MicroRNAs and Robustness in Biological Regulatory Networks. A Generic Approach with Applications at Different Levels: Physiologic, Metabolic, and Genetic - Systems Biology of Metabolic and Signaling Networks - page 68

2009a ). The substrates of the PFK and Aldolase are respectively fructose-6-phos-
phate and fructose-1,6-diphosphate, of which concentrations are x 1 and x 2 , respec-
tively. The corresponding 2-dimensional differential system is defined by the
following equations:

V 1 x 1 n x 1 n V 1 x 1 n x 1 n

d x 1 = d t

¼ J = ð K 1 þ Þ;

d x 2 = d t

¼ = ð K 1 þ Þ

2

V 2 x 2 2 x 2 2 = K 2 þ ; or by changing the variables y i ¼ð x i Þ

=

V 1 y 1 2 n y 1 2 n d y 1 = d t ¼ J = K 1 þ 2 y 1 ;

=

V 1 y 1 2 n y 1 2 n V 2 y 2 4 y 2 4 d y 2 = d t ¼ = K 1 þ = K 2 þ 2 y 2

We can find a potential-Hamiltonian decomposition for this new differential
system:

d y 1 = d t

¼@ P =@ y 1 þ @ H =@ y 2 þ R ð y 1 ; y 2 Þ;

d y 2 = d t

¼@ P =@ y 2 @ H =@ y 1

y 1 2 n y 2 2 where we have: P ¼ J Log ð y 1 Þ= 2 þ V 1 Log K 1 þ ð Þ= 4 n þ V 2 Log K 2 þ ð Þ= 4,

V 1 Ð y 1 2 n V 1 Ð y 1 2 n

2 y 2 2 ,
H and R being negligible in the domain where y 2 * is sufficiently large and y 1 *
sufficiently small, which corresponds for example to a large value of the V max ratio
V 1 / V 2 .

y 1 2 n y 1 2 n H ¼ ½ = ð K 1 þ Þ d y 1 = 2 y 2 and R ¼ ½ = ð K 1 þ Þ d y 1 =

4.3.5 Example of a Robust Network, the Neural
Hippocampus Network

In a toy model of the vegetative system, we can consider two simple networks, each
of them having a regulon structure, i.e., two nodes in interaction with a negative
circuit, one of them being auto-excitable (self-positive or auto-catalytic loop)
[cf. Fig. 4.4 , left and Elena et al. ( 2008 ) and Demongeot et al. ( 2002 )]. The first
regulon represents the vegetative control of the respiratory system with its inspira-
tory I and expiratory E neurons; the second one describes the vegetative control of
the cardiac oscillator with the cardiomodulator bulbar node CM ruling the activity
of the sinusal node S.
The corresponding dynamics is completely different if we consider the
2 regulons coupled or not, and noised or not. By modeling the transition of neuron
states between time t and ( t + dt ) by 2 coupled van der Pol differential equations:
d x /d t x 2 ) y , for the respiratory dynamics, where x

represents the activity of neurons E, y the activity of neurons I, and

¼ y ,d y /d t ¼ x + μ (1

μ is an

z 2 ) w + k ( y ) y , for the
cardiac dynamics, where z represents the activity of pace-maker cells S, w the
activity of neurons CM,

anharmonic parameter, and d z /d t ¼ w ,d w /d t ¼ z + η (1

the anharmonic parameter, and k ( y ) the coupling
intensity parameter between inspiratory neurons I and cardiomodulator neurons

η