Abstract
In quantitative genetics, there are two basic “conflicting” observations: abundant polygenic variation and strong stabilizing selection that should rapidly deplete that variation. This conflict, although having attracted much theoretical attention, still stands open. Two classes of model have been proposed: real stabilizing selection directly on the metric trait under study and apparent stabilizing selection caused solely by the deleterious pleiotropic side effects of mutations on fitness. Here these models are combined and the total stabilizing selection observed is assumed to derive simultaneously through these two different mechanisms. Mutations have effects on a metric trait and on fitness, and both effects vary continuously. The genetic variance (V_{G}) and the observed strength of total stabilizing selection (V_{s,t}) are analyzed with a rarealleles model. Both kinds of selection reduce V_{G} but their roles in depleting it are not independent: The magnitude of pleiotropic selection depends on real stabilizing selection and such dependence is subject to the shape of the distributions of mutational effects. The genetic variation maintained thus depends on the kurtosis as well as the variance of mutational effects: All else being equal, V_{G} increases with increasing leptokurtosis of mutational effects on fitness, while for a given distribution of mutational effects on fitness, V_{G} decreases with increasing leptokurtosis of mutational effects on the trait. The V_{G} and V_{s,t} are determined primarily by real stabilizing selection while pleiotropic effects, which can be large, have only a limited impact. This finding provides some promise that a high heritability can be explained under strong total stabilizing selection for what are regarded as typical values of mutation and selection parameters.
THE presence of genetic variation in quantitative traits is important for the selective breeding of domestic animals and crops, evolution, and adaptation (Charlesworthet al. 1982; Barton and Turelli 1989; Falconer and Mackay 1996; Barton and Keightley 2002). The existence of genetic variation is, however, paradoxical because stabilizing selection acting on the population usually depletes genetic variation (Wright 1935; Crow and Kimura 1970; Bürger and Gimelfarb 1999; Bürger 2000). As the ultimate source of genetic variation is mutation, an intuitively appealing explanation for the maintenance of polygenic variation is that there is an equilibrium between the input of new variation by mutation and its erosion by natural selection. For real stabilizing selection it is assumed that natural selection acts directly and solely on the metric trait, relative fitness having a quadratic relationship with the trait. Under the rareallele model and the assumption of Gaussian fitness function, predictions for the equilibrium genetic variance are given by the houseofcards approximation V_{G} = 4λ_{t}V_{s,r} (Turelli 1984; Bürger 2000), where λ_{t} is the average number of mutations of genes that affect the trait per generation per haploid genome, and V_{s,r} is the strength of real stabilizing selection, the “variance” of the fitness profile, with a large value of V_{s,r} implying weak selection. It is difficult to account for the observed high variance with this model for what are regarded as typical values of V_{s,r} (e.g., 20V_{e}), mutation rate per locus, and number of relevant loci (Turelli 1984; Falconer and Mackay 1996). Furthermore, simple genetic load arguments suggest that real stabilizing selection cannot operate independently on many characters (Robertson 1967; Turelli 1985; Barton 1990). In a recent review, however, Kingsolver et al. (2001) found that estimates of the strength of stabilizing selection vary greatly, and the typical selection may be much weaker than previously assumed.
In an alternative model, the pure pleiotropic model, natural selection is assumed not to act directly on the metric trait in question, but through pleiotropic side effects of mutant alleles on fitness (Robertson 1967; Hill and Keightley 1988). This model can generate apparent stabilizing selection, shown as a negative correlation between relative fitness and phenotypic deviation from the mean (Robertson 1967; Hill and Keightley 1988; Barton 1990; Gavrilets and de Jong 1993): Extreme individuals for the trait tend to carry more harmful mutations. There is observational evidence for apparent selection in nature (Kruuket al. 2002). This model can provide an explanation for observed levels of V_{G} but for only part of the strength of apparent stabilizing selection observed (V_{s,t}) and has the further defect that V_{G} increases without bound as the effective population size increases when the mutational effects are not completely correlated and the distribution of fitness effects is leptokurtic (Keightley and Hill 1990; Caballero and Keightley 1994). Such stabilizing selection induced solely by pleiotropic effects on fitness of mutations is referred to as “pleiotropic selection” in this article. There is a general relationship V_{s} ≥ V_{G}^{2}/V_{m} (Barton 1990; Kondrashov and Turelli 1992; Gavrilets and de Jong 1993; Zhanget al. 2002); therefore the pure pleiotropic model cannot in principle explain both the observed levels of genetic variances and typical estimates of strengths of stabilizing selection, provided the mutational variance V_{m} is of the order 10^{3}V_{e} as observed (Houleet al. 1996; Lynch and Walsh 1998; Lynchet al. 1999).
In addition to the above two hypotheses, many others such as overdominance (Wright 1935; Robertson 1956; Gillespie 1984; Barton 1990), frequencydependent selection (Slatkin 1979; Barton 1990), genotypebyenvironment interaction (Gillespie and Turelli 1989; Gimelfarb 1990; Zhivotovsky and Gavrilets 1992), and epistatic interaction (Zhivotovsky and Gavrilets 1992; Gavrilets and de Jong 1993) have been proposed to explain the maintenance of polygenic variation. All these models have their respective appeal and weaknesses in explaining the maintenance of polygenic variation.
Nevertheless, the real stabilizing selection and pleiotropic models are not mutually exclusive. Individual mutant alleles can have both deleterious pleiotropic effects on fitness and effects on the metric trait in question (Falconer and Mackay 1996). If the metric trait is not completely neutral, that is, the extreme phenotypes of the metric trait are less fit, natural selection takes place simultaneously through two different mechanisms: the deleterious pleiotropic effects on all other aspects of fitness and real stabilizing selection on the metric trait under study. Individuals that carry mutants are therefore selected against because of both deleterious pleiotropic effects of mutants (i.e., pleiotropic selection) and phenotypic deviations of the trait value from the optimum (i.e., real stabilizing selection). The strength of total stabilizing selection is therefore attributed to both kinds of natural selection. As Kondrashov and Turelli (1992, p. 615) noted, “A complete treatment should consider both direct and indirect selection on the quantitative trait.” Tanaka (1996) used a cohortofmutations model to combine both pleiotropic and real stabilizing selections and assumed that all mutations had an equal deleterious effect on fitness and a Gaussian distribution of effects on the target trait. However, this cannot readily account for both high heritabilities and strong stabilizing selection (Tanaka 1996, 1998). Although the assumption of an equal fitness effect for all mutations is a convenient way to obtain analytical approximations for V_{G} and V_{s,t} (Barton 1990; Kondrashov and Turelli 1992; Tanaka 1996), it lacks rigorous support, and experimental data illustrate the highly leptokurtic distribution of mutational effects on fitness (Mackayet al. 1992). As shown by Zhang et al. (2002), the shape of the distribution of mutational effects does affect the predictions of the pleiotropic model, so that it is necessary to take into account variation in effects of mutations both on the trait and on fitness.
In this study, a compound model of continuously varying effects of mutations on the trait and on fitness is constructed to investigate the maintenance of genetic variance and the observed strength of total stabilizing selection. The interaction between both kinds of selection and their overall impact on genetic variation and strength of total stabilizing selection are explored. We hope thereby to provide a possible explanation for the observations of both high genetic variance and the strong observed stabilizing selection.
MODEL
We assume additivity of gene action, linkage equilibrium, a randommating diploid population, and rare mutant alleles. In accordance with the model of real stabilizing selection (Turelli 1984, 1985), the relative fitness of individuals that have a phenotypic value P, the sum of the contributions from each locus plus a random independent environmental effect of mean zero, is assumed to be given by W(P) = exp(P^{2}/2ω^{2}). The mean fitness of individuals with genotypic value G = Σ_{i}a_{i} is W(G) = exp(G^{2}/2V_{s,r}) ≈ 1  G^{2}/2V_{s,r} with V_{s,r} =ω^{2} + V_{e} measuring the intrinsic strength of real stabilizing selection. V_{e} is the environmental variance and is scaled as a unit of variance.
It is assumed that there are infinitely many loci on each individual and at each locus there is a continuum of possible mutational effects, but each locus has the same mutation distribution and loci are exchangeable. There are at most two alleles segregating at each locus: the wild type, which is assumed to be at optimum, and the mutant. Mutations have effects on a metric trait (a) and pleiotropic deleterious effects on fitness (s ≥ 0), with a bivariate distribution h(a, s). If the metric trait undergoes real stabilizing selection due to mutations, the observed stabilizing selection would come from these two parts and the equivalent total selection coefficient within each individual is given by s˜ = s + (1  2x)a^{2}/(4V_{s,r}) (see appendix a), where x is the frequency of the mutant allele. The equivalent total selection coefficient is in general not independent of the frequency of mutant alleles in this compound model. It is therefore less tractable (see appendix a) than the pure pleiotropic model (Barton 1990; Keightley and Hill 1990), in which selection is assumed to act directly on the pleiotropic effect on fitness of each mutant allele and the coefficient is always independent of the frequency of the mutant allele. With the assumption of real stabilizing selection (Turelli 1984; Keightley and Hill 1988), however, selection acts on the total effect of all mutants within individuals and hence depends on the frequency of mutant alleles (Robertson 1956). Such frequency dependence of selection leads to multiple equilibria (Bulmer 1985; Barton 1986) but, unless population size is very small, mutant alleles cannot increase to a high frequency without passing through an intermediate frequency, against which there is selection. The frequency of mutant alleles therefore remains very low (Bulmer 1989). With rare mutant alleles, the equivalent total selection coefficient within each individual organism can therefore be approximated by
Although the properties of mutant effects on the metric trait and on fitness are crucial to evaluating V_{G} and V_{s,t}, the distribution of mutational effects is hard to estimate accurately (Mackay and Langley 1990; Hill and Caballero 1992; Mackayet al. 1992; Davieset al. 1999; Elena and Moya 1999; Keightleyet al. 2000; Shawet al. 2000; Imhof and Schlötterer 2001; Wlochet al. 2001). Even for Drosophila, for which there are many studies, the data seem to suggest a highly skewed and leptokurtic distribution of mutational effects (Mackay and Langley 1990; Hill and Caballero 1992; Mackayet al. 1992), but finescale information is still lacking. As in Keightley and Hill (1990), the distribution of mutant effects on the metric trait is assumed to be symmetrical about a = 0, and only deleterious effects of mutations on fitness are assumed to occur, in accord with the classical view (Falconer and Mackay 1996). The variability of the distribution of a is defined in terms of
RESULTS
Analytical approximations are obtained for some special cases for an infinite population and a rareallele approximation, and numerical calculations were performed to provide support and to extend the results to more general situations. Simple results for some special situations are also presented within Keightley and Hill’s (1990) framework using Kimura’s (1969) diffusion approximation.
Pure real stabilizing selection within a finite population, i.e., s = 0, thuss˜ = (1  2x)a^{2}/(4V_{s,r}): The observed strength of real stabilizing selection is V_{s,t} = V_{G2}/(2 Cov_{r}) = V_{s,r} and the genetic variance is given by (A2). Numerical calculation shows that, as the effective population size N_{e} increases, V_{G} increases and approaches the rareallele approximation 4λV_{s,r} (Turelli 1984; see Figure 2). Theoretically this is because the equilibrium frequencies of mutant genes, x, become very small and the heterozygosity can thus be approximated by H(s˜) = 4λ/s˜ as the effective population size N_{e} → ∞ and thus N_{e}s˜ ⪢ 1. Figure 2 also shows that the genetic variance maintained in a finite population depends on the distribution of mutational effects (cf. Keightley and Hill 1988). Further, if mutational effects on the trait follow a reflected gamma (½), V_{G} depends little on the mutation rates; whereas for equal mutational effects on the trait, V_{G} in small populations depends heavily on the mutation rates for given λV_{s,r}.
Pure pleiotropic effects, where the target trait is completely neutral in itself (i.e., V_{s,r} → ∞) and s˜ = s: With all mutants having equal pleiotropic effects, the genetic variance is V_{G} = 2V_{m}/s (Barton 1990), which is too small, given that the estimates of selection coefficients with detectable effects in the laboratory are in the range s = 0.020.08 (Crow and Simmons 1983; Keightley and Hill 1990; Caballero and Keightley 1994; Chavarras et al. 2001; Wlochet al. 2001). If, however, the pleiotropic effects vary among mutants, substantial variation can occur; indeed V_{G} becomes unbounded for an infinite population if neutral mutants predominate (see Keightley and Hill 1990; Caballero and Keightley 1994; Zhanget al. 2002). Moreover, the pure pleiotropic model can only partially account for the “typical” strength of stabilizing selection (Barton 1990; Kondrashov and Turelli 1992; Zhanget al. 2002).
Joint effects of both pleiotropic and real stabilizing selections within an infinite population, but assuming equal mutational effects on both the trait (ε_{a}) and fitness (s¯): From Equation 1,
Joint effects, but assuming mutations have an equal pleiotropic effect on fitness (s¯) and a continuous distribution f(a) of mutational effects on the metric trait: In this situation Kimura’s (1969) diffusion theory leads to H(s˜) = C(s˜) = 4λ/s˜ approximately, and K(s˜) = 0 approximately for an infinite population (Zhanget al. 2002). The genetic variance is
If the mean pleiotropic effect on fitness is much weaker than that from real stabilizing selection (i.e., s¯ ⪡ s¯_{r}), the genetic variance approaches the rareallele approximation V_{G} = 4λV_{s,r}. In general
Figure 3 clearly shows how both effects interfere and contribute to the overall outcome in V_{G} and V_{s,t}. When the mutation rate is very low (e.g., λ< 10^{5}) and each mutant has large effects on the trait relative to its effect on fitness, the results approach the houseofcards approximation (Turelli 1984). If the mutation rate is high (e.g., λ> 0.1) and each mutant has a relatively small effect on the trait, the pleiotropic effect must be widespread and becomes the main force of selection, the genetic variance tends to that of Barton (1990) but the strength of total stabilizing selection approaches
General case: As shown above and by previous work (Barton 1990; Kondrashov and Turelli 1992; Tanaka 1996), the equal fitness effect assumption cannot provide a simultaneous explanation for the observed high heritability and strong stabilizing selection. If mutational effects on fitness vary across loci in the absence of real stabilizing selection a huge genetic variance can be generated (Keightley and Hill 1990; Zhanget al. 2002), so it is important to investigate the influence of variation in fitness effects on V_{G} and V_{s,t}.
The first check is whether the unbounded V_{G} with increasing population size is avoided with the inclusion of a real stabilizing selection on the trait. The example in Figure 4 shows that with even a weak real stabilizing selection (e.g., V_{s,r} = 1000), the genetic variance increases with effective population size N_{e} when it is small, but asymptotes when N_{e} exceeds some large value. This asymptotic value of V_{G} depends on the value of V_{s,r}, with a high V_{G} for a weak real stabilizing selection (i.e., a large V_{s,r}). At the same time, the value of V_{s,t} also increases and approaches a limit that is less than V_{s,r}. This implies that selection becomes weaker as the effective population size increases, but the total stabilizing selection is stronger than the real stabilizing selection.
Suppose that mutational effects on the trait are Gaussian and mutational effects on fitness follow a gamma (½) with mean
Numerical results are shown in Figure 5 for a range of distributions of effects of mutations on the trait and on fitness such as equal, Gaussian, gamma (½), gamma (¼), and gamma (
In a realistic model, mutational effects on the trait and on fitness must be correlated (Keightley and Hill 1990). Although analytical treatment is never easy (if possible) when a correlation between mutational effects is included (e.g., Turelli 1985), it is important to consider the impact of such a correlation on the results for V_{G} and V_{s,t}. Using the method of Keightley and Hill (1990), the mutational effects a and s were sampled from a bivariate gamma (½) distribution. The numerical calculations show that when this correlation is only intermediate (ρ< 0.5), its impact on V_{G} and V_{s,r} is not large (see Figure 6). Unless the correlation between a and s is very high, the results based on the assumption of independent mutational effects apply approximately.
DISCUSSION
The assumptions for the origin of both kinds of selection are distinct. In models of real stabilizing selection, selection is assumed to arise solely from the deviations of the metric traits from their optimum due to mutational effects (i.e., phenotypic selection, selection directly acting on the trait), whereas in pure pleiotropic models the apparent stabilizing selection is assumed to arise as a consequence of direct effects of deleterious mutations on overall fitness, ignoring any effect on the trait itself (i.e., selection acting directly on genes). By assuming that the total stabilizing selection observed on individuals comes simultaneously from both kinds of selection, the joint effect model presented in this article includes the properties of both the real stabilizing selection (Turelli 1984) and the pure pleiotropic models (Keightley and Hill 1990). It is important to know whether new findings about the genetic variation and the stabilizing selection emerge from the analyses of the joint effect model.
The pure pleiotropic model (Keightley and Hill 1990; Kondrashov and Turelli 1992; Zhanget al. 2002) can account for substantial quantitative genetic variation, but the apparent stabilizing selection of strength
In contrast to Tanaka’s (1996, 1998) pleiotropic model, which includes both kinds of selection but assumes an equal deleterious effect on fitness for all mutants, the joint effect model presented here, which allows both mutational effects to vary, leads to quite different pictures of how both kinds of selection are responsible for V_{G}. As found by Tanaka (1996) and intuitively argued by Kondrashov and Turelli (1992), the total selection coefficient should be equal to a linear sum of that arising from real stabilizing selection and that solely attributable to pure pleiotropic effect: s_{T} = s¯_{r} + s¯_{p}. As in general s¯_{p} ⪢ s¯_{r} (Gillespie 1991; Kondrashov and Turelli 1992), the total selection coefficient is approximately equal to the pleiotropic effect s_{T} ≈ s¯_{p} and Kondrashov and Turelli (1992, p. 615) concluded that real stabilizing selection was “essentially irrelevant to the dynamics of the alleles responsible for variation in the trait.” Within the joint effect model, the total selection coefficient, which is a complicated function of both components [see (8) and appendix b], is >s¯_{r}, but ⪡s¯_{p} as well if s¯_{p} ⪢ s¯_{r}. This of course leads to a larger V_{G}. Therefore pleiotropic effects on fitness can be large but their impact on V_{G} is limited.
For a simple explanation of why a distribution of pleiotropic effects allows the model to generate high V_{G}, suppose that new mutations are divided into two equally possible classes: one with equal pleiotropic effect s_{1}, the other with s_{2}, but with both having the same effect on the trait (i.e., s¯_{r}). The two classes contribute to V_{G} as 2V_{m}/(s_{1} + s¯_{r}) and 2V_{m}/(s_{2} + s¯_{r}), respectively, from Tanaka (1996), and the total genetic variance maintained is then larger than if all mutations have the same mean pleiotropic effect (s_{1} + s_{2})/2 because [1/(s_{1} + s¯_{r}) + 1/(s_{2} + s¯_{r})]/2 > 1/[(s_{1} + s_{2})/2 + s¯_{r}]. The numerical results show that if a very small minimum total selection coefficient, say 10^{10}, is assumed, the genetic variance maintained is nearly the same as that without such minimum fitness effect. As the mutant alleles of large effects on fitness would be quickly eliminated from the population, the genetic variance is attributable primarily to mildly deleterious mutations. The huge genetic variation generated in the joint effect model of continuously varying pleiotropic effects on fitness, therefore, comes mainly from “a class of alleles with significant effects on the character, but very little effect on fitness” (Barton 1990, p. 779).
It is also interesting to compare the prediction of the joint effect model with the houseofcards approximation V_{G} = 4λ_{t}V_{s,r}, where λ_{t} refers solely to the total rate of mutations that affect the metric trait under study (Turelli 1984, 1985). When the pleiotropic effect is weak in relation to the effect on fitness from real stabilizing selection, the genetic variance can be approximated by the houseofcards approximation (see Equation 8, Table 1, and Figure 5); but if the pleiotropic effect is large, the genetic variance maintained is given by (9) for Gaussian effects on the trait and gamma (½) effects on fitness of mutations. As the genomewide mutation rate λ exceeds λ_{t}, our prediction of V_{G} may not be smaller than the houseofcards approximation (cf. Tanaka 1996, 1998). For the typical estimate of strength of real stabilizing selection, V_{s,r} = 20 (Turelli 1984),
The mutation rate λ assumed in this study is the genomewide mutation rate. Although all of the mutations may affect fitness to a varying degree, only a small fraction of them may be considered to appreciably affect the trait under study. It is, however, unrealistic to assume no effect and more appropriate to assume that the distribution of mutational effects on the trait is more leptokurtic than on fitness (see Robertson 1967; Keightley and Hill 1988; Hill and Caballero 1992). The analyses of the joint effect model show that the genetic variance maintained at mutationselection balance depends not only on the variance of mutational effects but also on their leptokurtosis. For a given distribution of mutational effects on fitness, a more leptokurtic mutational effect on the trait induces a smaller genetic variance, consistent with the results of Keightley and Hill (1988) who studied pure real stabilizing selection in finite populations. Even for this more realistic model, the joint effect model can still generate abundant genetic variation if mutational effects on fitness are sufficiently leptokurtic, say gamma (½), and the genomewide mutation rate is not <0.01 (see Figure 5).
The scanty data for multicellular eukaryotes are consistent with any value of λ between 0.1 and 100 (Charlesworthet al. 1990; Kondrashov and Turelli 1992; Lynchet al. 1999; Kumar and Subramanian 2002). Recent studies on Caenorhabditis elegans, however, show that the mutation rate for life history traits is ⪡1.0 and is of the order 10^{3} (Keightley and Caballero 1997; GarciaDoradoet al. 1999; Vassilieva and Lynch 1999). The best estimate of the average selection coefficient against heterozygous mutations is E[s/2] = 0.02 (Crow and Simmons 1983). Data for Drosophila bristle traits show that λ is in the range 0.091.0 and ε_{s} in the range 0.010.2 (Keightley and Hill 1990; Caballero and Keightley 1994). Data for competitive viability in Drosophila suggest that λ≥ 0.01 and E[s] ≤ 0.08 (Chavarriaset al. 2001). Data for yeast Saccharomyces cerevisiae show that λ is of the order 10^{3} and E[s/2] is in the range 0.010.05 (Wlochet al. 2001). Even with such large pleiotropic effects, our joint effect model, which assumes leptokurtic effects both on the trait and on fitness of mutations, predicts high heritabilities under strong total stabilizing selection unless λ is very small, say <0.01 (see Equation 8, Figure 5, and Table 1). But the estimates of mutation and selection parameters are not very reliable (Kondrashov 1998; Lynchet al. 1999; Kingsolveret al. 2001). Mutation rates are usually underestimated and mean fitness effects are usually overestimated as the effects of most mutants may be too small to be detected (Kondrashov and Turelli 1992; Davieset al. 1999; Lynchet al. 1999). The observation of high heritabilities and strong total stabilizing selection may then be interpreted in terms of the joint effect model of continuously varying mutational effects.
In summary, the joint effect model presented here shows that V_{G} and V_{s,t} are determined primarily by real stabilizing selection while pleiotropic effects, which can be large, have only a limited impact. With an abundant supply of mutations and leptokurtic mutational effects on fitness, the joint effect model can induce a significant amount of stabilizing selection as well as a substantial genetic variance, even with a mutational variance on the trait as low as V_{m} = 10^{3} V_{e} (cf. Barton 1990). Combining both kinds of selection and allowing mutational effects on the metric trait and on fitness both to vary change the picture of the mutationselection model and therefore enable the mutationselection balance to be a plausible cause of quantitative variation.
APPENDIX A
Let us assume that the gene action within and across loci is additive and loci are unlinked and in linkage equilibrium. A randommating diploid population is assumed. Mutations in a diploid individual have an effect on a metric trait z with a the difference in value between homozygotes and a net effect on fitness that includes pleiotropic effects on all other traits, with s the difference in fitness between homozygotes. There is therefore a bivariate distribution, h(a, s), of a and s for alleles affecting the trait. If there is real stabilizing selection, the total observed stabilizing selection would come from these two parts. Following the method of Falconer and Mackay (1996, p. 27), the mutant allele frequency within a singlelocus model is given by x_{1} = {x  x(1  x)[s/2 + a^{2}/(8V_{s,r})]}/w¯ with the mean fitness given by w¯ = 1  x(1  x)a^{2}/(4V_{s,r}) if the previous frequency is x. With weak selection (i.e., w¯ ≈ 1), the change in the mutant allele frequency is Δx = x_{1}  x ≈ x(1  x) [s/2 + (1  2x)a^{2}/(8V_{s,r})]. Thus the equivalent total selection coefficients are
For an infinite population, by using the approximations H(s˜) = C(s˜) = 4λ/s˜ and K(s˜) = 0, the expressions for the genetic variance and strength of the total stabilizing selection reduce to those given in (2) and (3). Equation 3 is obtained by noting that
APPENDIX B
We consider the evaluation of genetic variance assuming that the population is under stabilizing selection because of the joint effect of pleiotropic and real stabilizing selections and that both mutational effects are independent. If mutational effects on the trait and on fitness follow distributions g_{1}(a), where ∞ < a < ∞, and g_{2}(s), where 0 < s < ∞, respectively, then evaluation of V_{G} = 4λV_{s,r}I_{2} according to (2) is equivalent to the expectation,
Thus the expectation is determined only by the ratio of s¯_{p} to s¯_{r}.
Acknowledgments
We are grateful to Nick Barton, Brian Charlesworth, Peter Keightley, Jinliang Wang, and a referee for helpful comments and Ian White for help in proving Equation 8. This work was supported by a grant from the Biotechnology and Biological Sciences Research Council (R35396).
Footnotes

Communicating editor: J. B. Walsh
 Received April 16, 2002.
 Accepted June 10, 2002.
 Copyright © 2002 by the Genetics Society of America