STATE-SPACE MODEL WITH TIME DELAYS FOR GENE REGULATORY NETWORKS

Abstract

A gene regulatory network can be considered a dynamic cellular system which describes the behavior (development) of a living cell and depends completely on the current internal state plus any external inputs, if these exist. Although many details inside a cell are not precisely known, gene expression data on a genome scale provide useful insights into such a cellular system. With gene expression data, a wide variety of models, such as Boolean networks and differential/difference equations, have been proposed to model gene regulatory networks. In these previously proposed models, genes are viewed as the internal state variables of a cellular system. This viewpoint has suffered from the underestimation of the model parameters. In addition, these models ignore an important problem with a gene regulatory network — time delay. Instead, this paper proposes a state-space model with time delays for gene regulatory networks. The proposed model views genes as the observation variables, whose expression values depend on the current internal state variables and any external inputs. Bayesian information criterion (BIC) and probabilistic principal component analysis (PPCA) are used to estimate the number of internal state variables and their expression profiles from gene expression data. By constructing dynamic equations with time delays for the internal state variables and the relationships between the internal state variables and the observation variables (gene expression profiles), state-space models with time delays for gene regulatory networks are constructed. The parameters of the proposed model can be unambiguously identified from time-course gene expression data with a lower computational cost. The proposed model is applied to two time-course gene expression datasets, and two gene regulatory~networks are inferred, respectively. The analysis shows that the inferred gene regulatory networks have several features of the real gene regulatory networks, such as the stability, the robustness, and the periodicity. Further, compared to state-space models without time delays, the proposed model with time delays has better prediction accuracy.