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Goles, E., Lobos, F., Ruz, G. A., & Sene, S. (2020). Attractor landscapes in Boolean networks with firing memory: a theoretical study applied to genetic networks. Nat. Comput., 19(2), 295–319.
Abstract: In this paper we study the dynamical behavior of Boolean networks with firing memory, namely Boolean networks whose vertices are updated synchronously depending on their proper Boolean local transition functions so that each vertex remains at its firing state a finite number of steps. We prove in particular that these networks have the same computational power than the classical ones, i.e. any Boolean network with firing memory composed of m vertices can be simulated by a Boolean network by adding vertices. We also prove general results on specific classes of networks. For instance, we show that the existence of at least one delay greater than 1 in disjunctive networks makes such networks have only fixed points as attractors. Moreover, for arbitrary networks composed of two vertices, we characterize the delay phase space, i.e. the delay values such that networks admits limit cycles or fixed points. Finally, we analyze two classical biological models by introducing delays: the model of the immune control of the lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{69pt} \begin{document}$$\lambda $$\end{document}phage and that of the genetic control of the floral morphogenesis of the plant Arabidopsis thaliana.

Goles, E., Montalva, M., & Ruz, G. A. (2013). Deconstruction and Dynamical Robustness of Regulatory Networks: Application to the Yeast Cell Cycle Networks. Bull. Math. Biol., 75(6), 939–966.
Abstract: Analyzing all the deterministic dynamics of a Boolean regulatory network is a difficult problem since it grows exponentially with the number of nodes. In this paper, we present mathematical and computational tools for analyzing the complete deterministic dynamics of Boolean regulatory networks. For this, the notion of alliance is introduced, which is a subconfiguration of states that remains fixed regardless of the values of the other nodes. Also, equivalent classes are considered, which are sets of updating schedules which have the same dynamics. Using these techniques, we analyze two yeast cell cycle models. Results show the effectiveness of the proposed tools for analyzing update robustness as well as the discovery of new information related to the attractors of the yeast cell cycle models considering all the possible deterministic dynamics, which previously have only been studied considering the parallel updating scheme.
Keywords: Boolean networks; Attractors; Update robustness; Alliances; Dynamics

Gregor, C., Ashlock, D., Ruz, G. A., MacKinnon, D., & Kribs, D. (2022). A novel linear representation for evolving matrices. Soft Comput., Early Access.
Abstract: A number of problems from specifiers for Boolean networks to programs for quantum computers can be encoded as matrices. The paper presents a novel family of linear, generative representations for evolving matrices. The matrices can be general or restricted within special classes of matrices like permutation matrices, Hermitian matrices, or other groups of matrices with particular algebraic properties. These classes include unitary matrices which encode quantum programs. This representation avoids the brittleness that arises in direct representations of matrices and permits the researcher substantial control of the part of matrix space being searched. The representation is demonstrated on a relatively simple matrix problem in automatic content generation as well as Boolean map induction and automatic quantum programming. The automatic content generation problem yields interesting results; the generative matrix representation yields worse fitness but a substantially greater variety of outcomes than a direct encoding, which is acceptable when generating content. The Boolean map experiments extend and confirm results that demonstrate that the generative encoding is superior to a direct encoding for the transition matrix of a Boolean map. The quantum programming results are generally quite good, with poor performance on the simplest problems in two of the families of programming tasks studied. The viability of the new representation for evolutionary matrix induction is well supported.

Ruz, G. A., & Goles, E. (2013). Learning gene regulatory networks using the bees algorithm. Neural Comput. Appl., 22(1), 63–70.
Abstract: Learning gene regulatory networks under the threshold Boolean network model is presented. To accomplish this, the swarm intelligence technique called the bees algorithm is formulated to learn networks with predefined attractors. The resulting technique is compared with simulated annealing through simulations. The ability of the networks to preserve the attractors when the updating schemes is changed from parallel to sequential is analyzed as well. Results show that Boolean networks are not very robust when the updating scheme is changed. Robust networks were found only for limit cycle length equal to two and specific network topologies. Throughout the simulations, the bees algorithm outperformed simulated annealing, showing the effectiveness of this swarm intelligence technique for this particular application.

Ruz, G. A., Goles, E., Montalva, M., & Fogel, G. B. (2014). Dynamical and topological robustness of the mammalian cell cycle network: A reverse engineering approach. Biosystems, 115, 23–32.
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.

Ruz, G. A., Timmermann, T., Barrera, J., & Goles, E. (2014). Neutral space analysis for a Boolean network model of the fission yeast cell cycle network. Biol. Res., 47, 12 pp.
Abstract: Background: Interactions between genes and their products give rise to complex circuits known as gene regulatory networks (GRN) that enable cells to process information and respond to external stimuli. Several important processes for life, depend of an accurate and contextspecific regulation of gene expression, such as the cell cycle, which can be analyzed through its GRN, where deregulation can lead to cancer in animals or a directed regulation could be applied for biotechnological processes using yeast. An approach to study the robustness of GRN is through the neutral space. In this paper, we explore the neutral space of a Schizosaccharomyces pombe (fission yeast) cell cycle network through an evolution strategy to generate a neutral graph, composed of Boolean regulatory networks that share the same state sequences of the fission yeast cell cycle. Results: Through simulations it was found that in the generated neutral graph, the functional networks that are not in the wildtype connected component have in general a Hamming distance more than 3 with the wildtype, and more than 10 between the other disconnected functional networks. Significant differences were found between the functional networks in the connected component of the wildtype network and the rest of the network, not only at a topological level, but also at the state space level, where significant differences in the distribution of the basin of attraction for the G(1) fixed point was found for deterministic updating schemes. Conclusions: In general, functional networks in the wildtype network connected component, can mutate up to no more than 3 times, then they reach a point of no return where the networks leave the connected component of the wildtype. The proposed method to construct a neutral graph is general and can be used to explore the neutral space of other biologically interesting networks, and also formulate new biological hypotheses studying the functional networks in the wildtype network connected component.

Ruz, G. A., Zuniga, A., & Goles, E. (2018). A Boolean network model of bacterial quorumsensing systems. Int. J. Data Min. Bioinform., 21(2), 123–144.
Abstract: There are several mathematical models to represent gene regulatory networks, one of the simplest is the Boolean network paradigm. In this paper, we reconstruct a regulatory network of bacterial quorumsensing systems, in particular, we consider Paraburkholderia phytofirmans PsJN which is a plant growth promoting bacteria that produces positive effects in horticultural crops like tomato, potato and grape. To learn the regulatory network from temporal expression pattern of quorumsensing genes at root plants, we present a methodology that considers the training of perceptrons for each gene and then the integration into one Boolean regulatory network. Using the proposed approach, we were able to infer a regulatory network model whose topology and dynamic exhibited was helpful to gain insight on the quorumsensing systems regulation mechanism. We compared our results with REVEAL and BestFit extension algorithm, showing that the proposed neural network approach obtained a more biologically meaningful network and dynamics, demonstrating the effectiveness of the proposed method.

Timmermann, T., Gonzalez, B., & Ruz, G. A. (2020). Reconstruction of a gene regulatory network of the induced systemic resistance defense response in Arabidopsis using boolean networks. BMC Bioinformatics, 21(1), 16 pp.
Abstract: Background An important process for plant survival is the immune system. The induced systemic resistance (ISR) triggered by beneficial microbes is an important costeffective defense mechanism by which plants are primed to an eventual pathogen attack. Defense mechanisms such as ISR depend on an accurate and contextspecific regulation of gene expression. Interactions between genes and their products give rise to complex circuits known as gene regulatory networks (GRNs). Here, we explore the regulatory mechanism of the ISR defense response triggered by the beneficial bacterium Paraburkholderia phytofirmans PsJN in Arabidopsis thaliana plants infected with Pseudomonas syringae DC3000. To achieve this, a GRN underlying the ISR response was inferred using gene expression timeseries data of certain defenserelated genes, differential evolution, and threshold Boolean networks. Results One thousand threshold Boolean networks were inferred that met the restriction of the desired dynamics. From these networks, a consensus network was obtained that helped to find plausible interactions between the genes. A representative network was selected from the consensus network and biological restrictions were applied to it. The dynamics of the selected network showed that the largest attractor, a limit cycle of length 3, represents the final stage of the defense response (12, 18, and 24 h). Also, the structural robustness of the GRN was studied through the networks' attractors. Conclusions A computational intelligence approach was designed to reconstruct a GRN underlying the ISR defense response in plants using gene expression timeseries data of A. thaliana colonized by P. phytofirmans PsJN and subsequently infected with P. syringae DC3000. Using differential evolution, 1000 GRNs from timeseries data were successfully inferred. Through the study of the network dynamics of the selected GRN, it can be concluded that it is structurally robust since three mutations were necessary to completely disarm the Boolean trajectory that represents the biological data. The proposed method to reconstruct GRNs is general and can be used to infer other biologically relevant networks to formulate new biological hypotheses.
