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Circadian clocks play a pivotal role in orchestrating numerous physiologicaland developmental events. Waveform shapes of the oscillations of proteinabundances can be informative about the underlying biochemical processes ofcircadian clocks. We derive a mathematical framework where waveforms do revealhidden biochemical mechanisms of circadian timekeeping. We find that the costof synthesizing proteins with particular waveforms can be substantially reducedby rhythmic protein half-lives over time, as supported by previous plant andmammalian data, as well as our own seedling experiment. We also find thatpreviously-enigmatic, cyclic expression of positive arm components within themammalian and insect clocks allows both a broad range of peak time differencesbetween protein waveforms and the symmetries of the waveforms about the peaktimes. Such various peak-time differences may facilitate tissue-specific ordevelopmental stage-specific multicellular processes. Our waveform-guidedapproach can be extended to various biological oscillators, includingcell-cycle and synthetic genetic oscillators.

Waveforms of molecular oscillations reveal circadian timekeeping mechanisms