Local negative circuits and cyclic attractors in Boolean networks with at most five components

TL;DR

This paper proves that in Boolean networks with up to five components, the existence of cyclic attractors necessarily implies the presence of local negative circuits, linking network structure to dynamic behavior.

Contribution

It establishes a formal equivalence between cyclic attractors and local negative circuits for Boolean networks with at most five components, using satisfiability analysis.

Findings

Cyclic attractors imply local negative circuits for n ≤ 5.

Boolean satisfiability approach used to analyze network properties.

The relationship differs for networks with n ≥ 6 components.

Abstract

We consider the following question on the relationship between the asymptotic behaviours of asynchronous dynamics of Boolean networks and their regulatory structures: does the presence of a cyclic attractor imply the existence of a local negative circuit in the regulatory graph? When the number of model components $n$ verifies $n \geq 6$, the answer is known to be negative. We show that the question can be translated into a Boolean satisfiability problem on $n \cdot 2^n$ variables. A Boolean formula expressing the absence of local negative circuits and a necessary condition for the existence of cyclic attractors is found unsatisfiable for $n \leq 5$. In other words, for Boolean networks with up to $5$ components, the presence of a cyclic attractor requires the existence of a local negative circuit.