repressilator-muller-JMB06 - J Math Biol(2006 53:905937 DOI...

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J. Math. Biol. (2006) 53:905–937 DOI 10.1007/s00285-006-0035-9 Mathematical Biology A generalized model of the repressilator Stefan Müller · Josef Hofbauer · Lukas Endler · Christoph Flamm · Stefanie Widder · Peter Schuster Received: 8 June 2006 / Revised: 11 July 2006 / Published online: 2 September 2006 © Springer-Verlag 2006 Abstract The repressilator is a regulatory cycle of n genes where each gene represses its successor in the cycle: 1 a 2 a ··· a n a 1. The system is mod- elled 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 transcrip- tion 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 sin- gle-step kinetics, and mRNAs as well as proteins are assumed to be degraded by Frst 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 com- plemented by a detailed and complete stability analysis of all equilibria and of the heteroclinic cycle. Keywords Gene regulatory network · Negative feedback loop · Repressilator · Stability analysis · Hopf bifurcation · Heteroclinic cycle S. Müller ( B ) Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria e-mail: [email protected] J. Hofbauer Department of Mathematics, University College London, London WC1E 6BT, UK L. Endler · C. ±lamm · S. Widder · P. Schuster Institute for Theoretical Chemistry, University of Vienna, Währingerstraße 17, 1090 Wien, Austria
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906 S. Müller et al. 1 Introduction The seminal work of Jacob, Monod, and Changeaux [8,10] on the regulation of gene expression in the lac operon initiated early studies on gene regula- tion through repression by speciFc proteins which demonstrated the possibility of oscillations in some special systems with few genes [5]. Numerical integra- tion of differential equations with delay were used to model cyclic repression systems of the type 1 a 2 a ··· a n a 1 1 [4,14] and showed that cycles with odd numbers of genes exhibit oscillations over a wide range of conditions. Later work presented stability analysis of equilibria in cyclic repressor systems [1,2,15], and eventually the existence of oscillations resp. multiple stable steady states has been proven for cycles with odd resp. even numbers of genes [11]. The mathematical analysis of such feedback loops culminated in the establish- ment of a Poincaré–Bendixson theorem [9]. ±or a comprehensive summary of biological feedback loops we refer to the monograph by Thomas and D’Ari [13].
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repressilator-muller-JMB06 - J Math Biol(2006 53:905937 DOI...

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