Local Negative Circuits and Cyclic Attractors in Boolean Networks with at most Five 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 ≥ 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.

Isometries of the hypercube: A tool for Boolean regulatory networks analysis

“…The action of Γ n on boolean FDS was first introduced by L.Glass [5] for the classification of asynchronous STG.In [22,2], this action is studied with symbolical considerations, and the authors note that isometric STG are also isomorphic (in the sense of graph theory, renumbering of the nodes preserving adjacency). In Proposition 2 we emphasize a class of dynamics for which isomorphic STG are also isometric.…”

“…This point of view is not new and was first introduced by L.Glass [6]. It has been taken up in recent works [22,2].…”

Positive and negative cycles in Boolean networks

“…Ruet's counter example is built with another technic. Recently, Tonello, Chaouiya and Farcot [69] proved, using a SAT solver, that there is no counter example with n ≤ 5. Theorem 16 (Counter example to the local version of Thomas' second rule).…”