Analysis of robustness of oscillations in models of the mammalian circadian clock.

Abstract

The mammalian circadian clock mechanism features robust generation of ∼24 h rhythmicity under varied levels of the key clock proteins. To identify such robustness, we use bifurcation analysis to examine the models developed by Kim & Forger. At the core of their original ‘single negative feedback’ (SNF) model is a negative feedback loop whereby PER binds to and inhibits its transcriptional activator, BMAL1. For robust oscillations to occur, the dissociation constant of PER:BMAL1 complex, Kd, must be several orders of magnitude smaller than a reasonable expectation of 1-10 nM for this protein complex. We relax this constraint by two modifications: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for nuclear degradation of PER by a Michaelis-Menten rate law. These modifications lengthen time delays, thereby increasing the maximum allowable Kd for robust oscillations to ∼2 nM. In a third modification, we consider an alternative rate law for Per gene transcription to resolve an unrealistically large transcription rate at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) or a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger's observations of these extended models, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for ∼24 h period with Kd≥1 nM. Our results suggest crucial roles of the biochemical details of Per gene expression and provide testable predictions for future studies.