Gene regulation is a complicated process. The interaction of many genes and their products forms an intricate biological network. Identification of this dynamic network will help us understand the biological process in a systematic way. However, the construction of such a dynamic network is very challenging for a high-dimensional system. In this article we propose to use a set of ordinary differential equations (ODE), coupled with dimensional reduction by clustering and mixed-effects modeling techniques, to model the dynamic gene regulatory network (GRN). The ODE models allow us to quantify both positive and negative gene regulations as well as feedback effects of one set of genes in a functional module on the dynamic expression changes of the genes in another functional module, which results in a directed graph network. A five-step procedure, Clustering, Smoothing, regulation Identification, parameter Estimates refining and Function enrichment analysis (CSIEF) is developed to identify the ODE-based dynamic GRN. In the proposed CSIEF procedure, a series of cutting-edge statistical methods and techniques are employed, that include non-parametric mixed-effects models with a mixture distribution for clustering, nonparametric mixed-effects smoothing-based methods for ODE models, the smoothly clipped absolute deviation (SCAD)-based variable selection, and stochastic approximation EM (SAEM) approach for mixed-effects ODE model parameter estimation. The key step, the SCAD-based variable selection of the proposed procedure is justified by investigating its asymptotic properties and validated by Monte Carlo simulations. We apply the proposed method to identify the dynamic GRN for yeast cell cycle progression data. We are able to annotate the identified modules through function enrichment analyses. Some interesting biological findings are discussed. The proposed procedure is a promising tool for constructing a general dynamic GRN and more complicated dynamic networks.
