Semi-tensor product based long-run behavior estimation of generalized asynchronous Boolean networks with time delays

This paper investigates the long-run behavior estimation of generalized asynchronous Boolean networks with time delays, particularly the evolutionary trends of the attractor and its basin. First, the dynamic form of the original network is remapped into an equivalent augmented system based on the algebraic state-space representation, and a transition table is constructed. Second, definitions of delayed fixed points and delayed limit cycles are provided based on the augmented system. Then, according to the transition table created in the first part, some necessary and sufficient conditions for the delayed fixed points and delayed limit cycles of generalized asynchronous Boolean networks with time delays are provided. Third, basins of delayed fixed points and delayed limit cycles are found. Afterward, the state transition diagram of generalized asynchronous Boolean networks with time delays is drawn, and the overlap elimination for different basins is studied by providing the necessary and sufficient conditions. The effectiveness of the proposed scheme is demonstrated through a simplified model for the lac operon regulating lactose metabolic enzymes and the metabolites of other metabolic pathways in Escherichia coli.