Polygenic Variation Maintained by Balancing Selection: Pleiotropy, Sex-Dependent Allelic Effects and G × E Interactions
Michael Turelli, N. H. Barton

Abstract

We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G × E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model produces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleiotropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G × E can maintain variation at loci for which the coefficient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The critical value depends on the correlation between substitution effects at different loci. For large positive correlations (e.g., ρij2>34 ), even extreme fluctuations in allelic effects cannot maintain variation. Surprisingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms.

IT remains a challenge for evolutionary geneticists to understand the additive genetic variance observed for most traits in most populations. Given the ubiquity of additive genetic variation, it is natural to seek an explanation in terms of ubiquitous forces. Lande (1975) proposed mutation-selection balance. However, over the past 25 years, attempts to explain standing levels of quantitative genetic variation in terms of mutation-selection balance have been at best only partially successful (e.g., Caballero and Keightley 1994; Charlesworth and Hughes 2000; but see Zhang and Hill 2002). One alternative is that some form of balancing selection, unconnected to the trait of interest, may account for persistent polymorphism at the underlying loci (e.g., Robertson 1965; Bulmer 1973; Gillespie 1984; Barton 1990). In contrast to such pleiotropic explanations, balancing selection might arise from variation in the effects of alleles that contribute to the trait, for instance, through genotype-by-environment (G × E) interactions. Here we explore four scenarios in which variance-depleting stabilizing selection interacts with pleiotropic balancing selection, environment-dependent allelic effects (treating spatial and temporal heterogeneity separately), and sex-dependent allelic effects. The thread that unites these scenarios is that, under weak selection, each produces very similar allele-frequency dynamics and polymorphism conditions. An empirical motivation for these analyses is that alleles of intermediate frequency seem to contribute to phenotypic variation in natural populations (e.g., Mackay and Langley 1990; Longet al. 2000). Such polymorphisms are incompatible with mutation-selection balance for plausible levels of selection and mutation.

The mathematical motivation for our analyses is Wright’s (1935) demonstration that stabilizing selection tends to eliminate polygenic variation. Using a weak-selection approximation, he showed that at most one locus is expected to remain polymorphic at equilibrium. More recent analyses of strong selection (Nagylaki 1989; Bürger and Gimelfarb 1999) have found that two-locus polymorphisms can be stably maintained with sufficiently strong selection and sufficient interlocus variation in allelic effects. We provide new simulations that further illustrate the restrictive conditions needed to maintain even stable two-locus polymorphisms for additive traits under stabilizing selection and loose linkage.

Robertson (1965) proposed that additive variation may be maintained by pleiotropically induced overdominant selection, which counteracts the effects of stabilizing selection. His conjecture was explored analytically by Bulmer (1973) for diallelic loci and extended to multiple alleles by Gillespie (1984). Both assumed equal allelic effects across loci, symmetric overdominance of equal intensity at all loci, and that the population mean at equilibrium coincided with the optimal trait value. They found lower bounds on the intensity of overdominant selection required to maintain stable multilocus polymorphisms. Our diallelic analyses generalize theirs and that of Zhivotovsky and Gavrilets (1992), by allowing for unequal allelic effects and arbitrary overdominance across loci and by considering the simultaneous stability of alternative equilibria at which the population mean can depart from the optimum.

Gillespie and Turelli (1989) showed how balancing selection could arise at individual loci by averaging over randomly fluctuating allelic effects. In their symmetric model of G × E interactions, all alleles have essentially the same mean and variance of effects. With this extreme symmetry assumption, even slight fluctuations can maintain indefinitely many alleles at an arbitrary number of loci. However, the essential interchangeability of the alleles implies that there will be essentially no correlation between the phenotypes produced by a given genotype across unrelated environments (i.e., two environments chosen at random from the distribution of environments responsible for maintaining variation; Gillespie and Turelli 1989, 1990; Gimelfarb 1990). Genetic variation that shows so little consistency of effects would severely limit the resemblance between parents and offspring across different environments.

Below we explore the consequences of allowing appreciable differences in the mean effects of different alleles. We show that under simple forms of spatial and temporal variation in allelic effects, the conditions for the maintenance of variation become much more restrictive than those indicated by Gillespie and Turelli (1989). Nevertheless, a surprisingly simple necessary condition for the maintenance of variation emerges. Our weak-selection approximations apply to a broad class of selection regimes in which balancing selection acts on the loci that contribute to trait variation. In particular, we show that the approximate dynamics obtained for average allele frequencies under G × E interactions and stabilizing selection are very similar to those arising from pleiotropic balancing selection.

The consequences of sex-dependent allelic effects, as extensively documented by Mackay, Langley, and their collaborators (e.g., Laiet al. 1995; Longet al. 1996; Nuzhdinet al. 1997; Gurganuset al. 1998; Wayne and Mackay 1998; Vieiraet al. 2000; Dilda and Mackay 2002), are approximated by a special case of the model for spatial variation. Contrary to the expectation from single-locus analyses that such sex-dependent effects may promote the maintenance of variation, we show that sex-dependent allelic effects do not stably maintain polygenic variation for additive traits.

All of our analyses assume that selection is weak enough relative to recombination that linkage disequilibrium is negligible. We also assume diallelic loci. It is not clear to us how restrictive this assumption is. In models of mutation-selection balance, two-allele and continuum-of-allele models give similar results provided that the alleles responsible for variation are rare (Turelli 1984; Slatkin and Frank 1990). However, when loci are highly polymorphic (as might occur under balancing selection), continuum-of-allele models can give qualitatively different results (Bürger 1999; Waxman and Peck 1999; Waxman 2003). Nevertheless, we believe that models with two alleles are a better approximation to reality (where there will usually be a few discrete alleles) than are a continuum of alleles, particularly when pleiotropy is considered, because it takes an extraordinary number of discrete alleles to approximate even a two-dimensional continuum (Turelli 1985; Wagner 1989). Models with discrete alleles also preclude a particular multilocus genotype that produces the highest fitness under all conditions. By implicitly allowing such genotypes, Via and Lande (1987) concluded that G × E interactions could not maintain stable polygenic variation.

Another critical assumption is that the temporal and spatial scales of fluctuating allelic effects are sufficiently small, relative to the timescale of selection, that we can average over these fluctuations to approximate the allele-frequency dynamics with deterministic differential equations. Both the linkage equilibrium and averaging approximations were made by Gillespie and Turelli (1989), and we explore their validity numerically with temporal fluctuations in allelic effects (and sex-dependent allelic effects). We conjecture that more highly autocorrelated temporal fluctuations would maintain less variation (Gillespie and Guess 1978), whereas a coarser spatial variation can maintain more variation (Barton and Turelli 1989; Barton 1999).

Our analyses show that balancing selection can maintain variation at loci for which the intensity of balancing selection exceeds the strength of stabilizing selection. With pleiotropy, this follows from sufficiently strong balancing selection. In general, we find multiple alternative stable equilibria, but these tend to produce similar mean phenotypes and levels of variation. With fluctuating allelic effects, stable polymorphism requires sufficiently large fluctuations in the effects and sufficient independence of the fluctuations across loci. The restrictiveness of the conditions is illustrated by the fact that sex-dependent allelic effects cannot maintain stable polygenic variation. Although fluctuations of allelic effects that are extreme enough to maintain variation significantly limit the consistency of genotypic effects, this lack of consistency is apparent only if the genotypes are assayed across the entire range of environments responsible for maintaining variation. This may reconcile the polymorphism conditions with experimental observations.

MODELS AND APPROXIMATE ANALYSES

We analyze in turn pleiotropic balancing selection, G × E with spatial variation and complete mixing, sex-dependent allelic effects, and G × E with temporal variation. The connection uniting these alternative scenarios is that in the weak-selection limit, they lead to essentially identical allele-frequency dynamics and hence similar stability properties for equilibria. This is somewhat surprising, since temporal G × E leads to stochastic fluctuations in allele frequencies whereas pleiotropic balancing selection, spatial variation, and sex-dependent allelic effects are wholly deterministic (but see Gillespie and Turelli 1989 for motivation of this deterministic approximation and our results below for numerical support). We start with the simplest deterministic model to illustrate our stability analyses and then apply essentially the same analyses to “averaged” versions of more complex models involving environment- or sex-dependent allelic effects. We support our average-based analytical approximations with exact multilocus numerical analyses and also use numerical analyses to explore the properties of simultaneously stable alternative equilibria.

Pleiotropic balancing selection: Let Bi and bi denote the alleles at locus i. We let pi,t denote the frequency of Bi in generation t and set qi,t = 1 - pi,t. We assume that selection is sufficiently weak and linkage sufficiently loose that we can ignore linkage disequilibrium. We assume diploidy and random mating. Let βii) denote the additive contribution of Bi (bi) to the trait of interest. We set αiii, so that αi denotes the average effect of a substitution at locus i (Falconer and Mackay 1996, Chap. 7). (Table 1 provides a glossary of notation.) Assuming no dominance or epistasis for the trait, the population mean and additive genetic variance in generation t are zt=2i=1n(pi,tβi+qi,tγi)andVA,t=2t=1nαi2pi,tqi,t. (1)

We assume constant Gaussian stabilizing selection on this trait with optimum θ and strength S, so that the fitness assigned to genotypes producing mean phenotype G (averaged over nongenetic sources of variation) is w(G) = exp(-(S/2)(G -θ)2); this produces both dominance and epistasis for fitness. For weak selection, we can approximate w(G) by a linear function of S. In the weak-selection limit, the population’s mean fitness is w=1S2(VA+(zθ)2)+o(S), (2) where o(S) denotes a quantity that vanishes faster than S does as S → 0. At linkage equilibrium, the allele-frequency dynamics can be described by Δpi=piqi2lnwpi=Spiqi2(αi2(piqi)2αi(zθ))+o(S) (3) (Wright 1937). (The first equation above is exact for one locus; the approximation is the calculation of mean fitness for the one-locus genotypes.) Assuming weak selection, we can approximate (3) by dpidt=Spiqi2(αi2(piqi)2αi(zθ)). (4)

Note that loci other than i enter these dynamics only through their contribution to , and this is true for all of the models we consider. Note also that the allele-frequency dynamics depend on the allelic effects only through αiii and zθ=2i=1npiαi+2i=1nγiθ . Because the scale of measurement of our trait is arbitrary, θ can absorb any constants that enter the determination of the mean phenotype (such as the γi and the contributions of monomorphic loci not considered in our analyses). Thus, we are free to choose any values for the βi and γi that satisfy αiii.

Without loss of generality, we assume that βi=αi2andγi=αi2for alli, (5) so that z=i=1nαi(piqi). (6)

As shown initially by Wright (1935) from an approximation like (4) (cf. Bulmer 1971), stabilizing selection will generally eliminate additive polygenic variation (see Bürger and Gimelfarb 1999 for a recent review). To maintain variation, we assume that the loci experience balancing selection of some sort. The simplest such mechanism is overdominance, but our analysis also covers cases in which additive effects on fitness are linear functions of allele frequencies. This may be a good approximation for a wide range of models of negative frequency dependence, especially if allele frequencies are not perturbed too far from equilibrium.

We assume that the relative contributions to fitness from pleiotropic effects are 1 - sii, 1, and 1 - sii for BiBi, Bibi, and bibi, respectively, with 0 < si ⪡ 1, i = 1 -i, and 0 < i < 1 for all i. These fitnesses lead to a stable equilibrium at i, a fixed parameter in the model. For definiteness, we assume that these pleiotropic fitness effects are multiplicative across loci and that this pleiotropic selection acts before stabilizing selection in the life cycle, with both affecting viability. However, these assumptions are irrelevant in our weak-selection approximation. With weak selection, we can, like Bulmer (1973) and Gillespie (1984), superimpose the pleiotropic overdominant selection on the trait-induced selection to approximate the allele-frequency dynamics by dpidt=S2piqi(αi2(piqi)+2αi(zθ)+2siS(pip^i))=Sαi22piqi((piqi)2δi2vi(pip^i)), (7a) where vi=siαi2S,δi=Δαi,Δ=zθ. (7b) Given that a deviation from the optimum of αi reduces fitness by Sαi22 , vi quantifies the intensity of balancing selection at locus i relative to stabilizing selection. To make the stability analysis more transparent, we can rewrite (7a) as dpidt=Sαi2piqi(pi12+1αi(ziθ)+vi(pip^i)), (8) where zi=jiαj(pjqj) denotes the contribution to the mean phenotype from all loci but i.

View this table:
TABLE 1

Glossary of repeatedly used notation

Stability of fully polymorphic equilibria: From (7a) we see that each locus can fall into one of three possible equilibria: pi = 0, pi = 1, or pi, satisfying pi=2vip^i12δi2(vi1) (9) (note that δi depends on all of the allele frequencies), with vi and δi defined in (7b). [Because of the simple form (3) for our approximate dynamics, only point equilibria can occur; Bürger 2000, Appendix A3.] As expected, the polymorphic equilibrium (9) becomes i if vi is very large and becomes 1/2 if pleiotropic balancing selection is eliminated and the population mean is at the optimum (Δ=δi = 0). Condition (9) creates a system of linear equations for the pi that will generally have a unique solution for a fixed set of polymorphic loci. We first focus on the stability of fully polymorphic equilibria at which all n loci satisfy (9), but we show below that precisely the same stability conditions emerge whenever two or more loci are polymorphic. Conditions for the feasibility of fully polymorphic equilibria are discussed below, along with boundary equilibria at which some or all of the loci are fixed.

As shown in appendix a, stability of the fully polymorphic equilibrium is determined solely by the vi. To maintain a stable polymorphism, balancing selection must be sufficiently strong relative to stabilizing selection (Bulmer 1973). The stability of the fully polymorphic equilibrium depends on the eigenvalues of the Jacobian matrix A = (aij), with aii=pi(dpidt)=Spiqiαi2(1+vi) (10a) and aij=pj(dpidt)=2Spiqiαiαjforij, (10b) evaluated at allele frequencies that satisfy (9). The fully polymorphic equilibrium is locally stable if all of the eigenvalues of this matrix have negative real parts. In general, it is difficult to calculate the eigenvalues. Nevertheless, the stability conditions can be determined because of symmetries imposed by our model. We consider first a completely symmetric model, as discussed by Bulmer (1973), for which all of the eigenvalues and the equilibrium allele frequencies can be explicitly determined.

Suppose that the loci are interchangeable with αi = α, si = s, and i = ; then vi = v for all i and the equations (9) for the equilibrium allele frequencies have a unique solution: pi=p=2θ+(2n1)α+2vαp^2α(2n1+v). (11)

In this symmetric case, the stability matrix A has all diagonal elements equal and all off-diagonal elements equal. A has only two distinct eigenvalues, λ1=Spqα2(v1)andλ2=Spqα2(2n+v1), (12) where λ1 has multiplicity n - 1. Obviously, λ2 is always negative, but λ1 is negative if and only if v>1. (13)

Thus, as Bulmer (1973) found by assuming that z =θ, s2S is the lower bound on the intensity of balancing selection relative to stabilizing selection that must be exceeded to produce a stable polymorphism. Essentially the same constraint on vi arises for the general model (7a).

The necessary and sufficient conditions derived in appendix a for stability are that either vi>1for alli, (14) or one locus (locus 1, say) has ν1 < 1, but this locus obeys v1>11(12)+i=2n(1(vi1)). (15)

For large numbers of polymorphic loci, the sum in the denominator is large, and so this condition is barely different from the simpler sufficient condition (14). Indeed, as shown in appendix a, a necessary condition for stability is (vi+1)(vj+1)>4for allij, (16) so that condition (14) is not far from being both necessary and sufficient. Conditions (14) and (15) can be understood from Wright’s (1935) result that in the absence of balancing selection, i.e., vi = 0 for all i, at most one locus is expected to be polymorphic in the weak-selection limit. At loci with vi > 1, balancing selection is strong enough to maintain polymorphism. Our weak-selection analysis indicates that at most one such locus can be polymorphic with 1 > vi >-1.

Multiple characters: The model readily generalizes to multiple characters, but the resulting stability conditions involve an important difference that illuminates our sex-dependent model. We suppose that stabilizing selection of intensity Sω acts toward an optimum θω, independently across a set of characters, labeled ω, i.e., w(G) = Exp[-Σω(Sω/2)(Gω - θω)2] (corresponding to multiplying the Gaussian selection across characters). Following the arguments leading to (7a), we obtain dpidt=ST2piqi((piqi)2δi2vi(pip^i)), (17a) where ST=ΣωSωαi,ω2,δi=ΣωSωαi,ω2δi,ωST,δi,ω=zωθωαi,ω,vi=siST. (17b) The polymorphic equilibria can still be represented by (9) with vi replaced by i and δi replaced by δi .

However, the stability conditions are more complex than those for the one-dimensional model, because the stability-determining matrix A in (10) is replaced by aii=STpiqi(1+vi) (18a) and aij=2pipjωSωαi,ωαj,ωforij. (18b)

Because of the summation in (18b), the signs of the eigenvalues of (18) do not depend solely on the i. Unlike the one-character model in which at most one locus is expected to be polymorphic with vi < 1, for multiple characters with i = 0 for all i and equal allelic effects, the number of stably polymorphic loci can be as large as the number of traits (Hastings and Hom 1989) or larger with strong selection (Gimelfarb 1992). We return to this result when we consider sex-dependent allelic effects.

Stability, feasibility, and positions of alternative equilibria: Next, we consider equilibria for the one-character model in which some loci are monomorphic. Several complexities arise due to the possible simultaneous stability of multiple equilibria with different numbers of polymorphic loci and fixation of either Bi or bi at the monomorphic loci. First, consider the conditions for polymorphic equilibria to be feasible. The conditions will depend on whether vi > 1 (recall that at most one stably polymorphic locus can violate this). If vi > 1, (9) implies that 0 < pi < 1 only if vi>max(1+2δi2p^i,12δi2q^i). (19a)

If vi < 1, feasibility requires vi<min(1+2δi2p^i,12δi2q^i). (19b)

Unless Δ= 0, (19a) constrains at least all but one of vi to exceed 1 by an amount that depends on Δ. Conversely, if stability is achieved with one locus satisfying vi < 1, (19b) puts an upper bound on this vi that must be satisfied along with the lower bound given by (15). Overall, these feasibility conditions for polymorphisms and the conditions described next for stability of fixation equilibria require allele frequencies that make Δ very small.

Consider an equilibrium at which pi = 0 for all i in the set Ω0, pi = 1 for i in Ω1, and 0 < pi < 1 for i in Ωp. In this case, the stability matrix A can be partitioned into blocks corresponding to the fixed and polymorphic loci, because aij=0forijifiis in eitherΩ0orΩ1, (20a) whereas aii=Sαi22(2p^ivi12δi)foriΩ0, (20b) and aii=Sαi22(2q^ivi1+2δi)foriΩ1. (20c) Thus, the eigenvalues governing the stability of the fixed loci (i.e., i ∊ Ω0 222A; Ω1) are simply λi = aii, and the stability conditions for the subsystem of polymorphic loci (i.e., i ∊ Ωp) are just (14) and (15). Equations 20b and 20c show that the stability conditions for the fixed loci are vi<1+2δi2p^iifpi=0, (21a) and vi<12δi2q^iifpi=1. (21b) Hence, the conditions for the stability of the fixed equilibria are complementary to the feasibility conditions (19a) for the polymorphic equilibria with vi > 1. The implications of (21) can be seen by assuming, without loss of generality, that Δ = -θ is negative and all of the αi are positive. In this case, increasing pi at each locus moves the population mean closer to the optimum. Then, inequalities (21) with vi = 0 imply that, whenever possible, the multilocus system will equilibrate so that |Δ| is less than mini∊Ω0i/2}. Inequalities (21) imply that |Δ| is even smaller with vi > 0.

Because alternative multilocus equilibria will generally produce different values of Δ, and hence different δi for each locus, conditions (19) and (21) do not preclude a locus from having stable alternative fixation and polymorphic equilibria (cf. Hastings and Hom 1990). In particular, if vi is only slightly above one, the locus can be stably polymorphic at an equilibrium with Δ very near 0, but stably monomorphic at equilibria with larger |Δ|. This is illustrated numerically below.

Now consider Δ at equilibria. Assuming as above that pi = 0 for i ∊ Ω0, pi = 1 for i ∊ Ω1, and 0 < pi < 1 for i ∊ Ωp, we have Δ=iΩ1αiiΩ0αi+iΩpαi(piqi)θ. (22) Substituting expression (9) for the equilibrium allele frequencies and rearranging, we find that Δ=Δf1+2C, (23a) where Δf=iΩ1αiiΩ0αi+iΩpαivi(p^iq^i)vi1θ, (23b) and C=iΩp1vi1. (23c) The population mean would depart from the optimum by Δf in the absence of stabilizing selection returning the trait toward the optimum. [Without stabilizing selection, the terms in the final summation in (23b) reduce to αi(i - i).] In this sense, Δf represents a natural resting point of the system under balancing selection alone. Stabilizing selection generally reduces this deviation by a factor B = 1/(1 + 2C) (as noted in appendix b, the factor is negative if vi < 1 for one of the i in Ωp, but we ignore this special case). Thus, B is a cumulative measure of the strength of stabilizing selection relative to balancing selection. As noted above, we generally expect vi > 1 at stably polymorphic loci. For any fixed lower bound on the vi, (23c) shows that as the number of polymorphic loci increases, the population mean will converge to the optimum by slightly perturbing the polymorphic allele frequencies away from i as described by (9).

The stability conditions for the full system, including fixed and polymorphic loci, are detailed in appendix b. The qualitative conclusion is that, for a wide range of parameter values, loci with vi > 1 can be stably polymorphic and loci with vi < 1 are generally monomorphic. Moreover, although alternative equilibria may be simultaneously stable, they generally produce mean phenotypes very near the optimum. These generalizations are illustrated by our numerical examples below, which also suggest that the alternative equilibria produce similar equilibrium levels of genetic variation.

Consequences of G × E with spatial variation and complete mixing: Next we consider a deterministic model that involves only stabilizing selection on the trait, but allows for environment-specific allelic effects, which can produce balancing selection at individual loci (Gillespie and Turelli 1989). Following Levene (1953), we assume that each environment contributes a constant proportion to the random-mating pool that forms the next generation of zygotes. In our weak-selection limit, this means that we can simply average the equations that emerge in each environment, weighting each environment by its fractional contribution to the next generation (cf. Gillespie and Langley 1976). Let βi,ki,k) denote the effect of Bi (bi) in environment k. With weak selection, as in (4), we can approximate the allele frequency dynamics in this environment by dpidt=Spiqi2((piqi)(βi,kγi,k)2+2(βi,kγi,k)(zθ)). (24) We assume that S and θ remain fixed across environments. Averaging over environments, we define E(βiγi)=αiandVar(βiγi)=viαi2. (25) Thus, the effect of a substitution at locus i has mean αi and variance is viαi2 , so that vi is the square of the coefficient of variation of the substitution effect. We demonstrate that these vi play the same role in the stability analysis of this model as do the vi defined by (7b) for the pleiotropy model. Averaging over environments, as done in Gillespie and Turelli (1989), we obtain dpidt=Spiqi2(αi2(1+vi)(piqi)+2αi[E(z)θ]+2Cov(βiγi,z)). (26) As discussed after Equation 4, we can absorb constants that enter E() into θ, so we assume Ei) =αi/2 and Ei) =-αi/2, without loss of generality. Thus, E() = Σi(pi - qii. Rearranging (1), we have z=i=1n(piqi)(βiγi)+i=1n(βi+γi), (27) where the βi and γi have environment-specific values. Hence, the term Cov(βii, z¯) that enters (26) and the analyses below depends on the scaled covariances, ij and ij, defined by Cov(βiγi,βjγj)=cijαiαjρijαiαjvivj (28a) and Cov(βiγi,βj+γj)=dijαiαj, (28b) where ii = vi and ρij denotes the correlation of substitution effects at loci i and j. Note thatij = ij ≡ 0 for all ij if either the allelic effects at different loci fluctuate independently or we impose the symmetry constraints, Cov(βi, βj) = Cov(γi, γj) = Cov(γi, βj) = Cov(βi, γj) for all ij. Gillespie and Turelli (1989) assumed the latter. Under the less restrictive assumptions that Cov(βi, βj) = Cov(γi, γj) and Cov(γi, βj) = Cov(βi, γj), we have ij ≡ 0 for all i and j. In particular, ii = Var(βi) - Var(γi), so that ii = 0 if Var(βi) = Var(γi). Separating the terms in (26) that depend on locus i, we have dpidt=Spiqi2(αi2(1+vi)(piqi)+2αi[E(zi)θ]+2αi2dii+2Cov(βiγi,zi)), (29) where E(zi)=ji(pjqj)αj denotes the average contribution to the mean phenotype from the loci other than i. It is easy to see that Cov(βiγi,zi)=0ifcij=0 for all ij. Gillespie and Turelli (1989) assumed this and found that interlocus correlations did not affect their polymorphism condition. We show below that this conclusion depends critically on their symmetry assumption concerning interlocus correlations.

Analysis of fully polymorphic equilibria: At equilibrium, each locus must satisfy pi = 0, pi = 1, or pi=12αi[E(zi)θ]+αi2dii+Cov(βiγi,zi)αi2(1+vi). (30) Note that increasing ii, corresponding to raising the variance of effect for allele Bi relative to the variance for bi, decreases the equilibrium pi, consistent with the general principle that selection in variable environments tends to favor more homeostatic genotypes (Gillespie 1974). The stability of the fully polymorphic equilibrium depends on the eigenvalues of the Jacobian matrix A = (aij) with elements aii=pi(dpidt)=Spiqiα12(1+vi) (31a) and aij=pj(dpidt)=2Spiqiαiαj(1+cij)forij, (31b) evaluated at allele frequencies that satisfy (30), withij as defined in (28a). If Cov(βiγi,zi)=0 , the terms ij in (31b) vanish and the stability conditions are precisely those for the pleiotropy model, (14) and (15). In this case, the G × E model reduces to the pleiotropic balancing selection model (apart from ii, which does not affect stability of polymorphic equilibria) with i = 0.5 at all loci. Following the argument in appendix a, it is easy to see that in general the stability properties of (31) depend only on the vi and the ρij defined in (28a).

In general, positive correlations across loci are destabilizing in the sense that larger vi are needed to achieve stability with ρij > 0 than with ρij = 0. The destabilizing effect is dramatic. The necessary condition for stability, analogous to (16), is (vi+1)(vj+1)>4(1+ρijvivj)2for allij. (32) With ρij > 0 and ρij2>34 , (32) cannot be satisfied for three or more loci. The general stability conditions can be explicitly obtained following the procedure given in appendix a, but they seem too complicated to be informative. However, the qualitative effects of correlations across loci can be seen under the symmetry assumptions viv and ρij ≡ ρ. In this case, a feasible fully polymorphic equilibrium is stable if and only if ρ<12andv>112ρ. (33) Hence, polygenic variation can be stably maintained under this model of G × E interactions only if the loci experience at most moderate positive correlations among their fluctuating allelic effects, and the variance in effects is sufficiently large. As illustrated by our analysis of sex-dependent allelic effects, with only two environments, |ρij| = 1 and polymorphism condition (32) cannot be satisfied.

Stability, feasibility, and position of alternative equilibria: Consider an equilibrium with pi = 0 for i ∊ Ω0, pi = 1 for i ∊ Ω1, and 0 < pi < 1 for i ∊ Ωp. First note that if Cov(βiγi,zi)=0anddii=0 , we can use the results for the pleiotropic balancing selection model with the additional constraint i = 0.5 for all i. This generally simplifies the analysis. For instance, the stability conditions for the fixed loci reduce to vi<1+2δi,ifiΩ0,andvi<12δi,ifiΩ1, (34) where δi = [E(z¯) - θ]/αi. Numerical examples of pleiotropic overdominance presented below, which assume i = 0.5, illustrate the approximate equilibria and dynamics of this model.

New phenomena appear with Cov(βiγi,z1)0 and ii ≠ 0. As noted above, positive correlations across loci make stable polymorphisms more difficult to obtain and positive values of ii tend to lower pi. Hence, we expect that positive correlations between loci and ii > 0 will broaden the conditions for stability of pi = 0. As before, the stability matrix A can be partitioned into blocks corresponding to the fixed and polymorphic loci, because aij=0forijifiis in eitherΩ0orΩ1, (35a) whereas aii=Sαi22(vi12ei)foriΩ0, (35b) and aii=Sαi22(vi1+2ei)foriΩ1, (35c) with ei=δi+dii+Cov(βiγi,zi)αi2. (35d)

Thus, the eigenvalues governing the stability of the fixed loci (i.e., i ∊ Ω0 222A; Ω1) are simply λi = aii, and the stability conditions for the subsystem of polymorphic loci (i.e., i ∊ Ωp) are determined by the eigenvalues of (31), which depend only on the parameters for the polymorphic loci. Equations 35b and 35c show that the stability conditions for the fixed loci are vi<1+2ei,ifpi=0,andvi<12ei,ifpi=1. (36) As expected, positive values of Cov(βiγi,z1) and ii promote the stability of pi = 0.

Properties of polymorphic equilibria: One of our central motivations for allowing significant differences in the mean effects of alleles was to determine whether appreciable heritable variation could be maintained by G × E that would provide persistent selection response. We have shown that maintaining variation requires a sufficiently large coefficient of variation of the allelic effects and sufficient independence of the fluctuations across loci. At least two biologically interesting questions follow. First, how similar are the phenotypes of various relatives, for instance, parents and offspring, and second, how variable are the phenotypes produced by specific genotypes across the range of environments responsible for maintaining the variation (cf. Yamada 1962; Gillespie and Turelli 1990; Gimelfarb 1990). The second question is answered more easily than the first, because the similarity of relatives will depend on the similarity of their environments. Even if this is known, the correlations between relatives will depend on additional parameters, which do not enter the polymorphism conditions, that describe the covariance of the fluctuating effects of alleles within and across loci. These parameters also enter the variance for the mean phenotypes produced by specific genotypes across environments. To illustrate this, we calculate the expected variance of the mean phenotype of a randomly drawn genotype and then partition the equilibrium variance in mean phenotypes to quantify the consistency of genotypic differences across environments.

Let Gk(g) denote the average phenotype of a specific multilocus genotype g in a specific environment k (the same analyses apply to both spatial and temporal variation). Under our additivity assumption, Gk(g)=i=1nGi,k(g), (37) where Gi,k(g) denotes the contribution of the diploid genotype at locus i in environment k. Under the assumptions that lead to (24), the allele-frequency dynamics depend on the moments of allelic effects only through the means and variances of the substitution effects. Thus, we had to specify only Eii) =αi and Var(βiγi)=viαi2 . However, we will see that the variance of genotypic values depends separately on the variances and covariances of the allelic effects within and between loci. We assume that Var(βi)=Var(γi)=ciαi22andCov(βi,γi)=ρiciαi22, (38) so that Var(βiγi)=ciαi2(1ρi). (39) Hence, in the calculations above, e.g., Equation 8, vi = ci(1 -ρi). [Note that ρi describes correlations between allelic effects within loci, whereas ρij in (28a) describes correlations between substitution effects at different loci.]

First, assume uncorrelated fluctuating allelic effects across loci, so that Var[Gk(g)g]=i=1nVar[Gi,k(g)g],Eg{Var[Gk(g)g]}=i=1nEg{Var[Gi,k(g)g]}, (40) where Eg denotes averaging over the distribution of genotypes in the population. A central feature of all random environment models is that Var[Gi,k(g)|g] depends on whether genotype g is homozygous or heterozygous at locus i (Gillespie and Turelli 1989). Using (38), Var[Gi,k(g)g]=2ciαi2 if g is homozygous for either allele at locus i, and Var[Gi,k(g)g]=ciαi2(1+ρi) if g is heterozygous at locus i. Thus, Eg{Var[Gi,k(g)g]}=2ciαi2[1piqi(1ρi)]=2viαi2(11ρipiqi). (41) Note that this depends on both vi and ρi. With independent fluctuations across loci, we have Eg{Var[Gk(g)g]}=2i=1nviαi2(11ρipiqi). (42) To understand the implications of (42), we need a scale-independent quantification of this expected within-genotype variance. By analogy to broad-sense heritability, we can define an index for the stability of environment-dependent genetic effects as the fraction of the total genetic variance (across both genotypes and environments) attributable to the mean effects of different genotypes. In general, averaging over both environments and genotypes, we have Var[Gk(g)]=Varg{E[Gk(g)g]}+Eg{Var[Gk(g)g]}, (43) where, as indicated, the inner expectations on the right-hand side are taken over environments and the outer expectations are taken over genotypes. The first term is the variance of the mean genotypic values (i.e., the “main effect” of genotypes) and the second is the average across-environment variance for individual genotypes. We define the “consistency” of these genotypes as K=Var{E[Gk(g)g]}Var[Gk(g)]. (44) K near 0 implies that the differences among mean effects are small relative to the standard deviations of genotypes’ effects across environments, and K near 1 implies relatively large mean effects (e.g., K = 1 with constant allelic effects). Because K focuses on partitioning the variance of genotypic means within and among “macroenvironments” (e.g., spatial environmental “patches” or years), it bears no simple relationship to heritability estimates, which also account for “microenvironmental” variation (i.e., phenotypic differences among genetically identical individuals within the same spatial or temporal macroenvironment).

Our linkage equilibrium assumption and the definition of αi imply that irrespective of correlations in fluctuations within or among loci, Varg{E[Gk(g)g]}=2i=1nαi2piqi. (45) Hence, for independent fluctuations across loci, K=i=1nαi2piqii=1nαi2piqi+i=1nviαi2((1(1ρi))piqi). (46)

The qualitative implications of (46) are most easily seen with exchangeable loci (i.e., αi =α, ci = c, ρi =ρ, and pi = p), for which K=pq(1ρ)pq(1ρ)+v[1pq(1ρ)], (47) and the stability criterion is simply v > 1. Equation 47 implies that K decreases as ρ and v increase and as p departs from 0.5. Hence, for stable equilibria, K is maximized when ρ=-1, v = 1, and p = 0.5. At this point, K = 0.5. However, when the within-locus effects are uncorrelated, as the between-locus effects are assumed to be, K ≤ 0.25.

In general, positive between-locus covariances reduce K because the numerator remains constant but Eg{Var [Gk(g)|g]} in the denominator increases. The effects of these covariances depend not only on the covariances of substitution effects, i.e., Cov(βii, βjj) as described by (28), but also on the covariances between the individual alleles at each locus, i.e., Cov(βi, βj), Cov(γi, γj), and Cov(βi, γj). Nine different expressions for Cov{[Gi,k(g)|g], [Gj,k(g)|g]} are generated by the three genotypes at each locus. To illustrate the quantitative effects, we focus on the completely symmetrical case explored by Gillespie and Turelli (1989) and Gimelfarb (1990) with Cov(βi,βj)=Cov(γi,γj)=Cov(βi,γj)=Cov(βj,γi)=ρBciαi2cjαj22. (48)

As discussed following Equations 28, this implies that Cov(βii, βjj) = 0 for all ij, so that the allele frequency dynamics are still approximated by (29) with dii=Cov(βiγi,zi)=0 . In particular, in the symmetrical case with ci(1 -ρi) = v for all i, the polymorphic equilibrium is stable whenever v > 1. For exchangeable loci satisfying (48), (47) is replaced by K=pq(1ρ)pq(1ρ)+v[1pq(1ρ)+(n1)ρB]. (49)

Thus, for any positive ρB, K approaches 0 for large numbers of loci (see Gimelfarb 1990 for an analogous result). The implications of these upper bounds on K for the maintenance of variation by G × E interactions are considered in the discussion.

Sex-dependent allelic effects: A special case of this multiple-environment model approximates allele-frequency dynamics with sex-dependent allelic effects. In this case, the two sexes are the alternative “environments.” As first argued by Haldane (1926) and demonstrated rigorously for one locus by Nagylaki (1979), the dynamics of weak, sex-dependent viability selection can be approximated by simply averaging the fitnesses of each genotype over the two sexes. This is equivalent to averaging the allele-frequency dynamics as in (26), but now each random variable takes on only two values. This greatly simplifies and constrains the expressions for the coefficients of variation and the correlations of substitution effects across loci. For instance, if αf,im,i) denotes the effect of a substitution at locus i on females (males), we have vi=(αf,iαm,iαf,i+αm,i)2. (50)

The condition vi > 1 requires that αf,i and αm,i have different signs. Thus, if we use the convention that each Bi denotes the allele that increases the trait value in females, the polymorphism condition vi > 1 implies that each Bi must decrease the trait in males. By considering the symmetrical model with θ= 0, it is easy to see, however, that vi > 1 cannot suffice to maintain polymorphism. The multilocus recursions for the allele frequencies depend separately on the fitnesses assigned to each genotype in males and females. When θ= 0, our symmetrical selection model implies that altering the signs of all of the allelic effects in one sex will not change the fitnesses. Hence, for any assignment of allelic effects, identical dynamics must emerge if all of the signs of allelic effects in one sex are reversed. If the initial assignment of effects satisfies vi > 1 for all i, by reversing the signs of effects in one sex, we get identical dynamics but the new values of vi are the reciprocals of the old.

The additional constraint required for stable polymorphism involves the correlations in the fluctuating effects across loci. Our convention of labeling the alleles so that Bi and Bj increase the trait value in females implies that ρij=1for allij (51) (indeed, a two-valued bivariate random variable can have only correlations ±1). Thus, constraint (32) implies that a fully polymorphic equilibrium cannot be stable, although sex-dependent selection can readily maintain polymorphism at one locus (Kidwellet al. 1977).

Below, we present numerical analyses that support the qualitative conclusion that sex-dependent, additive allelic effects cannot maintain stable polygenic variation. From our sexes-averaged approximation, we expect that at most one locus can remain stably polymorphic under weak selection. In fact, however, several numerical examples described below indicate that up to two loci may be stably polymorphic for loose linkage. This shows that our averaging approximation misses some of the subtleties of sex-dependent selection, while accurately capturing its inability to maintain variation at many loci. The stable two-locus polymorphisms we find are reminiscent of Hastings and Hom’s (1989, 1990) results concerning pleiotropic effects on two characters. A more careful analysis of sex-dependent allelic effects requires distinguishing the allele frequencies in the two sexes, so that at linkage equilibrium the stability analysis for n loci involves 2n variables rather than n. This is discussed elsewhere.

Temporal variation and G × E: Finally, we apply our analyses to generalize the treatment by Gillespie and Turelli (1989) of temporally fluctuating allelic effects. Our approximation is based on averaging over the distribution of allelic effects to approximate the stochastic dynamics by a set of deterministic equations identical to those obtained for G × E with spatial variation. As noted below, this approximation relies on weak selection. Ultimately, the usefulness of our approximations depends on their ability to predict the maintenance of variation with biologically plausible levels of selection and environmental fluctuation. We present numerical results below, suggesting that our approximate polymorphism conditions are surprisingly accurate. Our deterministic analysis does not address the fluctuations of allele frequencies inherent in stable polymorphisms maintained by temporal fluctuations. This is explored numerically below.

As with spatial variation, under particular symmetry assumptions concerning the interlocus correlations in fluctuating allelic effects [see (28) and (29)], the deterministic approximation is precisely equivalent to the pleiotropy model analyzed above. As with spatial variation, the critical parameter governing the stability of polymorphism at each locus is just vi, the squared coefficient of variation of the substitution effect at that locus. Because we obtain identical approximations for temporal and spatial variation, we discuss the analytical approximations only briefly.

The assumptions of this model are the same as with spatial variation, except that the allelic-effect parameters, βi,t and γi,t, vary across generations. Note that (25) allows for arbitrary correlation between the fluctuating effects of the two alleles at a locus. Within-locus correlations of βi,t and γi,t do not explicitly affect the dynamics, because selection depends only on βi,ti,t. However, as discussed in the context of spatial variation, intralocus correlations can significantly affect the properties of the genetic variation maintained, depending on the time-scale of parameter variation. As before, the basic recursion is Δpi,t=Spi,tqi,t2((pi,tqi,t)(βi,tγi,t)2+2(βi,tγi,t)(ztθ))+o(S). (52)

A central assumption of our analysis is that selection is weak; i.e., S ⪡ 1. We also assume that the timescale of allele-frequency change is slower than the timescale of the environmental fluctuations, for instance, that the successive environments are independent or at most “weakly autocorrelated” (Gillespie and Guess 1978). These assumptions allow us to (i) average the random fluctuations over time and (ii) approximate the dynamics of the discrete-generation stochastic model (52) by a system of deterministic differential equations. [Because the leading term in (52) is proportional to S, the infinitesimal variance in a diffusion approximation vanishes, leaving a deterministic limit (cf. Gillespie and Turelli 1989).] Taking the expectation of the right-hand side of (52) over the fluctuations in allelic effects, ignoring the higher-order terms and going to the continuous-time limit, we obtain the time-averaged, weak-selection approximation (26) used above to discuss spatial variation with complete mixing. We have not made any explicit assumptions about the temporal correlations of the fluctuating parameters. However, our analysis implicitly assumes that autocorrelations decay faster than the timescale of allele frequency change (1/S).

The approximate polymorphism conditions are precisely those obtained for the spatial model considered above. Again, the central parameters governing the stability of polymorphisms are the vi, which describe the squared coefficient of variation of substitution effects at individual loci [see (25)], and the ρij, which describe the correlations between substitution effects at different loci [see (28a)]. In particular, when ρij = 0, we expect that loci satisfying vi > 1 will tend to remain polymorphic if the expected population mean is close enough to the optimum. Loci with vi < 1 will generally not be stably polymorphic, and they will fix for alleles that produce an expected population mean very near θ. As before, we predict from (32) that no stable multilocus polymorphisms can be maintained for loci with large positive ρij.

One important difference between the spatial and temporal models concerns the interpretation of the consistency index, K, defined by (44), and its relationship to empirical observations. The key point, as made by Gillespie and Turelli (1989, 1990), is that the temporal variation responsible for maintaining variation need not be observable over a few generations. For instance, even though the “true” value of K, obtained by averaging over all of the environments responsible for maintaining variation, may be quite small, the value observed in any one generation is one, since allelic effects are assumed to be fixed within any one generation. With high levels of positive autocorrelation for allelic effects, K would remain high even when averages are taken over several generations. Thus, temporal G × E may maintain high levels of genetic variation with consistent differences among genotypes over the timescale of reasonable experimental analyses. This point is elaborated below.

NUMERICAL ANALYSES

Pleiotropic balancing selection: The number of parameters in this model makes it impractical to explore equilibria and dynamics systematically across the parameter space. However, the qualitative features of alternative equilibria and dynamics are illustrated by the following two numerical examples. In both examples, we assume for simplicity that i = 1/2 at all loci. Loci with extreme values of i are less likely to be polymorphic because of the constraints associated with the feasibility conditions (19) and the increased influence of genetic drift. We concentrate on observable quantities such as mean fitness, deviation of the trait mean from the optimum, and genetic variance.

Example 1—alternative equilibria: To illustrate the implications of the stability and feasibility conditions, we first consider an example in which the αi and vi at 20 loci were drawn independently from gamma distributions (Table 2), and the optimum is θ= 0. Because of this symmetry, the equilibria come in pairs, the elements of which have Δ of opposite sign, produced by replacing each pi by 1 - pi. appendix c describes the procedure for finding the alternative stable equilibria. Note that since all of the equilibria discussed have some monomorphic loci, the polymorphic loci must produce a mean near an optimum different from 0, which cannot be achieved by multilocus heterozygotes or by any combination of homozygotes. This “effective optimum,” denoted θeff, is θeff=θiΩ1αi+iΩ0αi; (53) its values are given in Table 2.

For the parameter values in Table 2, there are seven pairs of distinct stable and feasible equilibria (or, more accurately, seven pairs of allele frequencies that satisfy the constraints for stable and feasible equilibria given by our approximations); only the stable allele frequencies that produce negative Δ are shown. All of the equilibria share similar properties: mean fitnesses are within 0.55S of each other, and the deviation from the optimum is |Δ| < 0.373 (relative to a mean allelic effect of α=1 and a range of genotypic values |G | < 21.07). Barring the most extreme pair of equilibria (presented in the first column of Table 2), mean fitnesses are within 0.0051S of each other, and the deviation from the optimum is |Δ| < 0.021. The main differences among the equilibria are in the number of polymorphic loci and in the combinations of fixed loci. As expected from the stability conditions for fixation equilibria (21), the polymorphic loci that differ among equilibria tend to be those of small effect, and the different combinations of polymorphic and monomorphic loci all bring the population mean very close to the optimum. This explains the similarity in overall fitness properties of the different equilibria. Given that selection is simply tending to climb toward local fitness optima, it is not clear whether equilibria with higher mean fitness will tend to be reached preferentially.

View this table:
TABLE 2

Multiple stable equilibria maintained by stabilizing selection on a single trait and overdominance

We can gain some insight by considering the feasibility conditions for the polymorphic equilibria, (19), and the stability conditions for the fixation equilibria, (21). For definiteness, we assume that Δ< 0 and αi > 0 for all i, so that directional selection will tend to increase the pi. As before, we let Ωp denote the set of polymorphic loci and Ω0 the set of loci with pi = 0. With i = 1/2, the feasibility condition (19) requires that for the loci in Ωp to avoid fixation at pi = 1, Δ<αi(vi1)2for alliΩp. (54)

Hence, for these loci to remain polymorphic, |Δ| must be less than mini∊Ωpi(vi - 1)/2}. Conversely, if the loci in Ω0 are to remain stably fixed, (21a) requires Δ<αi(1vi)2for alliΩ0. (55)

With multiple loci contributing to the character, (54) and (55) ensure a very close approach to the optimum. Because the deviation from the optimum is small, allele frequencies at polymorphic loci with significant effects are close to i. For small Δ, the allele frequencies should be near ½(1/(1 - vi)) + i(vi/(vi - 1)), which is a compromise between the unstable equilibrium under stabilizing selection (first term) and the stable equilibrium under balancing selection (second term). For i near 0.5, the genetic variance will be 12αi2 , where the sum is over those loci for which vi > 1, and which are hence polymorphic at Δ= 0. For example, the model of Table 2 gives a genetic variance that varies across equilibria from 41.2927 to 43.6166, whereas the value when all loci with vi > 1 are polymorphic with pi = 0.5 is 43.6166. If Δ deviates slightly from zero, then loci with small effects fix; however, such loci have little effect on the genetic variance.

In general, the mean relative fitness of the population is less than one because of stabilizing selection and balancing selection at the individual loci, which contributes an amount denoted LB to the genetic load. The stabilizing-selection component of the load can be partitioned into the portion attributable to departures of the population mean from the optimum, denoted LΔ, and variance in the population around the population mean, denoted LV. In our weak-selection limit, the contributions to the load are additive, i.e., L=1WLV+LΔ+LB, (56a) with LV=SVA2=SiΩpαi2piqi, (56b) LΔ=SΔ22, (56c) and LB=i=1nsi(p^iqi2+q^ipi2)=i=1nsi(p^ipi)2+i=1nsip^iq^i=LBS+LBB, (56d) where LBS in (56d) denotes the segregation load from balancing selection attributable to allele frequencies being perturbed by stabilizing selection from their balancing-selection equilibria, and LBB denotes the equilibrium segregational load under balancing selection alone. From Table 2, we see that LΔ, the loss in mean fitness due to Δ, is much smaller than that due to stabilizing selection (LV), which in turn is smaller than LB, the segregational load.

This numerical procedure was repeated with100 random choices of the set of allelic effects and strengths of balancing selection {αi, vi} at 20 loci. Although the quantitative outcomes depend on the arbitrary distributions chosen for these parameters, the qualitative features seem robust. In all but one case, the results were similar to those shown in Table 2. [In the one exceptional case, the largest allelic effect was much greater than the rest: α1 = 17.8, compared with a maximum of 1.78 among the remainder (results not shown). Thus, a single polymorphism was maintained by overdominance at this locus, with the remaining loci fixed for “1” alleles.] In the remaining discussion, we discard this outlier.

In the other 99 cases, multiple polymorphisms were maintained; on average, 6.8 loci were polymorphic [with standard deviation (SD) 2.3]. In 12% of cases, one of these polymorphic loci had vi < 1 as allowed by our stability condition. Conversely, 40% of the monomorphic loci had vi < 1; these loci had small allelic effects relative to the deviation from the optimum, so that αi(vi - 1) < 2|Δ|. Typically, many alternative equilibria were stable for any given set of {αi, vi}: on average, 17.7, with a range from 2 to 178. However, these equilibria had very similar properties. This is because the equilibria differ in whether loci with small effects [namely, αi(vi - 1) < 2|Δ|] are fixed for 0 or 1. Every allowable set of these loci leads to almost the same (small) deviation from the optimum, and this set contributes very little to the overall genetic variance.

Over our replicates, the mean deviates from the optimum by an average Δ= 0.17 (SD 0.28). This can be compared with a mean allelic effect α= 1 and with an average genetic variance of 31.1 (SD 39.9). The loss of mean fitness caused by this deviation is negligible (mean 0.06S, SD 0.17S) compared with the genetic load due to variation around the optimum (mean 15.5S, SD 20.1S) and due to balancing selection (mean 59.1S, SD 73.6S). In the great majority of cases, most of the genetic load is due to the perturbation of loci away from their equilibrium under balancing selection.

Example 2—response to changes in selection: Next we consider the consequences of varying the intensity of stabilizing selection and the position of the optimum. We explore how these alter the amount of variation maintained and the portions of the genetic load attributable to departures of the mean phenotype from the optimal phenotype [as described by LΔ (56c)], genetic variation about the mean [LV (56b)], and loss of fitness under balancing selection caused by stabilizing selection perturbing allele frequencies away from their balancing selection equilibria [LBS in (56d)]. In general, we expect that a moderate number of loci under balancing selection might affect a specific trait (10-100 with si > 0, say). There will be a distribution of strengths of balancing selection, si, and for each si, a distribution of allelic effects αi that may be correlated with the si. In Figure 1, we show a bivariate distribution of (αi, si) for 100 loci, where the si were chosen from a gamma distribution with mean 0.01 and coefficient of variation (CV) 1, and for each si, αi was chosen from a gamma distribution with mean 0.05 + 5si and CV = 1 and then made negative with probability 0.5 (i.e., a gamma distribution reflected around 0). There may also be many loci that affect the trait, but are not under balancing selection. These are not considered because they are expected to be fixed and thus will not affect the properties we discuss. For any given optimum, there may be many stable equilibria, as illustrated in Table 2. We circumvent this by considering equilibria that produce a specific departure from the optimum, Δ. Although alternative equilibria may exist even for fixed Δ (i.e., monomorphic loci may fix at either 0 or 1), the statistics we discuss have unique values because they depend only on polymorphic loci.

Figure 1.

—Values of αi and si for 100 loci separated by lines indicating their stable equilibria. A assumes Δ= 0 and B assumes Δ= 0.05; both assume S = 1. With Δ= 0, there are only two stability regimes; loci with {αi, si} above the parabola must be polymorphic, while those below the parabola can fix for 0 or 1. With Δ= 0.05, there are seven stability regimes, shown in B, as described in the text.

Suppose that the set of fixed loci is such that the natural resting point of the system coincides with the optimum; i.e., Δf = 0 [see (23b)]. In this case, the conditions for feasible and stable polymorphism are just vi>1orsi>Sαi2 ; that is, polymorphic loci must have (αi, si) that lie above the parabola in Figure 1A. Because the mean coincides with the optimum and i = 1/2 at all of the loci, the equilibrium allele frequency at each locus is 1/2, and the genetic variance is VA=12iΩpαi2 .

Figure 2A shows the genetic variance, VA, as a function of the strength of stabilizing selection: it decreases inversely with S. These calculations assume that Δ= 0 as in Figure 1A. Figure 2B shows LV, the load due to variation around the optimum, as a function of S. Initially, LV increases linearly. However, as polymorphic loci start to fix, LV decreases and remains almost constant for a wide range of S. It is not clear how general this pattern is, since it depends on the joint distribution of (αi, si).

Now, consider the short-term response to a decrease in the optimum. There will be an immediate increase in the “natural” deviation, to Δf > 0. Polymorphic allele frequencies will adjust rapidly, and some loci will fix. Overall, the new deviation from the optimum will be Δf(1+2C) [see (23)], which may be very much smaller than Δf. Note that the natural resting point will shift from Δf to Δf because some initially polymorphic loci have fixed, which also decreases C* from its initial value, C. In this new selection regime, the loci depicted in Figure 1A are scattered over seven stability regimes as shown in Figure 1B. Reading from left to right, we see: (i) loci that could fix at 0 or 1 under the old and new positions of the optimum; (ii) loci with αi < 0 that were originally fixed for 0 or 1 are now forced to fix for pi = 0, increasing the trait value; (iii) loci that were polymorphic are now forced to fix for pi = 0; (iv) loci that were polymorphic, and remain so; (v) loci that were polymorphic with αi > 0 are forced to fix for pi = 1; (vi) loci that were originally fixed for 0 or 1 are now forced to fix for pi = 1; and finally, (vii) loci that were fixed for 0 or 1 and remain so.

Figure 2.

—Additive genetic variance, VA, as a function of the intensity of stabilizing selection, S. We assume that Δ= 0 as in Figure 1A. VA declines inversely with the strength of stabilizing selection (A). However, the load due to variation around the optimum, LV = (S/2)VA, is almost constant for a wide range of S (B).

Figure 3 shows the short-term deviation from the optimum, Δ, as a function of Δf, the difference between the natural resting point and the optimum. Figure 3A shows the results on the original scale, and Figure 3B presents the same results as a log-log plot. In these calculations, we assume that the polymorphic loci are those found assuming Δ= 0, as in Figure 1A. The question addressed is how those specific polymorphic loci are expected to respond to changes in the optimum. As the optimum changes, all of the polymorphic allele frequencies adjust to produce a new value Δ, as described by Equation 23a. As Δf increases, there is remarkably little change in Δ, provided that Δf does not approach the maximum that can be compensated by shifts in polymorphic allele frequencies (corresponding to Δf = 2.38 in this example, see Figure 3A). For small deviations, Δ is only 0.01Δf1.85 (as determined by a least-squares fit of the log-log results in Figure 3B). The main effect of a small perturbation, Δf, is to reduce the genetic variance (Figure 4), as polymorphic allele frequencies shift from 0.5 and eventually fix. The efficiency with which selection optimizes the mean of a polygenic trait, despite underlying constraints, is seen in other models of this sort (Barton 1986, 1999).

Figure 3.

—The deviation of the mean from the optimum, Δ, as a function of Δf. The line Δ= Δf is also plotted. (A) The original scale and (B) a log-log scale are shown. This plot depends on the assumption that the polymorphic loci are those identified with Δ= 0 in Figure 1A.

Using the same procedure described for Figure 3, Figure 5 shows how three components of load, LΔ, LV, and LBS, and the total load, L (all scaled by the intensity of stabilizing selection S), increase as the mean is perturbed from the optimum. We ignore the contribution LBB to the total load, described in (56d), because this does not vary with changes in the polymorphic allele frequencies. As expected from the close adjustment of the trait mean to the new optimum (Figure 3), the load due to deviations of the mean, LΔ, is negligible until most variation is lost. The load due to variation around the optimum, LV, decreases as variation is lost, but this is compensated almost exactly by the load due to perturbing the allele frequencies away from their equilibria under balancing selection, LBS. The overall load, L, barely changes as the mean is perturbed slightly.

Figure 4.

—The genetic variance, VA, as a function of the deviation of Δf (defined by Equation 23b). This is based on the same assumptions as Figure 3. VA = 0.072 when Δf = 0 and declines to zero when the deviation is so large that all loci have fixed.

Long-term responses of the mean and the additive variance to changes in the intensity of selection or the position of the optimum are complex and depend on a large number of parameters describing the underlying genetics and the history of population size changes. However, the qualitative behavior is roughly as follows. If the strength of stabilizing selection increases, genetic variation will be rapidly lost, as some polymorphic loci become unstable (as illustrated in Figure 4); the timescale for this loss is set by Sαi2 . If stabilizing selection weakens again, then polymorphism will be recovered much more slowly, since the particular alleles under balancing selection must arise by mutation. Our assumption here is that the alleles would be lost for many plausible population sizes rather than retained at mutation-selection equilibrium. This follows because the mutation rates relevant to these particular alleles under balancing selection would be on the order of the pernucleotide mutation rate, say 10-8 or smaller. The chance that one such mutation at locus i will be fixed is twice the selection coefficient, siSαi2 ; hence, the timescale for recovery in a population of effective size N is set by 2Nμi(siSαi2) . Because μi is likely to be of the order of the per-nucleotide mutation rate, recovery of polymorphism could be very slow, unless the population is large enough to retain the previously polymorphic alleles through mutation-selection balance. For example, with an average si = 0.01 as assumed in the example of Figure 1, N = 106 and μ= 10-8, the rate of recovery of polymorphism at each locus individually is 2 × 10-4. To make this argument in a little more detail, suppose that a population with loci as illustrated in Figure 1 is initially under stabilizing selection with S = 1. Balancing selection then maintains genetic variance VA = 0.072, due to the pleiotropic effects of 36 polymorphic loci. If this variation is lost as a result of strong stabilizing selection, which then relaxes to its original intensity, the rate of increase in the number of polymorphic loci is 2Niμi(siSαi2)=0.008 ; thus, it will take ∼4500 generations to return to the original 36 polymorphic loci.

Figure 5.

—Genetic load, L, and its components, scaled by S, as functions of Δf. The total load is the solid line at the top, and the load due to deviations of the mean from the optimum is the thick solid curve that stays near 0. The portions of the load attributable to variation around the optimum (short dashes) and perturbations of the allele frequencies from their overdominant equilibria (long dashes) change relative magnitude as Δf increases.

Similarly, if the optimum changes, different responses occur on different timescales. After the first phase of response to a new optimum, in which allele frequencies adjust to approach the new optimum (Figure 3) and some polymorphic loci fix, there will be a loss of genetic variation (Figure 4). We now expect a second phase, in which the very many loci that may affect the trait but are not under balancing selection accumulate mutations, so as to bring the mean back toward the optimum. After this phase, we expect Δf =Δ= 0 to a very close approximation, and so the stability regimes return essentially to those depicted in Figure 1A. A third, and much slower, phase now occurs, in which variation at loci under balancing selection and lying above the parabola, si=Sαi2 , is recovered.

Sex-dependent allelic effects: To test the accuracy of our weak-selection approximations, calculations were performed using the full multilocus, diploid gamete-frequency recursions with selection, random mating, and recombination. Our goal was to test the prediction that sex-dependent allelic effects cannot maintain stable multilocus polymorphisms, at least with weak selection. Given that stable two-locus polymorphisms can be maintained under strong selection for sex-independent effects (Gimelfarb 1996; Bürger and Gimelfarb 1999), we performed comparable sets of simulations with sex-dependent and sex-independent allelic effects. We started a set of simulations by choosing independent random values for the allelic effects assigned to males and females. The effects were chosen using gamma-distributed, pseudo-random numbers, as described below. For each set of allelic effects and selection parameters, we generally used 10 randomly chosen initial conditions. We started at linkage equilibrium with each allele frequency chosen from a uniform distribution over the allowable allele frequencies (explained below). Iterations were stopped when the sum of absolute changes of gamete frequencies fell below a specified threshold. For all results reported here, with equilibrium allele frequencies restricted to (0.01, 0.99), a threshold of 10-9 or 10-10 proved adequate (the smaller value was used with weaker selection). For all cases in which two or more loci remained polymorphic, an output file was generated giving the parameter values and initial gamete frequencies. This allowed us to test the adequacy of our stopping criterion by lowering the threshold value and determining whether the same approximate equilibria were obtained. It also allowed us to reuse specific allelic effects and initial frequencies with different selection intensities, as discussed below.

Each set of simulations involved specifying the following: the number of loci; the number of replicate sets of allelic effects; the number of initial conditions for each set of allelic effects; the mean and CV of the allelic-effect parameters, βi,f, βi,m, γi,f, and γi,m (for simplicity we assumed the same mean and CV for each); the intensity of stabilizing selection, S; the optimal trait value, θ; the recombination rates between adjacent loci (assuming no interference); a threshold for minimum acceptable polymorphic allele frequencies (to avoid artifacts associated with slow convergence to fixation, we set this threshold at 0.01); and a threshold for the sum of the absolute changes in gamete frequencies. For the sex-dependent simulations, four independent pseudo-random, gamma-distributed deviates were chosen for each locus, denoted gi for i = 1,..., 4; and we set βi,f = g1, γi,f =-g2, βi,m = g3, and γi,m =-g4. For sex-independent effects, we set βi,fi,m = g1 and γi,fi,m =-g2. To reduce the dimensionality of the parameter space and facilitate investigating many sets of allelic effects, we used five unlinked loci, chose 10 sets of random initial allele frequencies for each set of allelic effects, assumed that each gi has mean 1, and set θ= 0 for all calculations. Our assignment of sex-specific allelic effects implies that the effects of substitutions have the same sign in both sexes. However, as explained above, with θ= 0, identical results are obtained by specifying βi,f = g1, γi,f =-g2, βi,m =-g3, and γi,m = g4, so that the effects of substitutions in the two sexes have different signs.

Several thousand sets of parameters were explored, all with unlinked loci. These led to four simple generalizations: (i) there were no stable equilibria involving three or more polymorphic loci; (ii) sex-dependent allelic effects facilitate stable two-locus polymorphisms; (iii) for both sex-dependent and sex-independent effects, choosing the effects from a distribution with larger CV facilitates stable two-locus polymorphisms; and (iv) for sex-dependent effects, unlike sex-independent allelic effects, stable two-locus polymorphisms can be found even with extremely weak selection. We briefly describe some results supporting these generalizations.

Over thousands of sets of allelic effects, each run with 10 sets of initial allele frequencies, no stable equilibria were found with more than two polymorphic loci. This is consistent with the numerical results of Bürger and Gimelfarb (1999) for sex-independent effects (see their Table 1). Although stable three-locus polymorphisms can be obtained even with sex-independent effects when recombination rates are low relative to selection (Gimelfarb 1996), such polymorphisms seem unlikely with loose linkage and weak selection.

View this table:
TABLE 3

Effects of sex dependence, CV of allelic effects, and intensity of stabilizing selection on the occurrence of stable two-locus polymorphisms with five-locus selection

Table 3 presents numerical results that illustrate the effects of sex dependence, interlocus variation in allelic effects, and the intensity of selection. The fact that larger CV for allelic effects produces more two-locus polymorphisms is expected from the analytical work of Nagylaki (1989) and Bürger and Gimelfarb (1999), showing that strong selection and significant asymmetries of effects across loci are required to maintain stable two-locus polymorphisms with stabilizing selection on an additive trait. The effect of asymmetries and the effect of sex dependence are illustrated with S = 0.2 in Table 3 by comparing results from CV = 0.5 vs. CV = 1 for sex-dependent vs. sex-independent allelic effects. In each case, larger CV produces significantly more sets of allelic effects leading to two-locus polymorphisms (under Fisher’s exact test, P < 10-9 for sex dependence and P < 10-4 for sex independence). Similarly, for each CV, sex dependence facilitates two-locus polymorphisms (P < 0.01 for CV = 0.05, P < 10-9 for CV = 1).

The qualitative difference between sex-dependent and sex-independent allelic effects with respect to maintaining stable two-locus polymorphisms under weak selection can be seen by concentrating on initial conditions and sets of allelic effects that produce stable two-locus polymorphisms with S = 0.2. Using 30 such sets of allelic effects and initial conditions, we set S = 0.02 and iterated to a new equilibrium. The results are shown in the third row of Table 3. For sex-independent selection, of the 30 sets that led to stable two-locus polymorphism with S = 0.2, only 1 produced a stable two-locus polymorphism with S = 0.02. (As expected, that example had one polymorphic locus with very large effects, β1 = 5.19973 and γ1 =-8.78685, and one with much smaller effects, β2 = 0.319015 and γ2=-0.144994.) When these same allelic effects and initial conditions were used with S = 0.002, only the locus of large effect remained polymorphic. In contrast, for sex-dependent effects, 17 of the 30 sets of allelic effects and initial conditions also produced a stable two-locus polymorphism with S = 0.02 (even though only a single initial frequency was used). Moreover, for all 17, a very similar two-locus polymorphism was also reached with S = 0.002 and S = 0.0002. As noted above, these stable two-locus polymorphisms obtained with sex-dependent allelic effects are analogous to those found by Hastings and Hom (1989, 1990) when alleles pleiotropically affect two characters under stabilizing selection. Although sex-dependent allelic effects do produce stable two-locus polymorphisms, our results suggest that they cannot maintain stable polygenic variation, even with strong selection.

Temporal variation and G × E: As with sex-dependent allelic effects, we tested our polymorphism conditions, based on weak-selection, deterministic approximations, by performing exact multilocus iterations with temporally varying allelic effects. The joint distribution of the fluctuating allelic effects depends on many parameters. As noted in our analytical approximations, intralocus correlations between allelic effects do not affect the polymorphism conditions, but interlocus correlations between substitution effects dramatically affect the levels of variation required to maintain polymorphism [see (33)]. To test our predictions concerning variances and interlocus correlations of substitution effects, we used symmetry assumptions to simplify the model description and our predictions. For all of our simulations, we assumed that θ= 0, Ei) =α/2, Ei) =-α/2, Var (βi) = Var(γi) = vα2, Cov(βi, γi) = 0, Cov(γi, βj) = Cov(βi, γj) = 0, Cov(βi, βj) = Cov(γi, γj), and no auto-correlation in the effects across generations. The allelic effects were chosen as multivariate pseudo-random, Gaussian deviates with the appropriate mean and covariance structure, which depends on only three parameters: α, the mean effect of a substitution at each locus; v, the squared CV of substitution effects [see (25)]; and ρ, the interlocus correlation in substitution effects [see (28a)]. Our calculations assumed six unlinked loci. With these symmetry assumptions, our approximate polymorphism criterion reduces to (33), namely ρ< 1/2 and v > 1/(1 - 2ρ). This prediction is independent of and S.

Figure 6.

—Stable equilibria under fluctuating allelic effects. (A and B) The mean allele frequency (across generations) and the standard deviation of allele frequencies for the “most polymorphic” and “least polymorphic” (explained in the text) of six diallelic loci. The average frequency of the least common allele is given for each locus. The dashed vertical lines indicate that the predicted critical values of the parameter varied. In A, stable polymorphism is expected only for values of v above the critical value; monomorphism is expected for smaller values of v. In B, stable polymorphism is expected only for values of ρ below the critical value.

Even with these symmetry assumptions, no attempt will be made to present simulations spanning the entire parameter space. Instead, we provide illustrative examples, using biologically plausible parameter values for selection intensity and average allelic effects, which focus on our predicted critical values for v and ρ. Figure 6 shows the effects of varying either v or ρ, holding all other parameters fixed. These simulations assume α= 0.7 and S = 0.05 (corresponding to the canonical values used in Turelli 1984 and many other articles to explore polygenic mutation-selection balance). To summarize the asymptotic behavior of the stochastically fluctuating allele frequencies, we first iterated the recursions for 500,000 generations starting with random initial allele frequencies and global linkage equilibrium. We then ran the recursions for an additional 500,000 generations, during which we calculated the mean and standard deviation of allele frequencies at each of the six loci. We report in Figure 6 the means and SDs of the two loci whose average allele frequencies depart least and most from 0.5. The former is called the “most polymorphic locus,” and the latter, the “least polymorphic.” To standardize the results, we report the average frequency of the less common allele.

Figure 6A shows the consequences of varying v with = 0.17. According to (33), all loci should remain stably polymorphic if v > 1.515 and all loci should become monomorphic if v < 1.515. The critical value for v is indicated by the dashed line in Figure 6A. As predicted, for v ≤ 1.4, all loci become monomorphic, with mean = SD = 0. Conversely, for v ≥ 1.6, all six loci remain polymorphic. For v = 1.5, very near the predicted threshold value, we see that at least one locus has become monomorphic, but at least one remains polymorphic. As expected given the high level of stochastic fluctuations, the polymorphic loci always show considerable fluctuations in allele frequencies. To put the observed SDs in perspective, note that if the allele frequencies fluctuated between 1 and 0 in an extremely rapid manner so that the allele frequency is essentially 1 with probability p and 0 with probability 1 - p, we would observe an average allele frequency of p and SD near the maximum value, p(1p) . When v = 1.5, the SD of the most polymorphic locus is ∼86% of this maximum value. In contrast, for v = 1.6, the SD for the most polymorphic locus is 78% of the maximum, and this ratio declines somewhat as v increases, despite the increasing intensity of the stochastic fluctuations.

Figure 6B considers varying ρ with v = 1.5. Prediction (33) implies that all loci should remain stably polymorphic if ρ< 0.167 (see the dashed line in Figure 6B), and all loci should become monomorphic if ρ> 0.167. As predicted, all loci become monomorphic when ρ≥ 0.18, and all loci remain polymorphic when ρ≤ 0.16. With ρ= 0.17, very near the predicted threshold, we see that at least one locus has become monomorphic, but at least one remains polymorphic.

As expected from the way the allele frequency dynamics in (29) depend on α2 and S, varying each of these parameters has a similar effect on polymorphisms. For instance, if we set v = 1.6, ρ= 0.17, and α= 0.07, the mean allele frequency at the most polymorphic locus remains very close to 0.5 for S = 0.01, 0.05, and 0.25, but the SD increases from 0.32 with S = 0.01 to 0.45 with S = 0.25. This reflects the fact that, with stronger selection, allele frequencies respond faster to changing selection forces produced by varying allelic effects. Similarly, if we set v = 1.6, ρ= 0.17, S = 0.05, and vary α2 from 0.09 to 2.56 (roughly a factor of 25, as with S above), again the mean stays very near 0.5 while the SD increases from 0.30 to 0.45.

Overall, our simulations suggest that our approximations provide useful guidelines concerning the maintenance of polygenic variation through fluctuating allelic effects. As shown in Figure 6, the simulations switch from stable multilocus polymorphisms to complete fixation near the predicted threshold values for v and ρ. As the parameters near the threshold values, interlocus differences become greater and allele-frequency fluctuations become more extreme.

DISCUSSION

Two basic classes of models explain the maintenance of stable polygenic variation: those that rely on mutation to maintain variation that would otherwise be largely eliminated by selection and those in which selection itself maintains variation. For a broad range of biologically reasonable parameter values, mutation-selection balance models imply that the variation maintained will generally be attributable to rare alleles at many loci (Turelli 1984). This remains true even for the most recent models of mutation-selection balance that consider both direct and pleiotropic selection (e.g., Zhang and Hill 2002). In contrast to this theoretical expectation, molecular studies suggest that variants at intermediate frequency contribute significantly to polygenic variation in natural populations (e.g., Mackay and Langley 1990; Longet al. 2000). This provides one of the primary empirical motivations for our study of alternative models for balancing selection, because such models generally lead to intermediate allele frequencies at the polymorphic loci. We discuss in turn the results from each of our models and then make some general comments, comparing alternative mechanisms for the maintenance of variation.

Pleiotropic balancing selection: Our analysis produces straightforward conditions under which balancing selection can maintain variation in a quantitative trait, despite stabilizing selection. At each locus, polymorphism can be maintained provided that two conditions are met. First, balancing selection must be stronger than stabilizing selection [vi=si(αi2S)>1] ; at most one polymorphic locus can violate this condition. Second, the net balancing selection must be stronger than the directional selection that arises when the trait mean deviates from its optimum. At equilibrium, the trait mean closely matches the optimum. The genetic variance is maintained by a set of loci that are highly polymorphic and is approximately equal to half the sum of squared allelic effects at these loci. At each polymorphic locus, the allele frequency is a compromise between the equilibrium favored by balancing selection and a slight shift that brings the overall trait mean close to the optimum. Our numerical results address the properties of alternative equilibria and the consequences of bouts of directional selection for variation, but we do not reiterate our findings here.

How likely is it that genetic variation is maintained as a pleiotropic effect of balancing selection? We have little idea how many balanced polymorphisms there might be, but it is plausible that variation is maintained by selection at a substantial fraction of genes (at some thousands of loci in multicellular eukaryotes, say). There is then no difficulty in accounting for high levels of genetic variance in any particular trait. However, this explanation faces two difficulties in explaining variation in most quantitative traits. First, balancing selection must be strong enough in total to counterbalance the stabilizing selection acting on all traits. Roughly speaking, we expect that the total strength of balancing selection, Σisi, should be greater than the net load due to variation of quantitative traits around their optima. Unfortunately, we do not know the magnitude of either of these quantities. If balancing selection is due to overdominance, then Σisi is proportional to the segregation load, which could in principle be measured as a component of inbreeding depression. However, if frequency-dependent selection predominates, it is hard to relate Σisi to observable quantities. The net genetic load due to deviation of traits under stabilizing selection from their optima is still harder to estimate; indeed, it is hard even to define the number of traits under stabilizing selection (though see Orr 2000). Despite these uncertainties, however, it is at least possible that there is sufficient balancing selection to counterbalance stabilizing selection on very many traits. For example, selection coefficients of 5% on 2000 loci would give Σisi = 100. This could counterbalance genetic loads of a few percent due to stabilizing selection on some thousands of independent traits.

More naively, we can ask how much balancing selection is required to maintain variation at a particular locus that affects a single trait under stabilizing selection. If we assume a heritability near 0.5 and scale genetic and environmental variance to 1, we can ask how much balancing selection is needed to maintain a polymorphism contributing ∼10% of the total genetic variance. From (1), we have αi2 near 0.2. Hence, if stabilizing selection is on the order of S = 0.05, we require balancing selection on the order of si = 0.01. If stabilizing selection is often much weaker than assumed (Kingsolveret al. 2001), the required intensity of balancing selection decreases.

A second constraint is that episodes of directional selection on quantitative traits must not eliminate variation at polymorphic loci. In our analysis, we assumed stabilizing selection toward a constant optimum. In reality, optima may vary, and so allele frequencies at the underlying loci will fluctuate. If directional selection is sufficiently strong for long enough, alleles will fix, and variation will be regenerated only when lost alleles are recovered. We consider such a scenario in detail above, at the end of our numerical analyses of pleiotropic balancing selection. Clearly, if balancing selection is sufficiently strong, and if traits depend on very many loci, then the mean can be adjusted by small changes at each locus, avoiding fixations. Moreover, if selection varies from place to place in a spatially subdivided population, alleles can be retrieved by migration rather than by mutation. Finally, it may be that balanced polymorphisms are usually transient [as seems to be the case for inversions in Drosophila (Andolfattoet al. 2001) and for human adaptations to malaria (Hamblinet al. 2002)] rather than maintained for very long times (as, for example, with incompatibility loci in flowering plants, or the human histocompatibility system in vertebrates; Hughes 1999). This idea is related to the transient maintenance of variation through fluctuating selection on traits themselves, as discussed below.

Relation with mutation-selection balance: Combining mutation with stabilizing selection leads to substantial mathematical complications. The polymorphic equilibria are now given by the solution to a system of cubic equations, and there may be multiple stable polymorphic equilibria for a fixed set of polymorphic loci. In contrast, the model analyzed here gives a unique polymorphic equilibrium for any fixed set of polymorphic loci. However, the two cases are similar; indeed, they must be because mutation represents a small perturbation of the model analyzed here and thus will give qualitatively similar results (e.g., Karlin and McGregor 1972). Both stabilizing selection and mutation-selection balance generally have the property that the mean can be adjusted to small changes in the optimum by slight changes in allele frequencies at many loci. In both models, many combinations of fixed or nearly fixed loci can give stable equilibria, all of which produce a population mean very close to the optimum.

Our numerical results suggest that with unequal allelic effects the properties of different equilibria become more similar to each other than do those in the case where loci are equivalent. The same may hold for mutation-selection balance. When allelic effects are equal, the variance can increase dramatically when the mean deviates above the optimum. This is because all the loci near p = 0 climb in frequency together, until a critical value is approached when some set of loci switch to the alternative equilibrium with p near 1. Near this critical value, the equilibrium genetic variance can greatly exceed that expected with the mean at the optimum (Barton 1986). However, with varying allelic effects and equilibrium allele frequencies, a single locus approaches the critical value and then fixes, greatly reducing the magnitude of the deviation from the optimum without giving much increase in genetic variance.

Fluctuating allelic effects: Under our models of temporal or spatial fluctuations, a necessary condition for the maintenance of polymorphism at a locus is that vi, the squared coefficient of variation of the effects of a substitution (across the distribution of environments), exceeds one. To address the biological plausibility of this condition, we first describe its mathematical implications by specifying distributions for the substitution effects. The range of implications can be illustrated by considering two particular distributions: Gaussian and gamma. The stability condition vi > 1 implies that the standard deviation of substitution effects exceed the mean. Under a Gaussian distribution, this implies that the sign of substitution effects at this locus must frequently change with environmental conditions. If we assume that the mean effect of a substitution is positive, the probability of a negative effect will be Φ(1vi) , where Φ denotes the cumulative distribution function of the standard normal distribution. This probability must be at least 0.16 for vi > 1, it is 0.24 for vi = 2, and it approaches 0.5 as vi increases. Such sign reversals are not necessary, however, because vi > 1 can also be achieved with a gamma distribution, which remains positive. In this case, vi > 1 puts a lower bound on the magnitude of fluctuations of the substitution effects. One way to quantify this is as the ratio of the 75th percentile of the distribution of substitution effects to the 25th percentile (which depends only on vi). This ratio must be at least 4.82 for vi > 1, it is 13.0 for vi = 2, and it approaches infinity rapidly as vi increases (e.g., it is 100 for vi = 4.01).

Are such dramatic fluctuations in substitution effects plausible? The most relevant data concerning the fluctuating effects of individual loci are the quantitative trait loci (QTL)-based G × E studies by Mackay and her collaborators (e.g., Gurganuset al. 1998; Vieiraet al. 2000; Dilda and Mackay 2002). These studies estimate the effects of individual QTL, which presumably correspond to one or a small number of closely linked loci, over a range of environmental conditions, such as alternative rearing temperatures, heat shock, and starvation; they also document sex dependence. It is important to recognize, however, that the genotypes used in these analyses are generally recombinant inbred lines derived from selection experiments or long-held laboratory stocks. Hence, the variation described may not be representative of variation in natural populations. Nevertheless, these studies demonstrate that QTL effects are generally sex or environment dependent. Scanning the data tables in these articles (see, for instance, Table 4 of Vieiraet al. 2000), examples can be found where statistically significant marker effects within a sex vary by more than a factor of 10 or change sign depending on the rearing environment. Hence, these data seem broadly compatible with the levels of variation required to maintain polymorphism in our analysis. However, Mackay and collaborators clearly recognize another important caveat (e.g., Dilda and Mackay 2002, p. 1671). We do not know whether the environments chosen for these laboratory experiments are representative of the environmental variation in nature that may be responsible for maintaining genetic variation. It seems reasonable, nevertheless, to assume that the range of conditions in nature would fluctuate in many more ways than considered in these experiments, with temperature, crowding, and food quality, for instance, all varying simultaneously in time and space.

Analyzing fluctuating allelic effects for individual loci is extremely difficult. A more traditional quantitative-genetic approach is to consider the “consistency index,” K (see Equation 44), the ratio of the variance of mean effects of genotypes to the total genetic variance, which includes mean effects plus interaction terms related to G × E. As noted above, the polymorphism conditions constrain K to be quite small, especially when the fluctuations in effects across loci are positively correlated (see Equations 47 and 49). Indeed, 0.25 is an upper bound for K if all fluctuating allelic effects, both within and across loci, are uncorrelated, but this bound quickly falls to values on the order of 0.1 or less under more plausible assumptions. Relevant data appear in several experimental studies of G × E that partition the total genetic variance observed across genotypes and environments into main effects of genotypes and interaction effects. For instance, Wayne and Mackay (1998) used three temperatures to study ovariole number and body size in mutation-accumulation lines of Drosophila melanogaster. Treating temperature and block as the environmental variables (see their Tables 1 and 2), we see that for ovarioles ∼57% of the newly arising variation for genotypes (lines) plus interactions between genotypes and environments is attributable to mean effects of genotypes. The comparable amount for body size is 42%. Given that these estimates come from a sample of newly arising variation, not all of which would be expected to remain stably polymorphic because of G × E, there is no reason to expect them to satisfy the constraints on K described above. Nevertheless, they demonstrate that mutation provides environment-sensitive variants that could plausibly satisfy the G × E polymorphism conditions. Similar data are available for life span (Vieiraet al. 2000) and bristle number (Gurganuset al. 1998), but in these studies, the genetic variation originates from recombinant inbred lines derived from long-held laboratory stocks. Again, these studies indicate the ubiquity of environment- (and sex-)dependent genetic effects. However, they cannot tell us whether the variation segregating in natural populations satisfies the constraints expected for stable G × E -maintained polymorphisms. Many other recent studies demonstrate G × E at the level of either QTL (e.g., Shook and Johnson 1999) or whole genotypes (Shawet al. 1995), but none provide data that allow us to estimate the relevant parameters.

One possible argument against the role of G × E in maintaining variation is that significant levels of additive variance are routinely found in laboratory populations, including long-established stocks, experiencing relatively homogeneous and constant environments (Weigensberg and Roff 1996). For populations recently established in the laboratory, two opposing forces affect VA: sampling, which tends to diminish variation, and relaxed selection, which tends to maintain variation that might be eroded under the more stringent selection expected in nature. To the extent that laboratory conditions minimize selection, levels of additive variance may approach a mutation-drift equilibrium that might be considerably higher than that maintained by either mutation-selection or balancing selection alone (Turelliet al. 1988). Because of the radical differences expected in the selection regimes, data on variation from laboratory populations cannot preclude a central role for G × E in maintaining variation in nature.

Sex-dependent allelic effects: We have shown for diallelic loci that sex-dependent additive allelic effects are no more effective at maintaining stable polygenic variation than is the classic sex-independent additive model investigated by Wright (1935). (Although sex-dependent effects can maintain variation at two loci whereas sex-independent effects maintain variation at only one, this distinction is negligible in the context of understanding polygenic variation.) In contrast, sexually antagonistic fitness effects can easily maintain single-locus polymorphisms (e.g., Kidwellet al. 1977). This distinction between the propensity of sex-dependent effects to facilitate one-locus polymorphism but inability to maintain polygenic variation is analogous to findings concerning antagonistic pleiotropic effects on life histories (compare Rose 1982 and Curtsingeret al. 1994).

Sex-dependent allelic effects have been extensively documented for several traits in D. melanogaster (summarized in Dilda and Mackay 2002). Our analytical and numerical results indicate that such effects per se cannot account for the maintenance of polygenic variation for traits under stabilizing selection. We do not yet know how such sex-dependent effects will interact with sex-dependent fitness regimes.

Comparisons to alternative mechanisms: We have considered models in which selection alone maintains quantitative variation. Many other models of this type have been proposed. For example, two-locus polymorphisms can be maintained by stabilizing selection on an additive trait if allelic effects are sufficiently different, even when selection is weak (Nagylaki 1989). When selection is strong enough that linkage disequilibrium is significant, polymorphism may be further facilitated (Gavrilets and Hastings 1994). However, Bürger and Gimelfarb (1999) used numerical investigations to show that when more than a few loci influence a trait, polymorphism at multiple loci becomes much less likely; moreover, the loci of smallest effect tend to be polymorphic, so that very little genetic variance is maintained. This pattern arises because with multiple loci the optimum can be closely matched by a homozygous genotype; selection then acts against deviations from this optimal genotype. With multiple traits, it is harder to match the trait mean to the optimum, and so relatively more polymorphism is expected: in Hastings and Hom’s (1989) model, there can be as many polymorphic loci as there are traits under stabilizing selection. A serious criticism of these models is that they apply to only outcrossing diploids and so cannot account for quantitative variation in haploid or selfing organisms that do not contain heterozygotes. Surprisingly little is known about polymorphism conditions in multilocus haploid models with recombination (e.g., Kirzhner and Lyubich 1997). On the basis of Rutschman’s (1994) two-locus analysis, it is reasonable to conjecture that epistatic selection cannot maintain variation in the absence of dominance interactions. In contrast, provided that balancing selection acts through negative frequency dependence rather than through overdominance, the pleiotropic balancing selection model we analyze is quite generally applicable. Fitness might depend on genotype frequencies through a number of selective mechanisms mediated by quantitative traits (see the theoretical analyses of Bulmer 1974, Slatkin 1979, and Bürger 2002 and the data analyzed by Bolnicket al. 2003). However, it seems simplest to treat variation in some arbitrary trait as being due to the pleiotropic effects of balanced polymorphisms, without detailing the causes of that balancing selection.

In the models just discussed, selection on the trait remains constant through time and we focus on stable equilibria. In contrast, Bürger (1999), Waxman and Peck (1999), and Bürger and Gimelfarb (2002) have recently shown that a changing optimum can generate substantially more genetic variance than would be generated in a balance between mutation and static stabilizing selection. Bürger (1999) and Waxman and Peck (1999) assumed a continuum of allelic effects at each locus, and it is unclear how far variation can be inflated with discrete alleles under their selection regimes. Kondrashov and Yampolsky (1996) demonstrate an increased genetic variance under fluctuating selection in a model with discrete alleles. However, at any one time, most of the variance in their model is contributed by a single locus as it sweeps between near fixation for alternative alleles. This does not seem to be an adequate explanation of polygenic variation. In the context of our pleiotropy model, adding balancing selection to these models might have rather little effect, because fluctuating selection would tend to eliminate the particular alleles that are required for balanced polymorphism. Bürger and Gimelfarb (2002) consider fluctuating optima under stabilizing selection for moderate numbers of diallelic loci. They demonstrate that considerably more variation can be maintained than expected under mutation-selection balance alone, but their results seem to depend on fairly extreme fluctuations in the position of the optimum relative to the width of the stabilizing selection function (cf. Turelli 1988). If, as Kingsolver et al. (2001) have recently argued, stabilizing selection is typically much weaker than usually assumed, the fluctuating-optimum hypothesis will be less credible.

Epistatic interactions have been widely documented (e.g., Shook and Johnson 1999; Dilda and Mackay 2002) and some numerical work has suggested that epistasis may help maintain polygenic variation (e.g., Gimelfarb 1989). However, recent analytical work by Hermisson et al. (2003) suggests that epistasis is likely to lower rather than raise the additive variance maintained by mutation-selection balance.

Future directions: We have shown that pleiotropic balancing selection and temporally or spatially varying allelic effects can maintain stable polygenic variation, but sex-dependent allelic effects cannot. Recent studies have championed varying selection pressures as a way to explain the maintenance of alleles at intermediate frequencies (e.g., Waxman and Peck 1999; Bürger and Gimelfarb 2002) and pleiotropic effects of deleterious alleles on weakly selected traits (Zhang and Hill 2002) as a way to explain abundant additive variance attributable to rare alleles at many loci. These models and ours present a daunting challenge to experimentalists to estimate the relevant parameters. As with molecular variation at individual loci, the explanation of polygenic variation may well depend on the simultaneous action of many alternative mechanisms, both across characters and across loci. Mutation-selection balance surely explains some of the variation we observe, but it is generally expected to explain only the persistence of rare alleles (but see Slatkin and Frank 1990). Intermediate allele frequencies might be explained by either some form of balancing selection, as we have discussed, or transient polymorphisms associated with fluctuating selection (or hitchhiking effects from linked sites under directional selection).

Studies of molecular variation at and near individual loci that contribute to polygenic variation may help unravel the relevant evolutionary forces. In particular, elevated levels of molecular variation in regions that contribute to quantitative variation are expected if balancing selection has maintained alleles for very long times (Hudsonet al. 1987). Such persistent polymorphism and excess variability have been seen at self-incompatibility loci in plants and around the MHC in mammals (Hughes 1999). However, it is likely that selection on balanced polymorphisms fluctuates over time, and such fluctuations will reduce neutral diversity over wide regions of the genome (with recombination rates comparable to the selection coefficient), whereas balancing selection is expected to increase diversity in only narrow regions (with recombination rates comparable to the mutation rate, μ) and then only when the same allele is maintained for times of order 1/μ. Thus, chromosomal inversions in Drosophila (Andolfattoet al. 2001) and polymorphisms that confer malaria resistance in humans (e.g., Sabetiet al. 2002) are associated with reduced—not increased—variability, because they are of relatively recent origin.

These arguments are based on the expected effects of balancing selection at single loci. Balancing selection at many loci can in principle maintain enormous diversity at linked sites, because it maintains many genotypes (defined by all possible combinations of selected alleles). However, in large but finite populations, the effect of increasing numbers of balanced polymorphisms reaches a limit, beyond which adding more polymorphisms does not further increase diversity (Navarro and Barton 2002). Thus, it may be difficult to detect even widespread balancing selection through its effects on neutral diversity. The most promising approach may be to ask whether variants that are actually under selection are often at intermediate frequency. This requires, however, that the precise targets of selection be identified and that other processes that can raise allele frequencies (for example, hitchhiking) can be ruled out. Progress in the near future will be limited to model systems, such as D. melanogaster, in which plausible candidate loci and tools for fine-scale genetic analyses are available. Recent data (De Lucaet al. 2003) suggest that the footprints of balancing selection may indeed be observable for polygenic traits; but the generality of this observation remains to be determined.

APPENDIX A: STABILITY CONDITIONS FOR POLYMORPHIC EQUILIBRIA

Here we establish that the eigenvalues of the stability matrix (10) are real and that necessary and sufficient conditions for them all to be negative are (14) and (15), whereas (16) is necessary for stability. Our demonstration rests on the fact that the eigenvalues of a real symmetric matrix are real (Gantmacher 1959, Sect. IX.13) and on some elementary properties of determinants and quadratic forms. First note that A can be written as A=SBC, (A1) with B a diagonal matrix whose ith diagonal element is 2piqi, the equilibrium heterozygosity at locus i [denoted B = diag(2piqi)], and C a symmetric, positive matrix whose elements are cij=αiαj[1+δij(vi12)]fori,j=1,,n, (A2) where δij denotes the Kronecker delta, with δii = 1 and δij = 0 for ij. Equation A1 implies that the eigenvalues of A are the eigenvalues of BC multiplied by -S. The remaining argument has four steps. First we show that the eigenvalues of BC are real, and then we show that they are all positive—implying that G × E maintains a stable polymorphic equilibrium—if and only if all of the eigenvalues of C are positive. Next, we reduce the problem further by showing that we can set all of the αi = 1 without loss of generality. The constraints on the vi that lead to stability then follow from properties of quadratic forms.

Let B12=diag(2piqi) and B12=diag(12piqi) . Note that BC = B-1/2(B1/2CB1/2)B1/2. Hence, BC and B1/2 CB1/2 are “similar” matrices and have identical eigenvalues (Gantmacher 1959, Sec. III.6). Because B1/2CB1/2 is real and symmetric, its eigenvalues are real.

Next we show that the eigenvalues of B1/2CB1/2 (and BC) are all positive if and only if the eigenvalues of C are positive. Real symmetric matrices have all positive eigenvalues if and only if they are positive definite (Gantmacher 1959, Secs. X.4-5). Hence the eigenvalues of B1/2CB1/2 are all positive if and only if for all nonzero vectors x xTB12CB12x>0. (A3)

This is equivalent to yTCy>0 (A4) for all nonzero y. Hence, stability of the polymorphic equilibrium is equivalent to determining when C is positive definite.

The problem can be simplified further by noting that C=DVD, (A5) where D = diag(α1, α2,..., αn) and V is the positive, symmetric matrix with elements vij=1+δij(vi12), (A6) with δij as in (A2). By the logic used in (A4), we see that C is positive definite if and only if V is. This demonstrates that the stability conditions depend on only the vi and are independent of the mean effects, αi, as well as the equilibrium allele frequencies, pi. When is V positive definite? It is easy to derive a sufficient condition by considering the quadratic form associated with V. Note that for any vector x, xTVx=(i=1nxi)2+i=1nxi2(vi12). (A7)

This is obviously positive for all nonzero x if vi>1for alli. (A8)

Hence (A8) suffices for stability. The necessary condition (18) can be obtained by considering V as a variance-covariance matrix. To be positive definite, all correlations must be less than one. Thus, a necessary condition for stability is (vi+1)(vj+1)>4 (A9) for all pairs with ij.

Necessary and sufficient conditions follow from the fact that a real symmetric matrix is positive definite if and only if all its “principal minors” (i.e., submatrices obtained by deleting rows and columns with identical indices) have positive determinants (Gantmacher 1959, Sec. X.4). To present the conditions for stability of the equilibria (9) concisely, assume that the vi are ordered from smallest, v1, to largest, vn. The necessary and sufficient conditions are i=1m(vi1)+2i=1mjim(vj1)>0, (A10) for all mn. Condition (A9), which corresponds to m = 2 in (A10), implies that vi > 1 for i ≥ 2. Consider (A10) with m = n. Then, if v1 < 1, we must have v1>11(12)+i=2n(1(vi1)). (A11)

Thus, if there are many loci, or if some loci have vi near 1, the lower bound on v1 will be not much below 1. Hence, for stable polygenic variation, the sufficient condition (A10), vi > 1 for all i, is effectively also necessary. This qualitative conclusion is supported by our analysis of boundary equilibria [see (21) and appendix b].

The special case in which only one locus is polymorphic deserves comment. In this case, the stability condition reduces to v1>1. (A12)

This trivially generalizes Wright’s (1935) result by showing that even underdominant selection can be balanced by stabilizing selection to retain one polymorphic locus if the heterozygote produces a near-optimal phenotype in a genetic background in which all other loci are fixed.

APPENDIX B: STABILITY CONDITIONS FOR BOUNDARY EQUILIBRIA

Let A = (aij) denote the stability matrix corresponding to an equilibrium with pi = 0 for i ∊ Ω0, pi = 1 for i ∊ Ω1, and 0 < pi < 1 for i ∊ Ωp. As shown by (20), the eigenvalues governing the stability of the loci fixed at 0 and 1 are simply λi = aii for i ∊ Ω0 222A; Ω1 (20b and 20c). Moreover, the eigenvalues governing the polymorphic loci are generated by a matrix whose elements are given by (10). The stability conditions for this subsystem are given by (14) and (15) if Ωp has at least two elements [or (A14) if there is just one polymorphic locus] and depend only on the vi. What remains is to find the conditions for stability of the fixed equilibria in an explicit form.

For definiteness, suppose that the mean lies above the optimum (Δ> 0) and that all αi > 0 (hence δi > 0); the argument is similar for the other cases. With Δ> 0, directional selection on the trait is tending to reduce the pi. First, suppose that the polymorphic loci all satisfy vi > 1. Then, from (23), Δf > 0. For the polymorphic equilibria to be feasible, (19a) implies that we must have αi(2p^ivi1)>2Δ=2Δf1+2C (B1a) and αi(2q^ivi1)>2Δ=2Δf1+2CforiΩp, (B1b) with Δf and C as in (23). Note that even with vi > 1, (B1a) implies that polymorphism may not be feasible for any Δf if i is too small (e.g., pˆi < 1/2vi). In contrast, if i = 1/2, then vi > 1 ensures that polymorphism is feasible if Δf is sufficiently small. Note that conditions (B1) are more easily satisfied with large C, corresponding to many polymorphic loci and/or relatively weak balancing selection (i.e., vi slightly >1). For the fixed loci to be stable, conditions (21) imply that αi(2p^ivi1)<2ΔforiΩ0, (B2a) and αi(2q^ivi1)<2ΔforiΩ1. (B2b)

Thus, for i = 1/2, loci with pi = 1 can be stable only if αi(vi - 1) <-2Δ, which requires vi < 1, but loci may fix for pi = 0 even if vi > 1, provided that αi(vi - 1) < 2Δ.

Now, consider an equilibrium with vI < 1 for one I ∊ Ωp. As before, we assume that Δ> 0. From stability condition (A9), we know that vi > 1 for i ∊ Ωp if iI. Then, from (A10), 1 + 2C < 0, with C as in (23c). Condition (A11) implies that vI>1(21+2j{ΩpI}(1(vj1)))>0. (B3)

Conditions (19b) require αI(2p^IvI1)<2ΔandαI(2q^IvI1)<2Δ. (B4) Overall, the constraints on the one polymorphic locus with vi < 1 are very restrictive.

APPENDIX C: FINDING ALTERNATIVE STABLE EQUILIBRIA

We restrict attention to the special case i = 0.5 and θ= 0 considered in our first numerical example concerning pleiotropic balancing selection. Because of the symmetry, we can restrict attention to equilibria with Δ> 0, as the complementary equilibria with Δ< 0 can be found by simply reversing the frequencies of Bi and bi at each locus. We seek an exhaustive list of multilocus equilibria that satisfy our stability and feasibility constraints. The key idea is to recognize that these alternative equilibria fall into classes that are determined by the relationship of 2Δ to the intervals defined by the sequence of values for ±αi(vi - 1). For any assignment of the αi and vi, we first order the sequence ±|αi(vi - 1)| and then find the possible equilibria that fall into each of these intervals, including the regions below mini {-|αi(vi - 1)|} and above maxi{|αi(vi - 1)|}.

The strategy is to start with a trial value of χ= 2Δ in one of these intervals, then to determine for this value the sets of loci that can be fixed for 0, fixed for 1, or polymorphic, and then from these to determine which values of Δ in the interval being considered can be realized. The crucial observation is that according to our stability and feasibility conditions, the qualitative equilibrium state of each locus depends on only the value of αi(vi - 1) relative to ±Δ. Because of this, we need consider only one trial value of χ in each interval.

First, consider equilibria where all polymorphic loci have vi > 1. A trial value of the threshold χ= 2Δ is chosen. If χ< 0, then the complementary case is considered, with χ> 0; loci fixed for 0 and 1 are then reversed. For fixed χ, all of the loci are sorted into three classes according to their value of αi(vi - 1). According to (B2), those loci with vi > 1 and αi(vi - 1) >χ must be polymorphic, and those with χ> αi(vi - 1) >-χ must fix at 0. Finally, those with -χ > αi(vi - 1) may fix at 0 or 1. Thus, the task reduces to considering all of the equilibria with the loci in this final class fixed for either 0 or 1 and determining which of these configurations produces values of Δ in the interval being considered. All that do produce them are feasible stable equilibria. By trying values of χ in each interval, all possible stable equilibria can be found.

Finally, for each χ, we consider possible stable equilibria at which one of the polymorphic loci has vi < 1. Let I denote such a locus. For it to be feasibly polymorphic, it must satisfy αI(vI - 1) <-χ (B2b), and it must also satisfy the stability criterion (B3). This condition can be written as 2Δ/θeff > 0. Jointly, these conditions are very restrictive. The algorithm is to find all loci that satisfy 2αI1+2j{ΩpI}(1(vj1))<αI(vI1)<χ, (C1) assign them individually as polymorphic, and then carry out the procedure described above with all of the remaining polymorphic loci satisfying vi > 1. Configurations of monomorphic loci that lead to values of 2Δ in the target interval are accepted as stable equilibria.

Acknowledgments

We thank R. Bürger, J. H. Gillespie, R. Haygood, T. F. C. Mackay, S. V. Nuzhdin, M. Slatkin, and an anonymous reviewer for helpful comments and discussion, and R. Haygood for implementing the multilocus iterations. We thank the Erwin Schrödinger Institute for Mathematical Physics at the University of Vienna for providing an excellent research environment in which to complete this work, and N.H.B. thanks the Center for Population Biology at University of California-Davis for its hospitality. This research was supported in part by National Science Foundation grant DEB-0089716 (M.T.) and plus grants GR3/11635 from the Natural Environment Research Council and MMI09726 from the Engineering and Physical Sciences Research Council (N.H.B.).

Footnotes

  • Communicating editor: W. Stephan

  • Received May 29, 2003.
  • Accepted October 17, 2003.

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