Journal
JOURNAL OF MATHEMATICAL BIOLOGY
Volume 53, Issue 6, Pages 905-937Publisher
SPRINGER
DOI: 10.1007/s00285-006-0035-9
Keywords
gene regulatory network; negative feedback loop; repressilator; stability analysis; Hopf bifurcation; heteroclinic cycle
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The repressilator is a regulatory cycle of n genes where each gene represses its successor in the cycle: 1 proves 2 proves center dot center dot center dot proves n proves 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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