Functional Mapping of Expression Quantitative Trait Loci that Regulate Oscillatory Gene Expression

Abstract

Genetic networks underlying many biological processes, such as vertebrate somitogenesis, cell cycle, hormonal signaling, and circadian rhythms, are characterized by oscillations in gene expression. It has been recognized that the frequency and amplitude of gene expression oscillations vary among individuals and can be controlled by specific expression quantitative trait loci (eQTLs). In this chapter, we develop a dynamic model for mapping and identifying such eQTLs by integrating mathematical aspects of oscillatory dynamics into the functional mapping framework. The model can determine whether and how eQTLs regulate individual genes’ activation kinetics and expression dynamics by estimating and testing Fourier series parameters for different eQTL genotypes. We incorporate a general autoregressive moving-average process of order (r,s), the so-called ARMA(r,s), to model the covariance structure for gene expression profiles measured in time course, broadening the applicability of the new dynamic model to mapping eQTLs in practice. The expectation-maximization algorithm (EM algorithm) was derived to estimate all parameters modeling the mean–covariance structures within a mixture model setting. Simulation studies were performed to investigate the statistical behavior of the model. The model will provide a powerful statistical tool for mapping eQTLs and their epistatic interactions that regulate oscillations in gene expression, helping to construct a regulatory genetic network for those periodic biological phenomena.

Appendix

Here, we give the procedure for estimating parameters in $\Lambda = (\Theta, \Psi )$ within the EM algorithm framework. In the E-step, the posterior expectation of z _ij is evaluated as:

$$ {P_{j|i}} = E[{z_{ij}}|\Lambda, {y_i}] = \Pr [{z_{ij}} = 1|\Lambda, {y_i}] = \frac{{{\omega_{j|i}}{f_j}({y_i};{\mu_j},\Sigma )}}{{\sum\nolimits_{j\prime = 0}^2 {{\omega_{j\prime |i}}{f_{j\prime }}({y_i};{\mu_j}^\prime, {\Sigma_i})} }}, $$

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where we assume that the covariance matrix is subject specific.

In the M-step, closed form solutions exist for ω (see Eqs. 8–11) and the parameters in $\Lambda$ except for τ and σ ².

Suppose the gene expression trajectory is approximated by the first K orders of the Fourier series, then ${\Lambda_j} = ({c_j},{\tau_j}),$ where ${c_j} = ({\alpha_{0j}},{\alpha_{1j}},{\beta_{1j}},\; \ldots, \;{\alpha_{Kj}},{\beta_{Kj}}).$ We have

$$ \frac{{\partial \log {L_c}(\Lambda |y)}}{{\partial {c_j}}} = \left[ {\frac{{\partial \log {L_c}(\Lambda |y)}}{{\partial {\mu_{ij}}}}} \right]\left[ {\frac{{\partial {\mu_j}}}{{\partial {c_j}}}} \right]. $$

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The parameter c _j can be updated by setting (Eq. 15) to zero. Since

$$ \frac{{\partial \log {L_c}(\Lambda |y)}}{{\partial {\mu_j}}} = \sum\limits_{i = 1}^n {{P_{j|i}}{{({y_i} - {\mu_j})}^T}\Sigma_i^{ - 1}} $$

and $\partial {\mu_j}/\partial {c_j} = {D_i}({\tau_j})$, where

$$ {D_i}({\tau_j}) = \left( {\begin{array}{*{20}{c}} 1 & {\cos \left( {\frac{{2\pi {t_{i1}}}}{{{\tau_j}}}} \right)} & {\sin \left( {\frac{{2\pi {t_{i1}}}}{{{\tau_j}}}} \right)} & \cdots & {\cos \left( {\frac{{2\pi K{t_{i1}}}}{{{\tau_j}}}} \right)} & {\sin \left( {\frac{{2\pi K{t_{i1}}}}{{{\tau_j}}}} \right)} \\ 1 & {\cos \left( {\frac{{2\pi {t_{i2}}}}{{{\tau_j}}}} \right)} & {\sin \left( {\frac{{2\pi {t_{i2}}}}{{{\tau_j}}}} \right)} & \cdots & {\cos \left( {\frac{{2\pi K{t_{i2}}}}{{{\tau_j}}}} \right)} & {\sin \left( {\frac{{2\pi K{t_{i2}}}}{{{\tau_j}}}} \right)} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & {\cos \left( {\frac{{2\pi {t_{i{m_i}}}}}{{{\tau_j}}}} \right)} & {\sin \left( {\frac{{2\pi {t_{i{m_i}}}}}{{{\tau_j}}}} \right)} & \cdots & {\cos \left( {\frac{{2\pi K{t_{i{m_j}}}}}{{{\tau_j}}}} \right)} & {\sin \left( {\frac{{2\pi K{t_{i{m_j}}}}}{{{\tau_j}}}} \right)} \\ \end{array} } \right), $$

we have

$$ {\hat{c}_j} = {\left[ {\sum\limits_{i = 1}^n {{P_{j|i}}{D_i}{{({\tau_j})}^T}\Sigma_i^{ - 1}{D_i}({\tau_j})} } \right]^{ - 1}}\left[ {\sum\limits_{i = 1}^n {{P_{j|i}}y_i^T\Sigma_i^{ - 1}{D_i}({\tau_j})} } \right]. $$

Since the analytical form of the inverse of ${\Sigma_i}$ is not available, we use the recursive method to calculate the inverse matrix of ARMA(p,q) through its association with ARMA(p,q − 1).

We can write ${\Sigma_i} = {\sigma^2}{R_i}$, where R _i is the correlation matrix that is entirely determined by the ARMA parameters ${\varphi_1},\; \ldots, \;{\varphi_p},\;{\theta_1},\; \ldots, \;{\theta_q}.$ The variance σ ² can be updated by:

$$ {\hat{\sigma }^2} = \frac{{\sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^J {{P_{j|i}}{{({y_i} - {\mu_j})}^T}R_i^{ - 1}({y_i} - {\mu_j})} } }}{{\sum\nolimits_{i = 1}^n {{m_i}} }}. $$

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Again $R_i^{ - 1}$ can be calculated by the method of Haddad (28).

Because there are no closed form solutions for τ _j and ARMA parameters ${\varphi_1},\; \ldots, \;{\varphi_p}$ and ${\theta_1},\; \ldots, \;{\theta_q}$, their estimates are updated using one-step Newton–Raphson method within each iteration. In particular, in the ${{(\nu + 1)th}}$ iteration, τ _j can be updated by:

$$ \tau_j^{\nu + 1} = \tau_j^\nu - \frac{{\partial /\partial {\tau_j}\log {L_c}(\Lambda |y){|_{\Lambda = {\Lambda^\nu }}}}}{{{\partial^2}/\partial \tau_j^2\log {L_c}(\Lambda |y){|_{\Lambda = {\Lambda^\nu }}}}}, $$

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where

$$ \frac{\partial }{{\partial {\tau_j}}}\log {L_c}(\Lambda |y) = \sum\limits_{i = 1}^n {{P_{j|i}}{{({y_i} - {\mu_j})}^T}\Sigma_i^{ - 1}} {\delta_{ij}} $$

with ${\delta_{ij}}$ being a ${m_i} \times 1$ vector whose components

$$ {\delta_{ijl}} = \sum\limits_{k = 1}^K {\left[ {{\alpha_{kj}}\sin \left( {\frac{{2\pi k{t_l}}}{{{\tau_j}}}} \right)\frac{{2\pi k{t_l}}}{{\tau_j^2}} - {\beta_{kj}}\cos \left( {\frac{{2\pi k{t_l}}}{{{\tau_j}}}} \right)\frac{{2\pi k{t_l}}}{{\tau_j^2}}} \right],} $$

and

$$ \frac{{{\partial^2}}}{{\partial \tau_j^2}}\log {L_c}(\Lambda |y) = \sum\limits_{i = 1}^n {\left[ { - {P_{j|i}}\delta_{ij}^T\Sigma_i^{ - 1}{\delta_{ij}} + {P_{j|i}}{{({y_i} - {\mu_j})}^T}\Sigma_i^{ - 1}\frac{{{\partial^2}}}{{\partial \tau_j^2}}{\mu_j}} \right]}. $$

Similarly, the parameters ${\varphi_1},\; \ldots, \;{\varphi_p}$ and ${\theta_1},\; \ldots, \;{\theta_q}$ can be updated by the one-step Newton–Raphson method outlined above. However, there are no analytical forms of the first and the second derivatives of the expected complete data log-likelihood with respect to the $\varphi {{\prime s}}$ and $\theta {{\prime s}}$, we use the numerical differentiation method to calculate these quantities. To ease the presentation of the method, denote the $(p + q)$ dimensional vector $\psi = ({\varphi_1},\; \ldots, \;{\varphi_p},\;{\theta_1},\; \ldots, \;{\theta_q}).$ The first and the second derivatives with respect to the Kth component in Ψ are approximated, respectively, by:

$$ \frac{{E\left[ {\log {L_c}({\Lambda_{ - \psi }},\psi + {h_n}{e_\kappa }|y)} \right] - E\left[ {\log {L_c}(\Lambda |y)} \right]}}{{{h_n}}}, $$

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and

$$ \frac{{E\left[ {\log {L_c}({\Lambda_{ - \psi }},\psi + {h_n}{e_\kappa }|y)} \right] - 2E\left[ {\log {L_c}(\Lambda |y)} \right] + E\left[ {\log {L_c}({\Lambda_{ - \psi }},\psi - {h_n}{e_\kappa }|y)} \right]}}{{h_n^2}}, $$

(19)

where we use E to represent the posterior expectation of the complete data log-likelihood with respect to, ${\Lambda_{ - \psi }}$ denotes the parameters in $\Lambda$ other than $\psi$, the $(p + q)$ vector e has unity length with the Kth component set to 1, and h _n is the bandwidth chosen by the investigator. When h _n is small enough, the numerical differentiation approximates the true derivatives adequately. On the other hand, if h _n is too small, the random errors from the numerical computation may deteriorate the results.