Local negative circuits and cyclic attractors in Boolean networks with at most five components
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We consider the following question on the relationship between the asymptotic behaviours 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 ≥ 6, the answer is known to be negative. We show that the question can be translated into a Boolean satisfiability problem on n · 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 ≤ 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.
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