Abstract
We extend current multilocus models to describe the effects of migration, recombination, selection, and nonrandom mating on sets of genes in diploids with varied modes of inheritance, allowing us to consider the patterns of nuclear and cytonuclear associations (disequilibria) under various models of migration. We show the relationship between the multilocus notation recently presented by Kirkpatrick, Johnson, and Barton (developed from previous work by Barton and Turelli) and the cytonuclear parameterization of Asmussen, Arnold, and Avise and extend this notation to describe associations between cytoplasmic elements and multiple nuclear genes. Under models with sexual symmetry, both nuclear-nuclear and cytonuclear disequilibria are equivalent. They differ, however, in cases involving some type of sexual asymmetry, which is then reflected in the asymmetric inheritance of cytoplasmic markers. An example given is the case of different migration rates in males and females; simulations using 2, 3, 4, or 5 unlinked autosomal markers with a maternally inherited cytoplasmic marker illustrate how nuclear-nuclear and cytonuclear associations can be used to separately estimate female and male migration rates. The general framework developed here allows us to investigate conditions where associations between loci with different modes of inheritance are not equivalent and to use this nonequivalence to test for deviations from simple models of admixture.
THE mixing of genetically divergent populations, and assortative mating within them, can generate strong associations (“linkage disequilibria”), even between unlinked genes. Associations between alleles can therefore be used to give estimates of migration rates and mating patterns in hybrid zones. This approach has been applied to estimate dispersal rates, using autosomal markers sampled from linear transects (Barton 1983; Szymura and Barton 1986, 1991; Malletet al. 1990); in these cases, random mating could be assumed. In parallel, associations between cytoplasmic and nuclear markers have been used to estimate both degree of assortment and migration rates, using samples from single populations subject to immigration (Asmussen et al. 1987, 1989; Arnoldet al. 1988; Asmussen and Arnold 1991). Under many circumstances, pairwise associations between cytoplasmic and nuclear markers evolve in the same way as those between unlinked nuclear markers. They differ, however, when there is some type of sexual asymmetry; this asymmetry is then reflected in the asymmetric inheritance of cytoplasmic markers. In addition, higher-order associations (e.g., between a cytoplasmic allele and heterozygosity at a neutral autosomal locus) can give information on asymmetries in mating preferences and differential dispersal between the sexes. For example, if females of one species hybridize more readily than females of the other, then the cytoplasm they pass on will tend to be found more often in individuals heterozygous for nuclear markers.
The array of genotypes in a hybrid zone contains a great deal of information; full analysis requires joint consideration of the deficit of heterozygotes; of discrepancies among associations involving autosomal, sex-linked, and cytoplasmic loci; of the relation between linkage disequilibrium and recombination rates; and of third- and higher-order associations. The large number of parameters involved for multilocus data, however, cannot generally be estimated without extremely large samples unless simplifying assumptions are made. Recent work by Pritchard et al. (2000) and Dawson and Belkhir (2001) has considered the problem of using multilocus genotype data from unlinked loci to infer population structure and assign individuals to (previously unknown) underlying populations. The general approach is to account for nonrandomness in genotype frequencies (departure from Hardy-Weinberg expectations and from linkage equilibrium) by partitioning the sampled individuals in some way. Both groups apply a Bayesian approach to this inference problem, with the evidence in favor of each possible partitioning or clustering of individuals represented by the posterior distribution of the populations and allele frequencies in those populations (Pritchardet al. 2000) or the partition of the sample (Dawson and Belkhir 2001). Other workers have also recently considered the use of multilocus genotypes to assign individuals to putative source populations, where the source populations were known a priori (Rannala and Mountain 1997; Cornuetet al. 1999). As pointed out in Dawson and Belkhir (2001), the problem of assigning alleles within a genotype to source populations is closely related to the problem of inferring the distribution of a hybrid index, recently addressed in Barton (2000) using a maximum-likelihood approach and considering unlinked autosomal loci. All of these approaches have the related aim of estimating a limited number of parameters (different orders of multilocus associations in the case of Barton 2000) that approximate the full genotype frequencies. These parameters may be the migration rates or strengths of assortment themselves, or they may be measures of disequilibria, averaged over loci, which can later be used to estimate migration rates or other parameters.
The main motive of this article is to set methods for estimating and interpreting disequilibria (Szymura and Barton 1986, 1991; Asmussen et al. 1987, 1989; Arnoldet al. 1988; Malletet al. 1990) in the general multilocus context of Kirkpatrick et al. (2002), thus allowing genetic information from hybridizing populations to be used more fully. Kirkpatrick et al. (2002) extended the multilocus notation of Barton and Turelli (1991; Turelli and Barton 1994) to describe the effects of migration, recombination, selection, and nonrandom mating on sets of genes with arbitrary modes of inheritance. This is implemented in a set of packages, written in Mathematica 4.0, which generate recursions for allele frequencies and linkage disequilibria (available at http://helios.bto.ed.ac.uk/evolgen/index.html). We use this framework to derive patterns of nuclear and cytonuclear associations under various migration models; comparison among the various nuclear and cytonuclear associations gives additional information not available from consideration of each type alone. We use simulated data to illustrate how simple two-way nuclear and cytonuclear associations can be used to estimate male and female migration rates when these differ.
MODEL
Notation: The notation used here is based on that introduced by Kirkpatrick et al. (2002), which extended and generalized the approach of Barton and Turelli (1991) to allow for arbitrary modes of inheritance, including diploidy, polyploidy, sex linkage, cytoplasmic inheritance, and genomic imprinting. Here, we explicitly consider diploids. Kirkpatrick et al. (2002) define the word “gene” to mean a particular copy of a nonrecombining sequence at some locus in some individual. The “context” of a gene includes such information as the life-history stage it is found in, the deme it is found in, the sex of its carrier, and whether it was inherited from the individual’s female or male parent (its “sex-of-origin”). A “position” is used to refer to a particular locus in a particular context. Consider a population of diploid individuals; a genome in such an individual is denoted by G , which is the set of all positions in an individual. For a single diploid autosomal locus i, for example, the genome of a male might be written as G={imm,imf} , where the first subscript indicates the “sex-of-carrier” (in this case, male), and the second subscript indicates the sex of the parent from which the allele was inherited, the sex-of-origin. Similarly, in a female, G={ifm,iff} . See Table 1 for frequently used notations.
The genotype of an individual at position i is represented by the indicator variable Xi . Each Xi takes some value that describes the allelic state; for example, if we have only two alleles per locus, the alleles can be represented by Xi=1 or Xi=0 ; for this special case, the frequency of allele 1 at position i is written as pi and the frequency of allele 0 as qi=1−pi . This representation generalizes to multiple alleles, where the indicator is taken to be a vector of length equal to the number of alleles; all entries are zero except for a 1 that indicates the allelic state.
The genetic state of the population is described by the set of statistical moments of Xi across loci, called “associations” by Kirkpatrick et al. (2002), which define the allele frequencies and linkage disequilibria. Following Barton and Turelli (1991), the association between the alleles at the positions in set A is defined as
DA=EX[ζA],
(1)
where
ζA=∏i∈Aζi,
(2)
ζi=Xi−pi , and the expectation E_{X}[] is taken over the distribution of genotype frequencies. Products over the empty set are defined to be 1, so that D_{Ø} = 1. The D’s are the same as the C’s in Barton and Turelli (1991). If these associations are defined relative to the actual allele frequency in the population (i.e., as central moments), then the Di are necessarily zero. However, it may be convenient to define these relative to allele frequencies at the start of the generation, in which case Di=Δpi , the difference between the current and the initial allele frequency. With two alleles per locus, moments involving repeated indices reduce to expressions involving the mean. For example, Dii=pipi if moments are defined relative to the current allele frequency.
We consider here two different measures of cytonuclear disequilibrium. In our notation, Dic is equivalent to the allelic cytonuclear disequilibrium, D, in Asmussen et al. (1987) and measures the nonrandom association of a nuclear locus (i ) and a maternally inherited cytoplasmic marker (ℂ). The third-order association can be written as Dii∗c , where ^{*} is an operator that changes sex-of-origin (for example, if the sex-of-origin for i is f, that for i∗ is m); it measures the association between a cytoplasmic marker and homozygosity at a nuclear locus. Dii∗c is equivalent to Weir and Wilson’s (1986) residual disequilibrium d and is a simple function of the genotypic cytonuclear disequilibria (D_{1}, D_{2}, and D_{3}) of Asmussen et al. (1987). With one nuclear and one cytoplasmic locus, both with two alleles, there are six distinct genotypes. Assuming that maternally and paternally derived alleles are equivalent, these genotype frequencies are determined by pi , pc , Dii∗ , Dic , and Dii∗c , where Dii∗ measures the heterozygote deficit at the nuclear locus.
TABLE 1
Frequently used notation
Nonrandom mating can generate associations across diploid mating pairs (for example, D_{{}_{i}_{mm,}_{j}_{mf,}_{c}_{mf,}_{k}_{fm,}_{l}_{ff}}); these in turn produce associations across genomes in the diploid offspring. If there is random mating, there are no associations across mating pairs (D_{{}_{i}_{mm,}_{j}_{mf,}_{c}_{mf,}_{k}_{fm,}_{l}_{ff}} = D_{{}_{i}_{mm,}_{j}_{mf,}_{c}_{mf}}D_{{}_{k}_{fm,}_{l}_{ff}}); if there is random union of gametes (no inbreeding), there are no associations across genomes (D_{{}_{i}_{mm,}_{j}_{mf,}_{c}_{mf}} = D_{{}_{i}_{mm}}D_{{}_{j}_{mf,}_{c}_{mf}}), which allows us to follow haploid genotypes, as in Barton and Turelli (1991).
Kirkpatrick et al. (2002) extended the notation of Barton and Turelli (1991) to describe associations among arbitrary sets of loci, which may be in different individuals and which may have unusual modes of inheritance. This does not change the form of the equations for selection (Equations 10 and 12 of Barton and Turelli 1991 or Equation 19 of Turelli and Barton 1994), provided that relative fitness is defined as the relative contribution of each gene combination to the next generation (see also Price 1970; Frank and Slatkin 1990). Nonrandom mating between diploid individuals can be described in just the same way as selection, by defining fitness as the relative contribution of a diploid pair. This is analogous to Barton and Turelli’s (1991) treatment of nonrandom mating between haploids.
Model of migration: We assume that migration of diploid individuals among demes occurs first, followed by random mating within demes (Figure 1). Initially, with only one migration rate for both males and females, we need follow only the sex-of-origin of each gene, denoted by the subsubscript f or m. Asmussen et al. (1989) calculate the allelic cytonuclear disequilibrium in diploid adults by considering the disequilibrium in the gametes that those adults would produce, given normal Mendelian segregation; in our notation, this is (D_{{}_{i}_{m,}_{c}_{f}} + D_{{}_{i}_{f,}_{c}_{f}})/2, where the f or m subsubscript refers to a maternally or paternally inherited allele. (Note that the two components D_{{}_{i}_{m,}_{c}_{f}} and D_{{}_{i}_{f,}_{c}_{f}} cannot be measured separately without knowing the origin of the autosomal alleles.) Therefore, census scheme 2 from Asmussen et al. (1989) corresponds to first allowing individuals to migrate and then considering the haploid gametes produced by those individuals after meiosis.
Migration is by diploid individuals; the location of an individual can be specified in the context in the same way that we specify sex-of-origin. The demes are labeled ω. Because we use central moments to describe disequilibria, which are measured relative to different reference points in each deme, we need an equation for admixture that takes into account the differences in allele frequency of each deme providing migrants (ω) and the deme under consideration (ν). Let Δpiω=(piω−piν′) be the difference between the allele frequency in the contributing deme ω before migration and that in the deme under consideration after migration, where the shorthand notation Aω or iω indicates the set of genes A or gene i located in deme ω. Note that this usage is inconsistent with the notation of Kirkpatrick et al. (2002), but we adopt it here for convenience. We can also express the difference in allele frequencies between each deme providing migrants and the deme of interest prior to migration,
diω=piω−piνanddUω=∏i∈U(piω−piν).
(3)
The allele frequency in deme ν after migration is given by
piν′=∑ωmωpiω,
(4)
where the migration rate m_{ω} gives the proportion of individuals contributed by deme ω. This is equivalent to Equation 29 of Kirkpatrick et al. (2002) if we allow Σ_{ω}m_{ω} = m for ω ≠ ν and m_{ν} = 1 - m. After migration, the associations in the deme ω relative to the allele frequencies in the deme of interest after migration are given by
DAω′=EX[∏i∈A(Xiω−piν′)]=EX[∏i∈A(Xiω−piω+Δpiω)]=∑U⊆ADA\Uω∏i∈UΔpiω,
(5)
where A\U stands for the positions in set A that are left after those in set U are taken away. Because genes change their contexts as they move from one deme to another, migration can be considered a form of transmission (an event that changes the contexts of genes). For this reason, (5) is simply (15) from Kirkpatrick et al. (2002), giving the associations after a change in reference values from piω to piν′ .
The associations in the deme of interest (ν) after migration are then a linear mixture of the associations of different demes, measured relative to the same reference point:
DAν′=∑ωmω∑U⊆ADA\Uω∏i∈UΔpiω.
(6)
If we instead use the allele frequencies in the deme of interest prior to migration as the reference point, so that we use the deme of interest before and after migration as the shift in reference, we require two changes of reference. When we again consider the deme of interest (ν), we find that (6) can be written as
DAν′=∑ωmωDAω′=∑ωmωEX[∏i∈A(Xiω−piν′)]=∑ωmωEX[∏i∈A(Xiω−∑xmxpix)]=mν∑U⊆ADA\Uν∏i∈U(∑x(−mx)dix)+∑ω≠νmω∑U⊆A∏i∈U(∑x(−mx)dix)∑V⊆A\UdVωD(A\U)\Vω.
(7)
If we consider only one migrant deme, and let mν=1−m,mω=m,DAν=DAR , and DAω=DAM , (7) is equivalent to Equation (30) from Kirkpatrick et al. (2002),
DA′=∑U⊆A(−m)∣U∣dU{(1−m)DA\UR+m∑V⊆A\UdVD(A\U)\VM}=(1−m)∑U⊆A(−m)∣U∣dUDA\UR+m∑U⊆A(−m)∣U∣dU∑V⊆A\UdVD(A\U)\VM
(8)
where ∣U∣ means the number of positions in set U .
All of the above assumes that male and female migration rates are equal; we generalize this later to include different male and female migration rates and discuss differences that arise in males versus females.
Description of recombination and random mating: Under the assumption that there is Mendelian segregation, with no genetic variation for the rules of transmission (e.g., no meiotic drive or modifiers of recombination), the effects of recombination and random mating on the associations and allele frequencies for a population can be described by a simple equation (Equation 12, Kirkpatricket al. 2002),
DA″=∑U:U=AtA←UDU′,
(9)
when the reference values are chosen to be equal for all positions at each locus. The summation is over all sets of positions U that could become set A following transmission. The notation “U : U = A” means that the genes in the sets U and A must be equal when the context information is stripped from them (i.e., when U = A). The transmission coefficient tA←U is defined as the probability that the positions in set A were inherited from positions in set U . As an example, for autosomal loci in a haploid population with two sexes, the transmission coefficient t_{{}_{i}_{m}_{,j}_{m}←{}_{i}_{f}_{,j}_{f}} is the probability that the genes at loci i and j in a male were both inherited from a female (the mother), which is (1 - r_{ij})/2, where r_{ij} is the recombination rate between loci i and j.
For transmission in diploids, the sex-of-origin for each position in set A must equal the sex-of-carrier for the corresponding position in set U , since that is the sex of the parent from which a gene in set A is descended. When A is a set of cytoplasmic loci or a mixture of cytoplasmic and nuclear genes, then certain transmissions do not occur; for example, under the assumption of a maternally inherited cytoplasmic locus (c) and an autosomallocus(i),t_{{}_{ix}_{m}_{,cx}_{m}←{}_{i}_{mm}_{,c}_{mf}} andt_{{}_{ix}_{m}_{,cx}_{m}←{}_{i}_{mf}_{,c}_{mf}}(wherex is f or m) are both zero, because a male cannot transmit his maternally inherited cytoplasmic allele to his sons or daughters.
As an example, consider the genotypic cytonuclear disequilibrium, Dii∗c , in female zygotes. Here, A={ifm,iff,cff} . These might have come from four possible sets of positions U in the previous generation of diploids, {i_{mf}, i_{ff}, c_{ff}}, {i_{mm}, i_{ff}, c_{ff}}, {i_{mf}, i_{fm}, c_{ff}}, and {i_{mm}, i_{fm}, c_{ff}}. The transition coefficients for all of these equal ^{1}/_{4}; to see this, consider t_{{}_{i}_{fm}_{,i}_{ff}_{,c}_{ff}←{}_{i}_{mf}_{,i}_{ff}_{,c}_{ff}}.This is the probability that, in a female, the locus i inherited from her male parent was inherited from that parent’s female parent, which is ^{1}/_{2}, times the probabilty that the locus i inherited from her female parent was inherited from that parent’s female parent, which also equals ^{1}/_{2}. The cytoplasmic locus must pass through the female lineage, with probability 1. Putting this together gives
Dii∗c″=14(Dimf,iff,cff′+Dimm,iff,cff′+Dimf,ifm,cff′+Dimm,ifm,cff′).
(10)
This shows that any associations between the maternally derived allele and the paternally derived allele at the same locus must have originated in associations between alleles from the different diploid parents (and, in each parent, could have involved the maternally derived or paternally derived gene). If we assume random mating, there are no associations between alleles in different diploid individuals (so that Dimf,iff,cff′=Diff,cff′Dimf′ , etc.), and Di′=0 (we have defined the associations relative to the actual allele frequencies in the population, as discussed earlier, and thus the disequilibria are central moments), so that Dii∗c″=0 .
We now consider the effects of this simple model of migration followed by meiosis and random mating on both the allelic cytonuclear disequilibrium (Dic ) and the genotypic cytonuclear disequilibrium (Dii∗c ).
Allelic cytonuclear disequilibrium, Dic : The allelic cytonuclear disequilibrium is Dic=E[(Xi−pi)(Xc−pc)]=E[XiXc]−pipc . In Asmussen et al.’s (1989) notation, Dic=D=D1+12D2 . We consider a specific model of continued immigration at rate ½m into a deme of interest (ν) from two source demes (ν- 1, ν+ 1) and consider the equilibrium under migration. Again, when considering only one migration rate for both males and females, we need follow only the sex-of-origin of each gene.
We use Equation 6 and note that the possible sets of positions for U include {i,c} , {i} , {c} , and {Ø}. We define D_{Ø} = 1 and Δp_{Ø} = 1, and, since we are using central moments, defined relative to each deme, Di=0 by definition, giving us
Dicν′=(1−m)Dicν+m2(Dicν−1+Dicν+1)+(1−m)Δpi{ν}Δpc{ν}+m2(Δpi{ν−1}Δpc{ν−1}+Δpi{ν+1}Δpc{ν+1}).
(11)
We can express this result in terms of deviations from moments and allele frequencies in the deme of interest (ν) by using (3) and allowing δDAω=DAω−DAν . (Note that these deviations are across demes at the same time point, whereas Δp_{i}_{{ω}} involves deviations across space and time.) The context of a gene now includes both its deme-of-origin and its sex-of-origin (in that order). For an i allele with a female sex-of-origin, we then have
Di{ν,f}c{ν,f}′=Di{ν,f}c{ν,f}+m2(δDi{ν−1,f}c{ν−1,f}+δDi{ν+1,f}c{ν+1,f})+m2(di{ν−1,f}dc{ν−1,f}+di{ν+1,f}dc{ν+1,f})−m24(di{ν−1,f}+di{ν+1,f})(dc{ν−1,f}+dc{ν+1,f}),
(12)
where gene context is given as {deme, sex-of-origin}.
If individuals within demes are formed by random union of gametes, the association between the paternal genotype and the maternally inherited cytotype must be zero. In that case, the association D_{i}_{{ω,m}}_{c}_{{ω,f}} (where m indicates a male sex-of-origin and f a female sex-of-origin) is initially zero before migration, leaving
Di{ν,m}c{ν,f}′=m2(di{ν−1,m}dc{ν−1,f}+di{ν+1,m}dc{ν+1,f})−m24(di{ν−1,m}+di{ν+1,m})(dc{ν−1,f}+dc{ν+1,f}),
(13)
where, again, context is given as {deme, sex-of-origin}. We next allow meiosis by applying Equation 9 and find
Di{ν,f}c{ν,f}″=Di{ν,m}c{ν,f}″=12(Di{ν,f}c{ν,f}′+Di{ν,m}c{ν,f}′).
(14)
Using (12) and (13), we find the full recursion for the allelic cytonuclear disequilibrium
Dicν″=12Di{ν,f}c{ν,f}+m4(δDi{ν−1,f}c{ν−1,f}+δDi{ν+1,f}c{ν+1,f})+m4(di{ν−1,f}dc{ν−1,f}+di{ν−1,m}dc{ν−1,f}+di{ν+1,f}dc{ν+1,f}+di{ν+1,m}dc{ν+1,f})−m28(di{ν−1,f}+di{ν+1,f}+di{ν−1,m}+di{ν+1,m})(dc{ν−1,f}+dc{ν+1,f}).
(15)
At equilibrium under this symmetric model of migration, d_{i}_{{ν}-_{1,f}} =-d_{i}_{{ν+1,f}} = d_{i}_{f}, d_{i}_{{ν-1,m}} =-d_{i}_{{ν+1,m}} = d_{i}_{m}, and d_{c}_{{ν-1,f}} =-d_{c}_{{ν+1,f}} = d_{c}_{f}, so that we have
D^icν=m[12(δDi{ν−1,f}c{ν−1,f}+δDi{ν+1,f}c{ν+1,f})+(dif+dim)dcf].
(16)
Note that we have calculated the change in a haploid association after migration and meiosis; this corresponds to census scheme 2 in Asmussen et al. (1989), for which they find
D^=m1+m[D(1)+D(2)+12(p1−p2)(x1−x2)].
(17)
Here, D^{(}^{i}^{)} gives the disequilibrium in deme i, demes 1 and 2 correspond to demes ν- 1 and ν + 1, and m_{1} = m_{2} = m/2. This is the result given in (16) if we assume p_{i}_{m} = p_{if}, since D^{(1)} = ½ (D_{i}_{{ν-1,f}}_{c}_{{ν-1,f}} + D_{i}_{{ν-1,m}}_{c}_{{ν-1,f}}), D^{(2)} = ½ (D_{i}_{{ν+1,f}}_{c}_{{ν+1,f}} + D_{i}_{{ν+1,m}}_{c}_{{ν+1,f}}) and, initially, D_{i}_{m}_{c}_{f} = 0. The (1 + m) term arises when we replace the deviations of moments (δDicω ) with the values of the moments in the neighboring demes (Dicω ). Note that the behavior of the disequilibrium between a nuclear allele and a cytoplasmic marker under random mating and migration is the same as that between two unlinked nuclear loci (Dij ); the pairwise disequilibrium caused by admixture is an average of the disequilibria in the source populations, plus an additional amount caused by the covariance between the allele frequencies in the source populations (Nei and Li 1973; Prout 1973).
Genotypic cytonuclear disequilibrium Dii∗c The genotypic cytonuclear disequilibrium measures nonrandom association between the same nuclear locus from two different genomes and a cytoplasmic locus. We use the definition Dii∗c=E[(Xif−pif)(Xim−pim)(Xcf−pcf)] , where the f subsubscript indicates female sex-of-origin, for diallelic loci with indicator Xi . We let Xi=1 if Pi is present at locus i and Xi=0 if Qi is present at locus i , and similarly for other loci. From the definitions of D_{1}, D_{2}, and D_{3} (Asmussenet al. 1989),
D1=ψ(PifPimPcf)−ψ(PifPim)ψ(Pcf)=ψ(PifPimPcf)−ψ(PifPim)pcfD2=ψ(PifQimPcf)−ψ(PifQim)pcfD3=ψ(QifQimPcf)−ψ(QifQim)pcf,
(18)
where, for example, Ψ(P_{i}_{f}P_{i}_{m}P_{c}_{f}) gives the frequency of P_{i}_{f}P_{i}_{m}P_{c}_{f}. Using these definitions and allowing p_{i}_{f} = p_{i}_{m} gives Dii∗c=D1−2piD=D3pi2−D2piqi+D1qi2 , where qi=1−pi . Note that, necessarily, D_{1} + D_{2} + D_{3} = 0, so that the genotypic cytonuclear disequilibria as defined by Asmussen et al. are not independent, as was originally noted by Asmussen et al. (1987).
To consider the case of admixture, we once again apply the migration model given earlier; (6) again gives the change in the disequilibrium due to admixture where, here, A={i,i∗,c} . Since D_{Ø} = 1, and Δp_{Ø} = 1, we have
Dii∗cν′=(1−m)Dii∗cν+m2(Dii∗cν−1+Dii∗cν+1)+(1−m)(Di{ν,f}i{ν,m}Δpc{ν,f}+Di{ν,f}c{ν,f}Δpi{ν,m}+Di{ν,m}c{ν,f}Δpi{ν,f})+m2(Di{ν−1,f}i{ν−1,m}Δpc{ν−1,f}+Di{ν−1,f}c{ν−1,f}Δpi{ν−1,m}+Di{ν−1,m}c{ν−1,f}Δpi{ν−1,f})+m2(Di{ν+1,f}i{ν+1,m}Δpc{ν+1,f}+Di{ν+1,f}c{ν+1,f}Δpi{ν+1,m}+Di{ν+1,m}c{ν+1,f}δpi{ν+1,f})+(1−m)(Δpi{ν,f}Δpi{ν,m}Δpc{ν,f})+m2(Δpi{ν−1,f}Δpi{ν−1,m}Δpc{ν−1,f}+Δpi{ν+1,f}Δpi{ν+1,m}Δpc{ν+1,f}),
(19)
where, again, we indicate context by the subsubscript {deme, sex-of-origin}. If individuals within demes are formed by random union of gametes, there will necessarily be no association between maternal and paternal loci (Dii∗c=DicDi∗=0 , etc.), and the equation above simplifies to
Dii∗cν′=m2(δDi{ν−1,f}c{ν−1,f}di{ν−1,m}+δDi{ν+1,f}c{n+1,f}di{ν+1,m})−m24(δDi{ν−1,f}c{ν−1,f}+δDi{ν+1,f}c{ν+1,f})(di{ν−1,m}+di{ν+1,m})−m(1−m)2(di{ν−1,f}di{ν−1,m}dc{ν−1,f}+di{ν+1,f}di{ν+1,m}dc{ν+1,f})−m2(1−m)4(di{ν−1,f}+di{ν+1,f})(di{ν−1,m}+di{ν+1,m})(dc{ν−1,f}+dc{ν+1,f}).
(20)
Thus, the genotypic cytonuclear disequilibrium after migration is due to interactions between the allelic disequilibrium and the nuclear allele frequency in males plus additional disequilibrium caused by the differences in allele frequencies in the migrant demes relative to those in the deme under consideration. Once again, cytoplasmic inheritance is irrelevant at this stage; the disequilibrium after migration for this model would be the same for a nuclear genotype and an unlinked nuclear locus (D_{i}_{f}_{i}_{m}_{j}_{f}) or (substituting k for i) for the threelocus association D_{i}_{m}_{j}_{f}_{k}_{f}.
As shown earlier, after meiosis,
Dii∗c″=14(Dimf,iff,cff′+Dimm,iff,cff′+Dimf,ifm,cff′+Dimm,ifm,cff′),
(21)
since any association involving a maternally derived allele and a paternally derived allele must originate in alleles that were in different individuals in the previous generation. Once again, if we assume random mating, there are no associations between alleles in different diploid individuals and Dii∗c″=0 . Therefore, at equilibrium, the disequilibrium generated by migration is lost after random mating and meiosis (D^ii∗c=0) . Note that the same is true under random mating when considering the association between the same nuclear locus from the maternal and paternal genomes and another nuclear locus (Dˆ_{i}_{f}_{i}_{m}_{j}_{f} = 0), but it is not true for three different nuclear loci on the same genome (Dˆ_{i}_{f}_{j}_{f}_{k}_{f}), since then all three alleles are derived from the same parent.
We have allowed individuals to migrate and then mate and are now considering associations in the new diploid individuals formed after mating; this corresponds to census 2 from Asmussen et al. (1989). As shown above, Dii∗c=D1−2piD , and from Equations B12 and B13 of Asmussen et al. (1989), D^1−2piD^=0 , so the two methods give the same result.
Recombination and random mating followed by migration: In the derivation above, we have assumed a model where individuals migrate and then mate; censusing the population after random mating means that associations between maternal and paternal genomes are lost and associations within genomes are reduced by random segregation and mating. If instead the population is censused after migration, we find
D^i{ν,f}c{ν,f}=m[δDi{ν−1,f}c{ν−1,f}+δDi{ν+1,f}c{ν+1,f}+(2dif+dim)dcf]
(22)
D^i{ν,m}c{ν,f}=mdimdcf
(23)
D^i{ν,f}i{ν,m}c{ν,f}=m2(δDi{ν−1,f}c{ν−1,f}−δDi{ν+1,f}c{ν+1,f})dim,
(24)
where now the δDAω and di are deviations from moments and allele frequencies in the deme of interest after recombination, d_{i}_{{ν-1,}_{x}_{}} =-d_{i}_{{ν+1,}_{x}_{}} = d_{ix}, where x is f or m, and we recall that, with random mating, D_{i}_{{ω,f}}_{i}_{{ω,m}}, D_{i}_{{ω,m}}_{c}_{{ω,f}}, and D_{i}_{{ω,f}}_{i}_{{ω,m}}_{c}_{{ω,f}} are all zero after recombination. The overall allelic cytonuclear disequilibrium will again be (Dˆ_{i}_{{ν,f}}_{c}_{{ν,f}} + D_{i}_{{ν,m}}_{c}_{{ν,f}})/2, which corresponds to Dˆ for census 1 in Asmussen et al. (1989), assuming that p_{i}_{m} = p_{i}_{f}, and is twice the allelic cytonuclear disequilibrium found under census 2.
Nuclear-nuclear vs. cytonuclear disequilibria: In the case of differences between females and males, the asymmetric inheritance of cytoplasmic markers becomes important. For example, with two unlinked nuclear loci, the disequilibrium after meiosis is given by (9) as
Dij″=(Difjf′+Dimjm′+Difjm′+Dimjf′)∕4,
(25)
where the f or m subsubscript indicates sex-of-origin. This will only be equivalent to that for the cytonuclear disequilibrium,
Dic″=(Difcf′+Dimcf′)∕2,
(26)
when there are no differences for within- and between-genome associations and allele frequencies between males and females (Difjf′=Dimjm′ and Difjm′=Dimjf′ ). Such differences could arise for a variety of reasons, such as imprinting, selection, or differences in allele frequencies or migration rates in males and females. We now consider the latter.
Unequal migration rates between the sexes: To incorporate different migration rates for males and females, we need to keep track of moments and allele frequencies separately in the two sexes. If we consider the allelic cytonuclear disequilibrium, in females after migration we have Di{f,f}c{f,f}′ and Di{f,m}c{f,f}′ as in (12) and (13) (where now the first subsubscript refers to sex-of-carrier and the second to sex-of-origin) and we replace the migration rate m with m_{f}, which refers only to migration of females. In males, Di{m,f}c{m,f}′ and Di{m,m}c{m,f}′ are given by equations identical to (12) and (13), with the exception that now all loci are in males and the migration rate is m_{m}. Because the mitochondrial locus cannot be paternally inherited, there are no associations such as D_{i}_{{f,m}}_{c}_{{f,m}}, D_{i}_{{m,m}}_{c}_{{m,m}}, D_{i}_{{f,f}}_{c}_{{f,m}}, or D_{i}_{{m,f}}_{c}_{{m,m}}.
We first consider migration followed by recombination and random mating, which corresponds to census scheme 2 of Asmussen et al. (1989). Recombination follows as before; any associations between the paternally inherited nuclear loci and the mitochondrial locus will be lost after the formation of gametes, so that Di{f,m}c{f,f}″=0 and Di{m,m}c{m,f}″=0 . This leaves
Di{f,f}c{f,f}″=(Di{f,f}c{f,f}′+Di{f,m}c{f,f}′)∕2
(27)
and
Di{m,f}c{m,f}″=(Di{f,f}c{f,f}′+Di{f,m}c{f,f}′)∕2
(28)
(any association between the maternally derived nuclear allele and the mitochondrial locus in males must have originated in the females of the previous generation). This means that the cytonuclear disequilibrium will depend only on female migration rates and at equilibrium will be equal in males and females,
D^i{ν,f,f}c{ν,f,f}=D^i{ν,m,f}c{ν,m,f}=D^i{ν,x,f}c{ν,x,f}=mf[12(δDi{ν−1,x,f}c{ν−1,x,f}+δDi{ν+1,x,f}c{ν+1,x,f})+(di{x,f}+di{x,m})dc{x,f}],
(29)
where the gene context is now given as {deme, sex-of-carrier, sex-of-origin), d_{i}_{{ν-1,}_{x}_{,}_{y}_{}} =-d_{i}_{{ν+1,}_{x}_{,}_{y}_{}} = d_{i}_{{}_{x}_{,}_{y}_{}} under a symmetric model of migration, m_{f} indicates the female migration rate, and x and y each are either f or m.
In contrast, for two unlinked nuclear loci, after meiosis the association between maternally derived alleles in both females and males is
Di{f,f}j{f,f}″=Di{m,f}j{m,f}″=(Di{f,f}j{f,f}′+Di{f,m}j{f,m}′+Di{f,f}j{f,m}′+Di{f,m}j{f,f}′)∕4
(30)
and so depends only on the female migration rate (m_{f}). The association for the paternally derived alleles in both females and males is
Di{f,m}j{f,m}″=Di{m,m}j{m,m}″=(Di{m,f}j{m,f}′+Di{m,m}j{m,m}′+Di{m,f}j{m,m}′+Di{m,m}j{m,f}′)∕4
(31)
and depends on the male migration rate (m_{m}). All of the cross-genome associations (Di{f,m}j{f,f}″,Di{f,f}j{f,m}″,Di{m,f}j{m,m}″,Di{m,m}j{m,f}″) would be zero under random mating. The two-locus disequilibrium measured in females is (D_{i}_{{f,f}}_{j}_{{f,f}} + D_{i}_{{f,m}}_{j}_{{f,m}})/2, while that in males is (D_{i}_{{m,f}}_{j}_{{m,f}} + D_{i}_{{m,m}}_{j}_{{m,m}})/2. In both cases, the disequilibrium would depend on both the female and male migration rates (m_{f} and m_{m}).
At equilibrium, the within-genome association in females equals the within-genome association in males,
(D^i{f,f}j{f,f}+D^i{f,m}j{f,m})∕2=(D^i{m,f}j{m,f}+D^i{m,m}j{m,m})∕2=mf+mm4[12(δDi{ν−1,∗,f}j{ν−1,∗,f}+δDi{ν+1,∗,f}j{ν+1,∗,f}+δDi{ν−1,∗,m}j{ν−1,∗,m}+δDi{ν+1,∗,m}j{ν+1,∗,m})+(di{∗,f}+di{∗,m})(dj{∗,f}+dj{∗,m})],
(32)
where, after random mating and assuming no selection, associations and allele frequencies must be equal in males and females, so that D_{i}_{{ω,f,f}}_{j}_{{ω,f,f}} = D_{i}_{{ω,m,f}}_{j}_{{ω,m,f}} = D_{i}_{{ω,*,f}}_{j}_{{ω,*,f}}, D_{i}_{{ω,f,m}}_{j}_{{ω,f,m}} = D_{i}_{{ω,m,m}}_{j}_{{ω,m,m}} = D_{i}_{{ω,*,m}j{ω,*,m}}, p_{i}_{{f,f}} = p_{i}_{{m,f}} = p_{i}_{{*,f}}, and p_{i}_{{f,m}} = p_{i}_{{m,m}} = p_{i}_{{*,m}}. Comparing (29) and (32), we see that the difference is due to inclusion of j_{{*,m)} terms; since the mitochondrial locus cannot be paternally inherited, there are no similar c_{{*,m}} terms in (29). Note that if we let m_{f} = m_{m}, once again D_{i}_{{*,f}}_{j}_{{*,f}} = D_{i}_{{*,m}j{*,m}}, and (32) takes the same form as (16).
If instead we consider recombination and random mating followed by migration (census scheme 1), the overall cytonuclear association will be made up of both within-genome and between-genome associations immediately following migration. At equilibrium, the between-genome association in females and males will depend on the female and male migration rates, respectively,
D^i{ν,f,m}c{ν,f,f}=mf2(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′)+mfdi{f,m}dc{f,f}
(33)
D^i{ν,m,m}c{ν,m,f}=mm2(δDi{ν−1,m,m}c{ν−1,m,f}′+δDi{ν+1,m,m}c{ν+1,m,f}′)+mmdi{m,m}dc{m,f}.
(34)
The within-genome association in females will also depend on the female migration rate only,
D^i{ν,f,f}c{ν,f,f}=mf[12(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′)+δDi{ν−1,f,f}c{ν−1,f,f}′+δDi{ν+1,f,f}c{ν+1,f,f}′]+mf(di{f,m}+2di{f,f})dc{f,f}
(35)
but the within-genome association in males will depend both on the male and the female migration rates,
D^i{ν,m,f}c{ν,m,f}=mf2(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′+δDi{ν−1,f,f}c{ν−1,f,f}′+δDi{ν+1,f,f}c{ν+1,f,f}′)+mm2(δDi{ν−1,m,f}c{ν−1,m,f}′+δDi{ν+1,m,f}c{ν+1,m,f}′)+mf(di{f,m}+di{f,f})dc{f,f}+mmdi{m,f}dc{m,f}.
(36)
Derivations for Equations 33, 34, 35, 36 are given in the appendix. In contrast to this asymmetry for within-genome cytonuclear associations, within-genome associations for unlinked nuclear loci take the same general form in males and females (see appendix).
Comparisons of nuclear-nuclear with cytonuclear disequilibria could therefore allow us to see if there are different migration rates for males and females and estimate those rates. However, differences between nuclear-nuclear and cytonuclear associations may arise due to selection on individual loci or assortative mating as opposed to migration. Consideration of a group of associations should average out selective effects on nuclear loci, but this would not rule out selection directly on the mitochondrial haplotype (e.g., MacRae and Anderson 1988; Hutter and Rand 1995).
ESTIMATING MIGRATION
One approach to estimating migration rates from disequilibrium measurements is to assume that the population reaches a “quasi-equilibrium” (Nagylaki 1976; Barton and Gale 1993) with the generation of associations due to migration or other forces such as selection balanced by their loss through recombination. After an initial period where large disequilibria break down quickly, linkage disequilibria change very slowly during the period when most allele frequency evolution takes place. In this case, the disequilibria in all of the demes converges to the same value, and the recursion in (14) gives
Dic=2mdidc,
(37)
where, under symmetric migration, diν−1=−diν+1=di and dcν−1=−dcν+1=dc , and assuming that there are no differences in allele frequencies for maternally inherited and paternally inherited alleles (d_{i}_{f} = d_{i}_{m} = d_{i}). Thus, for a nuclear marker and a cytoplasmic marker, the migration rate could be estimated from the differences in allele frequencies and the disequilibrium (as is true for any two unlinked loci; c.f. Kruuk 1997). This corresponds to the results for the center of a continuous population in which migration into the center is approximated by diffusion (σ^{2}; Barton 1986); then, for two unlinked loci i and j,
Dij=2σ2∂pi∂x∂pj∂x.
(38)
This also corresponds to an approximation of a stepping-stone model, where migration is between neighbors that are in a similar state (“short-range” migration model). This will differ from the exact equilibrium result, given by (15), when there are differences in disequilibria between the demes (δDicω nonzero).
Applications of exact equilibrium results to estimate migration rates have used the assumption that migrants come from two genetically homogeneous parental species, so that disequilibria in the source demes are zero (Arnoldet al. 1988; Asmussenet al. 1989). Under this “long-range” migration model, the equation corresponding to (37) is then
Dic=2m(1+m)didc,
(39)
for the case of migration followed by meiosis and random mating. Again, the cytonuclear disequilibrium is a function only of the migration rate and of differences in allele frequencies between the parental species.
A simple method for estimating multilocus moments from genotype frequencies assumes that associations within and between genomes can be disentangled by estimating Dii∗ from the deficit of heterozygotes at individual loci (Barton 2000). Let D_{0,2} = D_{i}_{{}_{x}_{,}_{y}_{}}_{j}_{{}_{x}_{,}_{y}_{}}, where x and y are f or m, stand for the usual measure of pairwise linkage disequilibrium between two genes derived from the same gamete, and let D_{1,1} = D_{i}_{{}_{x}_{,}_{y}_{}}_{i}_{{}_{x}_{,}_{y}_{*}}, where x and y are f or m and y^{*} indicates the sex opposite of y, be a measure of the deficit of heterozygotes (a measure of departure from Hardy-Weinberg equilibrium). We define an additive trait, z=∑i=1n(Xif+Xim) ; the variance is then
Var(z)=∑i∑j(Difjf+Difjm+Dimjf+Dimjm).
(40)
If we assume that between-genome associations are the same whether they involve the same or different loci [i.e., D_{i}_{{}_{x}_{,}_{y}_{}}_{i}_{{}_{x}_{,}_{y}_{*}} = D_{i}_{{}_{x}_{,}_{y}_{}}_{j}_{{}_{x}_{,}_{y}_{*}} = D_{1,1}, where again x and y are either f or m, and y^{*} indicates the sex opposite of y], as is always true for admixture models (Barton 2000), the component of var(z) due to between-genome associations can be separated from that due to associations within genomes (Barton and Gale 1993; Kruuk 1997; Barton 2000), and the above becomes
Var(z)=2[∑ipi(1−pi)+n2D1,1+n(n−1)D0,2]
(41)
since Dii=pi(1−pi) and assuming pif=pim . In this way, genotype and allele frequencies allow us to estimate D_{0,2} and D_{1,1}, which can be compared to the cytonuclear association. We give an example below, using simulated data.
This method can be extended to allow estimation of the whole matrix of moments, D_{ij} (where i is inherited from one gamete and j from the other) from a set of 3^{n} diploid genotype frequencies for all sets of loci, inherited maternally and paternally, assuming that the allele frequencies are known and are the same among male and female gametes (Barton 2000). The method can be straightforwardly extended to cytonuclear moments D_{ijc}; it gives satisfactory estimates of individual moments, but it is not a good method for estimating the overall genotypic structure given by the full set of moments, since individual genotype frequencies may be negative (see Barton 2000 for discussion).
Estimation of female and male migration rates from cytonuclear and nuclear-nuclear associations: Under census 2, where we allow migration followed by recombination and random mating, cross-genome associations are lost and D_{1,1} = 0. Comparison of within-genome with cytonuclear associations will allow an estimation of m_{f} and m_{m}. Let D_{1,}_{c} = (Dˆ_{i}_{{f,*}c} _{{f,*}} + Dˆ_{i}_{{m,*}c{m,*}})/2 (so that we are averaging over the sex-of-carrier), averaged over all nuclear loci. From (22) and (23), under the assumption of short-range migration, the δD = 0, allowing an estimate of the female migration rate as
mf=D1,c(di{f,∗}+di{m,∗})dc=D1,c2didc
(42)
and the male migration rate as
mm=4D0,2(di{f,∗}+di{m,∗})(dj{f,∗}+dj{m,∗})−D1,c(di{f,∗}+di{m,∗})dc=D0,2didj−D1,c2didc,
(43)
where, under the symmetric model of migration, d_{i}_{{ν-1,}_{x}_{,}_{y}_{}} =-d_{i}_{{ν+1,}_{x}_{,}_{y}_{}} = d_{i}_{{}_{x}_{,}_{y}_{}}, and we assume that d_{i}_{{}_{x}_{,f}} = d_{i}_{{}_{x}_{,m}} = d_{i}_{{}_{x}_{,*}}, where x is the sex-of-carrier and is either f or m, and we further allow the allele frequencies to be equal in the two sexes, so that d_{i}_{{f,*}} = d_{i}_{{m,*}} = d_{i}. We let D_{0,2} be
(D^i{f,f}j{f,f}+D^i{f,m}j{f,m}+D^i{m,f}j{m,f}+D^i{m,m}j{m,m})∕4,
(44)
averaged across all nuclear loci.
Assuming long-range migration, the disequilibria in the source demes are zero, and the equivalent equations to (42) and (43) are
mf=D1,c(di{f,∗}+di{m,∗})dc−D1,c=D1,c2didc−D1,c
(45)
and
mm=2D0,22didj−D0,2−D1,c2didc−D1,c.
(46)
Under census scheme 1, we census after migration, so that the between-genome associations are not zero. Equations A26, A30, and A34 from the appendix give the relationships between the measurable associations D_{1,}_{c}, D_{1,1}, and D_{0,2} and the two migration rates under short-range migration for census scheme 1, while Equations A38, A30, and A40 give these associations under long-range migration. When migration rates are equal in males and females, so that m_{f} = m_{m} = m, D_{1,1} and D_{0,2} under census scheme 1 are equivalent to D_{{}_{i}_{,}_{j}_{*}}(κ_{{}_{i}_{,}_{j}_{*}}) and D_{{}_{i}_{,}_{j}_{}}(κ_{{}_{i}_{,}_{j}_{}}), given in Equation 3 of Barton (2000).
Example using simulated data: To illustrate these results, we simulate a population receiving one-way migration from two source demes, deme ν- 1 and deme ν+ 1, as in Figure 1. Within each source deme, we have equal allele frequencies at n = 2, 3, 4, or 5 unlinked autosomal loci and also at a maternally inherited cytoplasmic locus; allele frequencies were 0 for deme ν- 1 (p_{i}_{ν-1} = p_{c}_{ν-1} = 0) or 0.8 for deme ν+ 1 (p_{i}_{ν+1} = p_{c}_{ν+1} = 0.8). We assume that male and female migration rates differ, but that male and female allele frequencies are equal (which will be true as long as the ratio of female to male migration rates is equal in each source deme). After migration, individuals were allowed to mate randomly each generation (census scheme 2). After 100 generations had elapsed, allowing deme ν to reach approximate equilibrium, 1000 individuals were sampled from the population. Nuclear-nuclear and cytonuclear associations (D_{ij}, D_{ic}, D_{ii}_{*}, and D_{ii}_{*}_{c}) were estimated using an extension of the moment-based method described in Barton (2000) that allows for cytoplasmic inheritance. Note that, under this census scheme, between-genome associations are zero at equilibrium (D_{ii}_{*} = D_{ii}_{*}_{c} = 0). Migration rates were estimated from these associations under the assumption of long-range migration (see above) using (45) for female migration rate (m_{f}) and (46) for male migration rate (m_{m}). The results are shown in Table 2 as the mean and standard deviation (SD) for 100 replicate samples for various values of m_{f} and m_{m}.
As one would expect, there is greater variance in the estimates of male migration rate (m_{m}) than in estimates of female migration rate (m_{f}; Table 2 and Figures 2 and 3), since both nuclear-nuclear and cytonuclear associations depend on the female migration rate, but the male migration rate influences only nuclear-nuclear associations. This is similar to the results found by Orive and Asmussen (2000) in considering pollen- or seed-mediated gene flow in plants with dicytonuclear inheritance of chloroplast and mitochondrial markers; in such a system, seed migration rates affect all types of associations, but pollen migration rates affect only nuclear and nuclear DNA-chloroplast DNA associations, making pollen migration rates more difficult to estimate. From (43) and (46), we see that m_{m} is estimated as the difference between two estimated disequilibrium values and so has an inherently larger error associated with it.
Increasing the number of autosomal loci decreases the standard deviation for male migration rate (shaded circles, Figure 3); the greatest decrease is seen in going from two unlinked autosomal loci to three. For the top of Figure 3, the increase in SD for female migration rate (solid circles) from four nuclear loci to five nuclear loci is not significant (F = 1.121, P > 0.25, one-tailed f-test of variances). Note that the change in SD for female migration rate is nearly flat for increasing numbers of unlinked autosomal loci analyzed (Figure 3, solid circles); the cytonuclear associations give the greatest amount of information for this migration estimate, and these are relatively unaffected by adding greater numbers of autosomal markers. These results imply diminishing returns on analysis of large numbers of unlinked autosomal markers for estimating migration rates in this manner. However, a large number of nuclear markers may allow one to detect the possible effects of selection on the nuclear markers or on loci linked to the nuclear markers, increasing our confidence that the disequilibria measured are the results of admixture (migration) and not underlying selection.
DISCUSSION
Recently, Kirkpatrick et al. (2002) presented a general framework for describing associations among multiple genes and their response to selection, nonrandom mating, and admixture. Multilocus associations are defined as moments of the distribution of allelic states. Here, we have used this framework to investigate the effects of migration on association among nuclear and cytoplasmic genes. In a simple model of admixture, where the two sexes are equivalent, the cytonuclear disequilibria (Dic and Dii∗c ) behave in the same way as the corresponding second- and third-order associations between unlinked nuclear alleles (Dij and Dii∗j ). This is not the case for more complex models involving some type of sexual asymmetry.
As a simple illustration, when migration rates differ between the sexes, the cytonuclear disequilibrium depends only on the female migration rate, whereas the association between two unlinked nuclear loci depends on the mean migration in males and females. This gives a simple way of separately estimating rates of migration in the two sexes. An example is given using simulated data where migration rates differ between the sexes but allele frequencies are equal; for this case, the momentsbased method of Barton (2000) can be used to estimate nuclear-nuclear associations. These can then be used to estimate the migration rates under simplifying assumptions about migration (long-range or short-range migration).
TABLE 2
Simulation results from 100 replicate samples of 1000 individuals
This is analogous to the use of the standardized variance in allele frequency (F_{ST}) of cytoplasmic and nuclear genes to estimate gene flow in the two sexes (or in plants, via seed or pollen; Dong and Wagner 1994; Ennos 1994; Latta and Mitton 1997). However, one should bear in mind that F_{ST} is proportional to the effective number of migrants (Nm), whereas linkage disequilibria are proportional to the proportion of migrants (m); since effective population size depends on the mode of transmission, Nm and m will vary between different modes in different ways. Different rates of migration between the sexes could also be inferred from differences in cline position and width between nuclear or phenotypic markers and cytoplasmic markers (Ferriset al. 1983; Harrison 1989; Young 1996; Rohweret al. 2001) or from differences in allele frequency or in linkage disequilibrium measures in males and females after migration (e.g., Kruuk 1997).
Figure 2.
—Mean estimates of female (m_{f}) and male (m_{m}) migration rates; n gives the number of unlinked autosomal loci. True values are (a) m_{f} = 0.05, m_{m} = 0.2 and (b) m_{f} = 0.2, m_{m} = 0.05 and are indicated by dashed lines in each graph.
The general framework developed here allows us to investigate the conditions under which various measures of nuclear-nuclear and cytonuclear disequilibria would not be equivalent and use this nonequivalence to test for deviations from the simple model of admixture presented above. Such deviations might be due (for example) to nonrandom mating or differences in fitness among first-generation reciprocal hybrids, as well as to differential dispersal. Other kinds of linkage disequilibria may also be used to distinguish among different evolutionary processes. For example, if a hybrid population is maintained by immigration from two distinct sources (perhaps via some network of intermediate demes), then all orders of linkage disequilibria are proportional to the product of the allele frequency differences at the loci involved (Barton 2000). This gives a testable null hypothesis that may be used to detect other processes such as selection that act on specific sets of loci. It also implies that long-range migration from highly divergent demes will maintain larger higher-order disequilibria than will short-range migration from genetically similar sources. Such genetically differentiated long-range migrants create leptokurtotic distributions resulting in larger higher-order moments. Hence, the relative magnitude of higher-order associations gives an estimate of the degree of divergence of the immigrants. Information may also come from comparison between heterozygote deficit Dii∗ and within-genome associations Dij ; in models of admixture, there is a definite relation between these.
Statistical analysis of multilocus data raises interesting (and difficult) issues. We require methods of testing for differences between different kinds of association and for making composite estimates of parameters such as migration rate on the basis of multilocus genotypes. One attraction of the cytonuclear associations defined by Asmussen et al. (1987) is that they can be estimated directly from the data. However, in general one must deal with the problem that the parental origin of alleles in a diploid is usually unknown and that cis and trans combinations cannot be distinguished. The likelihood-based methods developed in Barton (2000) may be appropriate to such problems.
Figure 3.
—Standard deviations (SD) of estimates of female (m_{f}) and male (m_{m}) migration rates as a function of the number of unlinked autosomal loci (n). Solid circles give values for m_{f}; shaded circles give values for m_{m}. Each point is calculated from 100 replicate samples of 1000 individuals (Table 2).
Another general difficulty is that, even with the large sample sizes used in these simulation examples (n = 1000 individuals), it is very difficult to accurately estimate disequilibria. The standard deviations for the migration estimates will scale approximately with 1∕n , where n is the number of individuals sampled. Generally, estimation of disequilibria requires large samples unless allele frequencies are intermediate or the disequilibria to be estimated is maximal for the set of allele frequencies (Brown 1975). To test for significance of migration rates estimated from multilocus disequilibria, one can either use simulations or find the sampling errors of the estimates of disequilibria used.
The main aims of this article have been to show the relation between different measures of multilocus association and to show how comparison among the various nuclear-nuclear and cytonuclear associations can be used in estimation of important population parameters. Combining nuclear-nuclear and cytonuclear associations should give more information than consideration of each alone. However, we have shown that there may be little gain with a small number of loci, since the majority of the associations are then with the cytoplasmic marker. Sampling of several nuclear genes and complete analysis of multilocus genotype data, including higher-order disequilibria and heterozygote deficit, may be especially useful in teasing apart the effects of selection and/or assortative mating from those of simple admixture.
APPENDIX: CYTONUCLEAR AND NUCLEAR ASSOCIATIONS UNDER RECOMBINATION FOLLOWED BY MIGRATION (CENSUS 1) WITH UNEQUAL MALE AND FEMALE MIGRATION RATES
Cytonuclear association: After recombination, the between-genome associations in both females and males are zero (Di{f,m}c{f,f}′=Di{m,m}c{m,f}′=0) , the within-genome associations are Di{f,f}c{f,f}′=Di{m,f}c{m,f}′=(Di{f,f}c{f,f}+Di{f,m}c{f,f})∕2 , and there are no associations such as Di{f,m}c{f,m}′ and Di{m,f}c{m,m}′ .
After migration, the cross-genome cytonuclear association in females is
Di{ν,f,m}c{ν,f,f}″=Di{ν,f,m}c{ν,f,f}′+mf2(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′)+mf2(di{ν−1,f,m}dc{ν−1,f,f}+di{ν+1,f,m}dc{ν+1,f,m})−mf24(di{ν−1,f,m}+di{ν+1,f,m})(dc{ν−1,f,f}+dc{ν+1,f,f}),
(A1)
where the full context for a gene is given as {deme, sex-of-carrier, sex-of-origin}. Since Di{ν,f,m}c{ν,f,f}′=0,di{ν−1,f,m}=−di{ν+1,f,m}=di{f,m} , and d_{c}_{{ν-1,f,f}} =-d_{c}_{{ν+1,f,f}} = d_{c}_{{f,f}} (under the symmetric model of migration), this becomes
Di{ν,f,m}c{ν,f,f}″=mf2(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′)+mfdi{f,m}dc{f,f}.
(A2)
At equilibrium, this is
D^i{ν,f,m}c{ν,f,f}=mf2(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′)+mfdi{f,m}dc{f,f}.
(A3)
Similarly, the cross-genome association in males will be
D^i{ν,m,m}c{ν,m,f}=mm2(δDi{ν−1,m,m}c{ν−1,m,f}′+δDi{ν+1,m,m}c{ν+1,m,f}′)+mmdi{m,m}dc{m,f}.
(A4)
The within-genome cytonuclear association in females after migration is
Di{ν,f,f}c{ν,f,f}″=Di{ν,f,f}c{ν,f,f}′+mf2(δDi{ν−1,f,f}c{ν−1,f,f}′+δDi{ν+1,f,f}c{ν+1,f,f}′)+mf2(di{ν−1,f,f}dc{ν−1,f,f}+di{ν+1,f,f}dc{ν+1,f,f})−mf24(di{ν−1,f,f}+di{ν+1,f,f})(dc{ν−1,f,f}+dc{ν+1,f,f})=12(Di{ν,f,f}c{ν,f,f}+Di{ν,f,m}c{ν,f,f})+mf2(δDi{ν−1,f,f}c{ν−1,f,f}′+δDi{ν+1,f,f}c{ν+1,f,f}′)+mfdi{f,f}dc{f,f}.
(A5)
Note that this depends on the female migration rate only. At equilibrium, this becomes
D^i{ν,f,f}c{ν,f,f}=mf[12(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′)+δDi{ν−1,f,f}c{ν−1,f,f}′+δDi{ν+1,f,f}c{ν+1,f,f}′]+mf(2di{f,f}+di{f,m})dc{f,f}.
(A6)
The within-genome association in males after migration is where the asymmetry inherent to this cytonuclear system comes into play,
Di{ν,m,f}c{ν,m,f}″=Di{ν,m,f}c{ν,m,f}′+mm2(δDi{ν−1,m,f}c{ν−1,m,f}′+δDi{ν+1,m,f}c{ν+1,m,f}′)+mm2(di{ν−1,m,f}dc{ν−1,m,f}+di{ν+1,m,f}dc{ν+1,m,f})−mm24(di{ν−1,m,f}+di{ν+1,m,f})(dc{ν−1,m,f}+dc{ν+1,m,f})=12(Di{ν,f,f}c{ν,f,f}+Di{ν,f,m}c{ν,f,f})+mm2(δDi{ν−1,m,f}c{ν−1,m,f}′+δDi{ν+1,m,f}c{ν+1,m,f}′)+mmdi{m,f}dc{m,f}.
(A7)
Note that associations found in females (the first term of the last line above) contribute after migration; thus at equilibrium, the within-genome association in males (unlike that in females) depends on both the female and male migration rates,
D^i{ν,m,f}c{ν,m,f}=12(D^i{ν,f,f}c{ν,f,f}+D^i{ν,f,m}c{ν,f,f})mm2(δDi{ν−1,m,f}c{ν−1,m,f}′+δDi{ν+1,m,f}c{ν+1,m,f}′)+mmdi{m,f}dc{m,f}=mf2(δDi{ν−1,f,m}c{ν−1,f,f}′+δDi{ν+1,f,m}c{ν+1,f,f}′+δDi{ν−1,f,f}c{ν+1,f,f}′+δDi{ν+1,f,f}c{ν+1,f,f}′)+mm2(δDi{ν−1,m,f}c{ν−1,m,f}′+δDi{ν+1,m,f}c{ν+1,m,f}′)+mf(di{f,f}+di{f,m})dc{f,f}+mmdi{m,f}dc{m,f}.
(A8)
Between-genome nuclear associations for unlinked loci: After recombination, all of the between-genome associationsare zero, Di{f,m}j{f,f}′=Di{f,f}j{f,m}′=Di{m,m}j{m,f}′=Di{m,f}j{m,m}′=0 . After migration between demes, the between-genome association in females is
Di{ν,f,m}j{ν,f,f}″=Di{ν,f,m}j{ν,f,f}′+mf2(δDi{ν−1,f,m}j{ν−1,f,f}′+δDi{ν+1,f,m}j{ν+1,f,f}′)+mf2(di{ν−1,f,m}dj{ν−1,f,f}+di{ν+1,f,m}dj{ν+1,f,m})−mf24(di{ν−1,f,m}+di{ν+1,f,m})(dj{ν−1,f,f}+dj{ν+1,f,f}).
(A9)
Since Di{ν,f,m}j{ν,f,f}′=0 and d_{i}_{{ν-1,f,m}} =-d_{i}_{{ν+1,f,m}} = d_{i}_{{f,m}}, etc. (under the symmetric model of migration), this becomes
Di{ν,f,m}j{ν,f,f}″=mf2(δDi{ν−1,f,m}j{ν−1,f,f}′+δDi{ν+1,f,m}j{ν+1,f,f}′)+mfdi{f,m}dj{f,f}.
(A10)
At equilibrium,
D^i{ν,f,m}j{ν,f,f}=mf2(δDi{ν−1,f,m}j{ν−1,f,f}′+δDi{ν+1,f,m}j{ν+1,f,f}′)+mfdi{f,m}dj{f,f}.
(A11)
Similarly,
D^i{ν,f,f}j{ν,f,m}=mf2(δDi{ν−1,f,f}j{ν−1,f,m}′+δDi{ν+1,f,f}j{ν+1,f,m}′)+mfdi{f,f}dj{f,m}.
(A12)
The equivalent equilibrium associations in males are
D^i{ν,m,m}j{ν,m,f}=mm2(δDi{ν−1,m,m}j{ν−1,m,f}′+δDi{ν+1,m,m}j{ν+1,m,f}′)+mmdi{m,m}dj{m,f}
(A13)
and
D^i{ν,m,f}j{ν,m,m}=mm2(δDi{ν−1,m,f}j{ν−1,m,m}′+δDi{ν+1,m,f}j{ν+1,m,m}′)+mmdi{m,f}dj{m,m}.
(A14)
Within-genome nuclear associations for unlinked loci: After recombination, the within-genome associations for maternally derived loci in females and in males depend on associations found in females in the previous generation,
Di{f,f}j{f,f}′=Di{m,f}j{m,f}′=(Di{f,f}j{f,f}+Di{f,m}j{f,m}+Di{f,f}j{f,m}+Di{f,m}j{f,f})∕4,
(A15)
while the within-genome associations for paternally derived loci in females and in males depend on associations found in males in the previous generation,
Di{f,m}j{f,m}′=Di{m,m}j{m,m}′=(Di{m,f}j{m,f}+Di{m,m}j{m,m}+Di{m,f}j{m,m}+Di{m,m}j{m,f})∕4.
(A16)
After migration, the within-genome nuclear association in females is
Di{ν,f,f}j{ν,f,f}″=Di{ν,f,f}j{ν,f,f}′+mf2(δDi{ν−1,f,f}j{ν−1,f,f}′+δDi{ν+1,f,f}j{ν+1,f,f}′)+mf2(di{ν−1,f,f}dj{ν−1,f,f}+di{ν+1,f,f}dj{ν+1,f,f})−mf24(di{ν−1,f,f}+di{ν+1,f,f})(dj{ν−1,f,f}+dj{ν+1,f,f}).
(A17)
Since d_{i}_{{ν-1,f,f}} =-d_{i}_{{ν+1,f,f}} = d_{i}_{{f,f}}, etc. (under the symmetric model of migration), this becomes
Di{ν,f,f}j{ν,f,f}″=14(Di{ν,f,f}j{ν,f,f}+Di{ν,f,m}j{ν,f,m}+Di{ν,f,f}j{ν,f,m}+Di{ν,f,m}j{ν,f,f})+mf2(δDi{ν−1,f,f}j{ν−1,f,f}′+δDi{ν+1,f,f}j{ν+1,f,f}′)+mfdi{f,f}dj{f,f}.
(A18)
At equilibrium, the within-genome association for maternally derived alleles in females is
D^i{ν,f,f}j{ν,f,f}=14(D^i{ν,f,f}j{ν,f,f}+D^i{ν,f,m}j{ν,f,m}+D^i{ν,f,f}j{ν,f,m}+D^i{ν,f,m}j{ν,f,f})+mf2(δDi{ν−1,f,f}j{ν−1,f,f}′+δDi{ν+1,f,f}j{ν+1,f,f}′)+mfdi{f,f}dj{f,f}.
(A19)
Similarly, for paternally derived alleles, the within-genome association in females is
D^i{ν,f,m}j{ν,f,m}=14(D^i{ν,m,f}j{ν,m,f}+D^i{ν,m,m}j{ν,m,m}+D^i{ν,m,f}j{ν,m,m}+D^i{ν,m,m}j{ν,m,f})+mf2(δDi{ν−1,f,m}j{ν−1,f,m}′+δDi{ν+1,f,m}j{ν+1,f,m}′)+mfdi{f,m}dj{f,m}.
(A20)
The equivalent equilibrium associations in males are
D^i{ν,m,f}j{ν,m,f}=14(D^i{ν,f,f}j{ν,f,f}+D^i{ν,f,m}j{ν,f,m}+D^i{ν,f,f}j{ν,f,m}+D^i{ν,f,m}j{ν,f,f})+mm2(δDi{ν−1,m,f}j{ν−1,m,f}′+δDi{ν+1,m,f}j{ν+1,m,f}′)+mmdi{m,f}dj{m,f}
(A21)
for maternally derived alleles and
D^i{ν,m,m}j{ν,m,m}=14(D^i{ν,m,f}j{ν,m,f}+D^i{ν,m,m}j{ν,m,m}+D^i{ν,m,f}j{ν,m,m}+D^i{ν,m,m}j{ν,m,f})+mm2(δDi{ν−1,m,m}j{ν−1,m,m}′+δDi{ν+1,m,m}j{ν+1,m,m}′)+mmdi{m,m}dj{m,m}
(A22)
for paternally derived alleles.
Short-range migration model: Under the short-range migration model, we assume that the generation of associations due to migration or other forces such as selection is balanced by their loss due to recombination. After an initial period, linkage disequilibria change very slowly compared to allele frequency changes, and the disequilibria in all of the demes converge to the same value. Under this “quasi-equilibrium” model of migration, the δD = 0, and (A3), (A4), (A6), and (A8) become
D^i{ν,f,m}c{ν,f,f}=mfdi{f,m}dc{f,f}D^i{ν,m,m}c{ν,m,f}=mmdi{m,m}dc{m,f}D^i{ν,f,f}c{ν,f,f}=mf(2di{f,f}+di{f,m})dc{f,f}D^i{ν,m,f}c{ν,m,f}=mf(di{f,f}+di{f,m})dc{f,f}+mmdi{m,f}dc{m,f}.
(A23)
Since, when we measure the cytonuclear association (D_{1,}_{c}), we are averaging over both maternally inherited and paternally inherited loci and also over males and females, our overall measure of cytonuclear association gives us
D1,c=(D^i{ν,f,m}c{ν,f,f}+D^i{ν,m,m}c{ν,m,f}+D^i{ν,f,f}c{ν,f,f}+D^i{ν,m,f}c{ν,m,f})∕4=3mf4(di{f,f}+di{f,m})dc{f,f}+mm4(di{m,f}+di{m,m})dc{m,f}.
(A24)
If d_{i}_{{}_{x}_{,f}} = d_{i}_{{}_{x}_{,m}} = d_{i}_{{}_{x}_{,*}}, where x is either f or m, this simplifies to
D1,c=3mf2di{f,∗}dc{f,∗}+mm2di{m,∗}dc{m,∗},
(A25)
where the sole subsubscript now indicates sex-of-carrier. And if we further assume that allele frequencies are equal in the two sexes, then d_{i}_{{f,*}} = d_{i}_{{m,*}} = d_{i}, and we get
D1,c=(3mf+mm2)didc.
(A26)
For between-genome nuclear associations, Equations A11, A12, A13, A14 become
D^i{ν,f,m}j{ν,f,f}=mfdi{f,m}dj{f,f}D^i{ν,f,f}j{ν,f,m}=mfdi{f,f}dj{f,m}D^i{ν,m,m}j{ν,m,f}=mmdi{m,m}dj{m,f}D^i{ν,m,f}j{ν,m,m}=mmdi{m,f}dj{m,m}.
(A27)
If we take D_{1,1} as the average of the associations Dˆ_{i{ν,f,m}j}_{{ν,f,f}} _{{ν,f,m} {ν,f,f}}, D_{i}_{{ν,f,f}}_{j}_{{ν,f,m}}, D_{i}_{{ν,m,m}}_{j}_{{ν,m,f}}, and D_{i}_{{ν,m,f}}_{j}_{{ν,m,m}} for each locus, we find
D1,1=mf4(di{f,m}dj{f,f}+di{f,f}dj{f,m})+mm4(di{m,m}dj{m,f}+di{m,f}dj{m,m}).
(A28)
If d_{i}_{{}_{x}_{,f}} = d_{i}_{{}_{x}_{,m}} = d_{i}_{{}_{x}_{,*}}, this further simplifies to
D1,1=mf2di{f,∗}dj{f,∗}+mm2di{m,∗}dj{m,∗},
(A29)
where, once again, the sole subsubscript now refers to sex-of-carrier. And if we further assume that allele frequencies are equal in the two sexes, we get
D1,1=(mf+mm2)didj.
(A30)
Note that, with an appropriate change of notation and letting m_{f} = m_{m}, this reduces to the result given in Equation 3 of Barton (2000) for D{i,j∗}short(κ{i,j∗}short) . Finally, for within-genome nuclear associations, Equations A19, A20, A21, A22 become
D^i{ν,f,f}j{ν,f,f}=14(D^i{ν,f,f}j{ν,f,f}+D^i{ν,f,m}j{ν,f,m}+D^i{ν,f,f}j{ν,f,m}+D^i{ν,f,m}j{ν,f,f})+mfdi{f,f}dj{f,f}D^i{ν,f,m}j{ν,f,m}=14(D^i{ν,m,f}j{ν,m,f}+D^i{ν,m,m}j{ν,m,m}+D^i{ν,m,f}j{ν,m,m}+D^i{ν,m,m}j{ν,m,f})+mfdi{f,m}dj{f,m}D^i{ν,m,f}j{ν,m,f}=14(D^i{ν,f,f}j{ν,f,f}+D^i{ν,f,m}j{ν,f,m}+D^i{ν,f,f}j{ν,f,m}+D^i{ν,f,m}j{ν,f,f})+mmdi{m,f}dj{m,f}D^i{ν,m,m}j{ν,m,m}=14(D^i{ν,m,f}j{ν,m,f}+D^i{ν,m,m}j{ν,m,m}+D^i{ν,m,f}j{ν,m,m}+D^i{ν,m,m}j{ν,m,f})+mmdi{m,m}dj{m,m}
(A31)
If we take D_{0,2} as the average of the associations Dˆ_{i{ν{ν,f,f }j}_{{ν,f,f}}, Dˆ_{i}_{{ν,f,m}} _{j} _{{ν,f,m}}, Dˆ_{i} _{{ν,m,f}} _{j}_{{ν,m,f}}, and D_{i}_{{ν,m,m}} _{j}_{{ν,m,m}} for each locus, using (A28), we find
D0,2=mf4(2di{f,f}dj{f,f}+di{f,m}dj{f,f}+di{f,f}dj{f,m}+2di{f,m}dj{f,m})+mm4(2di{m,f}dj{m,f}+di{m,m}dj{m,f}+di{m,f}dj{m,m}+2di{m,m}dj{m,m}).
(A32)
If d_{i}_{{}_{x}_{,f}} = d_{i}_{{}_{x}_{,m}} = d_{i}_{{}_{x}_{,*}}, where x is f or m, this further simplifies to
D0,2=3mf2di{f,∗}dj{f,∗}+3mm2di{m,∗}dj{m,∗},
(A33)
where, again, the remaining subsubscript refers to sex-of-carrier. And if we further assume that allele frequencies are equal in the two sexes, we get
D0,2=3(mf+mm)2didj.
(A34)
Again, allowing equal migration rates in males and females, this result reduces to that given in Barton (2000), Equation 3, for D{i,j}short(κ{i,j}short) . If we compare (A26), (A30), and (A34), we see that both the within- and between-genome nuclear associations should be proportional to the quantity m_{f} + m_{m}, while the overall cytonuclear association is proportional to 3m_{f} + m_{m}.
Long-range migration model: Under the long-range migration model, we assume that migrants come from source populations that are in linkage equilibrium, as would be the case if the sources are fixed for alternative alleles. Under this model of migration, the δDAν−1′=δDAν+1′=−DAν′ , and (A3), (A4), (A6), and (A8) become
D^i{ν,f,m}c{ν,f,f}=mfdi{f,m}dc{f,f}D^i{ν,m,m}c{ν,m,f}=mmdi{m,m}dc{m,f}D^i{ν,f,f}c{ν,f,f}=mf1+mf[(1−mf)di{f,m}+2di{f,f}]dc{f,f}D^i{ν,m,f}c{ν,m,f}=mf(1−mm)1+mf(di{f,f}+di{f,m})dc{f,f}+mmdi{m,f}dc{m,f}.
(A35)
Since, when we measure the cytonuclear association, D_{1,}_{c}, we are averaging over both maternally inherited and paternally inherited loci and also over males and females, our overall measure of cytonuclear association gives us
D1,c=(D^i{ν,f,m}c{ν,f,f}+D^i{ν,m,m}c{ν,m,f}+D^i{ν,f,f}c{ν,f,f}+D^i{ν,m,f}c{ν,m,f})∕4=(3−mm)mf4(1+mf)(di{f,f}+di{f,m})dc{f,f}+mm4(di{m,f}+di{m,m})dc{m,f}.
(A36)
If d_{i}_{{}_{x}_{,f}} = d_{i}_{{}_{x}_{,m}} = d_{i}_{{}_{x}_{,*}}, etc., this simplifies to
D1,c=(3−mm)mf2(1+mf)di{f,∗}dc{f,∗}+mm2di{m,∗}dc{m,∗},
(A37)
where the remaining subsubscript indicates the sex-of-carrier. And if we further assume that allele frequencies are equal in the two sexes, we get
D1,c=(3+mm)2(1+mf)didc.
(A38)
Between-genome nuclear associations are the same as for short-range migration, and the overall association is again given by (A28). For within-genome nuclear associations, Equations A19, A20, A21, A22 become
D^i{ν,f,f}j{ν,f,f}=(1−mf)4(D^i{ν,f,f}j{ν,f,f}+D^i{ν,f,m}j{ν,f,m}+D^i{ν,f,f}j{ν,f,m}+D^i{ν,f,m}j{ν,f,f})+mfdi{f,f}dj{f,f}D^i{ν,f,m}j{ν,f,m}=(1−mf)4(D^i{ν,m,f}j{ν,m,f}+D^i{ν,m,m}j{ν,m,m}+D^i{ν,m,f}j{ν,m,m}+D^i{ν,m,m}j{ν,m,f})+mfdi{f,m}dj{f,m}D^i{ν,m,f}j{ν,m,f}=(1−mm)4(D^i{ν,f,f}j{ν,f,f}+D^i{ν,f,m}j{ν,f,m}+D^i{ν,f,f}j{ν,f,m}+D^i{ν,f,m}j{ν,f,f})+mmdi{m,f}dj{m,f}D^i{ν,m,m}j{ν,m,m}=(1−mm)4(D^i{ν,m,f}j{ν,m,f}+D^i{ν,m,m}j{ν,m,m}+D^i{ν,m,f}j{ν,m,m}+D^i{ν,m,m}j{ν,m,f})+mmdi{m,m}dj{m,m}.
(A39)
Once again, we take D_{0,2} as the average of the associations Dˆ_{i}_{{ν,f,f}} _{j}_{{ν,f,f}}, Dˆ_{i}_{{,f,m}} _{j} _{ν {ν,f,m}}, D_{i}_{{ν,m,f}}_{j}_{{ν,m,f}}, and D_{i}_{{ν,m,m}} _{j}_{{ν,m,m}} for each locus. If we make the simplifying assumptions that d_{i}_{{}_{x}_{,f}} = d_{i}_{{}_{x}_{,m}} = d_{i}_{{}_{x}_{,*}} and d_{i}_{{f,*}} = d_{i}_{{m,*}} = d_{i}, this becomes
D0,2=[6−(mf+mm)](mf+mm)2(2+mf+mm)didj.
(A40)
If migration rates are equal for the two sexes, so that m_{f} = m_{m} = m, this reduces to the familiar result
D0,2=(3−m)m(1+m)didj,
(A41)
which, after a change in notation, is given as D{i,j}long(κ{i,j}long) in Equation 3 of Barton (2000).
Acknowledgments
The authors thank Toby Johnson for his helpful comments on this manuscript. This work was supported by a National Science Foundation NATO postdoctoral fellowship and National Science Foundation grants DEB-9813335 and DEB-0108242 to M.E.O.; N.H.B. gratefully acknowledges the support of the Darwin Trust of Edinburgh and the National Environmental Research Council.
- Received June 24, 2002.
- Accepted August 9, 2002.
- Copyright © 2002 by the Genetics Society of America