Identification of Boolean Network Models From Time Series Data Incorporating Prior Knowledge

Thomas Leifeld, Zhihua Zhang, Ping Zhang

Motivation: Mathematical models take an important place in science and engineering. A model can help scientists to explain dynamic behavior of a system and to understand the functionality of system components. Since length of a time series and number of replicates is limited by the cost of experiments, Boolean networks as a structurally simple and parameter-free logical model for gene regulatory networks have attracted interests of many scientists. In order to fit into the biological contexts and to lower the data requirements, biological prior knowledge is taken into consideration during the inference procedure. In the literature, the existing identification approaches can only deal with a subset of possible types of prior knowledge. Results: We propose a new approach to identify Boolean networks from time series data incorporating prior knowledge, such as partial network structure, canalizing property, positive and negative unateness. Using vector form of Boolean variables and applying a generalized matrix multiplication called the semi-tensor product (STP), each Boolean function can be equivalently converted into a matrix expression. Based on this, the identification problem is reformulated as an integer linear programming problem to reveal the system matrix of Boolean model in a computationally efficient way, whose dynamics are consistent with the important dynamics captured in the data. By using prior knowledge the number of candidate functions can be reduced during the inference. Hence, identification incorporating prior knowledge is especially suitable for the case of small size time series data and data without sufficient stimuli. The proposed approach is illustrated with the help of a biological model of the network of oxidative stress response. Conclusions: The combination of efficient reformulation of the identification problem with the possibility to incorporate various types of prior knowledge enables the application of computational model inference to systems with limited amount of time series data. The general applicability of this methodological approach makes it suitable for a variety of biological systems and of general interest for biological and medical research.