Hopf Bifurcation Analysis of the Repressilator Model

The repressilator is a genetic network that exhibits oscillations. The net-work is formed of three genes, each of which represses each other cyclically, creating a negative feedback loop with nonlinear interactions. In this work we present a computational bifurcation analysis of the mathematical model of the repressilator. We show that the steady state undergoes a transition from stable to unstable giving rise to a stable limit-cycle in a Hopf bifurcation. The nonlinear analysis involves a center manifold reduction on the six-dimensional system, which yields closed form expressions for the frequency and amplitude of the oscillation born at the Hopf. A parameter study then shows how the dynamics of the system are influenced for different parameter values and their associated biological significance.