Singularly impulsive model of genetic regulatory networks

In this paper we present singular and singularly impulsive model of genetic regulatory networks. Concrete example consists of two gene — two proteins simple synthetic network, and posses two fundamentally present parts in biochemical networks — positive and negative feedback loops. These are important structural parts of biochemical networks and are interesting for control theoreticians. By investigating purpose of positive and negative feedback presence, we see for example that they lead either to bistability (or multistability, in general) in case of positive feedback or generate oscillatory behaviour in case of negative feedback what we show. Mathematical model is derived in form of nonlinear two dimensional dynamical system which is further approximated to singular and singularly impulsive dynamical system. Importance of this example is singular systems approximation which in this particular example leads to interesting and well known phenomena — of relaxation oscillation. This phenomena is consequence of multiple-scale network, i.e. fast-slow decomposition. Jump phenomena that appears as a consequence of time scale differences is very different from jumps (impulsive behaviour) that we have in impulsive systems approximation of, for example, sigmoidal function response by piece-wise linear function in order to reduce complexity.