Intercellular delay regulates the collective period of repressively coupled gene regulatory oscillator networks

Most biological rhythms are generated by a population of cellular oscillators coupled through intercellular signaling. Recent experimental evidence shows that the collective period may differ significantly from the autonomous period in the presence of intercellular delays. The phenomenon has been investigated using delay-coupled phase oscillators, but the proposed phase model contains no direct biological mechanism, which may weaken the model's reliability in unraveling biophysical principles. Based on a published gene regulatory oscillator model, we analyze the collective period of delay-coupled biological oscillators using the multivariable harmonic balance technique. We prove that, in contradiction to the common intuition that the collective period increases linearly with the coupling delay, the collective period turns out to be a periodic function of the intercellular delay. More surprisingly, the collective period may even decrease with the intercellular delay when the delay resides in certain regions. The collective period is given in a closed-form in terms of biochemical reaction constants and thus provides biological insights as well as guidance in synthetic-biological-oscillator design. Simulation results are given based on a segmentation clock model to confirm the theoretical predictions.