  Circadian clocks play a pivotal role in orchestrating numerous physiological
and developmental events. Waveform shapes of the oscillations of protein
abundances can be informative about the underlying biochemical processes of
circadian clocks. We derive a mathematical framework where waveforms do reveal
hidden biochemical mechanisms of circadian timekeeping. We find that the cost
of synthesizing proteins with particular waveforms can be substantially reduced
by rhythmic protein half-lives over time, as supported by previous plant and
mammalian data, as well as our own seedling experiment. We also find that
previously-enigmatic, cyclic expression of positive arm components within the
mammalian and insect clocks allows both a broad range of peak time differences
between protein waveforms and the symmetries of the waveforms about the peak
times. Such various peak-time differences may facilitate tissue-specific or
developmental stage-specific multicellular processes. Our waveform-guided
approach can be extended to various biological oscillators, including
cell-cycle and synthetic genetic oscillators.
