A PBN is well known as a mathematical model of complex network systems such as gene regulatory networks. In Boolean networks, interactions between nodes (e.g., genes) are modeled by Boolean functions. In PBNs, Boolean functions are switched probabilistically. In this paper, for a PBN, a simplified representation that is effective in analysis and control is proposed. First, after a polynomial representation of a PBN is briefly explained, a simplified representation is derived. Here, the steady-state value of the expected value of the state is focused, and is characterized by a minimum feedback vertex set of an interaction graph expressing interactions between nodes. Next, using this representation, input selection and stabilization are discussed. Finally, the proposed method is demonstrated by a biological example.

In this paper, the optimal control problem of a probabilistic Boolean network (PBN), which is one of the significant models in gene regulatory networks, is discussed. In the existing methods of optimal control for PBNs, it is necessary to compute state transition diagrams with 2n nodes for a given PBN with n states. To avoid this computation, a polynomial optimization approach is proposed. In the proposed method, a PBN is transformed into a polynomial system, and the optimal control problem is reduced to a polynomial optimization problem. Since state transition diagrams are not computed, the proposed method is convenient for users.