Quantitative genetics of genomic imprinting: a comparison of simple variance derivations, the effects of inbreeding, and response to selection.

The level of expression of an imprinted gene is dependent on the sex of the parent from which it was inherited. As a result, reciprocal heterozygotes in a population may have different mean phenotypes for quantitative traits. Using standard quantitative genetic methods for deriving breeding values, population variances, and covariances between relatives, we demonstrate that although these approaches are equivalent under Mendelian expression, this equivalence is lost when genomic imprinting is acting. Imprinting introduces both parent-of-origin-dependent and generation-dependent effects that result in differences in the way additive and dominance effects are defined for the various approaches. Further, imprinting creates a covariance between additive and dominance terms absent under Mendelian expression, but the expression for this covariance cannot be derived using a number of the standard approaches for defining additive and dominance terms. Inbreeding also generates such a covariance, and we demonstrate that a modified method for partitioning variances can easily accommodate both inbreeding and imprinting. As with inbreeding, the concept of breeding values has no useful meaning for an imprinted trait. Finally, we derive the expression for the response to selection under imprinting, and conclude that the response to selection for an imprinted trait cannot be predicted from the breeder's equation, even when there is no dominance.

A gene is imprinted when its level of expression is dependent on the sex of the parent from which it was inherited. For example, insulin-like growth factor 2 (Igf2) is expressed only from the paternal allele in most fetal tissues of eutherian and marsupial mammals, while the maternally inherited allele is inactivated (DeChiara ; O’Neill ). More generally, imprinting results in nonequivalence of reciprocal heterozygotes, where inheriting an A1 allele from one’s mother and an A2 allele from one’s father gives a different phenotype, on average, than the reverse inheritance pattern. Complex processes of epigenetic regulation are necessary for the repression of one allele while the other is expressed. These processes include allele-specific modifications such as differential DNA methylation, chromatin structure and histone packing, and differences in replication timing of the maternally and paternally inherited genomes (Rand and Cedar 2003).

Approximately 234 imprinted genes have been identified in mammals, including 68 in humans, and many of these genes are thought to be involved in traits such as growth and development (Morison ). A publication predicting imprinted genes based on sequence characteristics suggests that imprinted loci in the human genome number as high as 156 (Luedi ). Recent years have seen an increasing number of statistical methods developed that aim to identify imprinting in quantitative traits. Using QTL mapping, for example, imprinting has been suggested for quantitative traits as diverse as carcass composition, growth, coat color and reproductive traits (de Koning ; de Koning ; Hager ; Hirooka ; Knott ; Lee ; Milan ; Quintanilla ; Rattink ), while general mixed models have demonstrated the involvement of imprinting in traits such as milk yield, litter size, and growth (de Vries ; Engellandt and Tier 2002; Essl and Voith 2002; Kaiser ; Schaeffer ; Stella ; Tier and Solkner 1993).

The inclusion of imprinting effects in these genetic methods highlights the significance of imprinting to a range of economically important livestock production traits and to human health and disease, as well as the importance of understanding the effect imprinting may have on traditional approaches to modeling quantitative genetic traits. Quantitative traits may be influenced by many genes, the environment, and any number of interactions between them, and models for these traits are correspondingly complex. Nevertheless, we here employ a one-locus, two-allele quantitative genetic model to demonstrate the differences in a number of standard approaches for theoretically defining breeding values, genotypic variance and covariances between relatives. In doing so, we show that genomic imprinting may have a significant effect on the assumptions made in these most minimal models, and is therefore likely to also influence more complex models involving many alleles and multiple genetic loci. We also compare the effects of imprinting and inbreeding on quantitative genetic parameters and predict the response to selection for an imprinted trait.

In Table 1, we list and define the important symbols used in this paper, following the convention of Nagylaki and Lou (2007). The reference is either to the equation closest to the definition of each symbol [thus (7), (7)+, and (7)- would mean equation (7), the text below (7), and the text above (7), respectively], or to or the relevant approach or table. The table is ordered alphabetically and split into Roman letters, Greek letters, equation simplifications, and subscripts and superscripts.

Table 1 

Important symbols used in this paper

Symbol	Reference	Definition
Roman letters
a	(1)−	Additive term
Ai	(1)−	Allele i
E	(19)+	Expectation
f	(5)−	Inbreeding coefficient
Gij	(1)−	Genotypic value of genotype AiAj
h2	(33)+	Narrow sense heritability
k	(9)−	Dominance term
ki	(1)−	Imprinting term
pi	(1)−	Frequency of allele i
S	(19)+	Selection differential
t	(18)+	Selection coefficient
wij	(18)+	Relative fitness of genotype AiAj
Greek letters
Δμ	(19)−	Response to selection
δ	(20)−	Difference between the mean genotypic values of offspring before and after selection
εi	Approach 3a	Average additive effect of allele Ai
ε_i•	Approach 2a	Average additive effect of inheriting an Ai allele from the mother
ε_•j	Approach 2a	Average effect of inheriting an Aj allele from the father
φij	(18)−	Absolute fitness of genotype AiAj
φ−	(18)+	Mean fitness
λij	Approach 2a	Dominance effect of genotype AiAj
μ	(1)−	Population mean
σG2	(2)−	Total genetic variance
σA2	Table 3	Additive variance
σD2	Table 3	Dominance variance
σ_AD	Table 3	Covariance between additive and dominance effects
Equation simplifications
α	(10)	a(1+k(p_1−p_2))
αf	(3)	a(1+k_1p_1−k_2p_2)
αm	(4)	a(1+k_2p_1−k_1p_2)
γ	(30)	12(σAf(1)2+σ_ADf(1)+σAm(1)2+σ_ADm(1))
ψ	(29)	σD(1)2+σ_ADf(1)+σ_ADm(1)
Subscripts and superscripts
f	(12)+	Female (maternal)
I	(5)−	Inbreeding model
ij	(1)−	Genotype AiAj
m	(12)+	Male (paternal)
*	(25)−	Next generation before selection

The Model

We here present an overview of a number of approaches for deriving quantitative genetic models for imprinting at one locus. Such models are the basis for many quantitative genetic approaches for dissecting genetic and environmental effects in quantitative traits. Following the approach of Spencer (2002), consider an autosomal diallelic locus subject to imprinting, with alleles A1 and A2 at frequency p1 and p2 (= 1 − p1) respectively in the population. Note that the population under consideration is static, without selection, migration or mutation operating. Assume that on some suitable scale, the genotypic value (G for genotype A) of A1A1 homozygotes is 0 and A2A2 homozygotes is 2a. Assuming no maternal effects, writing the maternally inherited allele first, A2A1 heterozygotes have genotypic value a(1+k1) and A1A2 heterozygotes have value a(1+k2), following the notation of Santure and Spencer (2006) (Figure 1).

Figure 1. 

Genotypic values (G) for genotypes A under genomic imprinting.

In general, imprinting is thought of as complete inactivation of one allele dependent on parental origin, corresponding to k1 = −1 and k2 = 1 (complete silencing of the maternal allele), or k1 = 1 and k2 = −1 (complete silencing of the paternal allele). More recently, however, imprinting has been treated as a quantitative trait, which implies that maternal or paternal alleles may only be partially inactivated (see, e.g., Naumova and Croteau 2004; Sandovici ; Sandovici ), and k1 and k2 may take any values in the range [−1,1]. Note that if k1 = k2 = 0 then the trait is purely additive, and both reciprocal heterozygotes have a genotypic value midway between the homozygotes. If k1 and k2 are equal but of opposite sign (for example, k1 = −1 and k2 = 1, giving complete maternal inactivation) then the locus is subject only to imprinting. However, in the most general case, where k1 and k2 take any values in the range [−1,1], we might consider that both imprinting and dominance are acting on the locus, as the mean genotypic value of heterozygotes is not the mean of the homozygotes.

With the help of Figure 1, the mean genotypic value over the population isand the total genetic variance iswhereand(Spencer 2002). We follow a number of approaches in calculating breeding values, components of variance and covariances between relatives. Doing so illustrates that various assumptions made in these approaches are not valid in the presence of imprinting.

Approaches

Five approaches are outlined in the Appendix, and the results of their partitioning breeding values, the corresponding calculation of variances and covariances, and the derivation of covariances between relatives are shown in Tables 2–4. In the absence of imprinting, all of these approaches give identical breeding values, variance components, and covariances between relatives. The expressions for these terms in the absence of imprinting are obtained by setting k1 = k2 = k. Importantly, it can be seen that it is only by modifying the least squares regression approach (Approach 2b) can the sex-specific additive and dominance values derived by Spencer (2002) be recovered (Santure and Spencer 2006). The other three approaches (Approaches 2a, 3a, and 3b) fail to incorporate sex effects, and give incorrect results when partitioning the variance components and calculating covariances between relatives.

Table 2 

Summary of breeding values for all approaches

	Genotype
	A_1A_1	A_2A_1	A_1A_2	A_2A_2
Approach 1 and 2b
Female	-2p_2α_f	α_f(p_1−p_2)	α_f(p_1−p_2)	2p_1α_f
Male	-2p_2α_m	α_m(p_1−p_2)	α_m(p_1−p_2)	2p_1α_m
Mean	-p_2(α_f+α_m)	12(p_1−p_2)(α_f+α_m)	12(p_1−p_2)(α_f+α_m)	p_1(α_f+α_m)
		mean=12(p_1−p_2)(α_f+α_m)
Approach 2a	-p_2(α_f+α_m)	p_1α_f−p_2α_m	-p_2α_f+p_1α_m	p_1(α_f+α_m)
		mean=12(p_1−p_2)(α_f+α_m)
Approach 3a	-p_2(α_f+α_m)	12(p_1−p_2)(α_f+α_m)	12(p_1−p_2)(α_f+α_m)	p_1(α_f+α_m)
		mean=12(p_1−p_2)(α_f+α_m)
Approach 3b	-p_2(α_f+α_m)	a(p_1−p_2+k_1−2p_1p_2(k_1+k_2))	a(p_1−p_2+k_2−2p_1p_2(k_1+k_2))	p_1(α_f+α_m)
		mean=12(p_1−p_2)(α_f+α_m)

Table 4 

Summary of covariances between relatives for all approaches

	Parent-offspring	Full sib	Half sib
Approach 1 and 2b1
Female	12p_1p_2α_f(α_f+α_m)	14p_1p_2(2(αf2+αm2)+a^2p_1p_2(k_1+k_2)^2)	12p_1p_2αf2
Male	12p_1p_2α_m(α_f+α_m)		12p_1p_2αm2
Approach 2a and 3b	12p_1p_2(αf2+αm2)	14p_1p_2(αf2+αm2)	14p_1p_2(2(αf2+αm2)+a^2p_1p_2(k_1+k_2)^2)
Approach 3a	14p_1p_2(α_f+α_m)^2	18p_1p_2(α_f+α_m)^2	14p_1p_2(α_f+α_m)^2+12a^2p_1p_2((k_1−k_2)^2+2p_1p_2(k_1+k_2)^2)

1These covariances between relatives were also derived by Dai and Weeks (2006) using an extension to the Li and Sacks (1954) method of calculating joint genotype probabilities between pairs of relatives. Dai and Weeks (2006) distinguish maternal and paternal genotypes in order to incorporate imprinting.

Inbreeding and Imprinting

An interesting aspect of the above variance decompositions is the similarity between inbreeding and imprinting, as inbreeding also introduces a covariance between additive and dominance effects (Harris 1964) that may not be partitioned if an incorrect method is used. To investigate this similarity, we incorporate inbreeding into Approach 2b. We represent an inbred population by dividing the population into two groups: a group that represents the expected Hardy-Weinberg proportions, comprising an overall proportion of (1 − f), and a completely homozygous group with no heterozygotes, comprising a proportion f of the population. Thus genotypic frequencies for A1A1, A2A1 and A1A2, and A2A2 genotypes are each and , respectively. Now the overall population mean incorporating both inbreeding (I) and imprinting is

When there is no inbreeding, f = 0, the population is in Hardy-Weinberg proportions, and the mean reduces to as expected. With no imprinting, the mean reduces to . We assume that the inbreeding coefficient f is stable across generations, so the proportion of heterozygotes does not change.

As in Approach 2b, male and female additive and dominance deviations may be calculated separately (Santure and Spencer 2006). For example, the additive effect of inheriting an A1 allele maternally isThe remaining additive effects are

Breeding values and dominance deviations may be calculated as in Approach 2b.

Genetic variance components

The total variance for an inbred population with imprinting iswhere is the total genetic variance for the case of imprinting only (2). When there is complete inbreeding (f = 1), the total variance isand for no inbreeding (f = 0), we recoverThe total variance may also be rewritten asand for the case of no imprinting (k1 = k2 = k), the total genetic variance becomes(Harris, 1964), whereand

Comparing equations (6) and (7), we can see that in the absence of imprinting and dominance (k1 = k2 = f = 0), the total variance for an inbred population is twice that of an outbred population (k1 = k2 = 0, f = 1), and indeed the effect of inbreeding is linear; an inbreeding coefficient of yields a total variance times the variance with no inbreeding. However, for dominance but no imprinting (9), the effect of increasing inbreeding is nonlinear; the population variance may increase or decrease relative to an outbred population depending on the allele frequencies and the value of the dominance coefficient. A similar effect is evident with imprinting; if there is no dominance (i.e., k1 = −k2), the population variance linearly increases with increasing inbreeding, while with both dominance and imprinting the population variance is a quadratic function of f. Thus, it is dominance but not imprinting which determines the relationship of the population variance with increasing inbreeding.

For a highly selfing species, the degree of imprinting may have a large effect on the total population variance. For example, consider a population with and . Setting k1 = −k2 (imprinting, no dominance), the total variance is 0.20 for and 0.25 for . Interestingly, the effect of imprinting becomes less pronounced as inbreeding levels increase; for , the total variance increases from 0.18 for to 0.25 for , while for the variance increases from 0.23 for to 0.25 for .

The female and male additive variances areandwhere and are the female and male additive variances calculated from Approaches 1 and 2b (Table 3). The female and male dominance variances areandwhere is the dominance variance (Approaches 1 and 2b, Table 3). Interestingly, and unlike the pure imprinting case, the dominance variance is different for males and females. The female and male covariances between additive and dominance terms areandwhere and are the female and male covariances between additive and dominance effects (Approaches 1 and 2b, Table 3). When there is no imprinting, the female and male covariances reduce to the same value:

Table 3 

Summary of variance components for all approaches

	Additive variance σA2	Dominance variance σD2	Covariance between additive and dominance effects σ_AD
Approach 1 and 2b
Female	σAf2=2p_1p_2αf2	a^2p_1p_2((k_1−k_2)^2+p_1p_2(k_1+k_2)^2)	σ_ADf=ap_1p_2α_f(k_2−k_1)
Male	σAm2=2p_1p_2αm2		σ_ADm=ap_1p_2α_m(k_1−k_2)
Approach 2a and 3b	p_1p_2(αf2+αm2)	(ap_1p_2(k_1+k_2))^2	0
Approach 3a	12p_1p_2(α_f+α_m)^2	12a^2p_1p_2((k_1−k_2)^2+2p_1p_2(k_1+k_2)^2)	0

Considering that we can see that and hence the covariance is strictly negative under inbreeding alone. Recall that under imprinting aloneand(Table 3). Now we may rearrange to give , so thatand hence , so the average of male and female covariances under inbreeding is also strictly negative. However, if k1 and k2 are of opposite sign, then one of or may be positive. Thus, although both imprinting and inbreeding introduce a covariance between additive and dominance effects, it is only the presence of imprinting that allows the covariance in one sex to be positive. Imprinting can therefore have a significant effect on the total genetic variance and on the sex-specific components of variance of an inbred population.

Response to Selection

We follow the approach of Heywood (2005) to investigate the response of an imprinted quantitative trait to natural selection. To include selection, let the absolute fitness of parent genotype A be and define the relative fitness w as whereis the mean fitness. Following Heywood (2005), we consider the special case with the linear fitness function , which gives . We denote mean offspring genotypic values after selection as . We can now write the change in mean trait value from the parent to the offspring generation (the response to selection; aswhere , the selection differential, is the covariance between parent relative fitness and genotypic value, is the change in mean trait value from parent to offspring, and the expectation is taken over parents (Heywood 2005; Price 1970; Price 1972).

Heywood (2005) defines as the mean genotypic value of offspring from parent A before selection, then sets and, after some algebra, restates (19) asor, alternatively,(Heywood 2005). We now apply this approach to an imprinted quantitative trait. As usual, we need to define both male (paternal) and female (maternal) terms. The absolute fitnesses of the four genotypes areand

Relative fitnesses are shown in Table 5, along with the frequency of each genotype and average values of offspring before and after selection. Note that the population mean, variances and covariances are the same as derived for Approaches 1 and 2b (Table 3).

Table 5 

Population values under selection model

Genotype	A_1A_1	A_2A_1	A_1A_2	A_2A_2
Genotypic value	0	a(1+k_1)	a(1+k_2)	2a
Frequency before selection	p12	p_2p_1	p_1p_2	p22
Fitness	1/φ−	(1+at(1+k_1))/φ−	(1+at(1+k_2))/φ−	(1+2at)/φ−
Average value of offspring before selection: maternal	ap_2(1+k_2)	12a(p_1(1+k_1)+p_2(3+k_2))	12a(p_1(1+k_1)+p_2(3+k_2))	a(p_1(1+k_1)+2p_2)
Average value of offspring before selection: paternal	ap_2(1+k_1)	12a(p_1(1+k_2)+p_2(3+k_1))	12a(p_1(1+k_2)+p_2(3+k_1))	a(p_1(1+k_2)+2p_2)
Frequency after selection	p12/φ−	p_2p_1(1+at(1+k_1))/φ−	p_1p_2(1+at(1+k_2))/φ−	p22(1+2at)/φ−
Average value of offspring after selection: maternal	ap2′(1+k_2)	12a(p1′(1+k_1)+p2′(3+k_2))	12a(p1′(1+k_1)+p2′(3+k_2))	a(p1′(1+k_1)+2p2′)
Average value of offspring after selection: paternal	ap2′(1+k_1)	12a(p1′(1+k_2)+p2′(3+k_1))	12a(p1′(1+k_2)+p2′(3+k_1))	a(p1′(1+k_2)+2p2′)

Now the allele frequencies after selection areand

For both female and male parents, the mean genotypic value of offspring before selection is equal to the mean genotypic value:

The mean values of offspring after selection for female and male parents are:

The difference between male and female offspring means after selection iswhich is zero when there is no imprinting (k1 = k2 = k). This result clearly demonstrates the difference between female and male parents in their effect on offspring means.

We derive the full set of covariances and expectations required for equations (20) and (21) in the Appendix. Now, the response to selection iswhereand

It is clear, therefore, that the response to selection is the same for males and females, and is, as expected, related to the population variances and covariances in addition to the selection coefficient t. In the absence of imprinting k1 = k2 = k, , where , and , and our total response to selection becomes(Heywood 2005) whereand

How does the magnitude of the response to selection compare to what we would predict if imprinting is ignored, and reciprocal heterozygotes are assumed to have the same genotypic value? Substituting into (31), we find that the expression for the response to selection is identical to the full expression derived with separate k1 and k2 terms (28). This suggests that even if imprinting is acting, the predicted response to selection is the same whether calculated using separate genotypic values, or using the average of the genotypic values for the two reciprocal heterozygotes. If k1 = −k2 so there is imprinting but no dominance (as the mean heterozygote genotypic value is midway between the homozygote genotypic values; ), expressions (28) and (31) become

Comparison to breeder’s equation

The response to selection according to the breeder’s equation iswhere the narrow sense heritability, h2, is the ratio between the additive and total genetic variance and as previously. For the case of imprinting, we can see that the breeder’s equationis only equal to the response to selectionwhen the dominance variance and the male and female covariances between the additive and dominance terms are zero, which requires k1 = k2 = 0. For the case of no imprinting, the breeder’s equation becomesand is equal to the response to selection (31) when . Therefore, we can see the well-known result that the response to selection and the breeder’s equation are equal only when the dominance variance is zero, and hence the breeder’s equation only predicts the response to selection in the absence of dominance, whether the locus is imprinted or not.

The difference between the breeder’s equation (34) and the predicted response to selection (28) is a function dependent on a, t, k1, k2 and p1 (= 1 − p2). For a dominant trait with no imprinting the true response to selection (31) is strictly less than that predicted by the breeder’s equation (35). Similarly, if k1 = −k2 so there is imprinting but no dominance (as the mean heterozygote genotypic value is midway between the homozygote genotypic values), the breeder’s equation becomeswhile the true response to selection is

Comparing equations (36) and (37), we can see that the breeder’s equation again overestimates the response to selection for the special case of imprinting but no dominance. For the case of complete inactivation of the maternal or paternal allele , the breeder’s equation predicts a response double that of the true response.

If we include both imprinting and dominance, and let , t = 1, , and , the response to selection (28) is also generally less than that predicted by the breeder’s equation. However, it is interesting to note that if the difference between k1 and k2 is less than , then the predicted response to selection may be the same as or slightly more than that predicted by the breeder’s equation. Therefore, very small differences in the genotypic values of reciprocal heterozygotes may result in the breeder’s equation underestimating the response to selection.

These results contrast with the derivation of de Vries , who from the covariance of parents and offspring predicted the response to selection for an imprinted trait aswhere is defined as the variance due to imprinted genes.

Discussion

We have demonstrated that a simple one-locus two-allele model of genomic imprinting produces large differences in predictions for additive (Table 2) and dominance terms from a number of standard approaches for partitioning the genotypic value of an individual. These approaches are equivalent in the absence of imprinting under standard Mendelian expression (where heterozygotes have equivalent genotypic values and hence k1 = k2). Although all approaches give identical total genetic variance, there are differences in the partitioning of the genetic variance into additive, dominance and covariance terms (Table 3).

The major differences in the approaches arise due to differences in how breeding values and additive effects are defined. Approaches 1 and 2b incorporate both sex- and generation-dependent terms, and breeding values are equivalent for these approaches (Table 2). However, Approaches 2a and the regression methods (Approaches 3a and 3b) are unable to partition separate male and female terms. Consider how breeding values are calculated for the different approaches. Approach 1 defines breeding values in terms of allelic contribution to offspring, and breeding values are the same for reciprocal heterozygotes. Genotypic values in Approach 2b are defined in terms of the male or female effect they pass on to offspring, and so include the same sex-specific generation effect as Approach 1. Breeding values are consequently equivalent for reciprocal heterozygotes. The single regression Approach 3a similarly forces genotypic equivalence for the predicted value of reciprocal heterozygotes. In contrast, the other two approaches define breeding values in terms of an individual’s own genotype and the parental origin of alleles in that genotype. As a consequence of imprinting, the parental origin of these alleles has an effect on the genotypic value of individuals and hence reciprocal heterozygotes have different breeding values (Table 2).

Under standard Mendelian expression, breeding values are expected to be equivalent whether defined as the sum of additive allelic effects (Approaches 2 and 3) or from the means of offspring (Approach 1). However, differences have been noted where alleles in the population are not in Hardy-Weinberg equilibrium (Ewens 1979), in relation to populations with nonrandom mating and inbreeding (Falconer 1985; Fisher 1941; Templeton 1987), and as a result of population subdivision (Goodnight 2000). Genomic imprinting represents a distinct phenomenon causing differences in the definition of additive effects between the approaches we have investigated. The difference of these approaches in predicted breeding values mirrors the conclusion of Falconer (1985), who found that, “the concept of breeding value [has] no useful meaning when mating is not random.” In addition, genomic imprinting introduces a covariance between breeding values and dominance deviations (Spencer 2002). This covariance between additive and dominance effects has only been noted previously when a population is inbred (Harris 1964).

In comparing these approaches, we assumed that Approach 1 gives us “correct” values for population parameters. Approach 1 is the most time-intensive method for partitioning genetic variance because it requires derivation of mating tables to give offspring mean values. However, this approach does allow separate calculation of male and female variances and covariances, which is of great value when considering offspring-parent and halfsib covariances in real populations.

Approach 2a was able to retrieve the additive variance, but the true additive-by-dominance covariance was included in the expression for the dominance variance. By defining additive terms specific to male and female inheritance, we were able to “rescue” this method to include separate breeding values and dominance deviations, and their corresponding variances, for the two sexes (Approach 2b). Of particular note is that Approach 2b was the only approach able to recover the Approach 1 covariance between additive and dominance effects. Defining separate male and female dominance terms and includes a “generation” effect that is not accounted for in Approaches 2a and 3. Approach 1 is based on calculating breeding values and dominance deviations that relate to the following generation because we use offspring means in their calculation. The equivalence of Approach 1 with Approach 2b is a reassurance that defining separate male and female dominance terms is an appropriate measure to include a sex and generation effect in this approach. A closer investigation of Approaches 1 and 2b is presented for a model including maternal genetic effects and genomic imprinting (Santure and Spencer 2006).

It is well known that parental effects may have a significant effect on the phenotype of offspring. It is important for methods to include such effects, but it is not easy to imagine how the linear regression models (Approaches 3a and 3b) could be extended to allow for parental effects such as imprinting and maternal genetic effects.

It is interesting to assess how different these approaches are in their estimation of variance and covariance components. The numerical examples in Table 6 contrast genetic variance components and resemblances between relatives for the different approaches for two scenarios, one where alleles are largely paternally inactivated, and one where maternally inherited alleles are largely inactivated. We assume that phenotypic (and hence genotypic) values range from 0 to 1 . We can see that, as one would expect, paternal inactivation increases the covariance between mothers and offspring and half sibs sharing a mother, relative to fathers and offspring and half sibs sharing a father respectively (and vice versa) (from the correct expressions using Approaches 1 and 2b). Approach 3a underestimates the true additive variance, while Approaches 2a, 3a, and 3b all underestimate the dominance variance. As discussed previously, Approaches 2a, 3a, and 3b are not able to calculate the covariance between additive and dominance effects (Table 6). This covariance between breeding values and dominance deviations is included in the expressions for resemblance between parents and offspring and full sibs and is likely to play a large role in identifying quantitative traits that are influenced by imprinted loci (Spencer 2002).

Table 6 

Values of variances and covariances for all approaches, given paternal and maternal inactivation

	Paternal inactivation p_1=12,p_2=12,a=12,k_1=910,k_2=-810			Maternal inactivation p_1=13,p_2=23,a=12,k_1=-710,k_2=95100
	Approaches 1 and 2b	Approaches 2a and 3b	Approach 3a	Approaches 1 and 2b	Approaches 2a and 3b	Approach 3a
Additive variance
Female	0.4278	0.2153	0.1250	0.0020	0.1777	0.1020
Male	0.0028			0.3534
Dominance variance	0.1808	0.0002	0.0905	0.1520	0.0008	0.0764
Additive by dominance covariance
Female	−0.1966	0	0	0.0122	0	0
Male	0.0159			−0.1635
Offspring-parent covariance
Female	0.1156	0.1077	0.0625	0.0071	0.0888	0.0510
Male	0.0094			0.0949
Half-sib covariance
Female	0.1070	0.0538	0.0313	0.0005	0.0444	0.0255
Male	0.0007			0.0883
Full-sib covariance	0.1077	0.1077	0.0851	0.0890	0.0890	0.0701

By using Approach 2b, we were able to extend the imprinting model to include inbreeding. As previously noted, inbreeding also creates differences in how breeding values are defined (see Falconer 1985) and creates a covariance between additive and dominance effects that is not present in a randomly mating population (Harris 1964). Interestingly, we have demonstrated that in the presence of both inbreeding and imprinting, the dominance variance is different for males and females. The covariance between additive and dominance terms is strictly negative under inbreeding alone, and is on average negative when averaged over males and females under imprinting alone. However, it is only imprinting that allows the covariance in one sex to be positive. The sex-based differences introduced by imprinting represent an important difference between the effects of inbreeding and imprinting on the derivation of quantitative genetic models.

Finally, we derived the full expression for the response to selection of an imprinted trait. For an imprinted trait, the breeder’s equation generally overestimates the true response to selection, a result well established when a trait is known to exhibit dominance. Excitingly, we have demonstrated that even in the absence of dominance, where on average reciprocal heterozygotes have a genotypic value midway between the two homozygotes, the breeder’s equation does not predict the true response to selection. This result has very great significance for predicting the reaction to selection in natural populations—if heterozygotes are not distinguished and we only measure additive variance, we are very likely to overestimate the expected change in mean trait values between generations.

Detecting genomic imprinting of a quantitative trait using, for example, covariances between relatives, is likely to be difficult given the large sampling variance of such covariances and the possibility of maternal effects increasing the covariance of offspring with their mothers (Santure and Spencer 2006; Spencer 2002). However, the derivations above do suggest that a number of different quantities may provide indicators for the influence of imprinting, such that if one approach lacks power to distinguish imprinting from nonimprinting, another avenue may provide fruitful. For example, 1) large differences in the covariance of offspring with their mothers compared to fathers (particularly if the covariance with fathers is greater), 2) the existence of a non-zero covariance between additive effects and dominance deviations (particularly if there is a difference in sign between male and female covariances), and 3) a smaller than expected response to selection based on the breeder’s equation (particularly when there is little evidence for dominance) all provide good evidence for the influence of genomic imprinting on a quantitative trait. A large range of methods is presently available for assessing the role of imprinting in complex and quantitative traits. These methods follow the broad spectrum of genetic approaches for dissecting complex traits, from general mixed models, use of covariances between relatives and identification of parent of origin effects in phenotype inheritance for traits without genotypic information available; to the marker-based approaches of linkage mapping, association studies and QTL mapping. A number of these approaches utilize variance component estimation, resemblances between relatives or differences in the phenotypic values of heterozygotes; quantities discussed in this manuscript. Such approaches are invaluable in the dissection of quantitative traits, and we encourage researchers to employ an approach that can successfully incorporate genomic imprinting into a model of the quantitative trait of interest.