Stability analysis of a reduced transcription-translation model of RNA polymerase

The aim of this paper is to analyse the dynamical behaviour of models of gene transcription-translation for the synthesis of RNA polymerase in a cell, with a closed loop from the produced RNA polymerase (end-product) to the transcription step (RNA polymerase is needed to transcribe its own gene). Using monotone system theory we study a reduced version of this model with two variables (mRNA and protein), and show that it has either a single globally stable trivial equilibrium in (0, 0), or has an unstable zero equilibrium and a globally stable positive one. This dynamical behaviour can be related to biological observations on the bacterium Escherichia coli.