MontalvaMedel, M., Ledger, T., Ruz, G. A., & Goles, E. (2021). Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements. Mathematics, 9(6), 600.
Abstract: In VelizCuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al., 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al., 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al., 2004.

Ruivo, E. L. P., MontalvaMedel, M., de Oliveira, P. P. B., & Perrot, K. (2018). Characterisation of the elementary cellular automata in terms of their maximum sensitivity to all possible asynchronous updates. Chaos Solitons Fractals, 113, 209–220.
Abstract: Cellular automata are fullydiscrete dynamical systems with global behaviour depending upon their locally specified state transitions. They have been extensively studied as models of complex systems as well as objects of mathematical and computational interest. Classically, the local rule of a cellular automaton is iterated synchronously over the entire configuration. However, the question of how asynchronous updates change the behaviour of a cellular automaton has become a major issue in recent years. Here, we analyse the elementary cellular automata rule space in terms of how many different onestep trajectories a rule would entail when taking into account all possible deterministic ways of updating the rule, for one time step, over all possible initial configurations. More precisely, we provide a characterisation of the elementary cellular automata, by means of their onestep maximum sensitivity to all possible update schedules, that is, the property that any change in the update schedule causes the rule's onestep trajectories also to change after one iteration. Although the onestep maximum sensitivity does not imply that the remainder of the timeevolutions will be distinct, it is a necessary condition for that. (C) 2018 Elsevier Ltd. All rights reserved.
