Biological oscillators coordinate individual cellular components to function coherently and collectively. They are typically composed of multiple feedback loops, and period mismatch between them is unavoidable in biological implementations. We investigated the advantageous effect of the period mismatch in terms of synchronization against external stimuli(or trainability) . Specifically, we numerically analyzed two fundamental genetic models , smooth and relaxation oscillators , on their trainability for different coupling strength and period ratio by the phase reduction and the Floquet multiplier analysis . We found that the period mismatch induces better entrainment in both oscillators ; the enhancement occurred in the vicinity of the bifurcation on limit cycle. The optimal period ratio in the smooth oscillator for the enhanced trainability coincided with the experimentally observed ratio, which suggested the biological exploitation of the period mismatch. Although the origin of multiple feedback loops is often accounted for by the notion of passive robustness against perturbation, we here studied active benefits of the period mismatch on the efficiency of  genetic oscillators. Our findings show  qualitatively different perspective on the essentiality and inherent advantage of multiple loops in genetic oscillators .
Biological oscillators coordinate individual cellular components so that they function coherently and collectively. They are typically composed of multiple feedback loops, and period mismatch is unavoidable in biological implementations. We investigated the advantageous effect of this period mismatch in terms of a synchronization response to external stimuli . Specifically, we considered two fundamental models of genetic circuits: smooth- and relaxation oscillators . Using phase reduction and Floquet multipliers, we numerically analyzed their entrainability under different coupling strengths and period ratios . We found that a period mismatch induces better entrainment in both types of oscillator ; the enhancement occurs in the vicinity of the bifurcation on their limit cycles. In the smooth oscillator , the optimal period ratio for the enhancement coincides with the experimentally observed ratio, which suggests biological exploitation of the period mismatch. Although the origin of multiple feedback loops is often explained as a passive mechanism to ensure robustness against perturbation, we study the active benefits of the period mismatch , which include increasing the efficiency of the genetic oscillators. Our findings show a qualitatively different perspective for both the inherent advantages of multiple loops and their essentiality .
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