The Journal of Heredity 2001:92(3)
© 2001 The American Genetic Association 92:274-276
Brief Communication |
Extension of the Castle–Wright Effective Factor Estimator to Sex Linkage and Haplodiploidy
From the Department of Biology, University of Rochester, ochester, NY, 14627. C. Jones is now at the Section of Evolution and Ecology, #1080, University of California, Davis, CA, 95616.
Address correspondence to the author at the current address above or E-mail: cojo{at}ucdavis.edu.
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Abstract |
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 Top Abstract IntroductionHaplodiploid SpeciesSex LinkageReferences |
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The Castle–Wright effective factor estimator gives a minimum estimate of the number of genes underlying complex traits. Because the Castle–Wright estimator does not rely on genetic markers, it is especially useful in genetically undeveloped species. In this article I describe two extensions of this estimator. The first corrects the estimator in heterogametic (XY) species with a partially sex-linked trait. In this case the traditional estimator underestimates gene number in F2 males and overestimates it in F2 females and backcross females and males. The second extension adapts the Castle–Wright equation to haplodiploid species.
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Introduction |
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TopAbstractIntroductionHaplodiploid SpeciesSex LinkageReferences |
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Since its creation by Wright (see Castle 1921), the Castle–Wright equation has become a widely used tool for estimating the number of genes affecting complex traits. Knowledge of the number of genes is important for two reasons. First, this information is important to quantitative genetics theory. As Lynch and Walsh (1998:231) point out, much of quantitative genetics theory assumes that many genes underlie a phenotypic trait. Therefore it is important to know how well and often this assumption is met. Second, accurate estimates of gene number may answer several important evolutionary questions. For example, current debate over the genetics of adaptation centers on the number of genes typically involved in adaptation (Bradshaw et al. 1998; Orr and Coyne 1992; Tanksley 1993) Not surprisingly, several recent studies of the genetics of adaptation have employed Wright's equation (Hatfield 1997; Sezer and Butlin 1998).
The Castle–Wright equation is especially useful in genetically undeveloped species because it does not require genetic markers. The data required are entirely phenotypic and derive from either an F2 cross or a backcross between two phenotypically divergent populations or species. For an F2 cross, the segregation variance of the F2(
S2)—which is estimated from the phenotypic variance—and the mean trait values of the parents
(
,
) are used to estimate the number of genes involved:
 | (1) |
This equation rests upon several simplifying assumptions. First, it assumes that all alleles behave additively (h = 1/2), all loci are unliked, and all alleles have equal effects. It also assumes the two parental strains are homozygous for alternative alleles at all loci affecting the trait and all chromosomes are diploid. If these assumptions are not met, the equation usually underestimates the true number of genes. For example, tightly linked loci are treated as a single effective factor by the Castle–Wright equation. Therefore the Castle–Wright estimate is considered a minimum estimate of gene number.
Despite these restrictive assumptions, a couple of recent genetic analyses of quantitative traits have suggested that the Castle–Wright estimator (and its improved descendants) can be fairly accurate. In a study of Populus, Wu et al. (1997) showed that estimates of gene number from the Castle–Wright equations were usually close to the number of factors found in quantitative trait loci (QTL) analysis. Likewise, Gurganus et al. (1999) found that the Castle–Wright equation accurately estimated of the number of factors underlying a difference in bristle number in Drosophila.
Many improvements have been made to the Castle–Wright estimator, relaxing most of its key assumptions (reviewed in Lynch and Walsh 1998). Wright (1968) made several improvements. He modified the estimator for backcross data, showed how to estimate the effect of the largest factor involved, modified the equation for special cases of dominant and unequal allele effects, and presented methods for improving the estimation of the segregation variance, such as eliminating environmental variation.
Later, Lande (1981) proved that the Castle–Wright equation could be applied to natural populations—that is, to populations not homozygous for all relevant factors. He also developed an equation for the variance of the estimator,
 | (2) |
1 - 2). Later, Cockerham (1986) presented a correction for the sampling variance of parental means. More recently, Zeng et al. (1990; Zeng 1992) studied the impact of linkage and unequal allelic effects on the estimator. Ollivier and Janss (1993) improved Wright's corrections for dominance. Finally, Wu (1996) corrected the estimator for parents with factors of both increasing and decreasing effects (gene dispersion). Despite these efforts, the effects of deviations from diploidy on the Castle–Wright equation have been ignored (but see Chovnick and Fox 1953). In this article I consider the effects of violating this assumption in heterogametic species (i.e., in taxa in which the X chromosome is hemizygous in one sex) and in haplodiploid species. Because the case of haplodiploidy is simpler, I consider it first.
Note that for simplicity, I illustrate these extensions using the traditional Castle–Wright equation. These extensions can be combined with other improvements to the Castle–Wright estimator detailed in Lynch and Walsh (1998).
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Haplodiploid Species |
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TopAbstractIntroductionHaplodiploid SpeciesSex LinkageReferences |
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In haplodiploid organisms (such as bees and wasps), females are diploid and males are haploid. Although F2 and backcross females can be treated as normal diploids, males cannot; F2 males inherit their chromosomes solely from their mother and therefore carry a random assortment of grandparental chromosomes. Thus male F2 progeny are hemizygous at all loci. This lack of heterozygotes inflates the F2 segregation variance and leads to an underestimate of the number of effective factors (nef).
Correcting for this effect in F2 males is analogous to correcting for F1 sexual haploids (see Chovnick and Fox 1953). Because the F2 variance of haplodiploids is twice that of diploids, it follows that
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). However, Lande's equation for the variance needs modification. By the delta method, it can be shown that an approximate equation for the variance is
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I provided Equation 3 to Weston et al. (1999), who applied it to data on the genetics of wing size differences between two species of parasitic wasps (Nasonia). Using the traditional estimator, they would have found only 0.8 (SD = 0.38) effective factors contributing to the difference between these species. The corrected estimator suggests that the actual number is 1.6 (SD = 0.78), a number supported by their introgression data.
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Sex Linkage |
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TopAbstractIntroductionHaplodiploid SpeciesSex LinkageReferences |
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Heterogametic species, such as humans and Drosophila, are effectively haplodiploid for their sex chromosomes (for simplicity, I assume males are the heterogametic sex). If some of the factors affecting a trait of interest are X linked, the Castle–Wright estimator is not appropriate. Because the X is never heterozygous in males, the variance of heterogametic F2 males will be inflated. This causes the Castle–Wright equation to underestimate nef. Figure 1 shows this underestimation is large when data from F2 males are considered, such as genetic studies of male secondary sexual characteristics (see True et al. 1997).
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In contrast, the traditional estimator overestimates nef when data derive from F2 females. This is because F2 females always get the same grand-maternal X chromosome from their father. The variance of the F2 will therefore be smaller than expected, leading to an overestimation of nef (Figure 1).
For similar reasons, estimates of gene number from backcross progeny of crosses using F1 males are also biased. In this case the Castle–Wright equation overestimates nef in both sexes (Figure 1). However, employing F1 females in backcrosses avoids these problems.
Correcting the estimator for sex linkage requires estimating the contribution of the X chromosome to the phenotype. This can be done by comparing the means of F1 males to those of the F1 females. Under an additive model, F1 males resemble their mothers more than F1 females, as males are hemizygous for the maternal X. Therefore the contribution of the X can be estimated as
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As in haplodiploids, the corrected estimator is given by
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 | (7) |
 | (8) |
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As an example of the use of these estimators consider Val's (1977) analysis of head shape differences in Hawaiian Drosophila. Male head width differs considerably between D. heteroneura and D. silvestris and a fraction of this difference was sex linked (
27%). Curiously Val found that in backcross data, nef depended on which sex was used in the backcross. Estimates based on backcrosses using F1 males were always greater than estimates based on crosses using F1 females. For example, nef = 4 in backcrosses to D. silvestris using females versus nef = 15 in backcrosses using males. Applying the above corrections to Val's backcross data makes nef almost the same regardless of the sex used in the backcross (after correction, 4 versus 5.8).
Similarly, in an F2 analysis of pteridine content in the heads of tsetse flies, McIntyre and Goodling (1996) underestimated the number of genes underlying this trait by 45–50% (cross 1, 2.3 versus 3.4; cross 2, 1.8 versus 2.6).
In this article I have pointed out that the widely used Castle–Wright equation estimates of the number of effective factors may be biased in cases of haplodiploidy and heterogamety. I then presented two simple extensions to the estimator to correct for these cases.
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Acknowledgments |
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I thank H. A. Orr and D. C. Presgraves for helpful discussion and commments on the manuscript. Also, I thank J. H. Werren for sharing his data. This work was supported by the David and Lucile Packard Foundation (H. A. Orr), National Institutes of Health grant GM-51932 (H. A. Orr), and a Caspari Fellowship from the University of Rochester (to C.D.J.).
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Footnotes |
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Corresponding Editor: Bruce S. Weir
Received February 28, 2000
Accepted October 3, 2001
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References |
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TopAbstractIntroductionHaplodiploid SpeciesSex LinkageReferences |
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