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P
2
gene u
gene v
gene w
Cell 2
P
1
AI
AI
gene u
AI
Cell 3
AI
P
3
Cell 1
AI
Cell 4
Fig. 3.8. Schematic diagram of the network of genetic relaxation oscillators. u; v and w denote
the genes, and P
1
, P
2
and P
3
the corresponding promoters.
3.3. Genetic Relaxation Oscillators
Dierent types of genetic circuit architectures, besides the repressilator, can give rise
to oscillations and dynamical behavior. We now consider a dierent kind of network,
consisting of coupled hysteresis-based genetic relaxation oscillators [Kuznetsov et al.
(2004)]. Studying this system allows the identication of the intercellular mecha-
nisms responsible for multirhythmicity in coupled genetic circuits. Additionally,
this system exhibits a dynamical behavior closely related to a known biological
problem, namely the existence of quantized cycles in cellular processes.
3.3.1. Dynamical regimes of coupled relaxators
Recently, Kuznetsov et al. (2004) proposed a model of hysteresis-based relaxation
genetic oscillators coupled via quorum-sensing. This oscillator can be constructed,
as shown in Fig. 3.8, by combining two engineered gene networks, the toggle switch
[Gardner et al. (2000)] and an intercell communication system, which have been
previously implemented experimentally in E. coli by Kobayashi et al. (2004), and
in V. scheri by Fuqua and Greenberg (2002), respectively. The synthesis of the
two repressor proteins, which constitute the toggle switch, are regulated such that
the expression of the two genes is mutually exclusive, which leads to bistability. The
second network is based on the dynamics of an AI, which on the one hand drives
the toggle switch through the hysteresis loop, and on the other hand provides an
intercell communication by diusion through the cell membrane. The time evolution
of the system is governed by the dimensionless equations [Kuznetsov et al. (2004)]:
du
i
dt
=
1
f(v
i
)u
i
+
3
h(!
i
)
(3.10)
