Dynamics of Multicellular Synthetic Gene Networks - Biological Networks - page 49

4

HB s2

LP 2

3

u

HB 2

BP 2

LP 1

2

HB s1

BP 1

HB s1 HB 1

1

HB s2

LP 1

LP 2

2.7 3 3.3 3.5 3.8

1

Fig. 3.10. Coexistence of ve dierent states for increased coupling strength d = 0:3. Other
parameters are: 2 = 5, 3 = 1, 4 = 4, = = = 2, d e = 1 and " = 0:05. Coexistence of the
OD and the in-phase oscillatory regime is also shown.

pare for instance Fig. 3.9B (5:3 distribution) with period T = 364:15 and Fig. 3.9C
(4:4 distribution) with period T = 256:27.
Another possible collective behavior of this system consists in asymmetric os-
cillations (for d < 0:003), when some of the oscillators in the system perform large
excursions, while the rest oscillate in the vicinity of a stable steady state with small
amplitude. This results in the presence of two oscillatory clusters, (Fig. 3.9D,E).
Again, the number of possible dierent distributions for a system of N oscillators
is N1, and each has dierent oscillation period: compare Fig. 3.9D (1:7) with
period T = 216:95 and Fig. 3.9E (4:4) with T = 141:01.
The oscillators in the system can be also ordered in multiple cluster regimes; we
present only two examples here: three (Fig. 3.9F) and ve (Fig. 3.9G) oscillatory
clusters. Again, dierent distributions of the oscillators between the clusters are
possible in this case. To illustrate this, we present here a 3:3:2 distribution when
three oscillatory clusters are formed (Fig. 3.9F), and a 1:2:2:2:1 distribution when
ve oscillatory clusters are created (Fig. 3.9G).

3.3.2. Bifurcation analysis

Bifurcation analysis can be used to identify and characterize the dierent dynamical
solutions described above. When applied to the case N = 2, it shows that already
two oscillators provide a large variety of possible regimes, as shown in Fig. 3.10.
The OD regime, similarly to the IHSS one, is a result of the symmetry breaking in
the system through a pitchfork bifurcation (labeled BP 1 in Fig. 3.10). The unstable
steady-state splits into two branches that gain stability through Hopf bifurcations,
denoted as HB s1 and HB s2 in Fig. 3.10. The solution coexists in the 1 -parameter
space with dierent oscillatory solutions, e.g. in-phase oscillations (marked with
dashed lines), as shown in Fig. 3.10. The true IHLC that emerges from HB s1 is
unstable in this model and not shown in Fig. 3.10.