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sic repressilator and the quorum sensing module, with respect to the case of the
previous Section, alters the coupling from its original reinforcing character to a
phase-repulsive one [Ullner et al. (2007)]. As a consequence, the previously favored
in-phase regime becomes now unstable, and many new dynamical regimes appear.
To create a phase-repulsive coupling, one can modify the initial scheme (Fig. 3.1)
by placing the gene luxI under inhibitory control of the repressilator protein TetR.
The proposed `rewiring' between the repressilator and the quorum sensing module
introduces a feedback loop that competes with the overall negative feedback loop
along the repressilator ring, resulting in a phase-repulsive intercellular coupling.
The mRNA and protein dynamics are described by Eqs. (3.1){(3.4) above.
In contrast to Section 3.2.1, we assume here dierent lifetime ratios for the pro-
tein/mRNA pairs, which results in a weak relaxator-like dynamics of the repres-
silator. The rewiring aects the equation of the AI concentration. Now the AI
concentration S
i
in cell i is generated at a rhythm proportional to B
i
:
S
i
=k
s0
S
i
+ k
s1
B
i
(S
i
S
e
) :
(3.9)
A moderate increase of the Hill coecient to n = 2:6, a value in agreement
with recent experimental measures [Rosenfeld et al. (2005)], together with dierent
lifetime ratios
a
= 0:85,
b
= 0:1, and
c
= 0:1, increase the nonlinear character
of the repressilator dynamics, leading to the appearance of two time scales in the
time series, with a fast concentration increase and a relative slow decay. The slower
protein decay increases the period of the repressilator by a factor of approximately
three.
3.2.2.1. Bifurcation analysis for two coupled repressilators
A rst glimpse into the eect of coupling on the dynamics of inter-cell genetic
networks can be obtained by investigating a minimal system of only two oscilla-
tors. Figure 3.3 shows representative time traces, obtained by direct numerical
calculations of a population of N = 2 coupled repressilators for increasing cou-
pling strength. The dierent dynamical regimes found are self-sustained oscillatory
solutions [Fig. 3.3(a)], inhomogeneous limit cycles (IHLC) [Fig. 3.3(b)], inhomo-
geneous steady states (IHSS) [Fig. 3.3(c)] and homogeneous steady states (HSS)
[Fig. 3.3(d)], all of which exist for biologically realistic parameter ranges.
A detailed bifurcation analysis allows to determine the origin of these dierent
solutions and the transition scenarios between them, thus providing deeper qualita-
tive and quantitative conclusions about the structure and dynamical behavior of the
system. This analysis can be performed with public software such as the XPPAUT
package [Ermentrout (2002)]. In the bifurcation analysis below we use the coupling
strength Q [Eq. (3.7)] as a biologically relevant parameter to obtain one-parameter
continuation diagrams. Starting from the homogeneous unstable steady state of
isolated oscillators (Q = 0), Fig. 3.4 shows the basic continuation curve containing
the homogeneous and inhomogeneous stable steady states.
