A Class of Switched Piecewise Quadratic Systems for Coupling Gene Expression with Growth Rate in Bacteria

In this paper we propose a novel qualitative formalism to model gene expression dynamics dependent on dilution due to growth rate of the cell. We extend the piecewise linear (PL) systems by keeping the use of step functions to model the interactions between the elements and adding a growth rate expression to model the dilution effect. Focusing on the global gene expression machinery in bacteria, we model the growth rate as the minimum of two limiting factors: RNA polymerase (RNAP) and ribosomes. The resulting system is a switched system with two piecewise quadratic (PQ) modes. We study the stability of such switched piecewise quadratic (SPQ) system starting from the stability analysis of the (PQ) modes. We also present and analyze by means of phase-planes a bidimensional SPQ model involving RNAP and ribosomes concentrations, which brings out the important differences with respect to PL systems. Finally, we qualitatively show that our growth rate expression acts well in different biological conditions.