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A generalized model of the repressilator

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Abstract

The repressilator is a regulatory cycle of n genes where each gene represses its successor in the cycle: \(1\,\dashv\,2\,\dashv\,\cdots\,\dashv\,n\,\dashv\,1\). 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.

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Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040, Linz, Austria

    Stefan Müller

  2. Department of Mathematics, University College London, London, WC1E 6BT, UK

    Josef Hofbauer

  3. Institute for Theoretical Chemistry, University of Vienna, Währingerstraße 17, 1090, Wien, Austria

    Lukas Endler, Christoph Flamm, Stefanie Widder & Peter Schuster

Authors
  1. Stefan Müller
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  2. Josef Hofbauer
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  3. Lukas Endler
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  4. Christoph Flamm
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  5. Stefanie Widder
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  6. Peter Schuster
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Correspondence to Stefan Müller.

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Müller, S., Hofbauer, J., Endler, L. et al. A generalized model of the repressilator. J. Math. Biol. 53, 905–937 (2006). https://doi.org/10.1007/s00285-006-0035-9

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  • DOI: https://doi.org/10.1007/s00285-006-0035-9

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