Mathematical analysis of a biochemical oscillator with delay

In this work we analyze a nonlinear delay differential equation (DDE) model with negative feedback and constant time delay. The model is constructed from a previously studied biochemical reaction network of gene transcription and protein synthesis. Linear analysis of the associated DDE model gives a critical time delay beyond which a periodic motion is born in a Hopf bifurcation. The method of multiple scales is then used to analyze the nonlinear system to obtain expressions for the amplitude and frequency of oscillation as a function of the system parameters. We use our closed form analytical expressions to study the importance of a well-balanced ratio between synthesis and degradation rates in the existence of periodic solutions. We show that our theoretical results are in agreement with numerical simulations and with experimental evidence found in the biological literature.