Synchronization dynamics of chemically coupled cells with activator-inhibitor pathways

Systems of interacting cells containing an activator-inhibitor pathway, regulating naturally in their inner parts their end-product concentrations through a sequence of biochemical reactions with feedback-loops: an end-product inhibition of the first substrate, and an autocatalytic activation of the end-product through an allosteric enzyme-mediated reaction are investigated. The individual cells are considered to be identical and are described by nonlinear differential equations recently proposed following the concerted transition model. The chemical and electrical coupling types, realized by exchange of metabolites across concentration of the cells are used in order to analyze the onset of phase and complete synchronization in the biochemical system. It is found that depending on the coupling nature and the range of coupling strength, cells enter into different synchronization regimes going from low-quality to high-quality synchronization. The synchronization manifold's stability is analyzed. The results are supported by numerical simulations using indicators such as the conditional Lyapunov exponents and the rate of change of the Lyapunov function. The results indicate that the system cannot completely synchronize under the single action of the chemical coupling. The combined effect of both chemical and electrical couplings is found to be of capital importance in the onset of complete synchronization and high quality synchronization.