Simulation of Probabilistic Sequential Systems

In this paper we introduce the idea of probability in the definition of Sequential Dynamical Systems, thus obtaining a new concept, Probabilistic Sequential System. The introduction of a probabilistic structure on Sequential Dynamical Systems is an important and interesting problem. The notion of homomorphism of our new model, is a natural extension of homomorphism of sequential dynamical systems introduced and developed by Laubenbacher and Paregeis in several papers. Our model, give the possibility to describe the dynamic of the systems using Markov chains and all the advantage of stochastic theory. The notion of simulation is introduced using the concept of homomorphisms, as usual. Several examples of homomorphisms, subsystems and simulations are given

Special homomorphisms between Probabilistic Gene Regulatory Networks

In this paper we study finite dynamical systems with $n$ functions acting on the same set $X$, and probabilities assigned to these functions, that it is called Probabilistic Regulatory Gene Networks (PRN. his concept is the same or a natural generalization of the concept Probabilistic Boolean Networks (PBN), introduced by I. Shmulevich, E. Dougherty, and W. Zhang, particularly the model PBN has been using to describe genetic networks and has therapeutic applications. In PRNs the most important question is to describe the steady states of the systems, so in this paper we pay attention to the idea of transforming a network to another without lost all the properties, in particular the probability distribution. Following this objective we develop the concepts of homomorphism and $\epsilon$-homomorphism of probabilistic regulatory networks, since these concepts bring the properties from one networks to another. Projections are special homomorphisms, and they always induce invariant subnetworks that contain all cycles and steady states in the network

A probabilistic regulatory network for the human immune system

In this paper we made a review of some papers about probabilistic regulatory networks (PRN), in particular we introduce our concept of homomorphisms of PRN with an example of projection of a regulatory network to a smaller one. We apply the model PRN (or Probabilistic Boolean Network) to the immune system, the PRN works with two functions. The model called ""The B/T-cells interaction"" is Boolean, so we are really working with a Probabilistic Boolean Network. Using Markov Chains we determine the state of equilibrium of the immune response.Comment: 9 page

Probabilistic Gene Regulatory Networks, isomorphisms of Markov Chains

In this paper we study homomorphisms of Probabilistic Regulatory Gene Networks(PRN) introduced in arXiv:math.DS/0603289 v1 13 Mar 2006. The model PRN is a natural generalization of the Probabilistic Boolean Networks (PBN), introduced by I. Shmulevich, E. Dougherty, and W. Zhang in 2001, that has been using to describe genetic networks and has therapeutic applications. In this paper, our main objectives are to apply the concept of homomorphism and $\epsilon$-homomorphism of probabilistic regulatory networks to the dynamic of the networks. The meaning of $\epsilon$ is that these homomorphic networks have similar distributions and the distance between the distributions is upper bounded by $\epsilon$. Additionally, we prove that the class of PRN together with the homomorphisms form a category with products and coproducts. Projections are special homomorphisms, and they always induce invariant subnetworks that contain all the cycles and steady states in the network. Here, it is proved that the $\epsilon$-homomorphism for $0<\epsilon<1$ produce simultaneous Markov Chains in both networks, that permit to introduce the concept of $\epsilon$-isomorphism of Markov Chains, and similar networks

A new algorithm for finding the nilpotency class of a finite p-group describing the upper central series

In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal p-subgroup N(G) of the automorphism group, Aut(G) of a bounded (or finite) abelian p-group G. This is the first part of two papers devoted to compute the nilpotency class of N(G) using formulas, and algorithms that work in almost all groups. Here, we prove that for p>2 the algorithm always runs. The algorithm describes a sequence of ideals of the Jacobson radical, J, and because N(G)=J+1, this sequence induces the upper central series in N(G).Comment: 15 pages, sending to a Journa

The reverse engineering problem with probabilities and sequential behavior: Probabilistic Sequential Networks

The reverse engineering problem with probabilities and sequential behavior is introducing here, using the expression of an algorithm. The solution is partially founded, because we solve the problem only if we have a Probabilistic Sequential Network. Therefore the probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions, whose imply that the distribution of probabilities in the systems are close. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given.Comment: Key words: simulation, homomorphism, dynamical system, sequential network, reverse engineering problem. Submitted to Elsevie

Introducing a Probabilistic Structure on Sequential Dynamical Systems, Simulation and Reduction of Probabilistic Sequential Networks

A probabilistic structure on sequential dynamical systems is introduced here, the new model will be called Probabilistic Sequential Network, PSN. The morphisms of Probabilistic Sequential Networks are defined using two algebraic conditions. It is proved here that two homomorphic Probabilistic Sequential Networks have the same equilibrium or steady state probabilities if the morphism is either an epimorphism or a monomorphism. Additionally, the proof of the set of PSN with its morphisms form the category PSN, having the category of sequential dynamical systems SDS, as a full subcategory is given. Several examples of morphisms, subsystems and simulations are given.Comment: 14 page

Probabilistic Regulatory Networks: Modeling Genetic Networks

We describe here the new concept of $\epsilon$-Homomorphisms of Probabilistic Regulatory Gene Networks(PRN). The $\epsilon$-homomorphisms are special mappings between two probabilistic networks, that consider the algebraic action of the iteration of functions and the probabilistic dynamic of the two networks. It is proved here that the class of PRN, together with the homomorphisms, form a category with products and coproducts. Projections are special homomorphisms, induced by invariant subnetworks. Here, it is proved that an $\epsilon$-homomorphism for 0 <$\epsilon$< 1 produces simultaneous Markov Chains in both networks, that permit to introduce the concepts of $\epsilon$-isomorphism of Markov Chains, and similar networks.Comment: 8 pages, 2 figures, International Congress of Matematicians, to be published in Proceedings of the Fifth International Conference on Engineering Computational Technology, 2006, Las Palmas, Gran Canari

Genetic Sequential Dynamical Systems

The whole complex process to obtain a protein encoded by a gene is difficult to include in a mathematical model. There are many models for describing different aspects of a genetic network. Finding a better model is one of the most important and interesting questions in computational biology. Sequential dynamical systems have been developed for a theory of computer simulation, and in this paper, a genetic sequential dynamical system is introduced. A gene is considered to be a function which can take a finite number of values. We prove that a genetic sequential dynamical system is a mathematical good description for a finite state linear model introduced by Brazma in www.ebi.ac.uk/microarray/biology-intro.html, EMBL-EBI, European Bioinformatics Institute, (2000)

Applications of Finite Fields to Dynamical Systems and Reverse Engineering Problems

We present a mathematical model: dynamical systems over finite sets (DSF), and we show that Boolean and discrete genetic models are special cases of DFS. In this paper, we prove that a function defined over finite sets with different number of elements can be represented as a polynomial function over a finite field. Given the data of a function defined over different finite sets, we describe an algorithm to obtain all the polynomial functions associated to this data. As a consequence, all the functions defined in a regulatory network can be represented as a polynomial function in one variable or in several variables over a finite field. We apply these results to study the reverse engineering problem