Abstract
A pleiotropic model of maintenance of quantitative genetic variation at mutationselection balance is investigated. Mutations have effects on a metric trait and deleterious effects on fitness, for which a bivariate gamma distribution is assumed. Equations for calculating the strength of apparent stabilizing selection (V_{s}) and the genetic variance maintained in segregating populations (V_{G}) were derived. A large population can hold a high genetic variance but the apparent stabilizing selection may or may not be relatively strong, depending on other properties such as the distribution of mutation effects. If the distribution of mutation effects on fitness is continuous such that there are few nearly neutral mutants, or a minimum fitness effect is assumed if most mutations are nearly neutral, V_{G} increases to an asymptote as the population size increases. Both V_{G} and V_{s} are strongly affected by the shape of the distribution of mutation effects. Compared with mutants of equal effect, allowing their effects on fitness to vary across loci can produce a much higher V_{G} but also a high V_{s} (V_{s} in phenotypic standard deviation units, which is always larger than the ratio V_{P}/V_{m}), implying weak apparent stabilizing selection. If the mutational variance V_{m} is ∼10^{3} V_{e} (V_{e}, environmental variance), the model can explain typical values of heritability and also apparent stabilizing selection, provided the latter is quite weak as suggested by a recent review.
MOST characters in morphology, behavior, and physiology vary continuously among individuals within populations (Falconer and Mackay 1996). These quantitative or metric traits are usually observed to have abundant genetic variation in natural populations, as reflected directly by the correlation between relatives and indirectly by sustained response to directional selection that takes the phenotype well beyond its original range (Endler 1986; Roff and Mousseau 1987). Such traits are often also found to be under apparent stabilizing selection, as evidenced by the reduced fitness of extreme phenotypes and the constancy of form over geological times and over vast geographic distances (Charlesworthet al. 1982; Endler 1986). The existence of genetic variation is thus paradoxical, because stabilizing selection usually depletes genetic variation (Wright 1935; Crow and Kimura 1970) although stabilizing selection in some extreme cases, e.g., in twolocus models with unequal effects, can maintain substantial variation (Bürger 2000, pp. 203216). How then is genetic variation in metric traits maintained in natural populations? This is a longstanding and fundamental question in evolutionary biology, the answer to which is important for understanding the genetic architecture of metric traits and the selective breeding of domestic animals and crops and comprehending the genetic basis of evolution and adaptation (Barton 1990; Falconer and Mackay 1996; Bürger 2000).
Since the ultimate source of genetic variation is mutation, an intuitively appealing explanation for the maintenance of polygenic variation is that there is equilibrium between the input of new variation by mutation and its erosion by natural selection [i.e., mutationselection balance (MSB)]. The question is whether mutations affecting a metric trait appear frequently enough and/or have large enough effects to provide sufficient new variation to counterbalance the depletion of variation by stabilizing selection. Empirical studies on various traits in different species show that mutational variance (V_{m}), the fresh genetic variance of a trait generated by mutation in one generation, is typically 10^{3}V_{e} (V_{e}, environmental variance), with a range from 10^{4} to 10^{2}V_{e} (Houleet al. 1996; Lynch and Walsh 1998). Assuming that the population is in MSB, a review of available data on V_{m} and standing genetic variance (V_{G}) showed that the persistence time of deleterious mutants, V_{G}/V_{m}, for life history traits is roughly onehalf that for morphological traits, consistent with the prediction of MSB that traits more closely related to fitness are under stronger selection and mutations affecting them are eliminated more quickly (Houleet al. 1996).
If MSB is accepted as the mechanism for the maintenance of polygenic variation in natural populations, the question remains of how natural selection acts on the trait or on the genes that affect it. Although the observation that intermediate phenotypes of a metric trait have the highest fitness indicates apparent stabilizing selection, it could have several distinct causes that lead to different predictions about the genetic architecture of metric traits in equilibrium populations (Robertson 1967).
A straightforward hypothesis is that natural selection acts directly and solely on the metric trait, the value of relative fitness having a quadratic relationship with the trait. This classical hypothesis, called “real” stabilizing selection (Falconer and Mackay 1996), has attracted extensive theoretical analyses, and two main classes of models have been developed. The two models make different extreme assumptions about allele effects and mutation rate per locus (u) and give quite different predictions of the equilibrium variance (Kimura 1965; Turelli 1984; Falconer and Mackay 1996; Bürger 2000): V_{G} = (2nV_{m}V_{s})^{1/2} for the continuumofalleles model and V_{G} = 4nuV_{s} for the rare allele model, where n is the number of potentially mutable loci affecting the trait and V_{s} is the strength of stabilizing selection, the “variance” of the fitness profile in phenotypic standard deviation units. The different approximations are a consequence of assumptions about the variance introduced by new mutations relative to the existing allelic variation (Turelli 1984; Bürger 2000). It is difficult to account for the observed high variance with either model for typical estimates of V_{m} (e.g., 10^{3} V_{e}) for what are regarded as typical values of V_{s} (e.g., 20V_{e}; Turelli 1984; Falconer and Mackay 1996). In a recent review, however, Kingsolver et al. (2001) concluded that the stabilizing selection might be substantially weaker than has been assumed. Further, simple genetic load arguments suggest that real stabilizing selection cannot operate independently on many characters (Robertson 1967; Turelli 1985; Barton 1990).
The hypothesis of the “pleiotropic model” is that natural selection does not act directly on the metric trait in question, but on the alleles affecting it through their pleiotropic side effects on fitness (Robertson 1967; Hill and Keightley 1988). With this model, an allele has a direct effect (a) on a specific trait, with its pleiotropic effects on all other traits condensed into its net effect (s) on fitness. The MSB under this model can also generate apparent stabilizing selection, shown as a negative correlation between fitness and phenotypic deviation from the mean, even for a trait that is purely neutral and irrelevant to fitness (Gavrilets and de Jong 1993). This is because, if unconditionally deleterious alleles can either increase or decrease the value of a specific trait, individuals carrying more deleterious alleles (Barton 1990; Kondrashov and Turelli 1992) or alleles of large deleterious effects if a and s are correlated (Keightley and Hill 1990) will tend to have more extreme phenotypes and low fitness. The model captures the conventional wisdom that essentially all mutations are pleiotropic and deleterious. The pleiotropic model is attractive, because it could explain both the appearance of stabilizing selection and the maintenance of polygenic variation as a consequence of the pleiotropic effects of mutations that are, in agreement with the classical view, unconditionally deleterious and highly pleiotropic.
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 weakness in explaining the maintenance of polygenic variation, and they and the real stabilizing selection and pleiotropic models are not mutually exclusive. For example, the epistatic model considered by Gavrilets and de Jong (1993) can explain both abundant variation and strong stabilizing selection, but experimental evidence is scanty as to whether such epistasis is common and strong enough (Elena and Lenski 1997; De Visser and Hoekstra 1998).
Given the ubiquity of mutations with deleterious pleiotropic effects, the pleiotropic model inevitably explains some polygenic variation and apparent stabilizing selection for a metric trait. The question is how much. Analytical models of the pleiotropic hypothesis, commonly assuming equal deleterious effect of mutations acting multiplicatively (Barton 1990) or synergistically (Kondrashov and Turelli 1992), can only partly explain the observed levels of polygenic variation and apparent stabilizing selection if V_{m} = 10^{3} V_{e}. Relaxing the assumptions makes the models more reasonable. Numerical methods (diffusion approximations and Monte Carlo simulation) can be used instead to tackle complex models (Keightley and Hill 1990; Caballero and Keightley 1994). With V_{m} = 10^{3} V_{e} but allowing deleterious effects to vary among alleles, numerical results indicate that the observed level of polygenic variance can be accounted for (Keightley and Hill 1990; Caballero and Keightley 1994), though the typical strength of stabilizing selection can be only partially explained (Barton 1990; Keightley and Hill 1990).
Here we construct a pleiotropic model on the basis of previous theoretical studies and empirical data available on new mutations and segregating alleles affecting metric traits and fitness to predict the polygenic variance (V_{G}) and strength of apparent stabilizing selection (V_{s}) for a metric trait in an equilibrium population. By applying a more general model of the relationship between a and s of mutations, we aim to find out what levels of V_{G} and V_{s} could be explained within the pleiotropic model. In particular, a weakness of models in which s has a continuous distribution is the unbounded increase of V_{G} with N_{e} (Keightley and Hill 1990). This was investigated and different assumptions were tried to decrease this dependence so that V_{G} can asymptote as N_{e} becomes large. We use both analytical and numerical approaches to obtain predictions from the model and compare them with observations. We hope thus to obtain a better understanding of the genetic properties of metric traits in an equilibrium population under the pleiotropic model. By comparing predictions with empirical observations, we also hope to ascertain how robust is the pleiotropic model as an explanation of polygenic variation.
MODEL AND ANALYSIS
Gene actions and contributions of mutants: Mutations in a diploid individual are assumed to have effects on a neutral metric trait z, with a the difference in value between homozygotes, and pleiotropic effects on fitness, with s the difference in fitness between homozygotes. The gene actions on the metric trait and on fitness within and across loci are assumed to be additive; that is, dominance and epistasis are ignored, which Barton and Keightley (2002) argue is an adequate model. The expected increase in variance in the metric trait each generation, the mutational variance, is given by V_{m} = ^{1}/_{2}λE[a^{2}] when the expected number of mutations per haploid genome per generation is λ. A population of N diploid individuals, with an effective population size N_{e} and random mating, is assumed. It is also assumed that the total number of loci per individual is so large and the mutation rate per locus is so low that mutations occurring at the same loci can be ignored.
To assess the genetic variation maintained and the strength of apparent stabilizing selection at MSB, we need to obtain the contribution from all mutant alleles. As the strength of apparent stabilizing selection is a function of the aggregate effects of alleles at all segregating loci in the population (Keightley and Hill 1990; Falconer and Mackay 1996), a multilocus model is analyzed. The results obtained in this subsection are valid in a quite general sense except that linkage disequilibrium between loci is ignored and gene action is additive.
Let the frequencies of the wildtype allele (A) and the mutant allele (a) at a given locus be 1  x and x, respectively. With a onelocus model, the conclusion about the contribution from mutant alleles is quite simple, while within a multilocus model, the conclusion is not straightforward. If mutations at any of n loci in a diploid individual can affect the neutral trait z and have pleiotropic effects on fitness, then the contribution from all these mutations can be described by the following properties. The genetic variance is
Bivariate distribution of mutant effects on the metric trait and on fitness and their simulations: To evaluate Equations 1, 2, 3, 4, we need to know the properties of mutant effects on the metric trait and on fitness, which vary between alleles. Although much effort has been made to quantify the features of mutant effects (Mackay and Langley 1990; Hill and Caballero 1992; Mackayet al. 1992; Davieset al. 1999; Elena and Moya 1999; Shawet al. 2000; Imhof and Schlötterer 2001; Wlochet al. 2001), detailed information about the marginal and joint distribution of mutant effects on the metric trait and fitness is unknown or indirect. Even for Drosophila, for which there are many studies, the data seem to suggest a highly skewed and leptokurtic distribution of mutation effects (Mackay and Langley 1990; Hill and Caballero 1992; Mackayet al. 1992), but the 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 mutations on fitness are assumed to occur, in accord with the classical view (Falconer and Mackay 1996). For a general model in which a wide relationship between effects and rates of mutations is included, it is assumed that the marginal distribution of effects on fitness, g(s), is a gamma distribution with scale parameter α_{s} and shape parameter β_{s},
Genetic variances and strength of apparent stabilizing selection were evaluated by Monte Carlo simulation and analytical methods. Effects of mutant alleles were sampled from a bivariate gamma distribution, h(a, s) with parameters ε_{a}, β_{a}, ε_{s}, β_{s}, and ρ using algorithm GTVR (Schmeiser and Lal 1982; see appendix a) and a from a by randomly allocating sign. Since selection acts on fitness, the conditional moments of the mutation effects on the metric trait given s are important in evaluating the properties of the populations at MSB,
The method of simulating a and s employed in this article is different from and more general than that in Keightley and Hill (1990) in which only a bivariate Γ distribution with β_{a} = β_{s} = ½ (i.e., Wishart distribution) was assumed for a and s.
Genetic variance maintained and strength of apparent stabilizing selection at MSB: Kimura’s (1969) diffusion approximations under the infinite independent loci model were used to obtain the equilibrium frequency distribution Φ(x; s) and other properties of a mutant with a specific fitness effect s in a large population at MSB. Since Φ(x; s)dx represents the expected number of loci in which the mutants of particular fitness effect s are in the frequency range x ∼ x + dx at equilibrium, the total expected number of mutants of fitness effect s is given by
As mutant alleles are unconditionally deleterious, the individuals with the more extreme genotypes with respect to the metric trait are less fit and thus selected against, which gives an appearance of stabilizing selection. Stabilizing selection is usually measured as the regression of relative fitness on squared phenotypic deviation from the optimum measured in phenotypic standard deviation units, given as
ANALYTICAL RESULTS
Let us consider an infinite population such that N_{e}s ⪢ 1, where the heterozygosity and other properties can be simplified to H = C = 4λ/s and K = 0. It is straightforward to obtain the following properties from (1416) and (67) if the distribution of mutation effects on fitness is not leptokurtic, i.e., β_{s} > 1,
If the distribution of mutation effects on fitness is leptokurtic, i.e., 0 <β_{s} ≤ 1, however, simple integration of Equations 13, 14, 15, 16 shows that whenever ρ< 1, V_{G} and V_{s} would be infinite for an infinite population, as shown in simulations of Keightley and Hill (1990). This intrinsic difficulty of the pleiotropic model of continuous fitness effects (Hill and Keightley 1988; Keightley and Hill 1990), which is also a consequence of the assumption of infinite independent loci, is avoided when equal fitness effects are assumed for all mutants (Barton 1990; Kondrashov and Turelli 1992; Tanaka 1996). If variation of the fitness effects of mutation is to be retained, then this difficulty can be avoided by assuming a minimum fitness effect for any mutation, as suggested by Mackay et al. (1992). A mutation can be defined as any change in the base sequence of DNA in the genome. Theoretically any such mutation change at the molecular level can affect the genome and thus the individual (Keightley and Hill 1988; Nielsen 2001). This minimum fitness effect exists in some way although it may be too small to be detected with current available equipment (Davieset al. 1999).
The relevant fundamental question is whether the distribution of the fitness effect of mutations is continuous or discrete. As Keightley (1991) argued, “The assumption of a continuous distribution of allelic deviations is also unrealistic because these are likely to fall into discrete classes.” These discrete classes are likely due to the fact that DNA sequence is discrete and that there may be a finite number of alleles and their effects could be discrete. As Max Planck did 101 years ago for avoiding “ultraviolet catastrophe” in black body radiation (e.g., Bransden and Joachain 1992), we assume here that fitness effects of mutations are discretized as nΔs, where n = 1, 2,..., and a “quantum” Δs > 0 is the possible minimum unit.
With the assumption of discretization of fitness effects of mutants, the properties for an infinite population can then be obtained by substituting (67) into (1316) and (11),
The covariance between the relative fitness and squared deviation is always independent of Δs, the correlation ρ, and shape parameter β_{s}, being equal to the negative of the mutational variance per generation (cf. Bürger 2000, p. 310). Combining (13) and (19) leads to a general constraint on V_{P} and V_{s},
It is interesting to compare our results with Barton’s (1990), who assumes equal fitness for all mutations and thus no correlation between the absolute values of mutant effects on the metric trait and on fitness. In our notation, this means s =Δs (i.e., the minimum effect of mutations is the exclusive effect) and g(s) =δ(s  Δs), where δ(·) is the Dirac delta function. The same results as Barton (1990) can be obtained from (1316). If the fitness effects vary across mutations and are distributed as a gamma random variable, then the results thus obtained are different from Barton’s (1990; see Table 1). In our results for V_{G} and V_{s}, there are two parameters relating to variation of mutation fitness effects: the minimum fitness effect, Δs, and variability of the fitness effects of mutations, ε_{s} (or equivalently the mean fitness effect E[s]). For shape parameter β_{s} > 1, the genetic variances are β_{s}/(β_{s}  1) times that of Barton’s (1990) for the same mean fitness effects (see Table 1). As β_{s} approaches 1, V_{G} tends to infinity. For β_{s} = 1 (i.e., exponential distribution), however, V_{G} is limited if Δs is finite. If the distribution of mutation effects on fitness is highly leptokurtic (i.e., β_{s} < 1), then the genetic variance can also be much larger than that of Barton’s (1990) for the same mean fitness effects (see Figure 2a). For instance, if β_{s} = ^{1}/_{4}, this increase is ∼12fold if Δs = E[s]/10^{2} and 69fold if Δs = E[s]/10^{3}. The results in Figure 2 show that the shape of distribution of fitness effects affects the genetic variance. At the limit β_{s} → 0, almost all mutations have the same effect Δs; whereas when β_{s} → ∞, the kurtosis of the distribution of mutation effects on fitness 3κ → 1 (i.e., equal effects). Both extreme situations return to Barton’s (1990) results (Table 1 and Figure 2a). Clearly β_{s} = 1 is a critical point.
An increasing correlation between a and s always reduces V_{G} (see Figure 2b). The marginal distributions of mutation effects on the trait and on fitness may be different, which does not cause any difference to the genetic variance if ρ= 0. If there is some correlation between the absolute values of mutation effects on the trait and on fitness, however, the genetic variance would be affected. The example shown in Figure 2b displays that when the correlation is intermediate, the impact is not large even with a large difference in distributions, i.e., β_{s}/β_{a} = ½, 2. For convenience, we consider only the situations where β_{s} =β_{a} in the following numerical investigation. As there exists a general relation (24), the high variation in the metric trait can be maintained only under weak selection; if mutation effects on fitness vary across loci substantially more genetic variance is induced, but the strong apparent stabilizing selection is still not achieved.
NUMERICAL RESULTS
The effective sizes of some natural populations, especially those of large vertebrates, are unlikely to be large and appear to be of the order of 10^{3}10^{4} (Frankham 1995; Yuet al. 2001). Furthermore, the distribution of mutation effects (s) could be leptokurtic, with most mutations having only slightly deleterious effects (Keightley 1994). Therefore N_{e}s may not always be much larger than unity and the dynamics of most mutations in natural populations may actually be dominated by drift rather than selection, so it is important to consider population size and genetic drift in models on the maintenance of polygenic variation.
The analytical expressions for V_{G} and V_{s} are difficult to obtain when N_{e}s ∼ 1, which might be typical values for most mutations in natural populations. The calculations using (1316) when the correlations between a and s are not unity show that if the continuous distribution of mutant effects on fitness is assumed, the genetic variance continues to increase as the effective population size increases. This is still true even for other distributions (e.g., normal) of fitness effects in which nearly neutral mutants are not predominant. If the mutant effects on fitness are discretized with the minimum effect Δs, then the genetic variance and the strength of apparent stabilizing selection would approach the asymptotes that were determined by Equations 17″ and 18″ and are shown in Figures 3 and 4. The minimum effect Δs in Figures 3 and 4 was set to 10^{6}. If it was changed to another value, e.g., 10^{8}, the trend of V_{G} and V_{s} with N_{e} for β_{s} ≤ 1 would be the same except that V_{G} and V_{s} would asymptote at a population size of ∼10^{8} rather than ∼10^{6}.
Experimental data regarding the joint distributions of both mutation effects on the metric trait and on fitness or even the marginal distributions are scant (e.g., Mackayet al. 1992; Keightleyet al. 2000). An important theoretical question is whether genetic variation and strength of apparent stabilizing selection are sensitive to the shape of distributions of mutation effects on the metric trait and fitness. For this purpose the situations of different β_{s} have been investigated, and the results are shown in Table 2. The difference is evident. If the correlation ρ< 1 and ε_{s} = 0.05, the genetic variance maintained at MSB decreases (e.g., from 0.87 to 0.13 for ρ= 0) and the selection becomes strong (e.g., V_{s} decreasing from 4580 to 1204 for ρ= 0) as the value of the shape parameter β_{s} increases (e.g., from 0.125 to 1.5). If ε_{s} = 0.005, the trend of V_{G} and V_{s} is the same as that for ε_{s} = 0.05 except for a maximum between β_{s} = 0.125 and 0.25 (cf. Figure 2a). If the correlation is equal to unity, the opposite trend occurs; i.e., V_{G} and V_{s} increase with β_{s} (cf. Table 1). It is clear in Figures 3 and 4 that V_{G} increases monotonically and approaches an asymptote as the population size increases for all the values of the shape parameter β_{s}. If β_{s} ≤ 1, V_{G} approaches an asymptote as the population size reaches ∼1/Δs. If β_{s} > 1, V_{G} approaches an asymptote independently of the size of the minimum fitness Δs assumed. This, in agreement with the analytical results above for infinite populations (Figure 2 and Table 1), results from the sharp difference between the distribution of mutation fitness effects of β_{s} > 1 and β_{s} ≤ 1. The former has few nearly neutral mutants while the latter has predominately mutants of s ∼ 0 (see Figure 1). Thus the differences between β_{s} > 1 and β_{s} ≤ 1 are qualitative, while the differences among values of β_{s} ≤ 1 or among values of β_{s} > 1 are only quantitative.
The way in which V_{s} approaches its asymptote with the effective population size N_{e} is more complicated (see Figure 4), as V_{s} can increase or decrease with N_{e}. For a finite population with a fixed flux of mutations, the strength of apparent stabilizing selection is determined by the interplay of the genetic drift and selection. Because the strength of selection is dependent on N_{e}s rather than purely on s, selection is predominant and genetic drift can be ignored only when the population size is large. When the population size is not large, the interplay between genetic drift and selection is complicated, which may lead to a different relationship between V_{s} and N_{e} for different values of other parameters [e.g., variability of fitness effects of mutations (ε_{s}), the correlation (ρ), and the mutation rate (λ)]. With ρ= 1, V_{s} decreases to its limiting value as N_{e} increases. For the situations with ρ< 1, the trend of V_{s} with N_{e} is complicated. If there are few nearly neutral mutants (i.e., β_{s} > 1), V_{s} decreases and asymptotes as N_{e} increases, due to the fact that N_{e}s increases with N_{e} and soon predominates over the genetic drift. If there are predominantly neutral mutants (i.e., β_{s} ≤ 1), V_{s} first decreases and approaches a minimum and then increases to a limiting value with N_{e}. In this case, N_{e}s may not significantly increase with N_{e} because most mutants are neutral or slightly deleterious. If the population size is too small, the dynamics of mutant alleles are controlled mainly by genetic drift. As N_{e} increases, selection first becomes stronger and then weakens and V_{s} finally asymptotes as N_{e}Δs > 1.
It is also evident in Figures 3 and 4 that V_{G} and V_{s} decrease as the correlation, ρ, or ε_{s} increases, in agreement with analytical results for infinite population (see Figure 2a and Table 1). As C(s) < 4λ/s, the absolute value of covariance of relative fitness and the squared deviation is <V_{m} for a finite population (cf. Equation 19), which sets up a constraint between V_{G} and V_{s} (see Equation 24). Figures 3 and 4 show that V_{G} can be high enough but V_{s} is always >10^{3} (i.e., the inverse of V_{m}).
As long as the mutational variance increment per generation is given as
DISCUSSION
Comparison with other models of maintenance of variation: A general pleiotropic model of variation maintained at MSB has been analyzed in this article. The mutants affecting the metric trait of interest also have a deleterious effect on the individual who carries them, and so, because extreme genotypes tend to be less fit, the metric trait appears to be under stabilizing selection. Assuming additive gene action across and within loci and linkage equilibrium, the genetic variance and strength of apparent stabilizing selection have been obtained. The mutation effects on the metric trait and on fitness both vary among loci and are assumed to be distributed as a bivariate gamma with any shape parameter. By employing diffusion approximations under the infinite independent loci model (Kimura 1969), formulas for V_{G} and V_{s} have been obtained, and results are in agreement with previous Monte Carlo simulations using a similar multilocus model (Keightley and Hill 1990). Compared to Keightley and Hill (1990) who assumed the heritability of a typical value ^{1}/_{3}, the heritability in this study varies with the genetic variance and is given by V_{G}/(V_{G} + V_{e}).
Analysis shows that the unlimited increase of V_{G} and V_{s} (Keightley and Hill 1990) disappears only when the shape parameter of distribution of the mutation effects on fitness is greater than one, in which there are few nearly neutral mutations. To avoid the unlimited increase of V_{G} for the highly leptokurtic distribution of mutation effects on fitness (i.e., β_{s} ≤ 1), fitness effects were assumed to have discrete values, s = nΔs, n = 1, 2,... with the minimum effect Δs > 0. Analysis for the infinite population limit reveals that V_{G} and V_{s} are proportional to the product of (Δs)^{β}^{s}^{1} and E(s)^{β}^{s}. Compared with Barton’s (1990) results, variation in fitness effects can induce much higher genetic variance at MSB for the population with the same mean fitness effect E[s]. If only a few deleterious alleles affecting the metric trait are segregating in the population, the apparent stabilizing selection is weaker than that of the houseofcards rare allele model (Turelli 1984; Barton 1990). If there are numerous mutant alleles segregating in the population, the constraint on V_{G} and V_{s} is similar to that of other pleiotropic models (Kondrashov and Turelli 1992; Gavrilets and de Jong 1993; Tanaka 1996), approximated by V_{s} = V_{P}/V_{m} (in the traditional definition of V_{s}: V_{s}/V_{G} = V_{G}/V_{m}).
The influence of the correlation between absolute values of mutation effects on the metric trait and fitness on V_{G} and V_{s} was assessed. In the extreme case of ρ=1, the selection becomes strongest and correspondingly the variance maintained in the metric trait shrinks. In general, an increase in the correlation leads to reduction in both V_{G} and V_{s}, as found by Keightley and Hill (1990). Numerical investigations on the dependence of V_{G} and V_{s} on the effective population size N_{e} show that V_{G} increases to an asymptote as N_{e} increases and reaches an order of 1/Δs. This prediction implies that the genetic variances depend greatly on N_{e} if N_{e} < O(1/Δs). In other words, 1/Δs would be a measure of the sensitive size of population to the genetic variances if leptokurtically distributed mutations were assumed.
As (1316) show, the genetic variance and strength of apparent stabilizing selection depend only on the squared deviations of mutation effects on the metric trait from the optimum. Thus the assumption of a symmetrical distribution of the mutation effects on the metric trait about zero is not of significance, confirmed by Monte Carlo simulations, but if the mean effect differs from zero, random genetic drift would lead to a directional change in the population mean. The impact of dominance of mutant alleles is not important if the degrees of dominance of the mutant effects on the metric trait and on fitness are the same. If they differ and if, for example, effects on fitness are purely recessive while effects on the metric trait are partially dominant, the genetic variance would increase and the apparent stabilizing selection would become weaker.
Comparison with observations: Although mutations are important to many phenomena and processes, including the maintenance of variability, estimates of mutation rate (λ) and average mutation effects (ε_{s}, ε_{a}) are imprecise. Data suggest that the total deleterious mutation rate is >1 in mammals and ∼1 in flowering plants (Kondrashov 1998). The scanty data for multicelluar eukaryotes are consistent with any value of λ between 0.1 and 100 (Kondrashov and Turelli 1992). 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; 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] in the range 0.01 and 0.05 (Wlochet al. 2001).
Estimates of the strength of natural selection in natural populations have been summarized by Turelli (1984), Endler (1986), Crow (1989), and Kingsolver et al. (2001). Endler’s (1986) survey indicates that stabilizing selection in natural populations is quite common and strong, with typical estimates of the strength V_{s} < 20V_{e} (Turelli 1984). The recent synthesis by Kingsolver et al. (2001) shows, however, the distribution of estimated quadratic selection gradients is symmetric about zero, so the mean is nearly zero, with the mean of the absolute values ∼0.1. Further, only ∼16% of the negative values of quadratic selection gradients are reported as significantly different from zero at p = 0.05 and in most cases where there are significant values there is also significant directional selection on the same trait. Thus the estimates provide little evidence for detectable stabilizing selection, given the limited power of the available evidence. Kingsolver et al. (2001) thus suggest that stabilizing selection is typically quite weak and not more common than disruptive selection.
To maintain abundant heritability the genetic variance should be of the order of 10^{3}V_{m} as experimental data typically show V_{m} ∼ 10^{3} V_{e} (Hill 1982; Houleet al. 1996; Lynch and Walsh 1998). This requires in Barton’s (1990) model that E[s](=Δs) ∼ 2 × 10^{3}, which is much smaller than the observed values ∼0.040.08. However, our model can produce abundant heritability for such values if the minimum fitness effect is assumed to be in the range of 1/10001/100 of the average effect (see Figure 2). These values of Δs, albeit being smaller than current experiments can detect (Wlochet al. 2001), do not seem unreasonable. Thus allowing deleterious effects to vary among alleles affects the results and gives a possible explanation of abundant heritability.
For a population of size N_{e} = 10^{3} and the mutation rate and average mutation effect on fitness in the ranges suggested by the experimental observations, the expectations of V_{s} (see Table 2 and Figures 4 and 5) are much larger than the typical values suggested by Turelli (1984). Moreover, there exists a general constraint, V_{s} ≥ V_{P}/V_{m} for infinite populations. This leads to the same conclusion as other pleiotropic models (Barton 1990; Kondrashov and Turelli 1992; Gavrilets and de Jong 1993; Tanaka 1996) that mutationselection balance cannot simultaneously explain both abundant heritability and strong stabilizing selection if V_{m} ∼ 10^{3} V_{e}.
Nevertheless, if the suggestion of Kingsolver et al. (2001) that the apparent stabilizing selection is “typically quite weak” is true, then this contradiction between the heritability and stabilizing selection disappears. The pleiotropic model studied here has no difficulty in providing a satisfactory explanation to maintenance of variation in the metric trait that is under quite weak apparent stabilizing selection. A “pure” pleiotropic model such as investigated here, in which there is no real stabilizing selection or other force maintaining the variance of the metric trait, would, however, lead to drift in its mean due to random fixation of mutants.
As pointed out by Kondrashov (1998) and Kingsolver et al. (2001), the estimates of mutation rates and selection effects are not very reliable. In the theoretical investigations of polygenic variation and stabilizing selection, both variances and strengths at MSB are assessed simultaneously. However, data on mutation rates and effects, heritabilities, and strengths of stabilizing selection that were used to compare with the theoretical predictions are collected, in most cases, from separate experiments. If the rates and effects of the base change in DNA sequences were universal, this might not cause any serious deviation. Although difficult, it would be desirable to design an experiment in which both mutation and selection parameters could be estimated with adequate precision at the same time.
APPENDIX A
Simulation of mutant effects on the metric trait and on fitness: Effects of mutant alleles are sampled from a continuous bivariate gamma distribution, h(a, s), with parameters ε_{a}, β_{a}, ε_{s}, β_{s}, and ρ using algorithm GTVR (Schmeiser and Lal 1982) as a = (Y_{1} + Y_{3})/α_{a} and s = (Y_{2} + Y_{3})/α_{s}, given three independent gamma random variables (Whittaker 1974): Y_{1}, Y_{2}, and Y_{3} with shape parameters
APPENDIX B
The functions ϕ_{i} of α_{s}Δs are defined as
For β_{s} > 1, Ψ_{i}(i = 1,..., 3) approximate C_{1}(α_{s}Δs)^{1β}^{s}, C_{1}(β_{s}  1)(α_{s}Δs)^{β}^{s}, C_{1}(β_{s}  1)β_{s}(α_{s}Δs)^{1β}^{s}, C_{1}(β_{s}  1)β_{s}(β_{s} + 1)(α_{s}Δs)^{2β}^{s}, and C_{1}(β_{s}  1)β_{s}(β_{s} + 1)(β_{s} + 2)(α_{s}Δs)^{3β}^{s}. Finally we have for β_{s} > 0, ϕ_{1} ≈ β_{s}/(α_{s}Δs), ϕ_{2} ≈ β_{s}(β_{s} + 1)/(α_{s}Δs)^{2}, ϕ_{3} ≈ β_{s}(β_{s} + 1)(β_{s} + 2)/(α_{s}Δs)^{3}, and ϕ_{1} ≈ [C_{0}/((1  β_{s})C_{1})](α_{s}Δs)^{β}^{s}  α_{s}Δs/(1 β_{s}) (0 < β_{s} < 1); ϕ_{1} ≈ α_{s}Δs ln(1/α_{s}Δs) (β_{s} = 1); ϕ_{1} ≈ α_{s}Δs/(β_{s}  1) (β_{s} > 1).
Acknowledgments
We are grateful to Nick Barton, Peter Keightley, Sergey Gavrilets, Ruth Shaw, and an anonymous reviewer for helpful comments. This work was supported by a grant from the Biotechnology and Biological Sciences Research Council (R35396).
Footnotes

Communicating editor: R. G. Shaw
 Received November 21, 2001.
 Accepted February 18, 2002.
 Copyright © 2002 by the Genetics Society of America