Saturation of enzyme kinetics in circadian clock models

From the mathematical study of simple models for circadian rhythm, the authors identified a clear effect of saturation in the enzyme kinetics on the promotion or suppression of a sustained oscillation. In the models, a clock gene (per gene) is transcribed to produce mRNAs, which are translated to produce proteins in the cytosol, which are then transported to the nucleus and suppress the transcription of the gene. The negative feedback loop with a long time delay creates sustained oscillation. All the enzymatic reactions (e.g., degradation, translation, and modification) are assumed to be of Michaelis-Menten type. The reaction rate increases with the amount of substrate but saturates when it is very large. The authors prove mathematically that the saturation in any of the reactions included in the feedback loop (in-loop reaction steps) suppresses the oscillation, whereas the saturation of both degradation steps and the back transport of the protein to cytosol (branch reaction steps) makes the oscillation more likely to occur. In the experimental measurements of enzyme kinetics and in published circadian clock simulators, in-loop reaction steps have a small saturation index whereas branch reaction steps have a large saturation index.