Endogenous circadian clocks (period around 24 h) are self-sustained biological oscillators that are present in many species. This rhythmic behavior is crucial for reliable regulation of biological activities. Disruption of circadian rhythms can decrease fitness and survival of both prokaryotes and eukaryotes. A common theme among the mechanisms of biological clocks is protein phosphorylation, a key regulator of the clock;;s period. This dissertation focuses on the mathematical modeling of protein phosphorylation and investigates the connections and distinctions among various systems of circadian clocks from cyanobacteria to mammals. First, we study the simplest circadian clock in cyanobacteria where three key clock proteins KaiA, KaiB, KaiC have been identified. Oscillations in KaiC phosphorylation level are present without any transcription or translation. KaiA activates the phosphorylation of KaiC while KaiB attenuates this process by restricting the activity of KaiA. Here we propose a mathematical model for the post-translational cyanobacterial clock and show that the sequestration mechanism in cyanobacteria involving KaiA, KaiB and KaiC is mathematically equivalent to the transcription regulation in mammalian circadian timekeeping involving the corresponding activators and repressors. We also find that an additional negative feedback loop keeps the molar ratio of clock proteins in balance and increases the robustness of the circadian clocks, which is another similarity shared between cyanobacteria and mammals. Therefore, similar dynamical principles regulating molecular timekeeping may have emerged in cyanobacteria and mammals through convergent evolution.Second, we focus on the sequential phosphorylation process of the key clock protein PERIOD 2 (PER2) in mammalian clocks. Mutation of a specific phosphorylation site on PER2 can shorten the period of circadian clocks, thus causing Familial advanced sleep phase (FASP). It is known that members of the casein kinase 1 (CK1) family are more efficient in phosphorylating the downstream sites of PER2 when the FASP site is already phosphorylated, yet the priming kinase targeting the FASP site has not been identified. Here, we incorporate into our mathematical model the new experimental result that the CK1 is indeed the kinase that works on both the priming FASP site and the downstream phosphorylation sites of PER2. Our modeling result suggests a robust yet fragile design of PER2 phosphorylation: the period of the circadian clock is robust to environmental variations but can be sensitive to regulatory changes in the tail behavior of CK1. Taken together, this presents a new mechanism for regulation of circadian period that is surprisingly divergent from that used in flies, where a separate priming kinase has been identified.Finally, we take a step further and study a much more general model of the multi-site phosphorylation process of proteins. One observation that motivates our study is that individual phosphorylation events (minutes) are typically much quicker than circadian timescales (hours), yet the changes in protein phosphorylation can affect the period of circadian clocks. Another motivation is that many interval timers are related to protein phosphorylation where a certain biological process can be paused for a fixed amount of time before it resumes. In our model, we show how kinases and phosphatases can work together to create an interval timer with a timescale much longer than individual phosphorylation events. We also show that product inhibition through sequestration on the kinase can be indispensable in sustaining the circadian rhythms.
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Mathematical Modeling of Phosphorylation in Circadian Clocks from Cyanobacteria to Mammals
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