Abstract
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.
Highlights
The repressilator is an artificial synthetic gene network created and named by Elowitz and Leibler [1]
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 biochemical details of the design principles for this oscillator are complicated, yet for the purposes of this work we present a brief background on the biology of the network: A gene undergoes a process called transcription, which consists in making a “copy” of the gene called messenger RNA
Summary
The repressilator is an artificial synthetic gene network created and named by Elowitz and Leibler [1]. The simplicity of the network’s structure and design makes the repressilator an ideal candidate for studies on the dynamic features of synthetic gene networks [2] [3]. The network exhibits negative feedback and nonlinear interactions, both of which are important features of dynamical systems exhibiting oscillations [4] [5] [6]. The biochemical details of the design principles for this oscillator are complicated, yet for the purposes of this work we present a brief background on the biology of the network: A gene undergoes a process called transcription, which consists in making a “copy” of the gene called messenger RNA (mRNA). Once mRNA is produced, it diffuses out of the nuc-
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