TBI-Preprint 06-05-002

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A Generalized Model of the Repressilator

Stefan Müller, Josef Hofbauer, Lukas Endler, Christoph Flamm, Stefanie Widder, Peter Schuster

Submitted to:
Journal of Nonlinear Science 2006

The repressilator is a regulatory cycle of n genes where each gene represses its successor in the cycle: 1 -| 2 -| . . . -| n -| 1. The system is modelled by ODEs for an arbitrary number of identical genes and arbitrarily strong repressor binding. A detailed mathematical analysis of the dynamical behavior is provided for two model systems: (i) a repressilator with leaky transcription and single-step cooperative repressor binding, and (ii) a repressilator with auto-activation and cooperative regulator binding. Genes are assumed to be present in constant amounts, transcription and translation are modelled by single-step kinetics, and mRNAs as well as proteins are assumed to be degraded by first order reactions. Several dynamical patterns are observed: Multiple steady states, periodic and aperiodic oscillations corresponding to limit cycles and heteroclinic cycles, respectively. The results of computer simulations are complemented by a detailed and complete stability analysis of all equilibria and of the heteroclinic cycle.

Gene regulatory network, negative feedback loop, repressilator, stability analysis, Hopf bifurcation, heteroclinic cycle

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