Distributed delays in Hes1 gene expression model

In the Hes1 gene expression system the protein (present as dimers) bounds to the promoter of its own DNA blocking transcription of its mRNA. This negative feedback leads to an oscillatory behavior, which is observed experimentally. Classical mathematical model of this system consists of two ordinary differential equations with discrete time delay in the term reflecting transcription. However, transcription takes place in the nucleus while translation occurs in the cytoplasm. This means that the delay present in the system is larger than transcription time. Moreover, in reality it is not discrete but distributed around some mean value. In this paper we present the model of the Hes1 gene expression system and discuss similarities and differences between the model with discrete and distributed delays. It turns out that in the case of distributed delays the region of stability of the steady state is larger than in the case of discrete delay. We also derive conditions that guarantee stability of the steady state for particular delay distributions.