Mathematical Analysis of Robustness of Oscillations in Models of the Mammalian Circadian Clock

ABSTRACTA wide variety of organisms possess endogenous circadian rhythms (~24 h period), which coordinate many physiological functions with the day-night cycle. These rhythms are mediated by a molecular mechanism based on transcription-translation feedback. A number of mathematical models have been developed to study features of the circadian clock in a variety of organisms. In this paper, we use bifurcation theory to explore properties of mathematical models based on Kim & Forger’s interpretation of the circadian clock in mammals. Their models are based on a simple negative feedback (SNF) loop between a regulatory protein (PER) and its transcriptional activator (BMAL). In their model, PER binds to BMAL to form a stoichiometric complex (PER:BMAL) that is inactive as a transcription factor. However, for oscillations to occur in the SNF model, the dissociation constant of the PER:BMAL complex, Kd, must be smaller than 10−3 nM, orders of magnitude below the limit set by the biophysics of protein binding. We have relaxed this constraint by introducing two modifications to Kim & Forger’s SNF model: (1) replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law, and (2) introducing a multistep reaction chain for posttranslational modifications of PER. These modifications significantly increase the robustness of oscillations, and increase the maximum allowable Kd to more reasonable values, 1—100 nM. In a third modification, we considered alternative rate laws for gene transcription to resolve an unrealistically large rate of PER transcription at very low levels of BMAL transcription factor. Additionally, we studied Kim & Forger’s extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). We found that the supplementary positive feedback loop—but not the supplementary negative feedback loop— provides additional robustness to the clock model.AUTHOR SUMMARYThe circadian rhythm aligns bodily functions to the day/night cycle and is important for our health. The rhythm originates from an intracellular, molecular clock mechanism that mediates rhythmic gene expression. It is long understood that transcriptional negative feedback with sufficient time delay is key to generating circadian oscillations. However, some of the most widely cited mathematical models for the circadian clock suffer from problems of parameter “fragilities”. That is, sustained oscillations are possible only for physically unrealistic parameter values. A recent model by Kim and Forger nicely incorporates the inhibitory binding of PER, a key clock protein, to its transcription activator BMAL, but oscillations in their model require a binding affinity between PER and BMAL that is orders of magnitude lower than the physical limit of protein-protein binding. To rectify this problem, we make several physiologically credible modifications to the Kim-Forger model, which allow oscillations to occur with realistic binding affinity. The modified model is further extended to explore the potential roles of supplementary feedback loops in the mammalian clock mechanism. Ultimately, accurate models of the circadian clock will provide predictive tools for chronotherapy and chrono-pharmacology studies.