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

The aim of this paper is to analyse the dynamicalbehaviour of models of gene transcription-translation for thesynthesis of RNA polymerase in a cell, with a closed loopfrom the produced RNA polymerase (end-product) to thetranscription step (RNA polymerase is needed to transcribe itsown gene). Using monotone system theory we study a reducedversion of this model with two variables (mRNA and protein),and show that it has either a single globally stable trivialequilibrium in (0, 0), or has an unstable zero equilibrium anda globally stable positive one. This dynamical behaviour can berelated to biological observations on the bacterium Escherichiacoli.