Scaling in Ordered and Critical Random Boolean Networks (original) (raw)
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Dynamical regimes in non-ergodic random Boolean networks
Random boolean networks are a model of genetic regulatory networks that has proven able to describe experimental data in biology. They not only reproduce important phenomena in cell dynamics, but they are also extremely interesting from a theoretical viewpoint, since it is possible to tune their asymptotic behaviour from order to disorder. The usual approach characterizes network families as a whole, either by means of static or dynamic measures. We show here that a more detailed study, based on the properties of system's attractors, can provide information that makes it possible to predict with higher precision important properties, such as system's response to gene knock-out. A new set of principled measures is introduced, that explains some puzzling behaviours of these networks. These results are not limited to random Boolean network models, but they are general and hold for any discrete model exhibiting similar dynamical characteristics.
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Random Boolean Networks (RBNs for short) are strongly simplified models of gene regulatory networks (GRNs), which have also been widely studied as abstract models of complex systems and have been used to simulate different phenomena. We define the “common sea” (CS) as the set of nodes that take the same value in all the attractors of a given network realization, and the “specific part” (SP) as the set of all the other nodes, and we study their properties in different ensembles, generated with different parameter values. Both the CS and of the SP can be composed of one or more weakly connected components, which are emergent intermediate-level structures. We show that the study of these sets provides very important information about the behavior of the model. The distribution of distances between attractors is also examined. Moreover, we show how the notion of a “common sea” of genes can be used to analyze data from single-cell experiments.
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The recently measured yeast transcriptional network is analyzed in terms of simplified Boolean network models, with the aim of determining feasible rule structures, given the requirement of stable solutions of the generated Boolean networks. We find that, for ensembles of generated models, those with canalyzing Boolean rules are remarkably stable, whereas those with random Boolean rules are only marginally stable. Furthermore, substantial parts of the generated networks are frozen, in the sense that they reach the same state, regardless of initial state. Thus, our ensemble approach suggests that the yeast network shows highly ordered dynamics.
Dynamics of unperturbed and noisy generalized Boolean networks
Journal of Theoretical Biology, 2009
For years, we have been building models of gene regulatory networks, where recent advances in molecular biology shed some light on new structural and dynamical properties of such highly complex systems. In this work, we propose a novel timing of updates in Random and Scale-Free Boolean Networks, inspired by recent findings in molecular biology. This update sequence is neither fully synchronous nor asynchronous, but rather takes into account the sequence in which genes affect each other. We have used both Kauffman's original model and Aldana's extension, which takes into account the structural properties about known parts of actual GRNs, where the degree distribution is right-skewed and long-tailed. The computer simulations of the dynamics of the new model compare favorably to the original ones and show biologically plausible results both in terms of attractors number and length. We have complemented this study with a complete analysis of our systems' stability under transient perturbations, which is one of biological networks defining attribute. Results are encouraging, as our model shows comparable and usually even better behavior than preceding ones without loosing Boolean networks attractive simplicity.
Robustness and fragility of Boolean models for genetic regulatory networks
Journal of Theoretical Biology, 2005
Interactions between genes and gene products give rise to complex circuits that enable cells to process information and respond to external signals. Theoretical studies often describe these interactions using continuous, stochastic, or logical approaches. We propose a new modeling framework for gene regulatory networks, that combines the intuitive appeal of a qualitative description of gene states with a high flexibility in incorporating stochasticity in the duration of cellular processes. We apply our methods to the regulatory network of the segment polarity genes, thus gaining novel insights into the development of gene expression patterns. For example, we show that very short synthesis and decay times can perturb the wild type pattern. On the other hand, separation of timescales between pre-and posttranslational processes and a minimal prepattern ensure convergence to the wild type expression pattern regardless of fluctuations.
Propagation of external regulation and asynchronous dynamics in random Boolean networks
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Boolean Networks and their dynamics are of great interest as abstract modeling schemes in various disciplines, ranging from biology to computer science. Whereas parallel update schemes have been studied extensively in past years, the level of understanding of asynchronous updates schemes is still very poor. In this paper we study the propagation of external information given by regulatory input variables into a random Boolean network. We compute both analytically and numerically the time evolution and the asymptotic behavior of this propagation of external regulation (PER). In particular, this allows us to identify variables which are completely determined by this external information. All those variables in the network which are not directly fixed by PER form a core which contains in particular all non-trivial feedback loops. We design a message-passing approach allowing to characterize the statistical properties of these cores in dependence of the Boolean network and the external condition. At the end we establish a link between PER dynamics and the full random asynchronous dynamics of a Boolean network.
Perturbation in Genetic Regulatory Networks: Simulation and Experiments
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Random boolean networks (RBN) have been proposed more than thirty years ago as models of genetic regulatory networks. Recent studies on the perturbation in gene expression levels induced by the knock-out (i.e. silencing) of single genes have shown that simple RBN models give rise to a distribution of the size of the perturbations which is very similar in different model network realizations, and is also very similar to the one actually found in experimental data concerning a unicellular organism (S.cerevisiae). In this paper we present further results, based upon the same set of experiments, concerning the correlation between different perturbations. We compare actual data from S. cerevisiae with the results of simulations concerning RBN models with more than 6000 nodes, and comment on the usefulness and limitations of RBN models.
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One aim of synthetic biology is to construct increasingly complex genetic networks from interconnected simpler ones to address challenges in medicine and biotechnology. However, as systems increase in size and complexity, emergent properties lead to unexpected and complex dynamics due to nonlinear and nonequilibrium properties from component interactions. We focus on four different studies of biological systems which exhibit complex and unexpected dynamics. Using simple synthetic genetic networks, small and large populations of phase-coupled quorum sensing repressilators, Goodwin oscillators, and bistable switches, we review how coupled and stochastic components can result in clustering, chaos, noise-induced coherence and speed-dependent decision making. A system of repressilators exhibits oscillations, limit cycles, steady states or chaos depending on the nature and strength of the coupling mechanism. In large repressilator networks, rich dynamics can also be exhibited, such as clu...
Interacting Random Boolean Networks
Random Boolean networks (RBN) have been extensively studied as models of genetic regulatory networks. While many studies have been devoted to the dynamics of isolated random Boolean networks, which may considered as models of isolated cells, in this paper we consider a set of interacting RBNs, which may be regarded as a simplified model of a tissue or a monoclonal colony. In order to do so, we introduce a cellular automata (CA) model, where each cell site is occupied by a RBN. The mutual influence among cells is modelled by letting the activation of some genes in a RBN be affected by that of some genes in neighbouring RBNs. It is shown that the dynamics of the CA is far from trivial. Different measures are introduced to provide indications about the overall behaviour. In a sense which is made precise in the text, it is shown that the degree of order of the CA is affected by the interaction strength, and that markedly different behaviours are observed. We propose a classification of these behaviours into four classes, based upon the way in which the various measures of order are affected by the interaction strength. It is shown that the dynamical properties of isolated RBNs affect the probability that a CA composed by those RBNs belongs to one of the four classes, and therefore also affects the probability that a higher interaction strength leads to a greater, or a smaller, degree of order.