Inferring latent gene regulatory network kinetics
-
Javier González
,
Ivan Vujačić
Abstract
Regulatory networks consist of genes encoding transcription factors (TFs) and the genes they activate or repress. Various types of systems of ordinary differential equations (ODE) have been proposed to model these networks, ranging from linear to Michaelis-Menten approaches. In practice, a serious drawback to estimate these models is that the TFs are generally unobserved. The reason is the actual lack of high-throughput techniques to measure abundance of proteins in the cell. The challenge is to infer their activity profile together with the kinetic parameters of the ODE using level expression measurements of the genes they regulate.
In this work we propose general statistical framework to infer the kinetic parameters of regulatory networks with one or more TFs using time course gene expression data. Our approach is also able to predict the activity levels of the TF. We use a penalized likelihood approach where the ODE is used as a penalty. The main advantage is that the solution of the ODE is not required explicitly as it is common in most proposed methods. This makes our approach computationally efficient and suitable for large systems with many components. We use the proposed method to study a SOS repair system in Escherichia coli. The reconstructed TF exhibits a similar behavior to experimentally measured profiles and the genetic expression data are fitted properly.
7 Appendix
7.1 Notation
For the proofs below we introduce some notation. We define a matrix
vectors Φk=(δk, θk, Σk, μ, αk), 
and functions
k=1, …, m.
Proof of (15). (Derivation of the influence matrix.)
We can write the penalized log-likelihood as
The maximizer of lλ,k with respect to αk is given by 
From (14) it follows
where Sλ,k=Kδk(Kδk+2λΣk)−1 is the influence matrix that depends on the regularization parameter λ.
Proof of (20). (Expectation step of EM algorithm.)
For every k=1, …, m we calculate 
Denote with ΣH,k and ΣO,k diagonal matrices whose diagonals are vectors of variances of observed and hidden observations of kth equation, respectively. Splitting the likelihood in two parts corresponding to the hidden and the observed observations we obtain that
where 
In the previous expression we have that
By using the properties of the expectation and the variance and by factorizing terms it is straightforward to obtain that
Replacing eq. (29) in the expected likelihood we obtain that
Define 
Grouping the terms we obtain that
By taking the sum over k’s and taking into account (18) it is straightforward to conclude the proof.
Proof of (22). (Maximization step of EM algorithm.) Denote by Then for fixed δk and Σk the maximum of 
is given for the vector 
By substituting αk into the expression and simplifying we obtain that
where 
By taking sum over k’s we obtain
as we aimed to prove.
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Artikel in diesem Heft
- Studying the evolution of transcription factor binding events using multi-species ChIP-Seq data
- Approximate Bayesian computation with functional statistics
- Monte Carlo estimation of total variation distance of Markov chains on large spaces, with application to phylogenetics
- Higher order asymptotics for negative binomial regression inferences from RNA-sequencing data
- Flexible pooling in gene expression profiles: design and statistical modeling of experiments for unbiased contrasts
- On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo
- Inferring latent gene regulatory network kinetics
Artikel in diesem Heft
- Studying the evolution of transcription factor binding events using multi-species ChIP-Seq data
- Approximate Bayesian computation with functional statistics
- Monte Carlo estimation of total variation distance of Markov chains on large spaces, with application to phylogenetics
- Higher order asymptotics for negative binomial regression inferences from RNA-sequencing data
- Flexible pooling in gene expression profiles: design and statistical modeling of experiments for unbiased contrasts
- On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo
- Inferring latent gene regulatory network kinetics
