A statistical model for dissecting genomic imprinting through genetic mapping

Abstract

As a result of nonequivalent genetic contribution of maternal and paternal genomes to offsprings, genomic imprinting or called parent-of-origin effect, has been broadly identified in plants, animals and humans. Its role in shaping organism’s development has been unanimously recognized. However, statistical methods for identifying imprinted quantitative trait loci (iQTL) and estimating the imprinted effect have not been well developed. In this article, we propose an efficient statistical procedure for genomewide estimating and testing the effects of significant iQTL underlying the quantitative variation of interested traits. The developed model can be applied to two different genetic cross designs, backcross and F₂ families derived from inbred lines. The proposed procedure is built within the maximum likelihood framework and implemented with the EM algorithm. Extensive simulation studies show that the proposed model is well performed in a variety of situations. To demonstrate the usefulness of the proposed approach, we apply the model to a published data in an F₂ family derived from LG/S and SM/S mouse stains. Two partially maternal imprinting iQTL are identified which regulate the growth of body weight. Our approach provides a testable framework for identifying and estimating iQTL involved in the genetic control of complex traits.

Appendices

Appendix A: EM algorithm for estimating the genetic parameters

The following algorithm is applied to Design I. Similar algorithm can be obtained for Design II with little modification and hence is omitted here.

The log-likelihood function of Eq. (5) can be written as

$$ \log L(\varvec{\Theta}\vert {\bf{M}},{\bf y}_1, {\bf y}_2)= \sum^{n_1}_{i=1}\log\left[\pi_{1\vert i}f_1(y_{1i})+\pi_{2\vert i}f_2(y_{1i})\right]+\sum^{n_2}_{i^{\prime}=1}\log\left[\pi_{3\vert i^{\prime}}f_3(y_{2i^{\prime}})+\pi_{4\vert i^{\prime}}f_4(y_{2i^{\prime}})\right] $$

with a derivative for a particular element Θ_ℓ,

$$ \begin{aligned} \frac{\partial}{\partial{\varvec{\Theta}}_{\ell}} \log L(\varvec{\Theta})\\ &=\sum_{i=1}^{n_1}\sum_{j=1}^2\frac{\pi_{j\vert i}\frac{\partial}{\partial{\varvec {\Theta}}_\ell}f_j(y_{1i})}{\sum_{j=1}^2\pi_{j\vert i}f_j(y_{1i})}+ \sum_{i^{\prime}=1}^{n_2}\sum_{j^{\prime}=3}^4\frac {\pi_{j^{\prime}\vert i^{\prime}}\frac{\partial}{\partial {\varvec{\Theta}}_\ell}f_{j^{\prime}}(y_{2i^{\prime}})} {\sum_{j^{\prime}=3}^4\pi_{j^{\prime}\vert i^{\prime}} f_{j^{\prime}}(y_{2i^{\prime}})}\\ &=\sum_{i=1}^{n_1}\sum_{j=1}^{2}\frac{\pi_{j\vert i}f_j(y_{1i})}{\sum_{j=1}^2 \pi_{j\vert i}f_j(y_{1i})}\frac{\partial}{\partial{\varvec{\Theta}}_\ell}\log f_j(y_{1i})+\sum_{i^{\prime}=1}^{n_2}\sum_{j^{\prime}=3}^{4} \frac{\pi_{j^{\prime}\vert i^{\prime}}f_{j^{\prime}}(y_{2i^{\prime}})} {\sum_{j^{\prime}=3}^4 \pi_{j^{\prime}\vert i^{\prime}}f_{j^{\prime}}(y_{2i^{\prime}})} \frac{\partial}{\partial{\varvec {\Theta}}_\ell}\log f_{j^{\prime}}(y_{2i^{\prime}})\\ &=\sum_{i=1}^{n_1}\sum_{j=1}^{2}\Pi_{j\vert i}\frac{\partial}{\partial {\varvec{\Theta}}_\ell}\log f_j(y_{1i})+ \sum_{i^{\prime}=1}^{n_2}\sum_{j^{\prime}=3}^{4}\Pi_{j^{\prime}\vert i^{\prime}} \frac{\partial}{\partial{\varvec{\Theta}}_\ell} \log f_{j^{\prime}}(y_{2i^{\prime}}) \end{aligned} $$

(A1)

where we define

$$ \Pi_{j\vert i}=\frac{\pi_{j\vert i}f_j(y_{1i})}{\sum^2_{j=1}\pi_{j\vert i}f_j(y_{1i})} \quad{\rm for} \ j=1,2 $$

(A2)

and

$$ \Pi_{j^{\prime}\vert i^{\prime}}=\frac{\pi_{j^{\prime}\vert i^{\prime}}f_{j^{\prime}}(y_{2i^{\prime}})}{\sum^4_{j^{\prime}=3}\pi_{j^{\prime}\vert i^{\prime}}f_{j^{\prime}}(y_{2i^{\prime}})}\quad {\rm for} \ j^{\prime}=3,4 $$

(A3)

which can be thought of as posterior probabilities of QTL genotypes given on the marker genotypes derived from the two backcross families. Given the initial values for the unknown parameters $\varvec{\Theta}$, we can update $\varvec{\Pi}_j=\Pi_{j\vert i}$ and $\varvec{\Pi}_{j^{\prime}}=\Pi_{j^{\prime}\vert i^{\prime}}$ (E-step). The estimated posterior probabilities are used to obtain the new MLEs of $\varvec{\Theta}$ (M step) based on the log-likelihood equations

$$ \hat{\mu}=\frac{{\bf y}_1-{\hat a}[{\bf 1}^{\prime}\varvec{\Pi}_1+{\bf 1}^{\prime}\varvec{\Pi}_2{\hat \gamma}_1]+{\bf y}_2-{\hat a}[{\bf 1}^{\prime}\varvec{\Pi}_3{\hat \gamma}_2+{\bf 1}^{\prime}\varvec{\Pi}_4]}{n} $$

(A4)

$$ \hat{a}=\frac{[{\bf \Pi}_1^{\prime}+\varvec{\Pi}_2^{\prime}{\hat \gamma}_1]({\bf y}_1-\hat{\mu})+[\varvec{\Pi}_3^{\prime}(1+{\hat \gamma}_2)-1]({\bf y}_2-\hat{\mu})}{{\bf 1}^{\prime}(\varvec{\Pi}_1+\varvec{\Pi}_2\hat{\gamma}_1+\varvec{\Pi}_3+\varvec{\Pi}_3\hat{\gamma}_2)} $$

(A5)

$$ \hat{\gamma}_1=\frac{\varvec{\Pi}_{2}^{\prime}({\bf y}_1-{\hat \mu})}{\hat{a}{\bf 1}^{\prime}\varvec{\Pi}_2} $$

(A6)

$$ \hat{\gamma}_2=\frac{\varvec{\Pi}_{3}^{\prime}({\bf y}_2-{\hat \mu})}{\hat{a}{\bf 1}^{\prime}\varvec{\Pi}_3} $$

(A7)

$$ \hat{\sigma}^2=\frac{\varvec{\Pi}_{1}^{\prime}({\bf y}_1-\hat{\mu}-\hat{a})+\varvec{\Pi}_{2}^{\prime}({\bf y}_1-\hat{\mu}-\hat{a}\hat{\gamma}_1)+\varvec{\Pi}_{3}^{\prime}({\bf y}_2-\hat{\mu}-\hat{a}\hat{\gamma}_2)+\varvec{\Pi}_{4}^{\prime}({\bf y}_2-\hat{\mu}+\hat{a})}{n} $$

(A8)

which are derived by letting the derivative in Eq. (A1) equal to zero. This iterative process is repeated between Eqs. (A2) and (A8) until the specified convergence criteria is satisfied. The values at convergence are regarded as the MLEs.

Appendix B: Assessing statistical significance by parametric bootstrap

To assess the statistical significance of the iTest 1–3, we use parametric bootstrap method. The detailed approach is described as follows:

• Step 1: Obtain the MLEs of parameters under the H ₀, denoted as $\hat{\varvec{\Theta}}_0$.

• Step 2: Simulate full phenotype information based on $\hat{\varvec{\Theta}}_0$. Note at this step, we only simulate phenotype data and keep the original marker information.

• Step 3: With the simulated phenotype and the original marker data, we calculate the LR statistic at the test position (specific chromosome location) and save the LR value denoted as LR ^*.

• Step 4: Repeat Step 2–3 for B times and obtain the bootstrap LR test statistics LR ^* ₁,...,LR ^* _B .

• Step 5: Calculate the bootstrap empirical P-value as

  $$ P\hbox{-value}=\frac{1}{B}\sum_{i=1}^B {\rm I}[LR^*_i > LR_{{\rm obs}}] $$

  where I is the indicator function with value 1 if LR ^* _i  >  LR _obs and 0 otherwise, and LR _obs is the observed LR test statistic calculated based on real data.

This empirical P-value is then compared with a significant level α. A general recommendation is to generate B = 1000 bootstrap samples through which the empirical P-value is calculated.