Nonlinear mathematical model of repressilator. Approximation to singular and singularly impulsive dynamical system

This paper concerns three genes model of synthetic network that carries out many essential functions in living cell. It is designed network of genes with mutual inhibition and/or activation resulting in triple negative, double negative (i.e. positive feedback) or classical negative feedback or positive feedback. Mathematical model which describes these networks is nonlinear. Approximating two main processes as fast and slow, the model results into multiple (specifically, two) time scale model, i.e. singular system which is of reduced dimension. Further, approximating Hill's nonlinear terms with switch-like function, the model results to impulsive or singularly-impulsive model. Also, having in mind that different species might have different number of molecules present, for case of small number of molecules discrete approximation is given, since by nature, the continuum approximation is not valid any more. Further, it is possible to have some species in small number of molecules with discrete approximation, while other species are present in excess so that continuum approximation holds, what results in mixed discrete-continuous model, what is another class of singularly impulsive, i.e. hybrid model. Stability results and responses are given for the biologically plausible set of parameter data.