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.