Recursive Self-Composite Approach Toward Structural Understanding of Boolean Networks

Boolean networks have been widely used in systems biology to study the dynamical characteristics of biological networks such as steady-states or cycles, yet there has been little attention to the dynamic properties of network structures. Here, we systematically reveal the core network structures using a recursive self-composite of the logic update rules. We find that all Boolean update rules exhibit repeated cyclic logic structures, where each converged logic leads to the same states, defined as kernel states. Consequently, the period of state cycles is upper bounded by the number of logics in the converged logic cycle. In order to uncover the underlying dynamical characteristics by exploiting the repeating structures, we propose leaping and filling algorithms. The algorithms provide a way to avoid large string explosions during the self-composition procedures. Finally, we present three examples–a simple network with a long feedback structure, a T-cell receptor network and a cancer network–to demonstrate the usefulness of the proposed algorithm.