An empirical Bayes approach to estimating dynamic models of co-regulated gene expression

Time-course gene expression datasets provide insight into the dynamics of complex biological processes, such as immune response and organ development. It is of interest to identify genes with similar temporal expression patterns because such genes are often biologically related. However, this task is challenging due to the high dimensionality of these datasets and the nonlinearity of gene expression time dynamics. We propose an empirical Bayes approach to estimating ordinary differential equation (ODE) models of gene expression, from which we derive a similarity metric between genes called the Bayesian lead-lag $R^2$ (LLR2). Importantly, the calculation of the LLR2 leverages biological databases that document known interactions amongst genes; this information is automatically used to define informative prior distributions on the ODE model's parameters. As a result, the LLR2 is a biologically-informed metric that can be used to identify clusters or networks of functionally-related genes with co-moving or time-delayed expression patterns. We then derive data-driven shrinkage parameters from Stein's unbiased risk estimate that optimally balance the ODE model's fit to both data and external biological information. Using real gene expression data, we demonstrate that our methodology allows us to recover interpretable gene clusters and sparse networks. These results reveal new insights about the dynamics of biological systems.