Fixed points and connections between positive and negative cycles in Boolean networks

We are interested in the relationships between the number of fixed points in a Boolean network $f:\{0,1\}^ n\to\{0,1\}^n$ and its interaction graph $G$, which is the signed digraph on $\{1,\dots,n\}$ that describes the positive and negative influences between the components of the network. A fundamental theorem of Aracena, suggested by the biologist Thomas, says that if $G$ has no positive (resp. negative) cycles, then $f$ has at most (resp. at least) one fixed point; the sign of a cycle being the product of the signs of its arcs. Here we generalize this result by taking into account the influence of connections between positive and negative cycles. In particular, we prove that if every positive (resp. negative) cycle of $G$ has an arc $a$ such that $G\setminus a$ has a non-trivial initial strongly connected component containing the final vertex of $a$ and only negative (resp. positive) cycles, then $f$ has at most (resp. at least) one fixed point. Besides, Aracena proved that if $G$ is strongly connected and has no negative cycles, then $f$ has two fixed points with Hamming distance $n$, and we prove that the same conclusion can be obtained under the following condition: $G$ is strongly connected, has a unique negative cycle $C$, has at least one positive cycle, and $f$ canalizes no arc of $C$.