Abstract
With the development of biotechnology, high-throughput studies on protein-protein, protein-gene, and gene-gene interactions become possible and attract remarkable attention. To explore the interactions in dynamic gene regulatory networks, we propose a single-index ordinary differential equation (ODE) model and develop a variable selection procedure. We employ the smoothly clipped absolute deviation penalty (SCAD) penalized function for variable selection. We analyze a yeast cell cycle gene expression data set to illustrate the usefulness of the single-index ODE model. In real data analysis, we group genes into functional modules using the smoothing spline clustering approach. We estimate state functions and their first derivatives for functional modules using penalized spline-based nonparametric mixed-effects models and the spline method. We substitute the estimates into the single-index ODE models, and then use the penalized profile least-squares procedure to identify network structures among the models. The results indicate that our model fits the data better than linear ODE models and our variable selection procedure identifies the interactions that may be missed by linear ODE models but confirmed in biological studies. In addition, Monte Carlo simulation studies are used to evaluate and compare the methods.
Highlights
Gene regulatory networks (GRN) are complex and dynamic systems in nature
In Section of Methods, we briefly describe the procedure for GRN construction with details for penalized profile least-squares (PPrLS) estimation and variable selection
We found the interactions identified by using single-index ordinary differential equation (ODE) were more accurate, i.e., the linear ODE models overlooked some confirmed regulator-regulator interactions [55]
Summary
Gene regulatory networks (GRN) are complex and dynamic systems in nature. They are composed of genes that interact with each other and with other substances inside cells, such as RNAs and proteins. ; p; ð1Þ dt where X(t) = (X1(t), Á Á Á, Xp(t))T represents gene expression levels at the time t of the p genes; F(Á, Á, Á) is a function which can be linear or nonlinear; and θ is an unknown parameter vector which quantifies the regulations or interactions among the genes in GRN. Once we can determine X(t), the gene expression levels which should be included in the ODE model (1), we can infer the interactions within a dynamic GRN. Parameter estimation and variable selection for single-index models have gained fruitful results, to the best of our knowledge, no method that couples single-index models with ODE to study dynamic GRN is available. The genes in a cluster (represented by a node) may play a common function in biological procession Such a network can single out regulator-regulator interactions which are helpful to avoid tedious experiments and to speed biological studies. All theory and associated technical details are given in the supporting materials (S1–S7 Files)
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