Justification of quasi-stationary approximation in models of gene expression of a self-regulating protein

We analyse a model of Hes1 gene transcription and protein synthesis with a negative feedback loop. The effect of multiple binding sites in the Hes1 promoter as well as the dimer formation process are taken into account. We consider three, possibly different, time scales connected with: (i) the process of binding to/dissolving from a binding site, (ii) formation and dissociation of dimers, (iii) production and degradation of Hes1 protein and its mRNA. Assuming that the first two processes are much faster than the third one, using the Tikhonov theorem, we reduce in two steps the full model to the classical Hes1 model. In the intermediate step two different models are derived depending on the relation between the time scales of processes (i) and (ii). The asymptotic behaviour of the solutions of systems are studied. We investigate the stability of the positive steady state and perform some numerical experiments showing differences in dynamics of the considered models.