This paper proposes and investigates a Boolean gossip model as a simplified but non-trivial probabilistic Boolean network. With positive node interactions, in view of standard theories from Markov chains, we prove that the node states asymptotically converge to an agreement at a binary random variable, whose distribution is characterized for large-scale networks by mean-field approximation. Using combinatorial analysis, we also successfully count the number of communication classes of the positive Boolean network explicitly in terms of the topology of the underlying interaction graph, where remarkably minor variation in local structures can drastically change the number of network communication classes. With general Boolean interaction rules, emergence of absorbing network Boolean dynamics is shown to be determined by the network structure with necessary and sufficient conditions established regarding when the Boolean gossip process defines absorbing Markov chains. Particularly, it is shown that for the majority of the Boolean interaction rules, except for nine out of the total @math possible nonempty sets of binary Boolean functions, whether the induced chain is absorbing has nothing to do with the topology of the underlying interaction graph, as long as connectivity is assumed. These results illustrate possibilities of relating dynamical properties of Boolean networks to graphical properties of the underlying interactions.
Gene Regulation . The evolution of gene expressions can be naturally described as a dynamical system where the two quantized levels, ON and OFF, are represented by logic states 1 and 0, respectively. Each gene normally would only interact with a small number of neighbouring genes Such number is two or three in Kauffman's original proposal @cite_11 . . Therefore, the proposed Boolean gossip network model at least serves as a good approximation for gene regulator networks, where a pair of genes interact at any given time and the Boolean function rules @math describe random outcomes of the interactions.
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Cells must respond to environmental changes to remain viable, yet the information they receive is often noisy. Through a biochemical implementation of Bayes's rule, we show that genetic networks can act as inference modules, inferring from intracellular conditions the likely state of the extracellular environment and regulating gene expression appropriately. By considering a two-state environment, either poor or rich in nutrients, we show that promoter occupancy is proportional to the (posterior) probability of the high nutrient state given current intracellular information. We demonstrate that single-gene networks inferring and responding to a high environmental state infer best when negatively controlled, and those inferring and responding to a low environmental state infer best when positively controlled. Our interpretation is supported by experimental data from the lac operon and should provide a basis for both understanding more complex cellular decision-making and designing synthetic inference circuits. Background: Organisms simplify the orchestration of gene expression by coregulating genes whose products function together in the cell. Many proteins serve different roles depending on the demands of the organism, and therefore the corresponding genes are often coexpressed with different groups of genes under different situations. This poses a challenge in analyzing wholegenome expression data, because many genes will be similarly expressed to multiple, distinct groups of genes. Because most commonly used analytical methods cannot appropriately represent these relationships, the connections between conditionally coregulated genes are often missed. Results: We used a heuristically modified version of fuzzy k-means clustering to identify overlapping clusters of yeast genes based on published gene-expression data following the response of yeast cells to environmental changes. We have validated the method by identifying groups of functionally related and coregulated genes, and in the process we have uncovered new correlations between yeast genes and between the experimental conditions based on similarities in gene-expression patterns. To investigate the regulation of gene expression, we correlated the clusters with known transcription factor binding sites present in the genes’ promoters. These results give insights into the mechanism of the regulation of gene expression in yeast cells responding to environmental changes. Conclusions: Fuzzy k-means clustering is a useful analytical tool for extracting biological insights from gene-expression data. Our analysis presented here suggests that a prevalent theme in the regulation of yeast gene expression is the condition-specific coregulation of overlapping sets of genes. This is the second paper of a series of two about the structural properties that influence the asymptotic dynamics of Random Boolean Networks. Here we study the functionally independent clusters in which the relevant elements, introduced and studied in our first paper [3], are subdivided. We show that the phase transition in Random Boolean Networks can also be described as a percolation transition. The statistical properties of the clusters of relevant elements (that we call modules) give an insight on the scaling behavior of the attractors of the critical networks that, according to Kauffman, have a biological analogy as a model of genetic regulatory systems. 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.