Generalized Boolean Networks

Abstract

Random Boolean Networks (RBN) have been introduced by Kauffman more than thirty years ago as a highly simplified model of genetic regulatory networks. This extremely simple and abstract model has been studied in detail and has been shown capable of extremely interesting dynamical behavior. First of all, as some parameters are varied such as the network’s connectivity, or the probability of expressing a gene, the RBN can go through a phase transition, going from an ordered regime to a chaotic one. Kauffman’s suggestion is that cell types correspond to attractors in the RBN phase space, and only those attractors that are short and stable under perturbations will be of biological interest. Thus, according to Kauffman, RBN lying at the edge between the ordered phase and the chaotic phase can be seen as abstract models of genetic regulatory networks. The original view of Kauffman, namely that these models may be useful for understanding real-life cell regulatory networks, is still valid, provided that the model is updated to take into account present knowledge about the topology of real gene regulatory networks, and the timing of events, without loosing its attractive simplicity. According to present data, many biological networks, including genetic regulatory networks, seem, in fact, to be of the scale-free type. From the point of view of the timing of events, standard RBN update their state synchronously. This assumption is open to discussion when dealing with biologically plausible networks. In particular, for genetic regulatory networks, this is certainly not the case: genes seem to be expressed in different parts of the network at different times, according to a strict sequence, which depends on the particular network under study. The expression of a gene depends on several transcription factors, the synthesis of which appear to be neither fully synchronous nor instantaneous. Therefore, we have recently proposed a new, more biologically plausible model. It assumes a scale-free topology of the networks and we define a suitable semi-synchronous dynamics that better captures the presence of an activation sequence of genes linked to the topological properties of the network. By simulating statistical ensembles of networks, we discuss the attractors of the dynamics, showing that they are compatible with theoretical biological network models. Moreover, the model demonstrates interesting scaling abilities as the size of the networks is increased.