Polygenic Variation Maintained by Balancing Selection: Pleiotropy, Sex-Dependent Allelic Effects and G x E Interactions

Corresponding author: Michael Turelli, University of California, 1 Shields Ave., Davis, CA 95616., mturelli{at}ucdavis.edu (E-mail)

Communicating editor: W. STEPHAN

	ABSTRACT

TOP ABSTRACT MODELS AND APPROXIMATE ANALYSES NUMERICAL ANALYSES DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G x 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 x 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., ²_ij > 3/4), 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 x 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 ; LONG et 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 ; BURGER 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 x 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 , GILLESPIE and TURELLI 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 x 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., LAI et al. 1995 ; LONG et al. 1996 ; NUZHDIN et al. 1997 ; GURGANUS et al. 1998 ; WAYNE and MACKAY 1998 ; VIEIRA et 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 (BURGER 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 x 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

TOP ABSTRACT MODELS AND APPROXIMATE ANALYSES NUMERICAL ANALYSES DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

We analyze in turn pleiotropic balancing selection, G x E with spatial variation and complete mixing, sex-dependent allelic effects, and G x 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 x 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 B_i and b_i denote the alleles at locus i. We let p_i,t denote the frequency of B_i in generation t and set q_i,t = 1 - p_i,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 ß_i (_i) denote the additive contribution of B_i (b_i) to the trait of interest. We set _i = ß_i - _i, so that _i 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

(1)

We assume constant Gaussian stabilizing selection on this trait with optimum 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 - )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

(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

(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

(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 _i = ß_i - _i and . Because the scale of measurement of our trait is arbitrary, can absorb any constants that enter the determination of the mean phenotype (such as the _i and the contributions of monomorphic loci not considered in our analyses). Thus, we are free to choose any values for the ß_i and _i that satisfy _i = ß_i - _i. Without loss of generality, we assume that

(5)

so that

(6)

As shown initially by WRIGHT 1935 from an approximation like (4) (cf. BULMER 1971 ), stabilizing selection will generally eliminate additive polygenic variation (see BURGER 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 - s_ii, 1, and 1 - s_ii for B_iB_i, B_ib_i, and b_ib_i, respectively, with 0 < s_i << 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

(7a)

where

(7b)

Given that a deviation from the optimum of _i reduces fitness by S²_i/2, v_i 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

(8)

where denotes the contribution to the mean phenotype from all loci but i.

Stability of fully polymorphic equilibria: From (7a) we see that each locus can fall into one of three possible equilibria: p_i = 0, p_i = 1, or p_i, satisfying

(9)

(note that _i depends on all of the allele frequencies), with v_i and _i defined in (7b). [Because of the simple form (3) for our approximate dynamics, only point equilibria can occur; BURGER 2000 , Appendix A3.] As expected, the polymorphic equilibrium (9) becomes _i if v_i is very large and becomes 1/2 if pleiotropic balancing selection is eliminated and the population mean is at the optimum ( = _i = 0). Condition (9) creates a system of linear equations for the p_i 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 v_i. 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 = (a_ij), with

(10a)

and

(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 _i = , s_i = s, and _i = ; then v_i = v for all i and the Equation 9 for the equilibrium allele frequencies have a unique solution:

(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,

(12)

where ₁ has multiplicity n - 1. Obviously, ₂ is always negative, but ₁ is negative if and only if

(13)

Thus, as BULMER 1973 found by assuming that , s = ²S 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 v_i arises for the general model (7a).

The necessary and sufficient conditions derived in Appendix A for stability are that either

(14)

or one locus (locus 1, say) has v₁ < 1, but this locus obeys

(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

(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., v_i = 0 for all i, at most one locus is expected to be polymorphic in the weak-selection limit. At loci with v_i > 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 > v_i > -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 S acts toward an optimum , independently across a set of characters, labeled , i.e., w(G) = Exp[-(S/2)(G - )²] (corresponding to multiplying the Gaussian selection across characters). Following the arguments leading to (7a), we obtain

(17a)

where

(17b)

The polymorphic equilibria can still be represented by (9) with v_i replaced by _i and _i replaced by _i.

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

(18a)

and

(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 v_i < 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 B_i or b_i at the monomorphic loci. First, consider the conditions for polymorphic equilibria to be feasible. The conditions will depend on whether v_i > 1 (recall that at most one stably polymorphic locus can violate this). If v_i > 1, (9) implies that 0 < p_i < 1 only if

(19a)

If v_i < 1, feasibility requires

(19b)

Unless = 0, (19a) constrains at least all but one of v_i to exceed 1 by an amount that depends on . Conversely, if stability is achieved with one locus satisfying v_i < 1, (19b) puts an upper bound on this v_i 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 very small.

Consider an equilibrium at which p_i = 0 for all i in the set ₀, p_i = 1 for i in ₁, and 0 < p_i < 1 for i in _p. In this case, the stability matrix A can be partitioned into blocks corresponding to the fixed and polymorphic loci, because

(20a)

whereas

(20b)

and

(20c)

Thus, the eigenvalues governing the stability of the fixed loci (i.e., i _{0 1}) are simply _i = a_ii, and the stability conditions for the subsystem of polymorphic loci (i.e., i _p) are just (14) and (15). Equation 20b and Equation 20c show that the stability conditions for the fixed loci are

(21a)

and

(21b)

Hence, the conditions for the stability of the fixed equilibria are complementary to the feasibility conditions (19a) for the polymorphic equilibria with v_i > 1. The implications of (21) can be seen by assuming, without loss of generality, that is negative and all of the _i are positive. In this case, increasing p_i at each locus moves the population mean closer to the optimum. Then, inequalities (21) with v_i = 0 imply that, whenever possible, the multilocus system will equilibrate so that || is less than min_i0{_i/2}. Inequalities (21) imply that || is even smaller with v_i > 0.

Because alternative multilocus equilibria will generally produce different values of , and hence different _i 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 v_i is only slightly above one, the locus can be stably polymorphic at an equilibrium with very near 0, but stably monomorphic at equilibria with larger ||. This is illustrated numerically below.

Now consider at equilibria. Assuming as above that p_i = 0 for i ₀, p_i = 1 for i ₁, and 0 < p_i < 1 for i _p, we have

(22)

Substituting expression (9) for the equilibrium allele frequencies and rearranging, we find that

(23a)

where

(23b)

and

(23c)

The population mean would depart from the optimum by _f 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 _i(_i - _i).] In this sense, _f 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 v_i < 1 for one of the i in _p, 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 v_i > 1 at stably polymorphic loci. For any fixed lower bound on the v_i, (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 v_i > 1 can be stably polymorphic and loci with v_i < 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 x 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 ß_i,k (_i,k) denote the effect of B_i (b_i) in environment k. With weak selection, as in (4), we can approximate the allele frequency dynamics in this environment by

(24)

We assume that S and remain fixed across environments. Averaging over environments, we define

(25)

Thus, the effect of a substitution at locus i has mean _i and variance is v_i²_i, so that v_i is the square of the coefficient of variation of the substitution effect. We demonstrate that these v_i play the same role in the stability analysis of this model as do the v_i defined by (7b) for the pleiotropy model. Averaging over environments, as done in GILLESPIE and TURELLI 1989 , we obtain

(26)

As discussed after Equation 4, we can absorb constants that enter E() into , so we assume E(ß_i) = _i/2 and E(_i) = -_i/2, without loss of generality. Thus, . Rearranging (1), we have

(27)

where the ß_i and _i have environment-specific values. Hence, the term Cov(ß_i - _i, ) that enters (26) and the analyses below depends on the scaled covariances, _ij and _ij, defined by

(28a)

and

(28b)

where _ii = v_i and _ij denotes the correlation of substitution effects at loci i and j. Note that _ij = _ij 0 for all i j if either the allelic effects at different loci fluctuate independently or we impose the symmetry constraints, Cov(ß_i, ß_j) = Cov(_i, _j) = Cov(_i, ß_j) = Cov(ß_i, _j) for all i j. GILLESPIE and TURELLI 1989 assumed the latter. Under the less restrictive assumptions that Cov(ß_i, ß_j) = Cov(_i, _j) and Cov(_i, ß_j) = Cov(ß_i, _j), we have _ij 0 for all i and j. In particular, _ii = Var(ß_i) - Var(_i), so that _ii = 0 if Var(ß_i) = Var(_i). Separating the terms in (26) that depend on locus i, we have

(29)

where denotes the average contribution to the mean phenotype from the loci other than i. It is easy to see that if _ij = 0 for all i j. 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 p_i = 0, p_i = 1, or

(30)

Note that increasing _ii, corresponding to raising the variance of effect for allele B_i relative to the variance for b_i, decreases the equilibrium p_i, 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 = (a_ij) with elements

(31a)

and

(31b)

evaluated at allele frequencies that satisfy (30), with _ij as defined in (28a). If , 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 x 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 v_i and the _ij defined in (28a).

In general, positive correlations across loci are destabilizing in the sense that larger v_i are needed to achieve stability with _ij > 0 than with _ij = 0. The destabilizing effect is dramatic. The necessary condition for stability, analogous to (16), is

(32)

With _ij > 0 and ²_ij > 3/4, (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 v_i v and _ij . In this case, a feasible fully polymorphic equilibrium is stable if and only if

(33)

Hence, polygenic variation can be stably maintained under this model of G x 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, |_ij| = 1 and polymorphism condition (32) cannot be satisfied.

Stability, feasibility, and position of alternative equilibria: Consider an equilibrium with p_i = 0 for i ₀, p_i = 1 for i ₁, and 0 < p_i < 1 for i _p. First note that if and _ii = 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

(34)

where . 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(ß_i - _i, ^*₁) 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 p_i. Hence, we expect that positive correlations between loci and _ii > 0 will broaden the conditions for stability of p_i = 0. As before, the stability matrix A can be partitioned into blocks corresponding to the fixed and polymorphic loci, because

(35a)

whereas

(35b)

and

(35c)

with

(35d)

Thus, the eigenvalues governing the stability of the fixed loci (i.e., i _{0 1}) are simply _i = a_ii, and the stability conditions for the subsystem of polymorphic loci (i.e., i _p) are determined by the eigenvalues of (31), which depend only on the parameters for the polymorphic loci. Equation 35b and Equation 35c show that the stability conditions for the fixed loci are

(36)

As expected, positive values of Cov(ß_i - _i, ^*_i) and _ii promote the stability of p_i = 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 x 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 G_k(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,

(37)

where G_i,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 E(ß_i - _i) = _i and . 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

(38)

so that

(39)

Hence, in the calculations above, e.g., Equation 8, v_i = c_i(1 - _i). [Note that _i describes correlations between allelic effects within loci, whereas _ij in (28a) describes correlations between substitution effects at different loci.]

First, assume uncorrelated fluctuating allelic effects across loci, so that

(40)

where E_g denotes averaging over the distribution of genotypes in the population. A central feature of all random environment models is that Var[G_i,k(g)|g] depends on whether genotype g is homozygous or heterozygous at locus i (GILLESPIE and TURELLI 1989 ). Using (38), if g is homozygous for either allele at locus i, and if g is heterozygous at locus i. Thus,

(41)

Note that this depends on both v_i and _i. With independent fluctuations across loci, we have

(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

(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

(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 _i imply that irrespective of correlations in fluctuations within or among loci,

(45)

Hence, for independent fluctuations across loci,

(46)

The qualitative implications of (46) are most easily seen with exchangeable loci (i.e., _i = , c_i = c, _i = , and p_i = p), for which

(47)

and the stability criterion is simply v > 1. Equation 47 implies that K decreases as and v increase and as p departs from 0.5. Hence, for stable equilibria, K is maximized when = -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 E_g{Var[G_k(g)|g]} in the denominator increases. The effects of these covariances depend not only on the covariances of substitution effects, i.e., Cov(ß_i - _i, ß_j - _j) as described by (28), but also on the covariances between the individual alleles at each locus, i.e., Cov(ß_i, ß_j), Cov(_i, _j), and Cov(ß_i, _j). Nine different expressions for Cov{[G_i,k(g)|g], [G_j,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

(48)

As discussed following Equation 28aEquation 28b, this implies that Cov(ß_i - _i, ß_j - _j) = 0 for all i j, so that the allele frequency dynamics are still approximated by (29) with . In particular, in the symmetrical case with c_i(1 - _i) = v for all i, the polymorphic equilibrium is stable whenever v > 1. For exchangeable loci satisfying (48), (47) is replaced by

(49)

Thus, for any positive _B, 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 x 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 _f,i (_m,i) denotes the effect of a substitution at locus i on females (males), we have

(50)

The condition v_i > 1 requires that _f,i and _m,i have different signs. Thus, if we use the convention that each B_i denotes the allele that increases the trait value in females, the polymorphism condition v_i > 1 implies that each B_i must decrease the trait in males. By considering the symmetrical model with = 0, it is easy to see, however, that v_i > 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 = 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 v_i > 1 for all i, by reversing the signs of effects in one sex, we get identical dynamics but the new values of v_i 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 B_i and B_j increase the trait value in females implies that

(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 (KIDWELL et 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 x 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 x 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 v_i, 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, ß_i,t and _i,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 ß_i,t and _i,t do not explicitly affect the dynamics, because selection depends only on ß_i,t - _i,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 timescale of parameter variation. As before, the basic recursion is

(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 v_i, which describe the squared coefficient of variation of substitution effects at individual loci [see (25)], and the _ij, which describe the correlations between substitution effects at different loci [see (28a)]. In particular, when _ij = 0, we expect that loci satisfying v_i > 1 will tend to remain polymorphic if the expected population mean is close enough to the optimum. Loci with v_i < 1 will generally not be stably polymorphic, and they will fix for alleles that produce an expected population mean very near . As before, we predict from (32) that no stable multilocus polymorphisms can be maintained for loci with large positive _ij.

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 , GILLESPIE and TURELLI 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 x 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

TOP ABSTRACT MODELS AND APPROXIMATE ANALYSES NUMERICAL ANALYSES DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

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 _i and v_i at 20 loci were drawn independently from gamma distributions (Table 2), and the optimum is = 0. Because of this symmetry, the equilibria come in pairs, the elements of which have of opposite sign, produced by replacing each p_i by 1 - p_i. 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 _eff, is