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Register a new userAttractors in continuous and Boolean networks
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Added by
Bj\\\"orn Samuelsson
Publication date
2007
fields
Biology
and research's language is
English
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We study the stable attractors of a class of continuous dynamical systems that may be idealized as networks of Boolean elements, with the goal of determining which Boolean attractors, if any, are good approximations of the attractors of generic continuous systems. We investigate the dynamics in simple rings and rings with one additional self-input. An analysis of switching characteristics and pulse propagation explains the relation between attractors of the continuous systems and their Boolean approximations. For simple rings, reliable Boolean attractors correspond to stable continuous attractors. For networks with more complex logic, the qualitative features of continuous attractors are influenced by inherently non-Boolean characteristics of switching events.
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Boolean networks are discrete dynamical systems for modeling regulation and signaling in living cells. We investigate a particular class of Boolean functions with inhibiting inputs exerting a veto (forced zero) on the output. We give analytical expressions for the sensitivity of these functions and provide evidence for their role in natural systems. In an intracellular signal transduction network [Helikar et al., PNAS (2008)], the functions with veto are over-represented by a factor exceeding the over-representation of threshold functions and canalyzing functions in the same system. In Boolean networks for control of the yeast cell cycle [Fangting Li et al., PNAS (2004), Davidich et al., PLoS One (2009)], none or minimal changes to the wiring diagrams are necessary to formulate their dynamics in terms of the veto functions introduced here.
Complex systems are often modeled as Boolean networks in attempts to capture their logical structure and reveal its dynamical consequences. Approximating the dynamics of continuous variables by discrete values and Boolean logic gates may, however, introduce dynamical possibilities that are not accessible to the original system. We show that large random networks of variables coupled through continuous transfer functions often fail to exhibit the complex dynamics of corresponding Boolean models in the disordered (chaotic) regime, even when each individual function appears to be a good candidate for Boolean idealization. A suitably modified Boolean theory explains the behavior of systems in which information does not propagate faithfully down certain chains of nodes. Model networks incorporating calculated or directly measured transfer functions reported in the literature on transcriptional regulation of genes are described by the modified theory.
One successful model of interacting biological systems is the Boolean network. The dynamics of a Boolean network, controlled with Boolean functions, is usually considered to be a Markovian (memory-less) process. However, both self organizing features of biological phenomena and their intelligent nature should raise some doubt about ignoring the history of their time evolution. Here, we extend the Boolean network Markovian approach: we involve the effect of memory on the dynamics. This can be explored by modifying Boolean functions into non-Markovian functions, for example, by investigating the usual non-Markovian threshold function, - one of the most applied Boolean functions. By applying the non-Markovian threshold function on the dynamical process of a cell cycle network, we discover a power law memory with a more robust dynamics than the Markovian dynamics.
Cancer forms a robust system and progresses as stages over time typically with increasing aggressiveness and worsening prognosis. Characterizing these stages and identifying the genes driving transitions between them is critical to understand cancer progression and to develop effective anti-cancer therapies. Here, we propose a novel model of the cancer system as a Boolean state space in which a Boolean network, built from protein interaction and gene-expression data from different stages of cancer, transits between Boolean satisfiability states by editing interactions and flipping genes. The application of our model (called BoolSpace) on three case studies - pancreatic and breast tumours in human and post spinal-cord injury in rats - reveals valuable insights into the phenomenon of cancer progression. In particular, we notice that several of the genes flipped are serine/threonine kinases which act as natural cellular switches and that different sets of genes are flipped during the initial and final stages indicating a pattern to tumour progression. We hypothesize that robustness of cancer partly stems from passing of the baton between genes at different stages, and therefore an effective therapy should target a cover set of these genes. A C/C++ implementation of BoolSpace is freely available at: http://www.bioinformatics.org.au/tools-data
Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. The topologies of random Boolean networks with one input per node can be seen as graphs of random maps. We introduce an approach to investigating random maps and finding analytical results for attractors in random Boolean networks with the corresponding topology. Approximating some other non-chaotic networks to be of this class, we apply the analytic results to them. For this approximation, we observe a strikingly good agreement on the numbers of attractors of various lengths. We also investigate observables related to the average number of attractors in relation to the typical number of attractors. Here, we find strong differences that highlight the difficulties in making direct comparisons between random Boolean networks and real systems. Furthermore, we demonstrate the power of our approach by deriving some results for random maps. These results include the distribution of the number of components in random maps, along with asymptotic expansions for cumulants up to the 4th order.
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