Abstract
Standard genetic analyses assume that reciprocal heterozygotes are, on average, phenotypically identical. If a locus is subject to genomic imprinting, however, this assumption does not hold. We incorporate imprinting into the standard quantitativegenetic model for two alleles at a single locus, deriving expressions for the additive and dominance components of genetic variance, as well as measures of resemblance among relatives. We show that, in contrast to the case with Mendelian expression, the additive and dominance deviations are correlated. In principle, this correlation allows imprinting to be detected solely on the basis of different measures of familial resemblances, but in practice, the standard error of the estimate is likely to be too large for a test to have much statistical power. The effects of genomic imprinting will need to be incorporated into quantitativegenetic models of many traits, for example, those concerned with mammalian birthweight.
THE expression of a gene at a genomically imprinted locus depends on the parent from which it was inherited. For example, in most fetal tissues of all eutherian and marsupial species examined to date, the maternal copy of the insulinlike growth factor II (Igf2) gene is inactive, only the paternal copy being transcribed (DeChiaraet al. 1991; Giannoukakiset al. 1993; Pedoneet al. 1994; Vranaet al. 1998; McLaren and Montgomery 1999; Nezeret al. 1999; O’Neillet al. 2000). For most (if not all) imprinted loci, however, both copies of the gene are expressed in some tissues at some stage of development. In the embryogenesis of rats and mice, for instance, standard biallelic or Mendelian expression of Igf2 occurs in the choroid plexus and leptomeninges (DeChiaraet al. 1991; Pedoneet al. 1994). Moreover, imprinting may not entail the complete inactivation of a gene. In a study of human IGF2 from peripheral blood leukocytes, for example, some 10% of a sample of phenotypically normal individuals (i.e., not cancer patients) showed biallelic expression, although in all cases the level of maternally derived gene product was lower than that of paternal (Sakataniet al. 2001). Hence, from the individual’s point of view, imprinting is not manifested as simple haploid (i.e., monoallelic) expression; rather, an imprinted gene shows diploid expression, with maternal and paternal copies having different levels of expression.
Under standard Mendelian expression, the number of phenotypic classes at a locus with k alleles is k(k + 1)/2. Complete inactivation of one allele would reduce the number of phenotypic classes to k. The more general view of imprinting outlined above, however, means that reciprocal heterozygotes need not have the same average phenotype. Consequently, the number of phenotypic classes is k^{2}, greater than under Mendelian expression. This increase has a number of implications for standard populationgenetic processes and phenomena. For instance, because it discriminates among different phenotypic classes, natural selection acts differently (Pearce and Spencer 1992; Anderson and Spencer 1999; Spencer 2000). In this article we examine the difference imprinting makes to the standard twoallele singlelocus model of quantitative genetics. In particular, we are concerned with how genetic variance may be partitioned and the consequences of this partitioning for resemblances among relatives.
This article thus adds to the literature on the effect of sex differences on quantitative characters, previous work considering models of sexlinked inheritance (Bohidar 1964; James 1973; Grossman and Eisen 1989), sexdependent expression (Grossman and Eisen 1989), and haplodiploidy (Liu and Smith 2000). The only previous work on the quantitative genetics of imprinting of which we are aware is that of Hill and Keightley (1988). Although their model considered multiple loci, they limited their analysis to the special case of complete inactivation.
MODEL
We consider a locus subject to imprinting, which has two alleles, A_{1} and A_{2}. By denoting a genotype A_{i}A_{j}, we mean that the A_{i} allele is maternally derived and the A_{j} allele is paternally derived. Imprinting means that reciprocal heterozygotes may differ in their average phenotypes. In the case of complete inactivation of the maternally derived gene, for instance, the average A_{1}A_{2} phenotype is the same as that of the A_{2}A_{2} homozygotes, whereas the average A_{2}A_{1} phenotype is the same as that of A_{1}A_{1} homozygotes. Since most cases of imprinting show some degree of biallelic expression in some tissues at some stage of development, however, we do not assume that heterozygotes are phenotypically equivalent to one or another homozygote.
Genetic components of variance: Following standard genetic models (see, e.g., Falconer and Mackay 1996; Roff 1997; Lynch and Walsh 1998), let us assume that on some suitable scale (see Figure 1) the mean phenotype (called the genotypic value) of A_{1}A_{1} homozygotes is a and that of A_{2}A_{2} homozygotes a. With Mendelian expression all heterozygotes have the same genotypic score, usually denoted as d. Under imprinting, however, the two classes of heterozygotes have different genotypic values, say, d_{1} for A_{1}A_{2} and d_{2} for A_{2}A_{1}. When the maternal allele is completely silenced we have d_{1} = a and d_{2} = a and vice versa for total paternal silencing. It also seems reasonable that a ≤ d_{1}, d_{2} ≤ a, since partial inactivation of a gene is unlikely to produce a more extreme phenotype than that of the homozygote for the unimprinted copy. Nevertheless, we do not make this assumption in most of what follows.
The mean genotypic value over the whole population is given by
We can now calculate the breeding values for each of the four genotypic classes. A breeding value is defined to be twice the difference between the mean genotypic value of that class’s offspring and the population mean (Falconer and Mackay 1996). Under imprinting, these deviations are different for males and females because the genotypic classes arising from reciprocal crosses may be different. For example, the breeding value for A_{1}A_{1} males involves the product of the probability that its A_{1} sperm fertilize an A_{2} egg, q, with the genotypic value of the resultant A_{2}A_{1}, d_{2}. The breeding value for A_{1}A_{1} females, however, involves the product of the probability that its A_{1} eggs are fertilized by A_{2} sperm, q, with the genotypic value of the resultant A_{1}A_{2}, d_{1}. Hence the mean genotypic value of offspring of A_{1}A_{1} males is p · a + q · d_{2} and so the breeding value is given by
The dominance deviation for a genotypic class is defined as the difference between the genotypic deviation and the breeding value. Since the latter differs for males and females, so too does the dominance deviation. A little algebra gives the values shown in Table 1; again their mean is zero and when d_{1} = d_{2}, the sex difference disappears, and we recover the standard Mendelian values. Note also that, as is the case without imprinting, these values are independent of a and are zero when d_{1} = d_{2} = 0.
The overall genetic variance of the population is the variance of the genotypic deviations:
The additive genetic variances for males and females are given by the respective variances of their breeding values,
The dominance genetic variance for each sex is, by definition, the variance of dominance deviations. For males, this variance is given by
With Mendelian expression the breeding values and dominance deviations are uncorrelated, but this result does not hold under imprinting. The covariance for males is given by
The male and female correlations between breeding values and dominance deviations are therefore given by, respectively,
Resemblance between relatives: We can now calculate various correlations between relatives in terms of the above components of variance, comparing the expressions with the standard, Mendelian formulas. Take, for instance, the covariance between the genotypic values of fathers and their offspring assuming random mating, σ_{OPm}. We follow the treatment of Falconer and Mackay (1996) for an unimprinted locus, considering deviations from the population mean. Since the mean genotypic deviation of offspring is, by definition, onehalf the breeding value of the father, we want to find the covariance between G (= A_{m} + D) and ^{1}/_{2}A_{m}, which, by elementary statistical theory, is
Hence, the regression coefficient of mean offspring phenotypes plotted against that of their fathers is given by
Similarly, the covariance between maternal and offspring genotypic values is given by
Interpreting differences between β_{OPm} and β_{OPf} is problematic, however, since if the latter is larger it may be due to a maternal effect rather than imprinting. This problem can be alleviated somewhat if the correlation among halfsibs is also calculated. The covariance of halfsibs can be found by the same logic as above. Remembering that the covariance among offspring who share a mother but not a father is the variance of the genotypic means of those halfsib groups and that these means are onehalf the breeding values of the mothers, we have that the covariance is onequarter the mothers’ additive variance:
This latter value should be unaffected by maternal effects provided mating has been at random (Roff 1997), and we can use it and the regression of offspring against their fathers to obtain an equation for σ_{ADm}:
The significance of the value of c obtained from an experiment can be deduced by considering the sampling distributions of b_{OPm} and r_{HSPm}. For instance, if n offspring from each of N families are used in assessing the regression of offspring means on their fathers and τ is the correlation among offspring within families, the variance of b_{OPm} is approximately
Unfortunately, although simulations show that values of c are very close to normally distributed, this test probably lacks sufficient power to be useful in many situations. For example, under the null hypothesis of no imprinting (d_{1} = d_{2} = d), the smallest standard errors arose when there was no environmental variance (i.e.,
DISCUSSION
At a genomically imprinted locus, the maternal and paternal copies are differentially expressed. Hence, the mean phenotypes of reciprocal heterozygotes, identical under the rules of standard Mendelian expression, need no longer be the same. We show that this loss of symmetry destroys much of the simplicity that occurs in the standard singlelocus models of quantitative genetics and their wellknown measures of genetic variance and resemblances among relatives. Under imprinting, breeding values and additive genetic variances are different for males and females. Although male and female dominance deviations are also different, dominance genetic variances for males and females are identical. Breeding values and dominance deviations are no longer uncorrelated, which means that the genetic variance cannot be partitioned into the usual additive and dominance variances.
Under imprinting, offspring will more closely resemble the parent that does not downregulate expression in the genes it transmits. In the case of Igf2, for instance, the maternal copy is silenced in a large number of fetal tissues and so offspring phenotypes resulting from Igf2 expression should be more similar to those of fathers than those of mothers. Sakatani et al.’s (2001) estimate of the mean ratio of maternal to paternal human IGF2 expression of 0.102 suggests that, if we assume additive contributions from each gene copy, plausible parameter values for the genotypic values in our model are a = 1, d_{1} =0.8, and d_{2} = 0.8. If two equally frequent alleles were present and the environmental variance is given by
We can contrast these values with various hypothetical nonimprinting examples, again assuming a = 1, two equally frequent alleles (p = q = 0.5), and an environmental variance,
The derivations above do not apply to Xlinked genes, of course. Ironically, however, the situation for sexlinked loci has been the subject of previous work. In marsupials (and eutherian placentas), dosage compensation is effected by condensation of the paternally derived X chromosome (Graves 1996), which amounts functionally to imprinting. In contrast, Xinactivation in placental mammals is random with respect to parental origin in embryonic cell lineages, which causes female eutherians to be mosaics for their active X chromosomes. Such modifications have been incorporated into the standard Xlinked models (James 1973; Grossman and Eisen 1989; Lynch and Walsh 1998). There are no obvious parallels between these models and those described above, however. Covariances between the breeding values and dominance deviations do not disappear in the autosomal imprinting model: even with complete paternal inactivation (d_{1} = a and d_{2} =a), for instance, σ_{ADf} =4a^{2} pq (although σ_{ADm} = 0). In sexlinked models, however, these covariances are always zero.
In principle, the nonzero correlation between breeding values and dominance deviations under imprinting may be used to test if imprinted loci are influencing a trait of interest. Continuing our numerical example above, the correlation among halfsibs sharing a father is
Unfortunately, the large standard error of this estimate of σ_{ADm} limits its use as a practical statistical test for the presence of imprinting. Only in cases with large data sets and large differences in the genotypic values of the reciprocal heterozygotes will a 95% confidence interval exclude zero, the value under the null hypothesis of standard expression. Again using the illustrative numbers above, 1000 simulations showed that, when obtaining both the regression and correlation estimates from families of two halfsibs of 100 fathers, 95% of the estimates of σ_{ADm} fell in the range from 2.136 to 0.625. Using 500 families sufficiently reduced this interval to (1.378, 0.156).
Of course, the model developed above is extremely simple in many ways. Most importantly, (i) it is concerned with a single locus only and so can ignore the complicating effects of epistasis; (ii) it assumes that there is no genotype by environment (G × E) interaction; and (iii) it avoids dealing with maternal effects. All of these aspects limit the direct applicability of the model. [The extension to several additive loci is presumably straightforward, however; see Hill and Keightley (1988) for a model of multiple, additive loci in which imprinting causes complete inactivation of one copy of each gene at the imprinted loci, e.g., d_{1} =a and d_{2} = a.] The importance of these limitations becomes apparent when we recall that many imprinted loci have effects on fetal growth (Bartolomei and Tilghman 1997) and this suite of phenotypes is clearly influenced by genes at several loci and maternal effects, not to mention probable G × E interactions. Nevertheless, incorporating the full range of features found in quantitative genetic models is not the objective of this article; rather, we aim to show how genomic imprinting requires the rederivation of even the simplest concepts in quantitative genetics. We suggest that such considerations will need to be built into more complex models of specific interest (e.g., those concerned with mammalian birthweight).
Acknowledgments
I thank A. G. Clark, K. G. Dodds, R. C. Lewontin, A. E. Weisstein, and two anonymous referees for helpful comments on the manuscript. This work was supported by the Marsden Fund of the Royal Society of New Zealand (contract UOO916).
Footnotes

Communicating editor: T. F. C. Mackay
 Received September 11, 2001.
 Accepted February 18, 2002.
 Copyright © 2002 by the Genetics Society of America