelic disequilibria. Census Method 2: After Reproduction and Before Intermigration

The dynamics when the two populations are censusedafter mating and reproduction are again simply related to the solutions for the first census method corresponding to the values immediately before the same mating. As in the continent-island model, only the gene frequency dynamics are independent of the census timing, the allelic disequilibrium trajectories are both reduced by half from those in (23) censused before mating, and, in contrast to the first census method, the values in each population exhibit the usual transient patterns found in isolated random-mating populations (Asmussen et al., 1987). The genotypic frequency and associations in population 1, for instance, are given at every census t 2 1 by the analogs of the leftmost equations in (15)-( 16), with Ul

*cl)= p;x, +p,d”’

+ d’i,

d:“’

= p,d”’ -+ 0

u:(‘)=2plq,x,+(q,-pr)d”‘~2~~~,

d;(‘)=(q,-p,)d”‘+O

wf(‘) = qfx, - q,d”’ + g’i etc.,

d*(‘) = -q,d”’ 3

-., 0,

where $ and i are given by (21) and its cytoplasmic analog. Thus, paralleling the continent-island model, constant intermigration will generate transient genotypic associations in a population censused after mating if and only if it generates transient allelic disequilibrium in that same population. Except for these noteworthy differences,the resulting time-dependent solutions for the genotypic disequilibria, (and cytonuclear frequencies) have the same basic form and other general properties as those under the first censusmethod. Although it is difficult to predict analytically the relative magnitudes of the genotypic disequilibrium solutions under the two census methods, numerical examples (with d(O)= D(O)= 0 and m, = m2) suggest that censusing after migration tends to generate higher homozygous disequilibria (i.e., greater maximum values for Id, 1, Id31, ID, 1,

294

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and lD3i) just as it generates greater allelic disequilibria. In contrast, the relative magnitudes of the maximal heterozygous disequilibria along a trajectory (i.e., the maximum values of In21 and ID21) can vary, depending on the initial gene frequencies in the two populations. Census timing thus has a complex effect on the genotypic associations detected (see Fig. 3), paralleling our results for the continent-island model.

DISCUSSION We have investigated how cytonuclear associations can arise as a result of continued gene flow from or between one or more populations. As the first step we derived admixture formulas for cytonuclear disequilibria, which, paralleling the case of nuclear linkage disequilibrium (Nei and Li, 1973; Prout, 1973), equal the average disequilibrium across the combining populations plus the covariance, across these populations, between the cytoplasmic frequency and the relevant nuclear frequency. The dynamical effects of both admixture and census timing were also examined under two different migration schemes with random mating in the censused population(s). In each case the transient and final cytonuclear structure are either largely (for censusing before mating) or entirely (for censusing after mating) driven by the effect of migration on the gene frequencies and allelic disequilibrium, where the latter are the same as those found for a system of two unlinked nuclear loci. In the case of the so-called continent-island model, cytonuclear dynamics are monitored within a population receiving migrants from one or more sources, whose overall genetic composition is fixed. In general, permanent cytonuclear disequilibria can develop on the “island” as long as there are nonzero associations in the overall “continent” (source) population. The equilibrium allelic association is always a simple fraction of the allelic disequilibrium in the migrants, while the determinants of the final genotypic associations and the conditions in which nonzero genotypic disequilibria are found depend strongly on the census time. If censusing occurs after reproduction by migrants, permanent genotypic associations develop if and only if the migrants have a nonrandom allelic association. When censusing is before mating, however, there are no simple rules for when nonzero genotypic disequilibria will be generated, for their ultimate values are functions of the corresponding association in the migrants as well as of the migration rate and the migrants’ allelic disequilibrium and nuclear gene frequency. Under both census methods, transient cytonuclear disequilibria may build up to levels significantly exceeding those found at equilibrium, whenever the migrants’ gene frequencies differ from the residents’ initial values.

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295

The continent-island formulation applies, for instance, to stable hybrid zones receiving migrants from two large, parental gene pools. An empirical example is provided by a recent cytonuclear survey of Atlantic eels (Avise et al., 1990). This species can be subdivided into two major taxa, one of North American ancestry and the other of European ancestry, which can be genetically distinguished through both their mitochondrial and nuclear genotypes. Hybridization appears to occur off the coast of Iceland, where, as predicted by the continent-island model, nonzero cytonuclear disequilibria are observed. Equilibria for both census timings were fitted to cytonuclear data from this hybrid zone in order to estimate the amount of gene flow from the parental populations (Avise et al., 1990). Under the best-fitting values (census after mating), the maximum likelihood estimates for the fraction of migrant North American eels per generation and migrant European eels per generation are 0.02 and 0.70, respectively. In the case of the so-called admixture model, cytonuclear dynamics are followed in two intermigrating (sub)populations. Here, long-term transient disequilibria can develop, although such admixture disequilibria always ultimately decay to zero as the continued bidirectional gene flow eventually blends the two (sub)populations into a single, random-mating population. Both the magnitude and duration of the transient associations depend in a predictable, albeit complicated, fashion on the amount of gene flow between the (sub)populations (Table II) as well as on the time of censusing. One empirical example of this second formulation is given by the admixture of different ethnic groups, such as Israeli Jews and Israeli Arabs (Bonn&Tamir et al., 1986). Both of the latter groups have distinct mitochondrial and nuclear frequencies. If admixture between these two groups is of recent origin, there is a strong possibility of nonzero cytonuclear disequilibria. Moreover, once joint cytonuclear data are available in this and other cases,gene flow estimates can be obtained by fitting the admixture model to the observed cytonuclear frequencies and disequilibria. Another, more complex example of the admixture situation is provided by the recent introduction of Africanized honeybees into Brazil and Uruguay. Based on allozyme data, the immigrant and resident specieswere initially genetically distinct, and genetic admixture is occurring in the zone of contact (Lobo et al., 1989). Here, it is very likely that significant admixture disequilibria exist in the hybrid population, since the introduction is quite recent. This situation is probably complicated, however, by the presence of assortative mating by the parental speciesand/or selection for the African genotypes. Such caseswill require an extension of the present models to include the effects of nonrandom mating (Arnold et al., 1988; Asmussen et al., 1989) and /or selection with different migration schemes

296

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(Hedrick et al., 1976; Felsenstein, 1976; Karlin, 1982; Takahata and Slatkin, 1983). Although we have focused on the generation of permanent nonrandom associations within the continent-island model, there is one, important circumstance in which this model can be directly compared to the admixture model. This is when the continent population has completely random cytonuclear associations. In such cases, the qualitative behavior of the cytonuclear disequilibria on the island parallels that in the subpopulations of the admixture model: all allelic and genotypic associations ultimately decay to zero, but significant and longlasting transient disequilibria can be generated during the initial stages of a trajectory. The admixture associations on the island are driven by the initial differences in nuclear and cytoplasmic allelic frequencies between the island and continent, in the same way that those in the admixture model are driven by the initial differences in allele frequencies between the subpopulations. The transient allelic disequilibrium with censusing before mating on the island, in fact, shows the same qualitative behavior found for (25) and illustrated in Table II for the symmetric admixture model, where m, = m2 and #O’ = (j(O)= 0 Two slight quantitative differences are nonetheless evident between the corresponding results for the two formulations. First, the duration of the increase (t,,,) is often one generation longer in the continent-island model, with allelic associations increasing for at least two generations for m < 0.22 (versus m, = m, < 0.15 for the symmetric admixture model). Second, the increase above the initial admixture value in the first generation and the subsequent normalized disequilibrium values are at least 5-12% greater in the island population, with the difference magnified by higher (inter)migration rates and longer periods of gene flow. A random-mating population receiving continued unidirectional migration from a source with random cytonuclear associations will thus both attain higher transient allelic associations and retain residual associations longer than a population experiencing continued bidirectional gene flow. In conclusion, our theoretical results show not only how cytonuclear disequilibria can be generated by admixture and how nuclear-cytoplasmic data can be used to estimate gene flow, but also how censustiming, relative to mating and migration, can strongly impact the ability to detect and utilize these associations to draw evolutionary inferences. Moreover, although estimates can be obtained from nuclear data alone, those from cytonuclear frequencies should prove more informative, becausethe maternal inheritance of the cytoplasmic marker makes cytoplasmic frequencies relatively insensitive to assortative mating (and possibly selection) but direct reflectors of migration (Asmussen et al., 1989).

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297

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A

APPENDIX

Explicit solutions for the genotypic disequilibriu and cytonuclear frequencies for the continent-island model: Substituting the solutions for pt, x,, and II(‘) from (9)-(12) into (8) shows that for all m # 4 the genotypic disequilibria are at any census t 2 1 before mating:

+c,(l -m)3’+c2[$(l

Di’)=fiI

-m)*]‘+2pb(l

+2@z[Jl -m)]‘+c,(l D~)=B,-2c,(l

-rn)‘<

-m)’

-m)3’-2c2[)(l

-m)*]‘+2(4-P)b(l

--WI)*’

+2(q-p)a[i(l-m)]‘+c,(l-m)’ D~)=fi’,+c,(l

-m)3’+c2[%(1-m)2]‘-2qb(l

-2fja[$(l

-m)]‘+cS(l

-m)”

-m)‘,

where b = 2m(po - P)h - 4 l-2m

a=D,-a-b,

c =(3-2m)(po-Ab 1 2(1-m)*

c* =

’

2(Pcl- Ha

l-m

cj=(Po-~)~+m(~*-d)(xo-,T) c4 = -2(p,

- p)6 + m(2@ - 6)(x, -2)

c5 = (pO - p)6 + m(Q* - W)(xO - 5)

and Ij, I?,, fi2, and L?, are as in (12)-(13). The time-dependent solutions for the cytonuclear genotypes are similarly obtained using the recursions in (6). For example, u(,‘)=li,+d,(l-m)3’+d2[~(1-m)2]‘+(~d3+d,)(l-m)2’

+2@2[4(1 -m)]‘+d,(l-m)’ u{“=C,-2d,(l

-m)“-2d,[i(l

+2(q-p)a[&(l

-m)‘]‘+

-m)]‘+d6(1

-m)’

wjr~=~,+d,(l-m)3’+d2[~(l-m)2]‘+(-~d,+d,)(l-m)2’

-2@2[$(1 -m)]‘+d,(l

-m)’

[(+P)d,-2d,](l

-m)2’

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ASMUSSEN

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for all m # 4 and t >, 1, where d, po-~)‘(.~~~--~l

(1 -2m)(l

/wdm

d

-m)“’

‘l ~2h-PK~“--~) 3 1 - 2m

2 d

’

1 -t72 = (PO-

4

PI’.?

1- m

d, = $(x0 - 2) -t 2F( /I0 - p),? + (p” - j5) B d, = 2&7(x, - x) + 2(L5- J?)(pO- /7)x - 2( p,, - p)6 d7=q2(xo-X)-22q(po-p)~~+((po-ij)~

and li,, 6,, and 6, are given in (14). The trajectories censused after mating are equivalent to those above under the first census method, with the following differences. In the case of the genotypic disequilibria, each L?, is replaced by fi: from (16); a is replaced throughout by a* = $a; b is replaced throughout by 6” = fb; c,=2(p,,-p)a*; and c3=cs= -&cq=2(p0--p)fi*. c,=2(p,-p)b*; For the cytonuclear frequencies, ti;, C,, bCi,etc., are replaced by a,*, 07, $17 from (15); a is again replaced throughout by a* = $7; d, is multiplied by (1 - m)‘; dZ, d,, and d4 are multiplied by 1 - m. The solutions for the special case where m = $ can be obtained in the same way, using the second solution for D”’ in (10).

APPENDIX

B

Gene ,fkequency and allelic disequilibrium trajectories under the admixture model censused after migration; The nuclear gene frequency dynamics in

(18) can be written in matrix form as p,+ , = Ap,, where p, = ( pt, P,)T with T denoting vector transpose, and the coefficient matrix is A=

l-m, m2

m,

l-m,

I’

(Bl)

Iterating this matrix equation immediately yields the solution, p, = A’p,. This reduces to the formulas in (20)-(21) after forming the spectral decomposition, A’ = PAY’, where

with I. r = 1 and AZ= 1 - m the eigenvalues of the coefficient matrix, A, and the columns of P the corresponding right eigenvectors.

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CYTONUCLEAR DISEQUILIBRIA

Given the gene frequency solutions, the allelic disequilibrium recursions can be written as a three-dimensional linear transformation of the variables, d,, D,, and z,= (P, - p,)(X, -x,) or, alternatively, as the twodimensional transformation in (22). The latter is of the form, d I + 1= Bd, + a’c,

033)

where d, = (d,, D,)T, B = iA, where A is the matrix in (Bl), a = (1 --wz)~, and c = (ci, c2)= is a constant vector with ci = mi( 1 - m,)(P, - pO)(XO- x0) for i= 1, 2. Iterating (B3) shows in conjunction with the spectral decomposition of B that at any census t > 1, I-l

d,=B’d,+

(

c

c

Bka’-lpk >

k=O

t-1

= (P/IF’)

do +a’-’

1 (PkY’) k=O

apk

c 1

(B4)

where A’, P, and Pp ’ are as in (B2), with in this case iI = f and 2, = i(l -m) the eigenvalues of the matrix B; and for i= 1,2

After performing the necessarymatrix multiplications in (B4), we obtain the general solutions d’) = 6”‘A;

+ (m,/m)(dO) - D”‘) 1”; + Fffi” + (ml/m)(cl

- cJf:”

DC’) = 6”‘1;

+ (m2/m)(D’o’ - do’) 2; + i;fl” + (m,/m)(c, - cl)f

:“,

(J36)

where D(O) is as in (23), ci= mi( 1 - m,)(Po- po)(Xo- x0) for i= 1, 2 are the elements of the constant vector c, and C= (m, c2+ m2cl)/m. ACKNOWLEDGMENTS This research was supported in part by National Science Foundation grants BSR-8819482 and BSR-8716804.We wish to thank M. T. Clegg, B. S. Weir, and the anonymous reviewers for helpful comments on the original draft.

65313913.5

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