Abstract: Deterministic Boolean networks have been used as models of gene regulation and other biological networks. One key element in these models is the update schedule, which indicates the order in which states are to be updated. We study the robustness of the dynamical behavior of a Boolean network with respect to different update schedules (synchronous, block-sequential, sequential), which can provide modelers with a better understanding of the consequences of changes in this aspect of the model. For a given Boolean network, we define equivalence classes of update schedules with the same dynamical behavior, introducing a labeled graph which helps to understand the dependence of the dynamics with respect to the update, and to identify interactions whose timing may be crucial for the presence of a particular attractor of the system. Several other results on the robustness of update schedules and of dynamical cycles with respect to update schedules are presented. Finally, we prove that our equivalence classes generalize those found in sequential dynamical systems. (C) 2009 Elsevier Ireland Ltd. All rights reserved.

Abstract: An evolutionary computation framework to learn binary threshold networks is presented. Inspired by the recent trend of binary neural networks, where weights and activation thresholds are represented using 1 and -1 such that they can be stored in 1-bit instead of full precision, we explore this approach for gene regulatory network modeling. We test our method by inferring binary threshold networks of two regulatory network models: Quorum sensing systems in bacterium Paraburkholderia phytofirmans PsJN and the fission yeast cell-cycle. We considered differential evolution and particle swarm optimization for the simulations. Results for weights having only 1 and -1 values, and different activation thresholds are presented. Full binary threshold networks were found with minimum error (2 bits), whereas when the binary restriction is relaxed for the activation thresholds, networks with 0 bit error were found.

Abstract: A common gene regulatory network model is the threshold Boolean network, used for example to model the Arabidopsis thaliana floral morphogenesis network or the fission yeast cell cycle network. In this paper, we analyze a logical model of the mammalian cell cycle network and its threshold Boolean network equivalent. Firstly, the robustness of the network was explored with respect to update perturbations, in particular, what happened to the attractors for all the deterministic updating schemes. Results on the number of different limit cycles, limit cycle lengths, basin of attraction size, for all the deterministic updating schemes were obtained through mathematical and computational tools. Secondly, we analyzed the topology robustness of the network, by reconstructing synthetic networks that contained exactly the same attractors as the original model by means of a swarm intelligence approach. Our results indicate that networks may not be very robust given the great variety of limit cycles that a network can obtain depending on the updating scheme. In addition, we identified an omnipresent network with interactions that match with the original model as well as the discovery of new interactions. The techniques presented in this paper are general, and can be used to analyze other logical or threshold Boolean network models of gene regulatory networks. (C) 2013 Elsevier Ireland Ltd. All rights reserved.