A mathematical model of endothelial nitric oxide synthase activation with time delay exhibiting Hopf bifurcation and oscillations

Nitric oxide (NO) is a gaseous compound that serves as a signaling molecule in cellular interactions. In the vasculature, NO is synthesized from endogenous agents by endothelial nitric oxide synthase (eNOS) where it plays key roles in several functions related to homeostasis, adaptation, and development. Recent experimental studies have revealed cycles of increasing and decreasing NO production when eNOS is stimulated by factors such as glucose or insulin. We offer a mathematical model of a generic amino acid receptor site on eNOS wherein this species is subject to activation/deactivation by a pair of interactive kinase and phosphatase species. The enzyme kinetic model is presented as a system of ordinary differential equations including time delay to allow for various intermediate, unspecified complexes. We show that under conditions on the model parameters, varying the delay time may give rise to a Hopf bifurcation. Properties of the bifurcating solutions are explored via a center manifold reduction, and a numerical illustration is provided.