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J. Math. Biol. (2006) 53:905–937
DOI 10.1007/s0028500600359
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
© SpringerVerlag 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 singlestep cooperative repressor binding, and (ii) a repressilator with
autoactivation and cooperative regulator binding. Genes are assumed to be
present in constant amounts, transcription and translation are modelled by sin
glestep 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
email: stefan.mueller@oeaw.ac.at
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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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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 Summer '06
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