Homomorphisms of 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) in [3]. This concept is the same or a natural generalization of the concept probabilistic Boolean networks (PBN), introduced by I. Shmulevich, E. Dougherty, and W. Zhang in [5], Particularly the model PBN has been using to describe genetic networks and has therapeutic applications, see [6]. In PRN 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 e-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 cycles and steady states.