Hopf bifurcation analysis for models on genetic negative feedback loops

Abstract

Negative feedback loops have been identified as genetic generation of oscillations in cells. The Goodwin model, a prototypical negative feedback oscillator, has been applied largely in the studies of biological rhythms. By applying the degenerate Routh-Hurwitz criterion, we perform a thorough Hopf bifurcation analysis on the Goodwin model and two other models on negative feedback loops. We not only derive the precise conditions for the Hopf bifurcation in these models, but also obtain the frequencies of the bifurcating periodic solutions. Such frequencies are acquired by factorizing the characteristic polynomials of the linearized systems at the bifurcation points and the bifurcation values. We also compute the higher-order terms which determine the stability of the bifurcating periodic solution and other properties of the Hopf bifurcation for the Goodwin model. Our result indicates that the minimal Hill coefficient does not increase and no further condition is required for the bifurcating periodic solution to be stable. We allocate the parameter values according to the present theories to illustrate limit-cycle oscillations in several numerical examples for these models.