Stabilization of Boolean Networks: A Model-Free Constrained State-Flipped Control Approach

In this paper, we explore the stabilization of Boolean Networks (BNs) under state-flipped control, incorporating three key constraints to address the challenges of disease treatment response times and the practical limitations of therapeutic interventions. Our approach includes: (i) a feasible flipped nodes constraint to ensure a subset of nodes can be realistically manipulated; (ii) a transient period constraint to align treatment with time-sensitive windows of therapeutic opportunity; and (iii) a constraint on the number of flip transitions to minimize resource consumption and potential side effects. Furthermore, an evolution selection rule is set for the BNs. Then, a necessary and sufficient condition is given for the stabilization under the above constraints. We present a modified Q-learning (QL) algorithm to design a policy for stabilization under constraints in the model-free case. Finally, BN models of metastatic melanoma network and T-cell large granular lymphocyte (T-LGL) network are considered to show the applicability of our results. Note to Practitioners—In this paper, we investigate the stabilization problem under three constraints for Boolean networks with state-flipped control. In gene regulatory networks, drug intervention usually only controls the activation and silencing of some or specific genes. Although the effect of drug intervention is significant in the short term, long-term application may cause cells to develop resistance. To solve these challenges, we consider the feasible flipped node constraint, transient period constraint and the number of flip transitions constraint. Current research on Boolean networks often overlooks various real-world constraints when considering state-flipped control, and our study addresses this oversight. We employ a model-free reinforcement learning approach to find the optimal state-flipped control policy under the given constraints, including timing and duration of drug use. The goal is to change the cell state from unstable to stable, which has important implications for improving the therapeutic effect of gene regulation. The solution proposed in this paper not only considers the dynamic characteristics of drug intervention, but also fully considers the biological feasibility and clinical application safety, which provides a new perspective and method for the state control of gene regulatory network.