Random Boolean networks (RBNs) have been a popular model of genetic regulatory networks for more than four decades. However, most RBN studies have been made with random topologies, while real regulatory networks have been found to be modular. In this work, we extend classical RBNs to define modular RBNs. Statistical experiments and analytical results show that modularity has a strong effect on the properties of RBNs. In particular, modular RBNs have more attractors, and are closer to criticality when chaotic dynamics would be expected, than classical RBNs.
There are studies where RBNs are generated in cells of a 2D lattice, similar to a cellular automaton, where each RBN is weakly coupled with its von Neumann neighbors @cite_35 @cite_29 @cite_48 . The goal is to model intercellular signaling in a tissue.
@cite_35 @cite_29 @cite_48
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Random boolean networks (shortly, RBN) have proven useful in describing complex phenomena occurring at the unicellular level It is therefore interesting to investigate how their dynamical behavior is affected by cell-cell interactions, which mimics those occurring in tissues in multicellular organisms It has also been suggested that evolution may tend to adjust the parameters of the genetic network so that it operates close to a critical state, which should provide evolutionary advantage ; this hypothesis has received intriguing, although not definitive support from recent findings It is therefore particularly interesting to consider how the tissue-like organization alters the dynamical behavior of the networks close to a critical state In this paper we define a model tissue, which is a cellular automaton each of whose cells hosts a full RBN, and we report preliminary studies of the way in which the dynamics is affected. The establishment and maintenance of precisely organized tissues requires the formation of sharp borders between distinct cell populations. The maintenance of segregated cell populations is also required for tissue homeostasis in the adult, and deficiencies in segregation underlie the metastatic spreading of tumor cells. Three classes of mechanisms that underlie cell segregation and border formation have been uncovered. The first involves differences in cadherin-mediated cell–cell adhesion that establishes interfacial tension at the border between distinct cell populations. A second mechanism involves the induction of actomyosin-mediated contraction by intercellular signaling, such that cortical tension is generated at the border. Third, activation of Eph receptors and ephrins can lead to both decreased adhesion by triggering cleavage of E-cadherin, and to repulsion of cells by regulation of the actin cytoskeleton, thus preventing intermingling between cell populations. These mechanisms play crucial roles at distinct boundaries during development, and alterations in cadherin or Eph ephrin expression have been implicated in tumor metastasis. Deciphering the influence of the interaction among the constituents of a complex system on the overall behaviour is one of the main goals of complex systems science. The model we present in this work is a 2D square cellular automaton whose of each cell is occupied by a complete random Boolean network. Random Boolean networks are a well-known simplified model of genetic regulatory networks and this model of interacting RBNs may be therefore regarded as a simplified model of a tissue or a monoclonal colony. The mechanism of cell-to-cell interaction is here simulated letting some nodes of a particular network being influenced by the state of some nodes belonging to its neighbouring cells. One possible means to investigate the overall dynamics of a complex system is studying its response to perturbations. Our analyses follow this methodological approach. Even though the dynamics of the system is far from trivial we could show in a clear way how the interaction affects the dynamics and the global degree of order. We describe a two-dimensional stochastic model of intercellular Ca2+ wave (ICW) spread in glia that includes contributions of external stimuli, ionotropic and metabotropic P2 receptors, exo- and ecto-nucleotidases, second messengers, and gap junctions. In this model, an initial stimulus evokes ATP and UTP release from a single cell. Agonists diffuse and are degraded both in bulk solution and at cell surfaces. Ca2+ elevation in individual cells is determined by bound agonist concentrations s and by number and features of P2 receptors summed with that generated by IP3 diffusing through gap junction channels. Variability of ICWs is provided by randomly distributing a predetermined density of cells in a rectangular grid and by randomly selecting within intervals values characterizing the extracellular compartment, individual cells, and interconnections with neighboring cells. Variability intervals were obtained from experiments on astrocytoma cells transfected to express individual P2 receptors and or the gap junction protein connexin43. The simulation program (available as Supplementary Material) permits individual alteration of ICW components, allowing comparison of simulations with data from cells expressing connexin43 and or various P2 receptor subtypes. Such modeling is expected to be useful for testing phenomenological hypotheses and in understanding consequences of alteration of system components under experimental or pathological conditions.