Common Attractors in Multiple Boolean Networks

Analyzing multiple networks is important to understand relevant features among different networks. Although many studies have been conducted for that purpose, not much attention has been paid to the analysis of attractors (i.e., steady states) in multiple networks. Therefore, we study common attractors and similar attractors in multiple networks to uncover hidden similarities and differences among networks using Boolean networks (BNs), where BNs have been used as a mathematical model of genetic networks and neural networks. We define three problems on detecting common attractors and similar attractors, and theoretically analyze the expected number of such objects for random BNs, where we assume that given networks have the same set of nodes (i.e., genes). We also present four methods for solving these problems. Computational experiments on randomly generated BNs are performed to demonstrate the efficiency of our proposed methods. In addition, experiments on a practical biological system, a BN model of the TGF-<inline-formula><tex-math notation="LaTeX">$\beta$</tex-math></inline-formula> signaling pathway, are performed. The result suggests that common attractors and similar attractors are useful for exploring tumor heterogeneity and homogeneity in eight cancers.