Abstract.This work describes the modeling, stability and regulation problem for a class of biological dynamical systems. The class of biological dynamical systems considered in this paper, are genomical systems described in terms of genetic regulatory networks. Genetic regulatory networks, connect genes by a set of boolean rules (switching conditions given in terms of concentration thresholds) in order to simulate the expression patterns presented in real cells. At the lowest level the evolution of proteins is continuous, discreteness arises when the concentration of enabling quantities is above the threshold, thus exhibiting an hybrid behavior. As a result, two modeling approaches are considered: the first one, based on place-transition Petri nets, describes the behavior of the protein concentration when there is a state change due to some concentration threshold, without being interested in the protein's concentration state at its lowest level. In the second approach, given in terms of dynamical colored Petri nets (DCPN), everything is taken in to consideration. Once the model is obtained the stability and regulation problems for genetic regulatory systems, employing Lyapunov methods are addressed.

Received: February 24, 2004

AMS Subject Classification: 93D30, 93A30, 93D05, 39A10, 39A11, 92D10, 92D15

Key Words and Phrases: biological dynamical systems, genomics, genetic regulatory networks, modeling, stability, regulation, switching inputs, Petri nets, Lyapunov methods

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 12
Issue: 3