Biology Reference
In-Depth Information
FIGURE 22.6
Bistable and oscillating scenarios for the core pluripotency networks. (A) Phase space of the model for three core pluripotency
regulators linked into fully connected triad (see Eqs
1
e
3
), the system is bistable at the selected parameter ranges; the two dynamic attractors are shown as
red dots. (B) The same phase space after incorporation of stochastic noise into the model. Dots correspond to cell states with respect to Nanog/Oct4/Sox2
concentrations. (C, D) Dynamic solutions for the same triad network, but with additional repressor links, corresponding, for instance, to repression of Oct4
by Tcf3, and repression of Nanog by Zfp281. The presence of repressors cause oscillations of Nanog, Oct4 and Sox2 concentrations. Depending on the
parameter setting, the system can either maintain its bistability (C, damped oscillator) or become a true oscillator with a limit cycle attractor type (D). Red
dots on D (finite simulation states) display the attractor states.
p
and the random variable
concentration-dependent action (dual regulation) of dual or
alternative transcriptional regulators.
Another important expansion of the model is the
incorporation of stochastic terms, necessary to describe
stochastic switching between metastable system states or
dynamic attractors. In our previous work we employed
a model for Nanog based on stochastic differential equa-
tions (SDE). Accordingly, the stochastic term is included in
the model:
approximated as
,whichis
the zero mean Gaussian 'white noise'. Such a stochastic
term is general, as it specifies no particular source of noise
in the system (
Figure 22.6
B). At the same time, at least two
potential sources of noise in transcriptional networks are
well known. First, the most obvious source is mRNA
degradation, which follows exponential decay. At the level
of the cell population this will produce a Gaussian distri-
bution for the number of mRNA molecules per cell. Other
sources of noise are related to mRNA synthesis because
the initiation of mRNA synthesis is inherently stochastic
owing to variations in the times required for the assembly
of large eukaryotic transcription initiation complexes.
Initiation complexes may also need to interact with
upstream enhancers, which is yet another stochastic
process.
The stochastic distributions for gene activity levels
(such as shown in the
Figure 22.6
B) fit well with the
actual experimental data obtained from cell sorting or
cell imaging. Bistable models can be fitted to the
intensity distribution data by adjusting three to four
x
d
dt
¼ að
½
x
ð
P
Nanog
Þð
P
Oct
Sox
ÞÞ b½
1
1
1
x
þ
g
ð
x
Þxð
0
;
sÞ
(3)
A system of three SDEs for the core network compo-
nents (x
Nanog, Oct4, Sox2), such as in Eq.
3
(repre-
sentative of one factor) incorporates both deterministic
and stochastic terms and will serve as the initial point
in the exploration of stochastic switches between meta-
stable states. The stochastic term in Eq.
3
incorporates
a concentration-dependent function g(x), which is often
¼
