Control of complex networks such as gene regulatory networks is one of the fundamental problems in control theory. A Boolean network (BN) is one of the mathematical models in complex networks, and represents the dynamic behavior by Boolean functions. In this paper, a solution method for the finite-time control problem of BNs is proposed using a BDD (binary decision diagram). In this problem, we find all combinations of the initial state and the control input sequence such that a certain control specification is satisfied. The use of BDDs enables us to solve this problem for BNs such that the conventional method cannot be applied. First, after the outline of BNs and BDDs is explained, the problem studied in this paper is given. Next, a solution method using BDDs is proposed. Finally, a numerical example on a 67-node BN is presented.

Boolean networks (BNs) are considered as popular formal models for the dynamics of gene regulatory networks. There are many different types of BNs, depending on their updating scheme (synchronous, asynchronous, deterministic, or non-deterministic), such as Classical Random Boolean Networks (CRBNs), Asynchronous Random Boolean Networks (ARBNs), Generalized Asynchronous Random Boolean Networks (GARBNs), Deterministic Asynchronous Random Boolean Networks (DARBNs), and Deterministic Generalized Asynchronous Random Boolean Networks (DGARBNs). An important long-term behavior of BNs, so-called attractor, can provide valuable insights into systems biology (e.g., the origins of cancer). In the previous paper [1], we have studied properties of attractors of GARBNs, their relations with attractors of CRBNs, also proposed different algorithms for attractor detection. In this paper, we propose a new algorithm based on SAT-based bounded model checking to overcome inherent problems in these algorithms. Experimental results prove the effectiveness of the new algorithm. We also show that studying attractors of GARBNs can pave potential ways to study attractors of ARBNs.

This paper addresses the designability of Boolean networks, i.e., the existence of a Boolean function satisfying an attractor condition under a given network structure. In particular, we present here a necessary and sufficient condition of the designability of Boolean networks with multiple attractors. The condition is characterized by the cyclicity of network structures, which allows us to easily determine the designability.

A Boolean network model is one of the models of gene regulatory networks, and is widely used in analysis and control. Although a Boolean network is a class of discrete-time nonlinear systems and expresses the synchronous behavior, it is important to consider the asynchronous behavior. In this paper, using a Petri net, a new modeling method of asynchronous Boolean networks with control inputs is proposed. Furthermore, the optimal control problem of Petri nets expressing asynchronous Boolean networks is formulated, and is reduced to an integer programming problem. The proposed approach will provide us one of the mathematical bases of control methods for gene regulatory networks.

The Boolean network (BN) can be used to create discrete mathematical models of gene regulatory networks. In this paper, we consider three problems on BNs that are known to be NP-hard: detection of a singleton attractor, finding a control strategy that shifts a BN from a given initial state to the desired state, and control of attractors. We propose integer programming-based methods which solve these problems in a unified manner. Then, we present results of computational experiments which suggest that the proposed methods are useful for solving moderate size instances of these problems. We also show that control of attractors is -hard, which suggests that control of attractors is harder than the other two problems.

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.

The impact degree is a measure of the robustness of a metabolic network against deletion of single or multiple reaction(s). Although such a measure is useful for mining important enzymes/genes, it was defined only for networks without cycles. In this paper, we extend the impact degree for metabolic networks containing cycles and develop a simple algorithm to calculate the impact degree. Furthermore we improve this algorithm to reduce computation time for the impact degree by deletions of multiple reactions. We applied our method to the metabolic network of E. coli, that includes reference pathways, consisting of 3281 reaction nodes and 2444 compound nodes, downloaded from KEGG database, and calculate the distribution of the impact degree. The results of our computational experiments show that the improved algorithm is 18.4 times faster than the simple algorithm for deletion of reaction-pairs and 11.4 times faster for deletion of reaction-triplets. We also enumerate genes with high impact degrees for single and multiple reaction deletions.