Abstract
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.
Highlights
Boolean networks (BNs) are discrete-time systems, whose variables can take only two possible values (i.e., 0 and 1)
The inference can be based on the connection of known biochemical reactions, like BN model for the yeast cell cycle in Davidich and Bornholdt (2008), or on experimental data, if the latter is the case it is called the identification problem
One of the first approaches to identify a BN was REVEAL which is based on mutual information (Liang et al, 1998)
Summary
Boolean networks (BNs) are discrete-time systems, whose variables can take only two possible values (i.e., 0 and 1). For the binarization several approaches can be found in the literature ranging from mixture model based clustering (Zhou et al, 2003) to more complex methods where the significance of a jump in the time series is estimated in Hopfensitz et al (2012). In Higa et al (2011) the data is considered as given constraint and the set of systems fulfilling the constraints is searched. This approach was further improved by reducing the sensitivity to noise in Ouyang et al (2014). An example of recent research is the identification of Boolean models for transient dynamics after perturbations from time course data with answer set programming (Ostrowski et al, 2016). Using the STP based matrix description of BCN several approaches for identifying BCN have been proposed (Cheng and Zhao, 2011; Fornasini and Valcher, 2014; Zhang et al, 2017a)
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