A Survey on Boolean Control Networks: A State Space Approach

Abstract

Boolean network is a proper tool to describe the cellular network. The rising of systems biology stimulates the investigation of Boolean (control) networks. Since the bearing space of a Boolean network is not a vector space, to use state space analysis to the dynamics of Boolean (control) network a proper way to describe the state space and its subspaces becomes a challenging problem. This paper surveys a systematic description of the state space of Boolean (control) networks. Under this framework the state space is described as a set of logical functions. Its subspaces are subsets of this set of logical functions. Using semi-tensor product of matrices and the matrix expression of logic, state space and each subspaces are connected to their structure matrices, which are logical matrices. In the light of this expression, certain properties of state space and subspaces, which are closely related to control problems, are obtained. Particularly, the coordinate transformation of state space, the regular subspaces, which is generated by part of coordinate variables; the invariant subspaces etc. are proposed and the corresponding necessary and sufficient conditions are presented to verify them.