The collective oscillation period of inter-coupled Goodwin oscillators

Many biological oscillators are arranged in networks composed of many inter-coupled cellular oscillators. However, results are still lacking on the collective oscillation period of inter-coupled gene regulatory oscillators, which, as has been reported, may be different from the oscillation period of an autonomous cellular oscillator. Based on the Goodwin oscillator, we analyze the collective oscillation pattern of coupled cellular oscillator networks. First we give a condition under which the oscillator network exhibits oscillatory and synchronized behavior, then we estimate the collective oscillation period based on a multivariable harmonic balance technique. Analytical results are derived in terms of biochemical parameters, thus giving insight into the basic mechanism of biological oscillation and providing guidance in synthetic biology design. Simulation results are given to confirm the theoretical predictions.