Stability and Time-Delay Modeling of Negative Feedback Loops

Negative feedback loops are a common cellular motif underlying many important gene regulation pathways, and therefore a clear understanding of their dynamics is important for the study of gene regulation. Here we analyze the linear stability of negative feedback loops with and without explicit time delays. When the degradation rates of each loop component are identical, we derive the analytical solution of the condition of marginal stability. In the most general case, when the degradation rates of each component differ, we derive a novel expansion of the condition of marginal stability in terms of the geometric mean and variance of the degradation rates. We use these results to demonstrate how the mathematical representation of a negative feedback loop can be simplified such that the stability characteristics are maintained. Finally we apply these results to study the stability structure of the p53-Mdm2 feedback loop. By simplifying a model involving nuclear and cytoplasmatic molecular species of mdm2 mRNA and Mdm2 protein to a model involving only nuclear components and a time delay (which summarizes transcriptional, nuclear import and export, and translational time scales), we demonstrate how our methods allow for an elucidation of the dynamic instabilities recently observed experimentally in the p53-Mdm2 system as the transcription of p53 changes from low to middle to high rates.