Finding the fixed points of a Boolean network from a positive feedback vertex set

In the modeling of biological systems by Boolean networks a key problem is finding the set of fixed points of a given network. Some constructed algorithms consider certain structural properties of the interaction graph like those proposed by Akutsu et al. in \cite{akutsu1998system,zhang2007algorithms} which consider a feedback vertex set of the graph. However, these methods do not take into account the type of action (activation, inhibition) between its components. In this paper we propose a new algorithm for finding the set of fixed points of a Boolean network, based on a positive feedback vertex set $P$ of its interaction graph and which works, by applying a sequential update schedule, in time $O(2^{|P|} \cdot n^2)$, where $n$ is the number of components. The theoretical foundation of this algorithm is due a nice characterization, that we give, of the dynamical behavior of the Boolean networks without positive cycles and with a fixed point. An executable file of \Afp made in Java and some examples of input files are available at: \href{http://www.inf.udec.cl/~lilian/FPCollector/}{\url{www.inf.udec.cl/~lilian/FPCollector/}}