Effect of transcriptional delay on ribosome abundance control

Time delays are inherent in biological processes such as transcription and translation, and they play a vital role in cellular control by altering timing of feedback loops and therefore stability of steady states. Standard ordinary differential equation (ODE) models neglect these delays, potentially eliminating important dynamic behavior. We present a mathematical model that explores the synthesis of ribosomal proteins and their mRNA in E. coli using both ODE and delay differential equation (DDE) models. Comparing these two models we discover that, while there can only be two equilibrium states in both the ODE and the DDE systems, the addition of delays destabilizes the internal equilibrium via the Hopf bifurcation, resulting in oscillatory dynamics. We show that there are parameter combinations where two stable periodic orbits coexist, and one of them terminates in a torus bifurcation giving rise to a stable torus of recurrent orbits. The existence of oscillatory behavior is consistent with experiments.