The Properties of Pseudo-Attractors in Random Boolean Networks

Random Boolean Networks are dissipative dynamical models of gene regulatory networks, which are older than fifty years but still raise considerable interest. In this paper we will rely on two key concepts which had been introduced in previous works, namely those of pseudo-attractors (which are obtained by projecting true dynamical attractors onto constant vectors) and of the “common sea” (de-fined as the set of nodes which take the same value in every pseudo-attractor of a given network realization). In particular, we will study the dependence of the number of pseudo-attractors and of the relative size of the common sea upon the values of some key parameters, like the average number of connections per node and the so-called bias of the set of Boolean functions, paying particular attention to dynamically critical networks. We will also comment on the relationship of these models with measured gene expression values in single-cell observations.