According to traditional Mendelian genetic theory, the maternally and paternally derived alleles of a gene should have a similar amount of expression because they carry the same DNA sequence. However, a growing number of studies suggest that alleles may be expressed from only one of the two parental chromosomes 
due to the difference of DNA methylation. Such genetic imprinting or parent-of-origin effects provide a possible source of phenotypic variation for complex traits in the absence of DNA sequence variants 
. Thus, to better elucidate the genetic architecture of complex traits and diseases for various organisms including humans, the magnitude and pattern of imprinting effects should be estimated and their impact on quantitative variation quantified.
The attempts to characterize imprinting effects are affected by our incapacity to discern the effect of DNA methylation variants from that of DNA sequence variants using a mapping study. This issue was, however, resolved by comparing two reciprocal crosses in which the maternally- or paternally-derived version of the same allele at a gene can be identified 
. Liu et al. 
incorporated identical-by-descent (IBD) sharing into a random-effect mapping model, allowing the characterization of the discrepancy of allelic transmission through different parents. Linkage mapping using controlled crosses or pedigrees with known parents has led to the genome-wide identification of imprinted quantitative trait loci (iQTLs) that affect body weight and growth in mice 
, physiological traits related to endosperm development in maize 
, and hip dysplasia in canines 
However, to study the precise genetic mechanisms through which chromatin dynamics alter quantitative variation, a simple test of imprinting effects of iQTLs is not adequate. Rather, a detailed understanding of whether and how imprinting effects are transmitted across generations is crucial for determining the contribution of epigenetic modification to heritable phenotypic variation for a complex trait. In this article, we present a new strategy for estimating and testing imprinting effects of iQTLs and their transgenerational transmission through two-generation reciprocal crosses leading to four epigenetically different F
families (Figure S1
). The new strategy displays two advantages compared with previous models. First, it provides a comprehensive elucidation of the genetic control mechanisms for a complex trait or disease in terms of traditionally defined additive and dominant effects, newly defined imprinted effects, and their interactions. Second, the strategy has power to detect the changes of imprinting effects from generation to generation, thus facilitating the modeling of transgenerational epigenetic variation and inheritance.
We formulated a mixture model-based likelihood for the imprinting effects of iQTLs flanked by markers in four epigenetically different F
families. A closed form of the EM algorithm was derived to estimate a high-dimensional set of genetic parameters that define the maternally- and paternally-imprinted genetic effects and their interactions in the F
, the additive, dominant, and imprinting effects in the F
, and the interactions of different orders between these effects expressed in different generations. The algorithm was tested through simulation studies from which the minimum heritability and sample size for reasonable estimates of each parameter are determined. Additional simulation studies were performed to test the power for the detection of imprinting effects at different levels. In general, the model shows reasonably low false positive rates for the data in which no imprinting effects exist. In an application of the new model for genetic mapping of iQTL in mice, we identified five significant QTLs on chromosomes 1, 4, 9, and 15 for the overall survival time to hyperoxic acute lung injury (HALI). Each of these QTLs displays remarked imprinting effects on HALI. The model was further used to test when and how these imprinting effects are activated to affect the expression of HALI. In general, all the iQTLs trigger marked imprinting effects in the F
estimates in ). During transmission into the next generation, these imprinting effects were observed to be shrunk (see
estimates in ). But highly significant imprinting effects in the F
generation can still be detected (; see also 
) when the interactions between the imprinting effects of the F
and main effects of the F
are jointly tested. This result suggests that imprinting effects detected from pure F
generations, as conducted in 
, may have confounded their interactions with other effects formed during transmission. The results from reanalyzing the mouse data with the new model shed light on the new inheritance and aetiology of HALI.
The model developed in this article will provide a useful tool for studying transgenerational imprinting inheritance and its impact on the variation in complex traits and diseases. As a first attempt of its kind, the model will need to be modified so as to broaden the scope of its application. Given its ubiquitousness in trait control, epistasis between different genes should be incorporated into the current model, helping to draw a comprehensive atlas of the genetic architecture for complex traits. Also, the expression of any genetic effects cannot be isolated from the environment in which organisms are reared 
. The interactions between different genetic effects and environmental factors should be modeled when a powerful imprinting model is developed. Genetic imprinting may be expressed at the DNA sequence level 
. Thus, the integration of haplotype diversity into the model will gain new insights into the genetic control mechanisms of complex traits. All these extensions, although straightforward in theory, will face with an increasing number of parameters being estimated. Statistical explorations for enhancing the efficiency of parameter estimation will be largely demanded. In sum, the development of the new strategy will facilitate our efforts to address many biological questions of fundamental importance in elucidating the genetic architecture of complex traits.