THE NUCLEAR DICYTOPLASMIC SYSTEM

We consider the nuclear-dicytoplasmic structure of a diploid plant population. The dicytonuclear system consists of an autosomal nuclear locus with two alleles, A and a, and two haploid cytoplasmic loci: a mitochondrial marker with two alleles (cytotypes) M and m and a chloroplast marker with two alleles (cytotypes) C and c. As is true in the Pinaceae (Mogensen 1996), we assume that the mitochondria are inherited maternally and chloroplasts paternally, with no cross-parental leakage of either organelle. The definitions of frequency and nonrandom association variables follow those of Schnabel and Asmussen (1989), and we use the notation of that article throughout, except where noted below.

Frequency variables: The frequencies of the 12 joint, three-locus genotypes are given in Table 1 as U_ij, V_ij, and W_ij, where the first index (i = 1, 2) indicates M or m alleles and the second index (j = 1, 2) indicates C or c alleles. The joint cytotype frequencies (X_ij) are obtained by summing across each row and the nuclear genotype frequencies (U, V, and W) by summing down each column. The nuclear allele frequencies are P=freq(A)=U+1∕2V (1a) and Q=freq(a)=W+1∕2V=1−P, (1b) where freq denotes “frequency of.” The cytotype frequencies for the two cytoplasmic markers are XM=freq(M)=X11+X12YM=freq(m)=X21+X22XC=freq(C)=X11+X21YC=freq(c)=X12+X22, (2) where Y_M = 1 − X_M and Y_C = 1 − X_C. It is also useful to specify the frequencies of the three-locus triallelic combinations where, for each joint cytotype combination i, j = 1, 2, Pij=Uij+1∕2Vij (3a) and Qij=Wij+1∕2Vij. (3b) For instance, P₁₁ = freq(A/M/C) is the probability that a randomly sampled individual has the joint cytotype MC and that a randomly selected nuclear allele from that individual is A. Note that although we normally use slashes (/) to separate different markers, we omit the slash between the two organellar loci when viewed as a joint cytotype.

The final frequency variables are those for the two-locus nuclear-cytoplasmic combinations within each of the two cytonuclear subsystems. The two sets of joint, two-locus genotypic frequencies are given in Table 2. The joint diallelic frequencies for the nuclear-mitochondrial subsystem represent the ovule frequencies, while those for the nuclear-chloroplast subsystem represent the pollen frequencies in the population (Table 3).

Two-locus disequilibria: We consider three sets of pairwise disequilibria among the three loci. These disequilibria are calculated in the same general way as are linkage disequilibria among nuclear loci and equal the difference between the frequency of each joint combination and that expected under random association between the two components involved. The first two involve the two-locus associations in each cytonuclear subsystem as defined in Asmussen et al. (1987). To facilitate the development of concise general formulas in the three-locus system, they are denoted here by a more informative notation that explicitly includes the nuclear and cytoplasmic components involved. The two allelic disequilibria DA∕M=freq(A∕M)−freq(A)freq(M)=P1M−PXMDA∕C=freq(A∕C)−freq(A)freq(C)=P1C−PXC (4) (D_M and D_c in Schnabel and Asmussen 1989) measure the nonrandom association between the nuclear alleles and each cytoplasmic marker. The six genotypic disequilibria DAA∕M=U1M−UXM,DAA∕C=U1C−UXCDAa∕M=V1M−VXM,DAa∕C=V1C−VXCDaa∕M=W1M−WXM,Daa∕C=W1C−WXC (5) similarly measure the nonrandom associations between each nuclear genotype and each cytoplasmic marker, where here we have substituted AA, Aa, aa for the usual genotype placeholders 1, 2, and 3. For example, D_AA/M = freq(AA/M) − freq(AA)freq(M) measures the nonrandom association between AA nuclear homozygotes and the mitochondrial marker M (D_{1 M} in Schnabel and Asmussen 1989). Within each of the two cytonuclear subsystems, each disequilibrium can be written as a linear combination of any two of the others via the relations DAA∕∗+DAa∕∗+Daa∕∗=0 (6a) and DA∕∗=DAA∕∗+1∕2DAa∕∗, (6b) where each * stands for M for the nuclear-mitochondrial associations and each * stands for C for the nuclear-chloroplast associations.

The third and final type of two-locus disequilibria is the cytoplasmic disequilibrium, DM∕C=freq(MC)−freq(M)freq(C)=X11−XMXC (7) (D_MC in Schnabel and Asmussen 1989), which is new to the three-locus system and measures the nonrandom association between alleles at the two cytoplasmic loci. The two-locus genotypic frequencies (U_iM, U_iC, etc.), joint diallelic frequencies (P_iM, P_iC, etc.), and joint cytotype frequencies (X_ij) can be written in terms of the marginal frequencies and the corresponding cytonuclear and cytoplasmic disequilibria, as shown in Tables 4, 5 and 6.

Three-locus disequilibria: The three-locus system can also result in higher-order associations that involve all three loci, both pairwise three-locus associations and full three-way associations. These are specified fully in Schnabel and Asmussen (1989) and for convenience are briefly summarized here. Throughout, we focus on the associations involving the M and C cytotypes; the interrelations in the three-locus system allow us to derive all of the other associations from these (Schnabel and Asmussen 1989). The basic three-locus disequilibria are the three joint genotypic disequilibria DAA∕MC=freq(AA∕MC)−freq(AA)freq(MC)=U11−UX11DAa∕MC=freq(Aa∕MC)−freq(Aa)freq(MC)=V11−VX11Daa∕MC=freq(aa∕MC)−freq(aa)freq(MC)=W11−WX11 (8) and the joint allelic disequilibrium DA∕MC=freq(A∕MC)−freq(A)freq(MC)=P11−PX11, (9) which, respectively, measure the nonrandom associations between the MC joint cytotype and the three genotypes and two alleles at the nuclear locus.

Several true three-way measures of nonrandom associations can also be defined that measure associations among the three markers (nuclear, mitochondrial, and chloroplast) after taking into account all of the possible two-way associations (nuclear-mitochondrial, nuclear-chloroplast, and mitochondrial-chloroplast). For the M/C cytotype, we have three three-way genotypic disequilibria DAA∕M∕C=U11−UXMXC−UDM∕C−XMDAA∕C−XCDAA∕MDAa∕M∕C=V11−VXMXC−VDM∕C−XMDAa∕C−XCDAa∕MDaa∕M∕C=W11−WXMXC−WDM∕C−XMDaa∕C−XCDaa∕M (10) and the three-way allelic disequilibrium DA∕M∕C=P11−PXMXC−PDM∕C−XMDA∕C−XCDA∕M, (11) which is analogous to the three-way gametic disequilibrium for three nuclear loci (Bennett 1954). The sets of four joint and four three-way associations each satisfy the relations given in (6a) and (6b), with * = MC for the joint disequilibria and * = M/C for the three-way disequilibria. We also note that each three-way disequilibrium can be written in terms of the corresponding joint dicytonuclear and two-locus disequilibria and the marginal cytotype frequencies, with DN∕M∕C=DN∕MC−XMDN∕C−XCDN∕M (12) for each nuclear component, N = A, AA, Aa, or aa.

Finally, only 11 independent variables are necessary to completely describe the full 12 joint genotype system. One such parameterization (Table 7) includes five two-locus disequilibria (Da/m, D_mc, Da/M, Dia/c, and Dm/c), two three-locus disequilibria (D_A/MC and D_Aa/MC), the nuclear, mitochondrial, and chloroplast allele frequencies (P, X_M, and X_C), and the nuclear heterozygote frequency (V). However, for completeness, our analysis below provides the results for an allelic, homozygote, and heterozygote association for each disequilibrium category: Da/mc, D_M/mc, and Dia/MC for the three-locus joint disequilibria and their counterparts for the three-way disequilibria.

ADMIXTURE FORMULAS

After migration by seeds (or adults), the dicytonuclear makeup of the new population will be a mixture of that of residents and migrants. We consider here the admixture effects for the general case where n genetically distinct sources contribute to a single population. Define the expected value of a variable Z across all n sources as E(Z)=∑i=1nmiZ(i), (13) where m_i is the fraction contributed by source i and Z⁽ⁱ⁾ is the value of the variable Z in source i. Frequencies after admixture are simply their weighted average (expectation) across all sources, ZT=E(Z), (14) where ^T indicates values in the total population after admixture.

Admixture has a more complicated effect on disequilibria since these can be generated in the total population by a two-locus Wahlund effect if the sources are genetically distinct (Nei and Li 1973; Asmussen and Arnold 1991; Goodisman and Asmussen 1997). Following admixture, the two-way disequilibria will be the weighted average of the disequilibria across the n sources plus the covariance across the sources between the frequencies of the two genetic components being considered (Asmussen and Arnold 1991). For example, the allelic and genotypic cytonuclear disequilibria after admixture are DN∕MT=E(DN∕M)+Cov(FN,XM) (15a) DN∕CT=E(DN∕C)+Cov(FN,XC), (15b) where N = A, AA, Aa, or aa, and F_N is the frequency of N. The cytoplasmic disequilibrium after admixture has the analogous form DM∕CT=E(DM∕C)+Cov(XM,XC) (16) while the joint allelic and joint genotypic disequilibria become DN∕MCT=E(DN∕MC)+Cov(FN,X11), (17) where, again, N = A, AA, Aa, or aa.

The derivation of these general formulas depends on the fact that Cov(X,Y) = E(XY) − E(X) E(Y) for any two random variables X and Y. As an example, we derive the joint allelic disequilibrium after admixture, using (9) and (14), as DA∕MCT=P11T−PTX11T=E(P11)−E(P)E(X11)=E(P11)−E(PX11)+Cov(P,X11)=E(P11−PX11)+Cov(P,X11)=E(DA∕MC)+Cov(P,X11), which is (17), where N = A and F_N = P.

For all of the two-way disequilibria, the covariance terms have a simple interpretation for the special case where there are only two source populations contributing to the total population. In this case the admixture formulas (15, 16 and 17) become DN∕MT=E(DN∕M)+m1(1−m1)(FN(1)−FN(2))(XM(1)−XM(2))DN∕CT=E(DN∕C)+m1(1−m1)(FN(1)−FN(2))(XC(1)−XC(2))DM∕CT=E(DM∕C)+m1(1−m1)(XM(1)−XM(2))(XC(1)−XC(2))DN∕MCT=E(DN∕MC)+m1(1−m1)(FN(1)−FN(2))(X11(1)−X11(2)), where ⁽ⁱ⁾ indicates source i and N = A, AA, Aa, or aa. One can see that the covariance term will be nonzero, causing an admixture effect, if and only if the two components both differ across the two source populations. In general, to have an admixture effect with more than two source populations, it is necessary but not sufficient that the frequencies of both components vary across the sources.

Three-way disequilibria after admixture are more complex than are two-way disequilibria, involving the weighted averages of the disequilibria and cytoplasmic frequencies, covariances between nuclear and cytoplasmic frequencies, and covariances between cytoplasmic frequencies and two-locus cytonuclear disequilibria. The general formula is DN∕M∕CT=E(DN∕M∕C)−E(XM)Cov(FN,XC)−E(XC)Cov(FN,XM)+Cov(FN,X11)+Cov(XM,DN∕C)+Cov(XC,DN∕M) (18) for N = A, AA, Aa, or aa, where, once again, F_N is the frequency of N. An admixture effect will be found for three-way associations only if at least one of the contributing covariances is nonzero. As an example of the derivation, we consider the three-way allelic disequilibrium, D_A/M/C. From (11) and (14), DA∕M∕CT=P11T−PTXMTXCT−PTDM∕CT−XMTDA∕CT−XCTDA∕MT, which, using (7) and (15), can be rewritten as DA∕M∕CT=P11T−PTX11T−XMTDA∕CT−XCTDA∕MT=E(P11)−E(P)E(X11)−E(XM)[E(DA∕C)+Cov(P,XC)]−E(XC)[E(DA∕M)+Cov(P,XM)]. Once again using the definition of the covariance given above, we have DA∕M∕CT=E(P11)−E(PX11)+Cov(P,X11)−E(XMDA∕C)+Cov(XM,DA∕C)−E(XM)Cov(P,XC)−E(XCDA∕M)+Cov(XC,DA∕M)−E(XC)Cov(P,XM). Finally, using the definition of the three-way allelic disequilibrium (D_A/M/C) in (11) gives us the formula corresponding to (18), with N = A and F_N = P, DA∕M∕CT=E(DA∕M∕C)−E(XM)Cov(P,XC)−E(XC)Cov(P,XM)+Cov(P,X11)+Cov(XM,DA∕C)+Cov(XC,DA∕M).

MODEL OF POLLEN AND SEED MIGRATION

The model of pollen and seed migration (summarized in Figure 1) represents a three-locus extension of the two-locus cytonuclear migration models considering only maternal or only paternal cytoplasmic inheritance in Asmussen and Schnabel (1991) and Schnabel and Asmussen (1992). We consider a population with nonoverlapping generations and no seed dormancy, from which we census adults. Migration is modeled via a continent-island framework with unidirectional migration, from the source population(s) to the population under consideration (Figure 2). We assume that the migration rates and the genetic composition(s) of the source population(s) are constant over time, and we ignore the effects of selection, mutation, and genetic drift. Mating is a mixture of selfing, which occurs at rate s, and outcrossing, which occurs at rate 1 − s in accordance with the mixed-mating model (Clegg 1980). During mating, pollen migration occurs at rate M, such that the fraction of outcrossed pollen that comes from the migrant pollen pool is M, and the fraction of outcrossed pollen that comes from within the population is 1 − M. The total fraction of migrant pollen per generation is thus M(1 − s). After pollination and seed maturation, seed migration occurs at a rate m, which means that each generation the seed population is a mixture of migrant seeds (a fraction m) and resident seeds (a fraction 1 − m). Following germination and growth, a new adult population is formed, completing the generation cycle.

We distinguish variables in the various life stages by letting uppercase letters represent variables in adults (e.g., P, U, X_M, X_C, D_M/C) and lowercase letters represent the corresponding variables in the interim seed population (e.g., p, u, x_M, x_C, d_M/C). Variables in the two migrant pools are distinguished by overbars, with lowercase letters again indicating seeds and uppercase letters now indicating pollen. Since migrant pollen carries only a haploid nuclear component and the chloroplast genome, it is characterized by its nuclear-chloroplast diallelic frequencies (P‒1C , P‒2C , Q‒1C , Q‒2C ) together with its nuclear and chloroplast allele frequencies (P‒ , X‒C ) and the nuclear-chloroplast allelic disequilibrium (D‒A∕C ). Migrant seeds, on the other hand, carry a full complement of all three genomes, and therefore have analogs of all the three-locus frequency and disequilibrium variables defined above (e.g., u‒ij , p‒ , x‒M , x‒C , d‒A∕M , d‒A∕C , d‒M∕C , d‒A∕MC , d‒A∕M∕C ). Disequilibria may occur in migrant seed and pollen if, for example, the source population itself is a mixture of genetically distinct populations or experiences appropriate forms of nonrandom mating (Arnoldet al. 1988) or selection (Babcock and Asmussen 1996, 1998).

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Figure 1.

Adult census model showing the two types of gene flow within a generation cycle. Pollen migration occurs first, at rate M, followed by seed migration at rate m. Mating is a mixture of selfing, which occurs at rate s, and outcrossing, which occurs at rate 1 − s.

The interim seed variables, recursion equations, and equilibrium values for the nuclear-mitochondrial and nuclear-chloroplast subsystems are equivalent to those for the cases of strictly maternal and strictly paternal cytoplasmic inheritance, respectively (Asmussen and Schnabel 1991; Schnabel and Asmussen 1992). To convert those results to the notation used here, we replace each cytoplasmic and cytonuclear variable z for maternal cytoplasmic inheritance with z_M (e.g., x becomes x_M,) and those variables for paternal cytoplasmic inheritance with z_C (e.g., x becomes x_C), with the nuclear component (A, AA, Aa, or aa) explicitly indicated in the cytonuclear disequilibria as above.

We develop the recursion equations for the frequencies and representative disequilibria new to the dicytonuclear system for the complete model with mixed mating and both pollen and seed migration in two steps, first finding the interim values in the seeds following pollen migration and fertilization, and then calculating the new adult values following seed migration. We also consider eight important cases subsumed within this general framework.

Interim seed values: The joint genotype frequencies at the interim seed stage can be calculated by considering the contribution to each joint genotype due to self-fertilization of resident ovules and that due to random outcrossing. The contribution due to self-fertilization depends only on the joint genotype frequencies of the resident adults (Table 1), while the contribution due to outcrossing is determined by the joint nuclear-mitochondrial allele frequencies in resident ovules and the joint nuclear-chloroplast allele frequencies in the pollen pool (Table 3), which contains both resident and migrant pollen. For example, consider the frequency of AA/M/C progeny. Such progeny result from the fertilization of a resident A/M ovule by A/C pollen. Under selfing (which occurs with probability s), the only individuals who can produce the right type of ovules and pollen are AA/M/C (frequency U₁₁, who always do) and Aa/M/C (frequency V₁₁, who produce each correct gamete type half of the time and the appropriate combination of gametes one-fourth of the time). The contribution due to outcrossing (probability 1 − s) is similarly straightforward to derive. In this case an A/M ovule (frequency P_1M) is fertilized by either migrant A/C pollen (probability M P_1C) or by resident A/C pollen [probability (1 − M) P_1C]. The frequency of AA/M/C progeny is then u11=s(U11+1∕4V11)+(1−s)P1M[MP‒1C+(1−M)P1C]. Repeating the same reasoning, we find that the frequencies for all of the interim seed genotypes are uij=s(Uij+1∕4Vij)+(1−s)PiM[MP‒jC+(1−M)PjC] (19a) vij=s2Vij+(1−s)PiM[MQ‒jC+(1−M)QjC]+(1−s)QiM[MP‒jC+(1−M)PjC] (19b) wij=s(Wij+1∕4Vij)+(1−s)QiM[MQ‒jC+(1−M)QjC], (19c) where i = 1 or 2 indicates M or m at the mitochondrial locus and j = 1 or 2 indicates C or c at the chloroplast locus.

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Figure 2.

Continent-island migration model with unidirectional pollen (M) and seed (m) migration.

From these interim dicytonuclear genotype frequencies, the marginal frequencies at the three loci can be found along with the interim two-locus cytonuclear disequilibria using the definitions in (4) and (5). Interim values for each of the two-locus subsystems that will be used in further derivations are given in appendix a, Equations A1, A2, A3, A4, A5, A6, A7 and A8. The interim seed values for the other new three-locus variables can then be derived from the joint genotype frequencies using Table 1 and (7–11). The joint cytotype frequencies are xij=sXij+(1−s)XiM[MX‒jC+(1−M)XjC], (20) where X_1M = X_M, X_2M = Y_M, X_1C = X_C, and X_2C = Y_C, and the two-locus cytoplasmic disequilibrium is dM∕C=sDM∕C. (21) The formulas for the interim three-locus joint and threeway disequilibria in seeds are more complex and are given in appendix a, Equations A9, A10, A11 and A12.

Recursion equations: Seed migration completes the life cycle. Using (13) and (14), each frequency variable in the new generation of adults is simply the weighted average of the corresponding value in migrant (m) and resident seeds (1 − m). The new frequencies of the joint genotypes and the joint cytotypes are then Uij′=mu‒ij+(1−m)uijVij′=mv‒ij+(1−m)vij Wij′=mw‒ij+(1−m)wij (22) and Xij′=mx‒ij+(1−m)xij, (23) where the corresponding interim seed frequencies are given by (19) and (20).

The disequilibria in the new adults are the result of admixture between resident and migrant seeds, which can be calculated using (15, 16, 17 and 18). The cytoplasmic and joint three-locus disequilibria in new adults, for example, will be the weighted average of the corresponding disequilibria in migrant and resident seeds plus the covariance across these two seed populations between the frequencies of the two genetic components being considered. From (16), the cytoplasmic disequilibrium after admixture is DM∕C′=md‒M∕C+(1−m)dM∕C+m(1−m)(xM−x‒M)(xC−X‒C), (24) while, using (17), the new joint three-locus disequilibria are found to be DN∕MC′=md‒N∕MC+(1−m)dN∕MC+m(1−m)(fN−f‒N)(x11−x‒11), (25) for N = A, AA, Aa, or aa, where f_N is the frequency of the nuclear component N in seeds. The corresponding three-way disequilibria are similarly obtained from (18) as DN∕M∕C′=md‒N∕M∕C+(1−m)dN∕M∕C+m(1−m)(fN−f‒N)[dM∕C−d‒M∕C+(2m−1)(xM−x‒M)(xC−x‒C)]+m(1−m)[(xM−x‒M)(dN∕C−d‒N∕C)+(xC−x‒C)(dN∕M−d‒N∕M)]. (26)

Multiple alleles: Extension of this general framework to include multiple alleles and cytotypes is straightforward for the joint genotype and marginal frequencies. If we have n₁ nuclear alleles (A₁, …, A_n1), n₂ mitochondrial cytotypes (M₁, …, M_n2), and n₃ chloroplast cytotypes (C₁, …, C_n3), for a total of n₁(n₁ + 1)n_2n3/2 joint genotypes, we can let F_ij,k,l indicate the frequency of adults with the A_iA_j nuclear genotype and the M_k mitochondrial and C_l chloroplast types. As an example of such a generalization, consider Equations 19a, 19b and 19c giving interim joint genotype frequencies in seeds (f_ij,k,l). These would be replaced by two general equations, one for homozygotes, fii,k,l=s(Fii,k,l+1∕4∑j≠1Fij,k,l)+(1−s)Pi,k[MP‒i,(l)+(1−M)Pi,(l)], and one for heterozygotes, fij,k,l=s2Fij,k,l+(1−s)Pi,k[MP‒j,(l)+(1−M)Pj,(l)]+(1−s)Pj,k[MP‒i,(l)+(1−M)Pi,(l)], where P_i,k is the frequency of resident A_i/M_k ovules, P_i,(l) is the frequency of resident A_i/C_l pollen, P‒i,(l) is the frequency of migrant A_i/C_l pollen, and i, j = 1, 2, …, n₁, k = 1, 2, …, n₂, and l = 1, 2, …, n₃. Note that we do not distinguish between the nuclear genotypes A_iA_j and A_jA_i, so that, for j > i, f_ij,k,l = f_ji,k,l and F_ij,k,l = F_ji,k,l. Substituting these interim seed values into the analogs of (22) then gives the adult genotypic recursions for the case of multiple alleles.

The corresponding increase in the number of disequilibria, however, is not as straightforward to analyze. Disequilibria could be defined, in the manner of Equations 4, 5, 6, 7, 8, 9, 10, 11 and 12, for each possible two-way or three-way association between the various alleles, genotypes, and cytotypes following the multiallelic approach for cytonuclear systems in Asmussen and Basten (1996). Although there is no difficulty in setting up such definitions, interpretation of such a complex system is left for further work. For the remainder of this article, we focus on the basic case of diallelic loci.

DICYTONUCLEAR EQUILIBRIUM STRUCTURE

The utility of cytonuclear and dicytonuclear data for decomposing and estimating gene flow in plant populations depends on the extent to which such data reflect the differential effects of pollen and seed migration. To address this issue, we now turn to the equilibrium state for the dicytonuclear system, which is determined by the equilibrium values of the frequency and disequilibrium variables within the parameterization shown in Tables 6 and 7. These are calculated by setting each value after one generation of migration and mating equal to its previous value (for example, P′ = P) and solving. Although not shown here, the stability of the three-locus equilibrium and the full dynamical behavior of the dicytonuclear system are determined by the explicit time-dependent solutions for the values of the independent populational variables in each generation t. The dynamical solutions for each of the two-locus cytonuclear systems are given in Asmussen and Schnabel (1991) and Schnabel and Asmussen (1992). Similar methods yield the explicit time-dependent solutions for the disequilibria needed to fully parameterize the three-locus dicytonuclear system. The expected form of these dynamical solutions with their multiple geometric terms shows that the resident population always converges to the unique three-locus equilibrium specified by the equilibrium equations given below.

We focus first on the outcome in mixed-mating populations receiving both pollen and seed migration (0 < s, m, M < 1); the distinctive features of populations that are purely selfing or random mating or experience only one form of gene flow are treated in the subsequent section as special cases. Two practical points should be kept in mind when interpreting the equilibrium values under these different biological conditions. First, although this is a continent-island model, the resident population does not simply become an exact replica of the source population because of the two distinct forms of gene flow and the various levels of nonrandom associations these can generate among the three markers. Second, we are particularly interested in when permanent disequilibria are produced. Although not strictly necessary for gene flow estimation, permanent disequilibria should increase the conditions under which the equilibrium for a cytonuclear or dictyonuclear system reflects, and can be used to estimate, the rate of pollen (M) or seed (m) migration.

Final cytonuclear variables and marginal frequencies: The equilibria for the two-locus subsystems are given in Asmussen and Schnabel (1991) and Schnabel and Asmussen (1992) and are provided in the current notation in appendix b, Equations B1, B2, B3 and B4. Because of their key roles in the new three-locus formulas, we reiterate the equilibrium formulas for the marginal frequencies and the nuclear genotype frequencies here. The final nuclear and chloroplast allele frequencies, P^=2mp‒+(1−m)M(1−s)P‒2m+(1−m)M(1−s) (27) and X^C=mx‒C+(1−m)M(1−s)X‒Cm+(1−m)M(1−s), (28) are weighted averages of the frequencies in migrant seeds and pollen, while the equilibrium mitochondrial frequency, X^M=x‒M, (29) is simply the frequency in migrant seeds, since only migrant seeds, and not migrant pollen, carry the mitochondrial locus. Note that in each case a polymorphic equilibrium is required to generate permanent disequilibria involving that marker. The frequency of nuclear heterozygotes converges to a complicated convolution of the frequency of heterozygotes in migrant seeds and the nuclear allele frequencies in the two migrant pools, V^=2mv‒+2(1−m)(1−s)[M(P^Q‒+Q^P‒)+2(1−M)P^Q^]2−(1−m)s, (30) with the final nuclear homozygote frequencies given by U^=P^−1∕2V^ and W^=Q^−1∕2V^ .

The final marginal frequencies are those of the joint cytotypes, which are new to the three-locus system. From (20) and (23), we find that the equilibrium frequency of each joint cytotype X^ij=mx‒ij+(1−m)(1−s)X^iM[MX‒jC+(1−M)X^jC]1−(1−m)s (31) (where X_1M = X_M, X_2M = Y_M, X_1C = X_C, and X_2C = Y_C) is the weighted average of the corresponding frequency in migrant seeds (mx‒ij ) and the frequency in resident outcrossed seeds formed using either migrant pollen [(1−m)M(1−s)X^iMX‒jC] or resident pollen [(1−m)(1−M)(1−s)X^iMX^jC] .

Final two-way disequilibria: Turning to disequilibrium measures new to the three-locus system, we find that the equilibrium value for the cytoplasmic disequilibrium is D^M∕C=m1−(1−m)sd‒M∕C. (32) From (32), we see that pollen migration has no effect on the final association between alleles at the two cytoplasmic loci since pollen carry only the chloroplast and not the mitochondrial marker. In fact, there will be permanent disequilibrium between the two cytoplasmic loci if and only if the mitochondrial and chloroplast cytotypes are nonrandomly associated in migrant seeds. Such nonrandom associations among these or other markers could reflect disequilibria generated by selection or other nonrandomizing forces in the source population; migrant seed (or pollen) disequilibria would also be expected if seeds (or pollen) are derived from multiple, genetically distinct sources, as would occur in hybrid zones or other areas of admixture (see Equations 15, 16, 17 and 18).

We now consider the representative three-locus disequilibria. At equilibrium, the joint allelic disequilibrium is a linear combination of six factors, D^A∕MC=c1d‒A∕MC+c2d‒A∕M+c3d‒A∕C+c4D‒A∕C+c5(P‒−p‒)d‒M∕C+c6(P‒−p‒)(X‒C−x‒C), (33) where the c_i are constants that depend on the migration and selfing rates and the cytotype frequencies in the migrant seeds and pollen; these are given in (C1) in appendix c. We see immediately that joint allelic disequilibrium can be produced by the corresponding disequilibrium in migrant seeds (d‒A∕MC ) or by allelic cytonuclear disequilibria in the migrant pools within either the nuclear-mitochondrial or nuclear-chloroplast subsystems (d‒A∕M , d‒A∕C , D‒A∕C ). Joint allelic associations can also be generated by intermigrant interactions involving unequal nuclear allele frequencies in migrant pollen and seeds coupled either with cytoplasmic disequilibrium in migrant seeds (P‒≠p‒ and d‒M∕C≠0 ) or with un-equal chloroplast frequencies in the two migrant pools (P‒≠p‒ and X‒C≠x‒C ). Different nuclear or chloroplast frequencies in the migrant pollen and seeds could be a result of differences in average distance for these two types of gene flow, resulting in different sources for pollen and seeds or differences in the relative contribution of multiple source populations to the two migrant pools (e.g., source 1 might contribute more pollen than seeds). Selection or other evolutionary forces within the life cycle of the source population(s) could also result in different nuclear or chloroplast frequencies in the two migrant pools. While joint nuclear-chloroplast allele frequency differences also generate nuclear-chloroplast allelic disequilibria [D^A∕C which is D^ in Schnabel and Asmussen (1992)], the intermigrant interaction of nuclear allele frequency differences with cytoplasmic disequilibria in migrant seeds is a new factor unique to the three-locus system.

The final three-locus joint genotypic disequilibria can be written as D^AA∕MC=c1d‒AA∕MC+c2D^Aa∕MC+c3d‒A∕M+c4d‒A∕C+c5D‒A∕C+c6d‒A∕Md‒A∕C+c7d‒A∕MD‒A∕C+c8d‒M∕C+c9(X‒C−x‒C)+c10(P‒−p‒)(X‒C−x‒C)+c11(P‒−p‒)(X‒C−x‒C)d‒A∕M (34a) D^Aa∕MC=c1d‒Aa∕MC+c2d‒A∕M+c3d‒A∕C+c4D‒A∕C+c5d‒A∕Md‒A∕C+c6d‒A∕MD‒A∕C+c7d‒M∕C+c8(X‒C−x‒C)+c9(P‒−p‒)(X‒C−x‒C)+c10(P‒−p‒)(X‒C−x‒C)d‒A∕M, (34b) where the c_i coefficients for each disequilibrium are given in (C2) and (C3), respectively. Similarly to the joint allelic disequilibrium, joint genotypic disequilibria can be produced by joint disequilibria in migrant seeds (d‒AA∕MC and d‒Aa∕MC for the homozygote disequilibrium, d‒Aa∕MC alone for the heterozygote disequilibrium) or by allelic cytonuclear disequilibria in the migrant pools (d‒A∕M,d‒A∕C,D‒A∕C and crossproducts of these). However, joint genotypic disequilibria can also be generated directly by cytoplasmic disequilibria in migrant seeds (d‒M∕C ) without the need for unequal nuclear allele frequencies in the two migrant pools. Finally, joint genotypic disequilibria can be generated via intermigrant interactions involving unequal chloroplast frequencies in migrant pollen and seeds (X‒C≠x‒C ), either alone or in conjunction with other intermigrant factors (P‒≠p‒ with or without d‒A∕M≠0 ).

Final three-way disequilibria: The three-way associations are generated in fewer ways than the three-locus joint disequilibria. Allelic cytonuclear disequilibria in the migrant pools (d‒A∕M,d‒A∕C , or D‒A∕C≠0 ) or “simple” intermigrant admixture effects due to allele frequency differences in migrant pollen and seeds (X‒C≠x‒C , or P‒≠p‒ and X‒C≠x‒C ) are insufficient to generate true three-way associations. For example, the three-way allelic disequilibrium is found at equilibrium to be D^A∕M∕C=c1d‒A∕M∕C+c2(P‒−p‒)d‒M∕C+c3(X‒C−x‒C)d‒A∕M, (35) where the c_i are given in (C4). This can be generated in only three ways: by the corresponding disequilibria in migrant seeds (d‒A∕M∕C ) or by either of two types of intermigrant interactions unique to the dicytonuclear system. The first couples unequal nuclear allele frequencies with cytoplasmic disequilibria in migrant seeds (P‒≠p‒ and d‒M∕C≠0 ), which also enters into the joint allelic disequilibrium (33). The second is a new, three-locus interaction involving unequal chloroplast frequencies in migrant pollen and seeds together with nuclear-mitochondrial allelic disequilibria in migrant seeds (X‒C≠x‒C and d‒A∕M≠0 ).

Finally, at equilibrium, the three-way genotypic disequilibria are D^AA∕M∕C=c1d‒AA∕M∕C+c2D^Aa∕M∕C+c3d‒M∕C+c4d‒A∕Md‒A∕C+c5d‒A∕MD‒A∕C+c6(X‒C−x‒C)d‒A∕M+c7(P‒−p‒)(X‒C−x‒C)d‒A∕M+c8(X‒C−x‒C)d‒AA∕M (36a) D^Aa∕M∕C=c1d‒Aa∕M∕C+c2d‒M∕C+c3d‒A∕Md‒A∕C+c4d‒A∕MD‒A∕C+c5(X‒C−x‒C)d‒A∕M+c6(P‒−p‒)(X‒C−x‒C)d‒A∕M+c7(X‒C−x‒C)d‒Aa∕M, (36b) where the c_i are given in (C5) and (C6), respectively. Paralleling the joint genotypic disequilibria in (34), these can be generated by three-way genotypic associations in migrant seeds (d‒AA∕M∕C or d‒Aa∕M∕C for the homozygote disequilibria, d‒Aa∕M∕C alone for the heterozygote disequilibria) and by cytoplasmic disequilibria in migrant seeds (d‒M∕C ). However, as for the three-way allelic association, the allelic cytonuclear disequilibria in the migrant pools contribute only in conjunction with other terms. For example, nuclear-chloroplast allelic disequilibrium in either migrant pool can generate three-way genotypic associations only in conjunction with nuclear-mitochondrial allelic disequilibria in migrant seeds (d‒A∕M≠0 plus d‒A∕C≠0 or D‒A∕C≠0 ). Additionally, nuclear-mitochondrial allelic disequilibria in migrant seeds can contribute when combined with unequal chloroplast frequencies in the migrant pools, with or without unequal nuclear allele frequencies (X‒C≠x‒C and d‒A∕M≠0 ; or P‒≠p‒,X‒C≠x‒C , and d‒A∕M≠0 ). Finally, another form of intermigrant interaction, new to the three-locus system, can also produce three-way genotypic disequilibria: unequal chloroplast frequencies in the migrant pools coupled with the corresponding nuclear-mitochondrial genotypic disequilibrium in migrant seeds (e.g., X‒C≠x‒C and d‒AA∕M≠0 for D^AA∕M∕C ). The dependence of the final three-way homozygote disequilibria on the heterozygote value (D^Aa∕M∕C ) means that the former can also be generated by unequal chloroplast frequencies in conjunction with nonrandom associations between heterozygotes and the mitochondrial marker in seeds (X‒C≠x‒C and d‒Aa∕M≠0 ).

Equilibrium three-locus genotype frequencies: To complete the specification of the equilibrium state for the dicytonuclear system, we must calculate the final three-locus genotype frequencies (U^ij , V^ij , W^ij ). These can be obtained by substituting the relevant equilibrium formulas [(27, 28, 29 and 30), (32, 33 and 34), (B1, B2, B3 and B4), (C1), and (C2)] into the decompositions given in Tables 6 and 7.

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Figure 3.

Trajectories of permanent nonzero disequilibria. Initial conditions were U₁₁ = 1.0; parameter values were s = 0.1, m = 0.05, M = 0.2, , , and . Disequilibria in the migrant pools were , (where * indicates M, C, or MC), , and .

Numerical examples: Numerical examples allow us to compare the cytoplasmic and three-locus disequilibria generated by pollen and seed migration in the full nuclear-mtDNA-cpDNA system with the disequilibria in each of the two cytonuclear subsystems. Previous work has found that 0.1 is roughly the minimal detectable level of two-way disequilibria, given reasonable sample sizes and marginal frequencies (Basten and Asmussen 1997). Generally, both permanent and transient disequilibria generated via pollen migration alone are smaller in magnitude than disequilibria generated via a comparable amount of seed migration alone, as was found previously for the cytonuclear subsystems by Schnabel and Asmussen (1992). Higher selfing rates increase the magnitude of disequilibria and slow the decay of transient disequilibria. Both permanent and transient three-way disequilibria tend to be smaller than the corresponding two-locus or joint disequilibria, as would be expected of measures for higher-order effects.

Figure 3 shows an example where the resident population is initially monomorphic at each marker (U₁₁ = 1.0) and receives genetically distinct migrant pollen and seeds (P‒=0 , p‒=0.7 , X‒C=1.0 , x‒C=0.7 ) with nonrandom cytonuclear associations in the seeds. Such a situation might arise, for example, if a genetically homogeneous population received migrant pollen from a source fixed for a different nuclear marker but received seeds from both this source and another source, differing in both nuclear and chloroplast frequencies. These differences in nuclear and chloroplast frequencies in the migrant pools and cytonuclear associations in migrant seeds generate permanent nonzero values for all the disequilibria. In the absence of actual values of seed and pollen migration rates and selfing rates, this example is used to illustrate cases with a larger pollen migration rate (M = 0.2) than seed migration rate (m = 0.05) as would be expected for most plant species with wind-dispersed pollen. The joint allelic and genotypic disequilibria for the three-genome system (Figure 3C) are similar in the shape of their trajectories to the disequilibria from the nuclear-mitochondrial subsystem (Figure 3A), although somewhat greater in magnitude. The nuclear-chloroplast and cytoplasmic disequilibria also have similar shapes but are smaller in magnitude (Figure 3, B and D). The three-way allelic and genotypic disequilibria for this example (Figure 3D) both have slightly smaller magnitudes and behave very differently than the two-way disequilibria. For instance, the three-way allelic and homozygote disequilibira have signs opposite those of the corresponding joint associations, while the three-way heterozygote disequilibrium differs from all other heterozygote disequilibria in this example by not changing sign along its trajectory. This example corresponds to a mainly outcrossing species (s = 0.1); increasing the selfing rate to s = 0.9 increases the magnitude of all the disequilibria except the heterozygote nuclear-mitochondrial (D_Aa/M) and joint (D_Aa/MC) disequilibria (all parameters other than s as in Figure 3, data not shown). Most of the other disequilibrium measures are increased by a factor of ~4 to 8, but some increase by a great deal more. For example, the homozygote nuclear-chloroplast disequilibrium (D_AA/C) increases by a factor of 20, from ~0.004 (s = 0.1) to 0.08 (s = 0.9).

Figure 4 gives a case where the resident population receives only migrant seeds and no migrant pollen (M = 0) and is initially fixed at the nuclear and mitochondrial loci (P = X_M = 1.0), but polymorphic at the chloroplast locus (X_C = 0.8). In contrast to the population shown in Figure 3, here the migrant seeds have no cytonuclear disequilibria, and there can be no intermigrant admixture effects with only one type of migration; therefore, no permanent nonzero disequilibria are generated. However, transient disequilibria are generated by differences in the nuclear, mitochondrial, and chloroplast frequencies between the original population and the migrant seeds (p‒=x‒M=0.0 , x‒C=0.4 ). In this example with high selfing (s = 0.9), the transient nuclear-mitochondrial disequilibria (Figure 4A) and joint dicytonuclear disequilibria (Figure 4C) that are generated can reach quite high values (>0.15 in magnitude) before dissipating; additionally, these can persist for relatively long periods of time (>0.05 in magnitude for almost 40 generations), increasing the probability of detection, particularly for long-lived organisms where generation times are long. However, if we consider lower selfing rates corresponding to predominantly outcrossing species, both the maximum level and the duration of the disequilibria are much reduced. For example, with a selfing rate of s = 0.1 (all other parameters as in Figure 4, data not shown), the transient nuclear-mitochondrial and joint dicytonuclear disequilibria do not reach 0.1 in magnitude before dissipating and remain above 0.05 in magnitude for <10 generations. Even with a high selfing rate (s = 0.9), neither the nuclear-chloroplast nor the cytoplasmic disequilibria (Figure 4, B and D) exceed 0.08 before dissipating. The three-way measures for this example are much smaller in magnitude than the other associations (Figure 4D) and are interesting in that all but the heterozygote association show a sign change in their trajectories.