Generalized Boolean Networks: How Spatial and Temporal Choices Influence Their Dynamics

Random Boolean Networks (RBN) have been introduced by Kauffman more than thirty years ago in a landmark paper as a highly simplified model of genetic regulatory networks. This extremely simple and abstract model has been studied in detail by analysis and by computer simulations of statistical ensembles of networks and it has been shown to be capable of extremely interesting dynamical behavior. First of all, it has been found that, as some parameters are varied such as the network's connectivity $K$, or the probability $p$ of expressing a gene, i.e. of switching on the corresponding node's state, the RBN can go through a phase transition. Indeed, for every value of $p$, there is a critical value of connectivity $K$ such that for values of $K$ below this critical value the system is in the ordered regime, while for values of $K$ above this limit the system is said to be in the chaotic regime. 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. From the structural and topological point of view, random networks with fixed connectivity degree $K$ were a logical generic choice in the beginning, since the exact couplings in networks were generally unknown. Today it is more open to criticism since it does not correspond to what we know about the topology of biological networks. According to present data, many biological networks, including genetic regulatory networks, seem, in fact, to be of the scale-free type or hierarchical and not random as suggested, among others, by Barabasi and coworkers. In addition, Aldana has recently presented a detailed analysis of Boolean networks with scale-free topology. He has been able to define a phase space diagram for Boolean networks, including the phase transition from ordered to chaotic dynamics, as a function of the power law exponent. From the point of view of the timing of events, standard RBN update their state synchronously. This assumption simplifies the analysis, but it is open to discussion when dealing with biologically plausible networks. In particular, for genetic regulatory networks, this is certainly not the case, as many recent experimental observations tend to prove. Rather, 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. Moreover, in some cases like the gene regulatory network controlling embryonic specification in the sea urchin, we can clearly see the presence of an activation sequence of genes. In view of the above shortcomings of RBN as an abstract description of genetic regulatory networks we conclude that neither fully synchronous nor completely random asynchronous network dynamics are suitable models. 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. Another interesting aspect of RBN is their resilience to perturbations and faults. Kauffman and coworkers have explored the robustness of fully synchronous RBN under different types of failures. In this chapter we also propose to investigate the robustness of our new model to perturbations.