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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.

Using single-index ODEs to study dynamic gene regulatory network