On robust stability and sensitivity of circadian rhythms

We develop an approach to the robustness analysis of circadian rhythms from the viewpoints of (local) robust stability of periodic trajectories and their period-sensitivity, based on mathematical models. Our attention is directed to the core molecular model generating circadian rhythms in Drosophila, proposed by Goldbeter [1], for our approach to be illustrated. We consider two cases when the circadian rhythms are subjected to periodic forcing by light/dark (LD) cycles and when the circadian rhythms are under continuous darkness (DD). First, we propose conditions/algorithms for (local) robust stability, and it is shown that the circadian rhythms generated by the core molecular model are (locally) robustly stable, in both LD and DD cases. Second, in DD case, we derive a formula of period sensitivity, and assess the effects of (infinitesimal) parameter-variations on the period of the circadian rhythms.