Abstract
It has been observed repeatedly that the distribution of new mutations of a quantitative trait has a kurtosis (a statistical measure of the distribution's shape) that is systematically larger than that of a normal distribution. Here we suggest that rather than being a property of individual loci that control the trait, the enhanced kurtosis is highly likely to be an emergent property that arises directly from the loci being mutationally nonequivalent. We present a method of incorporating nonequivalent loci into quantitative genetic modeling and give an approximate relation between the kurtosis of the mutant distribution and the degree of mutational nonequivalence of loci. We go on to ask whether incorporating the experimentally observed kurtosis through nonequivalent loci, rather than at locus level, affects any biologically important conclusions of quantitative genetic modeling. Concentrating on the maintenance of quantitative genetic variation by mutationselection balance, we conclude that typically nonequivalent loci yield a genetic variance that is of order 10% smaller than that obtained from the previous approaches. For large populations, when the kurtosis is large, the genetic variance may be <50% of the result of equivalent loci, with Gaussian distributions of mutant effects.
EXPERIMENTAL measurements of mutant effects on a polygenic trait have consistently found that the distribution of mutant effects is leptokurtic, with a kurtosis (fourth central moment divided by the squared variance) that is in excess of the value 3 associated with a normal distribution. A prominent finding was the work on Pelement insertions affecting Drosophila bristle number (Mackayet al. 1992; Lymanet al. 1996). This work yielded mutant distributions that were highly leptokurtic—with a kurtosis of order 40. In a recent review, GarciaDorado et al. (1999), while confirming this result, concluded that the extreme kurtosis of the sternopleural bristle mutations is not typical of other quantitative traits. However, all of the fitness and morphological traits they reviewed had distributions of mutant effects more leptokurtic than a normal distribution. A similar pattern appeared in the nine Drosophila characters assayed by Keightley and Ohnishi (1998). The range of experimental protocols and statistical techniques used in this work supports the notion that this pattern is not an experimental artifact and that a leptokurtic distribution of mutant effects is a real phenomenon.
A number of theoretical treatments have dealt with the implications of this kurtosis for biologically important quantities. The two most notable are the amount of genetic variance maintained by populations under mutationselection balance (Fleming 1979; Keightley and Hill 1988; Bürger and Lande 1994; Bürger 1998) and the variance of the genetic variance among replicate lines and thus the predictability of dynamics under selection (Keightley and Hill 1989; Bürger and Lande 1994; Mackayet al. 1994). We note that an assumption, underlying all of these models, is that there are identical distributions of mutant effects at each locus. These distributions are necessarily leptokurtic, to yield the empirically observed kurtosis of the overall mutant distribution.
Here, by contrast, we propose an alternative model that suggests that the observed kurtosis of the distribution of mutant effects may be a property that emerges only at the trait level, regardless of the distribution of mutant effects at individual loci. The model relies crucially on the empirically motivated assumption that the loci contributing to a trait have different mutational effects and thus are nonequivalent.
There is abundant evidence suggesting that quantitative trait loci (QTL) are mutationally nonequivalent. Studies have shown that the proportion of phenotypic variance contributed by different QTL can vary widely (Falconer and Mackay 1996, ch. 21). More specifically, the evidence suggests that mutations at the overwhelming majority of QTL contribute small fractions of the phenotypic variance, while only a small number make more substantial contributions (Bostet al. 1999). Despite the evidence, such nonequivalence has been incorporated only rarely into population/quantitative genetic modeling. This is surely because parameterizing each locus separately makes models unwieldy, and overly cumbersome models often obscure the biological point being made.
We present a new method of incorporating mutational nonequivalence of loci that avoids these problems. This is achieved by choosing the various mutational properties of each locus, at random, from a particular probability distribution. We then argue that certain observable quantities, such as the distribution of mutant trait effects, are “self averaging.” As such, we can replace these observable quantities by their average over the randomly chosen mutational properties. This reduces the number of free parameters in the problem to the (assumed small) set required to specify the probability distribution of mutational properties of individual loci. In addition to an economy of description, the small set of free parameters can also be thought of as encapsulating the degree of nonequivalence of loci. Thus by exposing the influence of these parameters we can make explicit the influence of nonequivalent loci. In the first part of this article, we show explicitly the influence of these parameters on the distribution of new mutations. Having demonstrated that the kurtosis of the distribution of mutant effects may emerge through nonequivalent loci, we follow this up, in the second part of this article, by exploring other biological implications of our way of incorporating mutational kurtosis into the model. In particular we compare the implications for the level of genetic variance maintained at mutationselection balance with the method of incorporating mutational kurtosis used in the theoretical articles cited above.
MODEL AND RESULTS
Distribution of mutant effects: We use the continuumofalleles model introduced by Crow and Kimura (1964) and analyzed in the context of the maintenance of genetic variation by, among others, Kimura (1965), Lande (1976), and Turelli (1984). In Crow and Kimura's model of allelic mutation, the effect of a mutated offspring's allele, y′, is given by the sum of the parental allelic effect, y, and a mutation effect x, thus y′= y + x. The effect of each new mutation at locus i is drawn from a continuous probability distribution, f_{i}(x). It is assumed that the quantitative trait in question is controlled by n additively contributing diploid loci. Thus an individual's genotypic value, G, is given by
The distribution of single mutation effects for the trait, F(x), is a weighted sum over the mutant distributions at each locus, with the weights proportional to the allelic mutation rate at each locus. In terms of the allelic mutation rate at the ith locus, μ_{i}, the mutation rate of the trait is
For simplicity, we assume that the distribution of mutant effects at locus i, f_{i}(x), is a parameterization of a “reference distribution,” g(z). This reference distribution has the properties that it (i) is normalized to unity, (ii) has unit variance, and (iii) has zero mean. The results given below apply for a range of distributions with these properties, but for concreteness, we introduce a specific form of the reference distribution, the Gaussian. This is the form of f_{i}(x) adopted by Crow and Kimura (1964) and is given by
We derive the allelic mutation distribution at locus i, f_{i}(x), from g(z) by incorporating a parameter b_{i} (where ∞ > b_{i} >–∞) and a parameter v_{i} (where v_{i} > 0), as
Initially, let us confine ourselves to the case of nonbiased (or uniformly biased) mutation. As such, we set all the b_{i} to zero and confine ourselves to variation only in the v_{i} (the more general case is discussed below).
To obtain the distribution of mutant effects, we substitute Equation 3 into Equation 1, yielding
Let us now allow variation in the mutational biases, b_{i}, to be taken into account. For independently chosen b's, with no correlation with other parameters, we find (details not given) a kurtosis of mutant trait effects of κ ≃ (κ_{0}[1 + CV^{2}(v)] + 6β + κ_{b} β^{2})/(1 +β)^{2}, where β= Var(b)/v and κ_{b} is the kurtosis of the distribution of the b's. When β ≪ 1, the effect of differences in biases across loci is negligible and we recover the result for the kurtosis of mutational effects given in Equation 7. However, in the opposite limit, β ≫ 1, the differences in bias dominate and the kurtosis is approximately equal to that of the b's. For intermediate values of β, the dependence of κ on β is nonmonotonic when (κ_{b} + κ_{0}[1 + CV^{2}(v)] – 6)/(κ_{b} – 3) ≥ 1. In the following, we assume that β ≪ 1 and that mutational biases have little effect on the results. This, although plausible, is mainly a convenience, as we have little empirical evidence to guide us as to an appropriate form for their distribution. Observations such as Clayton and Robertson's (1964) finding that Drosophila bristlenumber mutations do not change the trait mean can tell us little about the bias at any particular locus.
In contrast to the biases, there is some empirical evidence available for the distribution of the mutational variances, P(v). The aforementioned results from quantitative trait loci (QTL) analysis suggest that the vast majority of QTL contribute a very small proportion of phenotypic variance, while a much smaller number contribute a substantial proportion (Falconer and Mackay 1996; Bost et al. 1999, 2001). This suggests that the distribution of mutational variances may be L shaped; a candidate is the onesided gamma distribution, P_{gamma}(v; q, λ), that vanishes for v < 0, and for v > 0 is given by
Using Equation 8, we can go further and evaluate Equation 4, the averaged distribution of mutant effects, yielding the exact result
Nonequivalent loci and the maintenance of genetic variance: It is now appropriate to ask whether generating the empirically observed kurtosis using nonequivalent loci, rather than incorporating it at locus level, via equivalent loci, has a significant effect on other quantities of biological interest. We concentrate on the maintenance of quantitative genetic variation in a single phenotypic trait, through the balance between mutation and stabilizing selection.
Keightley and Hill (1988) suggested that increasing the mutational kurtosis could have a dramatic effect on the amount of genetic variance maintained in small populations, but their claim was disputed by Bürger and Lande (1994) whose simulation results suggested that it had very little effect. We concentrate on very large, effectively infinite populations and compare the results of three classes of mutant distributions. The first case is mutationally equivalent loci each with a Gaussian distribution, henceforth abbreviated to EG; the second case is equivalent leptokurtic loci, henceforth EL; and the third is nonequivalent Gaussian loci, NG. Table 1 summarizes the differences between the three cases.
The classic analyses of Crow and Kimura's model (Kimura 1965; Lande 1976; Turelli 1984) deal only with EG loci, while extended analyses by Fleming (1979) and Bürger (1998) treat EL loci to some extent, showing how mutational kurtosis enters, when small, as a correction to the approximations given in the earlier articles. Since substantial values of the kurtosis make analytical treatment difficult, we have solved the relevant equations numerically for all three cases.
We assume randomly mating populations, with discrete generations, and no sexual dimorphism. Furthermore, we follow all of the relevant articles cited above, by making the approximation of global linkage equilibrium (cf. Turelli and Barton 1990). Under these assumptions, the equilibrium genetic variance associated with the trait, V_{G}, can be determined by calculating the variance maintained at a single haploid locus, which, for locus i, we denote by
If the average fitness of an individual with genotypic value G is given by 1 – sG^{2}, we can find
To allow a meaningful comparison of results for the three classes of mutant distribution, they were generated as follows. First we generated a sample of n mutational variances, (v_{1}, v_{2},..., v_{n}) from the gamma distribution (Equation 8), where n is the number of loci. For all three classes of mutant distribution, we assumed, for simplicity, that the mutation rates at all loci were equal and the biases were all zero.
For the NG loci, the distribution of mutant allelic effects at locus i, namely f_{NG,}_{i}(x), was Gaussian, with a variance v_{i} (see Equation 3, with all b_{i} set to zero). As such, the overall (i.e., trait) mutant distribution,
For the EL loci, the distributions of allelic effects at each locus were constructed to be exactly equal to the overall (i.e., trait) mutant distribution in the NG case. Thus, the distribution of mutant allelic effects at locus i is given by f_{EL,}_{i}(x) = F_{NG}(x) for all i. As such, the overall (i.e., trait) mutant distributions in EL and NG cases are identical, F_{NG}(x) = F_{EL}(x), although resulting from very different distributions at locus level.
For the EG loci, each locus had the same mutational variance, which was set equal to
As a result of the way the distributions of mutations were determined, the amount of variation contributed by new mutations, V_{M}, was identical in all three cases and was given by V_{M} = 2nμ〈v〉.
Rather than present a full numerical investigation, we make our point with a series of examples. Since V_{M} is one of the most wellcharacterized parameters in quantitative genetics, we chose the other parameters such that V_{M} was set to the “typical level” of V_{M} = 10^{–3}, with the environmental variance set to unity throughout (Lynch 1988; Houleet al. 1996). To aid comparison with previous work, we set the strength of selection, s, to equal 0.025 (Turelli 1984) and took the expected value of mutational variance, v, to equal 0.05. This last value, often used in theoretical work, stems from Lande's (1976) extrapolation from the data of Russell et al. (1963). Since V_{M} = 2nμ〈v〉 = 10^{–3}, the value 〈v〉≈ 0.05 approximately requires 2nμ= 0.02 and this left us the choice of generating the required V_{M} through either an implausibly large number of loci or an implausibly high mutation rate. With this in mind, we examined two regimes, first n = 2000 and μ= 10^{–5}, and second, n = 200 and μ= 10^{–4}. See Turelli (1984) and Lynch and Walsh (1998, Chap. 12) for a full discussion of these and other parameter values.
The results given in Figure 2 involved drawing the v_{i} values from the distribution P_{gamma}(v; q, v/q) for the three values of the shape parameter q used in Figure 1 that encompass the range of experimentally observed kurtoses (GarciaDoradoet al. 1999).
These results and all other combinations we tried suggest strongly that V_{G}(NG) < V_{G}(EL) < V_{G} (EG), where V_{G}(EG) denotes the genetic variance maintained by EG loci and likewise for the other cases. In the most extreme case considered, however, the result for equivalent leptokurtic loci, V_{G}(EL), is only ∼12% smaller than the value of V_{G}(NG) that followed from nonequivalent loci (via the method presented in this work). Thus while there are differences in the genetic variances of “equivalent leptokurtic” and “nonequivalent Gaussian” loci, these are not particularly large. There does thus not seem to be a significant sensitivity of the genetic variance on the precise way mutational leptokurtosis is incorporated into the model.
A useful benchmark result is the house of cards approximation (Turelli 1984), which is closely related to a scheme of mutation introduced by Kingman (1978). This approximation applies to loci with vs/μ ≫ 1 and when applicable, yields a genetic variance of 2nμ/s. It works tolerably well for the EG loci in the regime where n = 2000 and μ= 10^{–4} and extremely well for the EG loci in the regime where n = 2000, μ= 10^{–5} (see Figure 2).
For both EL and NG loci, the genetic variance can, in some cases, be <50% of the genetic variance of EG loci and thus of the house of cards approximation. The reason for this is different in the two cases.
For EL loci Bürger (1998) proved that the house of cards approximation will always be an overestimate when the locus distributions are leptokurtic. We have demonstrated here that the correction can be substantial when the distributions are highly leptokurtic (but still within the empirically observed range of kurtoses).
For NG loci, Bürger's results do not apply, since at each locus, the distribution of mutant effects is itself Gaussian. There are, however, a range of mutational variances present in the loci controlling the trait, and the genetic variance is a sum over the genetic variances arising from loci with different mutational variances. Turelli's result applies well only to loci for which v ≥ 10μ/s. In this work, the expected proportion of loci lying outside the house of cards regime is
DISCUSSION
There is a tradition within quantitative genetic modeling of assuming that all loci can be treated as fully equivalent “average” loci. Although this may be adequate for many practical purposes, in some cases, it can have a significant effect on the results of the analyses (Gimelfarb 1986; Hastings and Hom 1990). A second approach, which avoids the assumption of equivalence, is to categorize loci as either “major” or “minor,” i.e., of having alleles with large or small phenotypic effect, and to treat each in a qualitatively different fashion (Lande 1983). Here, we have presented a third strategy that treats major and minor loci in a unified fashion, as extremes of a continuum. We could call those loci with the highest c_{i} values major loci in our model, although due to continuity of f_{i}(x), they are still capable of generating mutations with close to zero phenotypic effect. The existence of “isoalleles” at major loci offers empirical support for this (Lynch and Walsh 1998, pp. 322–323). We have shown that the reduced set of free parameters needed for this model can be viewed as encapsulating the degree of nonequivalence of loci. Thus, by exposing the influence of these parameters we can make explicit the implications of nonequivalent loci.
With this model, we have shown that the observed kurtosis in the distribution of mutant effects can plausibly be attributed to variation in the mutational properties of the loci, rather than to leptokurtic distributions at each locus. Conversely, we suggest that the distributions at each locus [denoted here by f_{i}(x)], are likely to have lower kurtoses than that of the overall distribution, F(x). Thus, to the extent that any biological prediction depends on high levels of kurtosis at locus level, that prediction would have to be revised in the direction of smaller effects from kurtosis. This conclusion does require that the assumptions leading to Equation 7 hold, at least roughly, and perhaps the most cautious conclusion is that knowledge of the distribution of mutations on the trait allows very little to be inferred about the distribution of mutant effects at individual loci.
We went on to examine the maintenance of genetic variance by mutationselection balance, since the role of mutational kurtosis here has been controversial. Our findings show that, in large populations, substantial differences are possible. In particular, for values of mutational kurtosis measured empirically, the reduction can be >50%. Furthermore, incorporating this kurtosis through nonequivalent loci, rather than at locus level, leads to a further reduction. However, our findings indicate that the differences between the results for “nonequivalent Gaussian” loci and “equivalent leptokurtic” loci are not large, typically ∼10%.
In conclusion, although a substantial proportion of standing genetic variation may result from mutationselection balance (Bürger and Lande 1994), our findings make it even more difficult to reconcile the results of simple models with the high heritabilities and strong selection observed in nature (see Turelli 1984). As such, our results make alternative explanations likely. Prominent candidates include the assertion that much of the observed stabilizing selection is merely “apparent,” the result of deleterious pleiotropic effects of mutations affecting quantitative traits (see Barton 1990; Gavrilets and DeJong 1993; Nuzhdinet al. 1995), and the suggestion that population subdivision (e.g., Goldstein and Holsinger 1992; Lythgoe 1997) or environmental change (Bürger 1999; Waxman and Peck 1999) may play an important role.
Acknowledgments
We thank James Crow, John Maynard Smith, and Adam EyreWalker for helpful discussions and the two anonymous reviewers for useful suggestions. This research was supported by the Biotechnology and Biological Sciences Research Council (United Kingdom) under grant 85/G11043 and by the University of Sussex under its Graduate Teaching Assistantship Scheme.
APPENDIX
Here, we specialize to the case of nonbiased mutation, where all b_{i} are set to zero. We estimate the typical error incurred by using the v averaged distribution of mutations, F(x), in place of F(x). We approach this by focusing on averaged deviations of moments. Let M_{a} denote the ath moment of
APPENDIX B
Here, we show the general validity of the inequality κ ≥ κ_{0} that relates the kurtosis of the distribution of mutant trait effects, κ, and the kurtosis associated with the distribution of mutant effects at a single locus,
To prove the inequality for the distribution
Footnotes

Communicating editor: D. Charlesworth
 Received September 10, 2001.
 Accepted March 11, 2002.
 Copyright © 2002 by the Genetics Society of America