1742-4682-5-6 1742-4682 Research <p>Functional mapping imprinted quantitative trait loci underlying developmental characteristics</p> Cui Yuehua cui@stt.msu.edu Li Shaoyu lishaoyu@stt.msu.edu Li Gengxin ligengxi@stt.msu.edu

Department of Statistics & Probability, Michigan State University, East Lansing, MI 48824, USA

 Theoretical Biology and Medical Modelling 1742-4682 2008 5 1 6 http://www.tbiomed.com/content/5/1/6 18346281 10.1186/1742-4682-5-6 18 1 2008 17 3 2008 17 3 2008 2008 Cui et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Background

Genomic imprinting, a phenomenon referring to nonequivalent expression of alleles depending on their parental origins, has been widely observed in nature. It has been shown recently that the epigenetic modification of an imprinted gene can be detected through a genetic mapping approach. Such an approach is developed based on traditional quantitative trait loci (QTL) mapping focusing on single trait analysis. Recent studies have shown that most imprinted genes in mammals play an important role in controlling embryonic growth and post-natal development. For a developmental character such as growth, current approach is less efficient in dissecting the dynamic genetic effect of imprinted genes during individual ontology.

Results

Functional mapping has been emerging as a powerful framework for mapping quantitative trait loci underlying complex traits showing developmental characteristics. To understand the genetic architecture of dynamic imprinted traits, we propose a mapping strategy by integrating the functional mapping approach with genomic imprinting. We demonstrate the approach through mapping imprinted QTL controlling growth trajectories in an inbred F2 population. The statistical behavior of the approach is shown through simulation studies, in which the parameters can be estimated with reasonable precision under different simulation scenarios. The utility of the approach is illustrated through real data analysis in an F2 family derived from LG/J and SM/J mouse stains. Three maternally imprinted QTLs are identified as regulating the growth trajectory of mouse body weight.

Conclusion

The functional iQTL mapping approach developed here provides a quantitative and testable framework for assessing the interplay between imprinted genes and a developmental process, and will have important implications for elucidating the genetic architecture of imprinted traits.

Background

Hunting for genes underlying mendelian disorders or quantitative traits has been a long-term effort in genetical research. Most current statistical approaches to gene mapping assume that the maternally and paternally derived copies of a gene in diploid organisms have a comparable level of expression. This, however, is not necessarily true as revealed by recent studies, in which some genes show asymmetric expression, and their expression in the offspring depends on the parental origin of their alleles 123. This phenomenon, termed genomic imprinting, results from the modification of DNA structure rather than changes in the underlying DNA sequences. As one type of epigenetic phenomenon, genomic imprinting has greatly shaped modern research in genetics since its discovery. Some previously puzzling genetic phenomena can now be explained by imprinting theory. However, little is known about the size, location and functional mechanism of imprinted genes in development.

The selective control of gene imprinting is unique to placental mammals and flowering plants. There is increasing evidence that many economically important traits and human diseases are influenced by genomic imprinting 3456. More recent studies have shown that genomic imprinting might be even more common than previously thought 7. Despite its importance, the study of genomic imprinting is still in its early infancy. The biological function of genomic imprinting in shaping an organism's development is still unclear. Recent publications have shown that the majority of imprinted genes in mammals play an important role in controlling embryonic growth and development 89, and some involve in post-natal development, affecting suckling and metabolism 910. The malfunction of imprinted genes at any developmental stage could lead to substantially abnormal characters such as cancers or other genetic disorders. It is therefore of paramount importance to identify imprinted genes and to understand at which developmental stage they function, to help us explore opportunities to prevent, control and treat diseases therapeutically. With the development of new biotechnology coupled with computationally efficient statistical tools, it is now possible to map imprinted genes and understand their roles in disease susceptibility.

Several studies have shown that the effects of imprinted quantitative trait loci (iQTL) can be estimated and tested in controlled crosses of inbred or outbred lines 61112131415. These approaches are designed on the traditional QTL mapping framework where a phenotypic trait is measured at certain developmental stage for a mapping subject, ignoring the dynamic features of gene expression. As a highly complex process, genomic imprinting involves a number of growth axes operating coordinately at different development stages 16. Changes in gene expression at different developmental stages reflect the dynamic changes of gene function over time. They also reflect the response of an organism to either internal or external stimuli, so it can redirect its developmental trajectory to adapt better to environmental conditions, and thereby to increase its fitness 17. For this reason, incorporating such information into genetic mapping should provide more information about the genetic architecture of a dynamic developmental trait.

When a developmental feature of an imprinted trait is considered, traditional iQTL mapping approaches that only consider the phenotypic trait measured at a particular time point will be inappropriate for such an analysis. In fact, for a quantitative trait of developmental behavior, the genetic effect at time t (denoted as Gt) is composed of the genetic effect at time t - 1 (denoted as Gt-1) and the extra genetic effect from time t - 1 to t (denoted as GΔt) 18. Therefore, the phenotypic trait measured at time t reflects the cumulative gene effects from initial time to t, and is highly correlated with the trait measured at time t - 1. The correlations among traits measured at different time periods (i.e., different developmental stages) thus provide correlation information about gene expressions, and hence tell us how genes mediate to respond to internal and external stimuli. Current imprinting QTL (iQTL) mapping approaches, by ignoring the correlations among traits measured at different developmental stages, could therefore potentially overestimate the number and the effective size of iQTLs, and lead to wrong inferences.

Although conditional QTL analysis can reduce bias and increase detecting power by partitioning the genetic effect in a conditional manner 18, analysis of traits at each measurement time point is still less powerful and less attractive than analysis by considering measurements at different developmental stages jointly 19. The recent development of functional mapping brings challenges as well as opportunities for mapping genes responsible for dynamic features of a quantitative trait 171920. Functional mapping is the integration between genetic mapping and biological principles through mathematical equations. The relative merits of functional mapping in biology lie in the strong biological relevance of QTL detection, and its statistical advantages are that it reduces data dimensions and increases the power and stability of QTL detection. By incorporating various mathematical functions into the mapping framework, functional mapping has great flexibility for mapping genes that underlie complex dynamic/longitudinal traits. It provides a quantitative framework for assessing the interplay between genetic function and developmental pattern and form.

In this article, we extend our previous work of interval iQTL mapping to functional iQTL mapping by incorporating biologically meaningful mathematical functions into a QTL mapping framework. We illustrate the idea through an inbred line F2 design, although it can be easily extended to other genetic designs. To distinguish the genetic differences between the two reciprocal heterozygous forms derived from an F2 population, information about sex-specific differences in the recombination fraction is used. Monte Carlo simulations are performed to evaluate the model performance under different scenarios considering the effect of sample size, heritability and imprinting mechanism. A real example is illustrated in which three iQTLs affecting the growth trajectory of body weight in an F2 family derived from two different mouse strains are identified through a genome-wide linkage scan.

Methods

Functional QTL Mapping

Statistical methods for mapping QTL underlying developmental characteristics such as growth or HIV dynamics have been developed previously 1920. The so called functional mapping approach has been recently applied to mapping QTL underlying programmed cell death 2122. Functional mapping is derived under the finite mixture model-based likelihood framework. In the mixture model, each observation y is modelled as a mixture of J (known and finite) components. The distribution for each component corresponds to the genotype category depending on the underlying genetic design. For an F2 design, there are three mixture components (J = 3). The density function for each genotype component is assumed to follow a parametric distribution (f) such as Gaussian, which can be expressed as:

y ~ p ( y | π → , ϕ → , η ) = π 1 f 1 ( y ; ϕ 1 , η ) + π 2 f 2 ( y ; ϕ 2 , η ) + π 3 f 3 ( y ; ϕ 3 , η ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6F86@

where π→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaa8HaaeaacqaHapaCaiaawEniaaaa@2F46@ = (π1, π2, π3) is a vector of mixture proportions which are constrained to be non-negative and sum to unity; ϕ→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaa8HaaeaacqaHvpGAaiaawEniaaaa@2F55@ = (ϕ1, ϕ2, ϕ3) is a vector for the component specific parameters, with ϕj being specific to component j; and η contains parameters (i.e., residual variance) that are common to all components.

For an F2 design initiated with two contrasting homozygous inbred lines, there are three genotypes at each locus. Suppose there is a putative segregating QTL with alleles Q and q that affects a developmental trait such as growth. In a QTL mapping study, the QTL genotype is generally considered as missing, but can be inferred from the two flanking markers. The missing QTL genotype probability πj can be calculated as the conditional probability of the QTL genotype given the observed flanking marker genotypes. For a population with structured pedigree like an F2 population, it can be expressed in terms of the recombination fractions, whereas for a natural population, it can be expressed as a function of linkage disequilibria. The derivations of the conditional probabilities of QTL genotypes can be found in the general QTL mapping literature 23.

In functional mapping, the parameters ϕ→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaa8HaaeaacqaHvpGAaiaawEniaaaa@2F55@ = (ϕ1, ϕ2, ϕ3) specify the underlying developmental mean function (m). For an F2 design, there are three sets of mean functions corresponding to three QTL genotypes. To reduce the number of parameters and enhance the interpretability of functional mapping, the mean process is modelled by certain biologically meaningful mathematical functions, either parametrically or nonparametrically. Suppose that the phenotypic traits are acquired from n individuals, and that t measurements are made on each individual i. Let the response of individual i at time t be denoted by yi(t), i = 1, ⋯, n; t = 1, ⋯, τ. Then the response can be modelled as

yi(t) = f (t) + ei(t),

where f(t) is a linear or nonlinear function evaluated at time t, depending on the underlying developmental pattern; ei(t) is the residual error, which is assumed to be normal with mean zero and variance σ2(t). The intra-individual correlation is specified as ρ, which leads to the covariance for individual i at two different time points, t1 and t2, expressed as cov(yi(t1), yi(t2)) = ρσt1σt2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqyWdiNaeq4Wdm3aaSbaaSqaaiabdsha0naaBaaameaacqaIXaqmaeqaaaWcbeaakiabeo8aZnaaBaaaleaacqWG0baDdaWgaaadbaGaeGOmaidabeaaaSqabaaaaa@36B1@. Assuming multivariate normal distribution, the density function for each progeny i who carries genotype j can be expressed as

f j ( y i | Ω q , Ω r ) = 1 ( 2 π ) τ / 2 | Σ | 1 / 2 exp ⁡ [ − 1 2 ( y i − m j ) Σ − 1 ( y i − m j ) T ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOzay2aaSbaaSqaaiabdQgaQbqabaGccqGGOaakieqacqWF5bqEdaWgaaWcbaGaemyAaKgabeaakiabcYha8HGabiab+L6axnaaBaaaleaacqWFXbqCaeqaaOGaeiilaWIae4xQdC1aaSbaaSqaaiab=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@6F73@

where mj = [mj(1), ⋯, mj(τ)] is the mean vector common for all individuals with genotype j, which can be evaluated through function f in Model (2). The unknown parameters that specify the position of QTL within a marker interval are arrayed in Ωr. The parameters that define the mean and the covariance functions are arrayed in Ωq.

Since we do not observe the QTL genotype, the distribution of y is modelled through a finite mixture model given in Model (1). At a particular time point (say t), the genetic effect can be obtained by solving the following equations

a ( t ) = 1 2 ( m 1 ( t ) − m 3 ( t ) )  and  d ( t ) = m 2 ( t ) − 1 2 ( m 1 ( t ) + m 3 ( t ) ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6434@

where a(t) and d(t) are the additive and dominant effects at time t, respectively.

Functional iQTL Mapping

Modelling the imprinted mean function

In an F2 population, three QTL genotypes are segregated. The three QTL genotypes may have different expressions which result in three different mean trajectories. Considering the imprinting property of an iQTL, we introduce the notation for the parental origin of alleles inherited from both parents. Let QM and qM be two alleles inherited from the maternal parent, and QP and qP be two alleles derived from the paternal parent. The subscripts M and P refer to maternal and paternal origin, respectively. These four parentally specific alleles form four distinct genotypes expressed as QMQP, QMqP, qMQP, and qMqP. In contrast, in a regular QTL mapping study without distinguishing the allelic parental origin, the two reciprocal heterozygotes, QMqP and qMQP, are collapsed to one heterozygote. When a QTL is imprinted, the four QTL genotypes show different gene expressions, which result in different developmental growth trajectories. For a maternally (or paternally) imprinted QTL, the allele inherited from the maternal (or paternal) parent is not expressed. Thus, two growth trajectories would be expected. By testing the differences of the four growth trajectories, one can test whether there is a QTL, and whether the QTL is imprinted.

For simplicity, we use numerical notation to denote the four parent-of-origin-specific genotypes, i.e., QMQP = 1, QMqP = 2, qMQP = 3, and qMqP = 4. The mean functions of these genotypes are denoted as mj, (j = 1, ⋯, 4). We know that for an imprinted gene, the expression of an allele depends on its parental origin. On a developmental scale, the two reciprocal heterozygotes, QMqP and qMQP, may present different mean trajectories. The degree of imprinting of an iQTL can thus be assessed by the genotype-specific parameters. Through testing the difference between the mean functions of the two reciprocal heterozygotes, we can assess the imprinting property of a QTL. An overlap of the two trajectories for the two reciprocal heterozygotes indicates no sign of imprinting.

For a developmental characteristic such as growth, it is well known that the underlying trajectory can be described by a universal growth law, which follows a logistic growth function 24. At a developmental stage, say time t, the mean value of an individual carrying QTL genotype j can be expressed by

m j ( t ) = α j 1 + β j e − γ j t , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyBa02aaSbaaSqaaiabdQgaQbqabaGccqGGOaakcqWG0baDcqGGPaqkcqGH9aqpjuaGdaWcaaqaaiabeg7aHnaaBaaabaGaemOAaOgabeaaaeaacqaIXaqmcqGHRaWkcqaHYoGydaWgaaqaaiabdQgaQbqabaGaemyzau2aaWbaaeqabaGaeyOeI0Iaeq4SdC2aaSbaaeaacqWGQbGAaeqaaiabdsha0baaaaGccqGGSaalaaa@43D0@

where the growth parameters (αj, βj, γj) describe asymptotic growth, initial growth and relative growth rate, respectively 25. With estimated growth parameters, we can easily retrieve the genotypic means at every time point by simply plugging t into Equation (5). This modelling approach can significantly reduce the number of unknown parameters to be estimated, especially when the number of measurement points is large 19.

At a particular time point (say t), the mean expression of an individual carrying QTL genotype j can be evaluated through the three growth parameters (αj, βj, γj). On the basis of the univariate imprinting model given in 12, we can partition the genetic effects at time t as the allele-specific effects, i.e.

a M ( t ) = 1 2 ( m 1 ( t ) + m 2 ( t ) − m 3 ( t ) − m 4 ( t ) ) a P ( t ) = 1 2 ( m 1 ( t ) − m 2 ( t ) + m 3 ( t ) − m 4 ( t ) ) d ( t ) = m 1 ( t ) − m 2 ( t ) − m 3 ( t ) + m 4 ( t ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@942C@

where aM and aP refer to the additive effects of alleles inherited from mother and father, respectively; d refers to the allele dominant effect.

To illustrate the idea, we use the growth trait to demonstrate the mapping principle. The idea can be easily extended to other developmental characteristics. For developmental characteristics other than growth, different mathematical functions should be developed. Some flexible choices include nonparametric regressions based on smoothing splines or orthogonal polynomials 21.

Modelling the covariance structure

To understand how QTL mediate growth, it is essential to take correlations among repeated measures into account 19. The repeated measures provide correlation information on gene expression. Hence, dissection of the intra-individual correlation will help us to understand better how genes function over time. One commonly used model for covariance structure modelling is the first-order autoregressive (AR(1)) model 26, expressed as

σ2(1) = ⋯ = σ2(τ) = σ2

for the variance, and

σ ( t k , t k ′ ) = σ 2 ρ | t k ′ − t k | ( t k ′ > t k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4WdmNaeiikaGIaemiDaq3aaSbaaSqaaiabdUgaRbqabaGccqGGSaalcqWG0baDdaWgaaWcbaGafm4AaSMbauaaaeqaaOGaeiykaKIaeyypa0Jaeq4Wdm3aaWbaaSqabeaacqaIYaGmaaGccqaHbpGCdaahaaWcbeqaamaaemaabaGaemiDaq3aaSbaaWqaaiqbdUgaRzaafaaabeaaliabgkHiTiabdsha0naaBaaameaacqWGRbWAaeqaaaWccaGLhWUaayjcSdaaaOGaeiikaGIaemiDaq3aaSbaaSqaaiqbdUgaRzaafaaabeaakiabg6da+iabdsha0naaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaaa@4F76@

for the covariance between any two time points tk and tk', where 0 <ρ < 1 is the proportion parameter with which the correlation decays with time lag.

For a developmental characteristic such as growth, the inter-individual variation generally increases as time increases, which leads to a nonstantionary variance function. Since the AR(1) covariance model assumes stationary variance, it can not be applied directly. To stabilize the variance at different measurement time points, we apply a multivariate Box-Cox transformation to stabilize the variance 27, which has the form

z i ( t ) = { y i ( t ) λ ( t ) − 1 λ ( t ) , if λ ≠ 0 log ⁡ ( y i ( t ) ) , if λ = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@621D@

The Box-Cox transformation ensures the homoscedasticity and normality of the response y. For repeated measures or longitudinal studies, a reasonable constraint is to set λ(t) = λ for all t. Then the optimal choice of λ can be estimated from the data. To preserve the interpretability of the estimated mean parameters, Carroll and Ruppert 28 proposed a transform-both-sides (TBS) model in which the same transformation form is applied to both sides of Model (2). For a log-transformation, this results in logyi(t) = logf(t) + ei(t). Wu et al. 29 later showed the favorable property of this approach in functional mapping. For the modelling purpose of stabilizing variances, we simply adopt the log-transformation in the current setting.

Alternatively, one can model the covariance structure nonstationarily without transforming the original data. Among a pool of choices, the structured antedependence (SAD) model 30 displays a number of favorable merits. The SAD model of order p for modelling the error term in Eq. (2) is given by

ei(t) = φ1ei(t - 1) + ⋯ + φpei(t - r) + εi(t)

where εi(t) is the "innovation" term assumed to be independent and distributed as N(0,σt2) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8xdX7KaeiikaGIaeGimaaJaeiilaWIaeq4Wdm3aa0baaSqaaiabdsha0bqaaiabikdaYaaakiabcMcaPaaa@3F40@. Therefore, the variance-covariance matrix can be expressed as

Σ = AΣεAT,

where Σε is a diagonal matrix with diagonal elements being the innovation variance; A is a lower triangular matrix which contains the antedependence coefficient φr. The SAD order (p) can be selected through an information criterion 31. The SAD(r) model has been previously applied in functional mapping of programmed cell death 21.

Parameter Estimation

Assuming inter-individual independence, the joint likelihood function is given by

L ( Ω | z , ℳ ) = ∏ i = 1 n ∑ j = 1 4 π j | i f j ( z i | Ω , ℳ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGccceGae8xQdCLaeiiFaWhcbeGae4NEaONaeiilaWYenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae03mH0KaeiykaKIaeyypa0ZaaebCaeaadaaeWbqaaiabec8aWnaaBaaaleaacqWGQbGAcqGG8baFcqWGPbqAaeqaaOGaemOzay2aaSbaaSqaaiabdQgaQbqabaGccqGGOaakcqGF6bGEdaWgaaWcbaGaemyAaKgabeaakiabcYha8jab=L6axjabcYcaSiab9ntinjabcMcaPaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaeGinaqdaniabggHiLdaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0Gaey4dIunaaaa@609E@

where zi = [zi(1), ⋯, zi(τ)] is the observed log-transformed trait vector for individual i (i = 1, ⋯, n) over τ time points; fj is the multivariate normal density function with log-transformed mean for QTL genotype j; πj|i (j = 1, ⋯, 4) is the mixture proportion for individual i with genotype j, which is derived assuming a sex-specific difference in recombination rate and can be found in 12. The unknown parameters in Ω comprise three sets, one defining the co-segregation between the QTL and markers and thereby the location of the QTL relative to the markers, denoted by Ωr, and the other defining the distribution of a growth trait for each QTL genotype, denoted by Ωq = (Ωm, Ωv), where Ωm=(Ωm1,Ωm2,Ωm3,Ωm4) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacceGae8xQdC1aaSbaaSqaaiabd2gaTbqabaGccqGH9aqpcqGGOaakcqWFPoWvdaWgaaWcbaGaemyBa02aaSbaaWqaaiabigdaXaqabaaaleqaaOGaeiilaWIae8xQdC1aaSbaaSqaaiabd2gaTnaaBaaameaacqaIYaGmaeqaaaWcbeaakiabcYcaSiab=L6axnaaBaaaleaacqWGTbqBdaWgaaadbaGaeG4mamdabeaaaSqabaGccqGGSaalcqWFPoWvdaWgaaWcbaGaemyBa02aaSbaaWqaaiabisda0aqabaaaleqaaOGaeiykaKcaaa@4589@ defines the mean vector for different genotypes and Ωv defines the covariance parameters.

We implement the EM algorithm to obtain the maximum likelihood estimates (MLEs) of the unknown parameters. The first derivative of the log-likelihood function, with respect to specific parameter ϕ contained in Ω, is given by

∂ ∂ Ω ϕ log ⁡ ℓ ( Ω | z , ℳ ) = ∑ i = 1 n ∑ j = 1 4 π j | i ∂ ∂ Ω ϕ f j ( z i | Ω , ℳ ) ∑ j ′ = 1 4 π j ′ | i f j ′ ( z i | Ω , ℳ ) = ∑ i = 1 n ∑ j = 1 4 π j | i f j ( z i | Ω , ℳ ) ∑ j ′ = 1 4 π j ′ | i f j ′ ( z i | Ω , ℳ ) ∂ ∂ Ω ϕ log ⁡ f j ( z i | Ω , ℳ ) = ∑ i = 1 n ∑ j = 1 4 Π j | i ∂ ∂ Ω ϕ log ⁡ f j ( z i | Ω , ℳ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeaabqGaaaaabaaabaqcfa4aaSaaaeaacqGHciITaeaacqGHciITiiqacqWFPoWvdaWgaaqaaiabew9aQbqabaaaaOGagiiBaWMaei4Ba8Maei4zaCMaeS4eHWMaeiikaGIae8xQdCLaeiiFaWhcbeGae4NEaONaeiilaWYenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae03mH0KaeiykaKcabaGaeyypa0dabaWaaabCaeaadaaeWbqcfayaamaalaaabaGaeqiWda3aaSbaaeaacqWGQbGAcqGG8baFcqWGPbqAaeqaamaalaaabaGaeyOaIylabaGaeyOaIyRae8xQdC1aaSbaaeaacqaHvpGAaeqaaaaacqWGMbGzdaWgaaqaaiabdQgaQbqabaGaeiikaGIae4NEaO3aaSbaaeaacqGFPbqAaeqaaiabcYha8jab=L6axjabcYcaSiab9ntinjabcMcaPaqaamaaqadabaGaeqiWda3aaSbaaeaacuWGQbGAgaqbaiabcYha8jabdMgaPbqabaGaemOzay2aaSbaaeaacuWGQbGAgaqbaaqabaGaeiikaGIae4NEaO3aaSbaaeaacqGFPbqAaeqaaiabcYha8jab=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L6axjabcYcaSiab9ntinjabcMcaPaWcbaGaemOAaOMaeyypa0JaeGymaedabaGaeGinaqdaniabggHiLdaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoaaaaaaa@09F4@

where we define

Π j | i = π j | i f j ( z i | Ω , ℳ ) ∑ j ′ = 1 4 π j ′ | i f j ′ ( z i | Ω , ℳ ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@690A@

The MLEs of the parameters contained in (Ωm, Ωv) are obtained by solving

∂ ∂ Ω ϕ log ⁡ ℓ ( Ω | z , ℳ ) = 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqGHciITaeaacqGHciITiiqacqWFPoWvdaWgaaqaaiabew9aQbqabaaaaOGagiiBaWMaei4Ba8Maei4zaCMaeS4eHWMaeiikaGIae8xQdCLaeiiFaWhcbeGae4NEaONaeiilaWYenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae03mH0KaeiykaKIaeyypa0JaeGimaadaaa@4C57@

Direct estimation is unavailable since there is no closed form for the MLEs of parameters. The EM algorithm is applied to solve these unknowns iteratively.

E-step: Given initial values for (Ωm, Ωv), calculate the posterior probability matrix Π = {Πj|i} in Eq. (8).

M-step: With the updated posterior probability Π, we can update the parameters contained in Ωq. The maximization can be implemented through an iteration procedure or through the Newton-Raphson or other algorithm such as simplex algorithm 32.

The above procedures are iteratively repeated between (8) and (9), until a certain convergence criterion is met. For details of the EM algorithm, one can refer to 19. The converged values are the MLEs of the parameters. The initial values under the alternative hypothesis are generally set as the estimated values under the null. Note also that in the above algorithm, we do not directly estimate the QTL-segregating parameters (Ωr). In general, we use a grid search approach to estimate the QTL location by searching for a putative QTL at every 1 or 2 cM on a map interval bracketed by two markers throughout the entire linkage map. The log-likelihood ratio test statistic for a QTL at a testing position is displayed graphically to generate a log-likelihood ratio plot called LR profile plot. The genomic position corresponding to a peak of the profile is the MLE of the QTL location.

We have found that the algorithm is sensitive to initial values, particularly the mean values of the two reciprocal heterozygotes. To make sure the parameters are converged to the "correct" ones, we normally give different initial values for the two reciprocal heterozygotes and check which one produces the highest likelihood value. The ones which produce higher likelihood value are considered as the MLEs.

Hypothesis Testing

Global QTL test

Testing whether there is a QTL affecting the developmental trajectory is the first step toward understanding of genetic architecture of an imprinted trait. Once the MLEs of the parameters are obtained, the existence of a QTL affecting the growth curve can be tested by formulating the following hypotheses

{ H 0 : Ω m 1 ≡ ⋯ , ≡ Ω m 4 H 1 : The equalities above do not hold , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaiqaaeaafaqaaeGadaaabaGaeeisaG0aaSbaaSqaaiabicdaWaqabaaakeaacqGG6aGoaeaaiiqacqWFPoWvdaWgaaWcbaGaemyBa02aaSbaaWqaaiabigdaXaqabaaaleqaaOGaeyyyIORaeS47IWKaeiilaWIaeyyyIORae8xQdC1aaSbaaSqaaiabd2gaTnaaBaaameaacqaI0aanaeqaaaWcbeaaaOqaaiabbIeainaaBaaaleaacqaIXaqmaeqaaaGcbaGaeiOoaOdabaGaeeivaqLaeeiAaGMaeeyzauMaeeiiaaIaeeyzauMaeeyCaeNaeeyDauNaeeyyaeMaeeiBaWMaeeyAaKMaeeiDaqNaeeyAaKMaeeyzauMaee4CamNaeeiiaaIaeeyyaeMaeeOyaiMaee4Ba8MaeeODayNaeeyzauMaeeiiaaIaeeizaqMaee4Ba8MaeeiiaaIaeeOBa4Maee4Ba8MaeeiDaqNaeeiiaaIaeeiAaGMaee4Ba8MaeeiBaWMaeeizaqMaeiilaWcaaaGaay5Eaaaaaa@6C20@

where H0 corresponds to the reduced model, in which the data can be fit by a single curve, and H1 corresponds to the full model, in which there exist different curves to fit the data. The above test is equivalent to test

{ H 0 : α 1 = α 2 = α 3 = α 4 , β 1 = β 2 = β 3 = β 4 , γ 1 = γ 2 = γ 3 = γ 4 H 1 : The equalities above do not hold , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@899B@

The statistic for testing the hypotheses is calculated as the log-likelihood (LR) ratio of the reduced to the full model

LR = − 2 [ log ⁡ L ( Ω ˜ | z , ℳ ) − log ⁡ L ( Ω _ |