Generalized Ordinary Differential Equation Models

<div><p>Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method. Supplementary materials for this article are available online.</p></div

Modeling Genome-Wide Dynamic Regulatory Network in Mouse Lungs with Influenza Infection Using High-Dimensional Ordinary Differential Equations

<div><p>The immune response to viral infection is regulated by an intricate network of many genes and their products. The reverse engineering of gene regulatory networks (GRNs) using mathematical models from time course gene expression data collected after influenza infection is key to our understanding of the mechanisms involved in controlling influenza infection within a host. A five-step pipeline: detection of temporally differentially expressed genes, clustering genes into co-expressed modules, identification of network structure, parameter estimate refinement, and functional enrichment analysis, is developed for reconstructing high-dimensional dynamic GRNs from genome-wide time course gene expression data. Applying the pipeline to the time course gene expression data from influenza-infected mouse lungs, we have identified 20 distinct temporal expression patterns in the differentially expressed genes and constructed a module-based dynamic network using a linear ODE model. Both intra-module and inter-module annotations and regulatory relationships of our inferred network show some interesting findings and are highly consistent with existing knowledge about the immune response in mice after influenza infection. The proposed method is a computationally efficient, data-driven pipeline bridging experimental data, mathematical modeling, and statistical analysis. The application to the influenza infection data elucidates the potentials of our pipeline in providing valuable insights into systematic modeling of complicated biological processes.</p></div

Overview of the temporal variations of DE genes.

<p>(a) Correlation matrix between every pair of time points indicates three major transcriptional phases. (b) Estimates for the first two eigenfunctions (first-black solid, second-red dashed) for the DE genes.</p

The road map of the proposed pipeline for reconstructing genome-wide dynamic GRNs.

<p>The road map of the proposed pipeline for reconstructing genome-wide dynamic GRNs.</p

Parameter Estimation and Variable Selection for Big Systems of Linear Ordinary Differential Equations: A Matrix-Based Approach

<p>Ordinary differential equations (ODEs) are widely used to model the dynamic behavior of a complex system. Parameter estimation and variable selection for a “Big System” with linear ODEs are very challenging due to the need of nonlinear optimization in an ultra-high dimensional parameter space. In this article, we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares (SLS). Simulation studies demonstrate that the proposed matrix-based SLS method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ODE system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector-based two-stage method that are currently available. We applied this new method to two real datasets—a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the Standard & Poor 1500 index stock price data with 1250 dimensions and 1,563,750 unknown parameters—to illustrate the utility and numerical performance of the proposed parameter estimation and variable selection method for big systems in practice. Supplementary materials for this article are available online.</p

Intra-module regulatory relationships for four modules M1 (a), M4 (b), M6 (c) and M7 (d).

<p>TFs are shown in aquamarine and target genes are shown in green. Isolated TFs and genes are not shown.</p

The inward and outward regulations in the module-based regulatory network.

<p>The negative sign indicates a negative coefficient in the linear ODE model; otherwise the coefficient is positive. The underlined modules are hub modules with the most outward regulations.</p

Temporal expression profiles of DE genes.

<p>(a) The DE genes are clustered into 20 modules (M1–M20) and the number of genes in each module is displayed in the parentheses. Shown are standardized gene expressions normalized to day 0. (b) The smoothed mean expression curve obtained from (6) (red solid) for each module overlaid with the refined estimate from the linear ODE model (blue dashed).</p

Research methodology overview.

Research methodology overview.</p

Relations between publications and RFAs through project number.

Relations between publications and RFAs through project number.</p