  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.
