High-Dimensional ODEs Coupled With Mixed-Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification

Abstract

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 processes in a systematic way. However, the construction of 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 regulation as well as feedback effects of genes in a functional module on the dynamic expression changes of 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 nonparametric 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, 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. This article has supplementary material online.