  Biochemical circadian rhythm oscillations play an important role in many
signalling mechanisms. In this work, we explore some of the biophysical
mechanisms responsible for sustaining robust oscillations by constructing a
minimal but analytically tractable model of the circadian oscillations in the
KaiABC protein system found in the cyanobacteria $S. \ elongatus$. In
particular, our minimal model explicitly accounts for two experimentally
characterized biophysical features of the KaiABC protein system, namely, a
differential binding affinity and an ultrasensitive response. Our analytical
work shows how these mechanisms might be crucial for promoting robust
oscillations even in sub optimal nutrient conditions. Our analytical and
numerical work also identifies mechanisms by which biological clocks can stably
maintain a constant time period under a variety of nutrient conditions.
Finally, our work also explores the thermodynamic costs associated with the
generation of robust sustained oscillations and shows that the net rate of
entropy production alone might not be a good figure of merit to asses the
quality of oscillations.
