Stabilizing Large-Scale Probabilistic Boolean Networks by Pinning Control

Set stabilization of logical control networks: A minimum node control approach

In network systems, control using minimum nodes or pinning control can be effectively used for stabilization problems to cut down the cost of control. In this paper, we investigate the set stabilization problem of logical control networks. In particular, we study the set stabilization problem of probabilistic Boolean networks (PBNs) and probabilistic Boolean control networks (PBCNs) via controlling minimal nodes. Firstly, an algorithm is given to search for the minimum index set of pinning nodes. Then, based on the analysis of its high computational complexity, we present optimized algorithms with lower computational complexity to ascertain the network control using minimum node sets. Moreover, some sufficient and necessary conditions are proposed to ensure the feasibility and effectiveness of the proposed algorithms. Furthermore, a theorem is presented for PBCNs to devise all state-feedback controllers corresponding to the set of pinning nodes. Finally, two models of gene regulatory networks are considered to show the efficacy of obtained results.

Necessary and Sufficient Conditions for Event-Triggered Set Stabilizability of Markovian Jump Logical Networks

This technical paper utilizes the Lyapunov theory to characterize the event-triggered set stabilizability of Markovian jump logical control networks (MJLCNs). Whereas the existing result for checking the set stabilizability of MJLCNs is only sufficient, this technical paper further establishes its necessary and sufficient condition. First, the Lyapunov function is established to describe the set stabilizability of MJLCNs necessarily and sufficiently by combining recurrent switching modes and desired state set. Then, the triggering condition and the input updating mechanism are designed regarding the value change of the Lyapunov function. Finally, the effectiveness of theoretical results is demonstrated by a biological example concerning the lac operon in Escherichia coli.

Dynamical methods for target control of biological networks

Estimating the influence that individual nodes have on one another in a Boolean network is essential to predict and control the system's dynamical behaviour, for example, detecting key therapeutic targets to control pathways in models of biological signalling and regulation. Exact estimation is generally not possible due to the fact that the number of configurations that must be considered grows exponentially with the system size. However, approximate, scalable methods exist in the literature. These methods can be divided into two main classes: (i) graph-theoretic methods that rely on representations of Boolean dynamics into static graphs and (ii) mean-field approaches that describe average trajectories of the system but neglect dynamical correlations. Here, we compare systematically the performance of these state-of-the-art methods on a large collection of real-world gene regulatory networks. We find comparable performance across methods. All methods underestimate the ground truth, with mean-field approaches having a better recall but a worse precision than graph-theoretic methods. Computationally speaking, graph-theoretic methods are faster than mean-field ones in sparse networks, but are slower in dense networks. The preference of which method to use, therefore, depends on a network's connectivity and the relative importance of recall versus precision for the specific application at hand.