Stable periodic solutions of a delayed reaction-diffusion model of Hes1-mRNA interactions

Hes1 (Hairy and enhancer of split 1) is a transcriptional repressor that plays a fundamental role in the regulation of embryogenesis and cell lineage specification. The temporal dynamics of Hes1 mRNA and Hes1 protein expression are known to exhibit sustained oscillations. However, many existing mathematical models can reproduce these oscillations only transiently, eventually dampening toward a steady state. This limits their biological fidelity, as sustained oscillations are observed in vitro and in vivo under physiological conditions. To address these limitations, we propose a more biologically realistic framework by incorporating both transcriptional/translational time delays and spatial diffusion effects into a Reaction-Diffusion (RD) system with discrete time delays. The model describes the spatiotemporal dynamics of Hes1 mRNA and protein concentrations in the cytoplasm and nucleus. We establish the conditions under which the RD model undergoes a delay-induced Hopf bifurcation, leading to the emergence of stable periodic solutions. Furthermore, our analysis establishes explicit criteria on the delay and diffusion coefficients that ensure the existence of sustained oscillatory patterns. Numerical simulations are conducted to validate the theoretical predictions, demonstrating the persistence and stability of oscillations under a range of biologically plausible parameters.