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

Abstract

Boolean finite dynamical systems (FDS) are commonly used in systems biology to model the dynamics of intracellular regulatory networks and interpret the emergence of cellular behaviors. Given a Boolean FDS, we can compute the corresponding regulatory network, that is a directed signed graph representing all the interactions between components (genes), endowed with logical rules specifying the dynamical behavior of the system. We consider the asynchronous trajectories generated by this Boolean FDS, supported by the hypercube. The exploration and analysis of this dynamics is a challenging task because of the combinatorial explosion that we face. A way to approach this problem is to exploit the links between the regulatory graph and the dynamics.We use the isometries of the hypercube to define classes gathering all the isometric Boolean FDS. Thus, we classify the set of Boolean FDS on the basis of those isometries, and emphasize their common features through regulatory graphs and logical rules. We can then restrict the dynamical analysis of all the Boolean FDS to one representative per class, and thereby considerably improve the efficiency of analysis of all the Boolean FDS. Relying on invariants properties, we propose a constructive method to provide, given a FDS, a representative regulatory graph of its class.We illustrate the efficiency of the method in concrete situations. For instance, the motif analysis (Remy et al., 2003; Remy et al., 2016; Didier and Remy, 2012) is strongly improved thanks to this classification. We also revisit the negative Thomas’ rule (Remy and Ruet, 2008; Richard, 2010) by establishing a new demonstration.