Article
Maximum number of fixed points in regulatory Boolean networks.
Departamento Ingeniería Matemática, Universidad de Concepción, Av. Esteban Iturra s/n, Casilla 160C, Concepción, Chile.
Bulletin of Mathematical Biology (Impact Factor: 2.02). 08/2008; 70(5):1398409. DOI: 10.1007/s1153800893047 Source: PubMed

Article: On the Relationship of Steady States of Continuous and Discrete Models Arising from Biology.
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ABSTRACT: For many biological systems that have been modeled using continuous and discrete models, it has been shown that such models have similar dynamical properties. In this paper, we prove that this happens in more general cases. We show that under some conditions there is a bijection between the steady states of continuous and discrete models arising from biological systems. Our results also provide a novel method to analyze certain classes of nonlinear models using discrete mathematics.Bulletin of Mathematical Biology 10/2012; · 2.02 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: It has been proved, for several classes of continuous and discrete dynamical systems, that the presence of a positive (resp. negative) circuit in the interaction graph of a system is a necessary condition for the presence of multiple stable states (resp. a cyclic attractor). A positive (resp. negative) circuit is said to be functional when it "generates" several stable states (resp. a cyclic attractor). However, there are no definite mathematical frameworks translating the underlying meaning of "generates." Focusing on Boolean networks, we recall and propose some definitions concerning the notion of functionality along with associated mathematical results.Bulletin of Mathematical Biology 03/2013; · 2.02 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Finite dynamical systems (e.g. Boolean networks and logical models) have been used in modeling biological systems to focus attention on the qualitative features of the system, such as the wiring diagram. Since the analysis of such systems is hard, it is necessary to focus on subclasses that have the properties of being general enough for modeling and simple enough for theoretical analysis. In this paper we propose the class of ANDNOT networks for modeling biological systems and show that it provides several advantages. Some of the advantages include: Any finite dynamical system can be written as an ANDNOT network with similar dynamical properties. There is a onetoone correspondence between ANDNOT networks, their wiring diagrams, and their dynamics. Results about ANDNOT networks can be stated at the wiring diagram level without losing any information. Results about ANDNOT networks are applicable to any Boolean network. We apply our results to a Boolean model of Thcell differentiation.Applied Mathematics & Information Sciences 11/2012; · 1.23 Impact Factor
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