Simplification And Control Problems Of Probabilistic Boolean Network

The application of DNA microarray technology and protein mass spectrometry promote the study of the biological phenomena from the view of the information science. Gene regulatory network is one of the hot spots in the post genomic era. By using bioinformatics method, one can construct and analyze the gene regulatory network. This network provides insights about how several genes working together to achieve some biological targets. Probabilistic Boolean network is one of the mathematical models for gene regulatory network. A key aspect of the analysis of the probabilistic Boolean network is to figure out its steady-state distribution. This would help to obtain important information such as the long-run influence of a gene on another gene and the long-run joint probabilistic behavior of several target genes. Since the size of state space grows exponentially in the number of nodes, it is difficult to obtain the steady-state distribution for the large PBN. Therefore, it is essential to simplify the original PBN.The paper aims to simplify the PBN by eliminating the unimportant nodes and analyze the PBN on the basis of semi-tensor product. The results of this paper include the following four aspects:(1) The application of semi-tensor product, including the random generation of canalizing functions, the calculation of the influence matrix, the design of the research platform for the study of PBN and so on; (2) A new Boolean function class, blocking functions, is proposed, which contains the canalizing functions and has the similar property of maintaining the stability of Boolean networks as canalizing functions. The conception of blocking functions, nested blocking functions, and generalized canalizing functions provide theoretical basis for the study of Boolean network; (3) An efficient method is proposed to identify unimportant nodes of a PBN. Then the PBN is simplified by eliminating the unimportant nodes and modifying the predictors that regulated by the removed nodes. The simplified PBN can facilitate the analyze and control of the gene regulatory network. The numerical experiment on synthetic PBNs suggest that the simplification of the PBN only has a small effect on the dynamics of the PBN; (4) The design of the research platform for the study of the PBN. This platform can be used in analyzing the real gene regulatory network and the synthetic PBN.