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 84

where:

=

P ¼ y 2 2 V 2 y 3 2 Log y 2 = y 3 2

J Log y 2 = 2 þ V 2 Log 1 þ 4 L α 21 þ

=

y 3 2 V 3 y 3 2 Log y 4 = y 3 2

þð V 3 þ L α V 2 Þ Log 1 þ 4 21 þ

L 0 V 3 y 4 2 þ ð V 4 þ Þ Log 1 þ = 4

and
H ¼ V 2 y 3 2 Log y 2 = y 3 2 L ' V 3 y 4 2 Log y 3 = y 4 2 V 3 y 3 2 Log y 4 = L α 41 ð þ Þþ 21 ð þ Þ

y 3 2

21 ð Þ:

In this case, parameters appearing only in P *, like V 4 , are modulating the
localization of the fixed point; hence, the values of the stationary concentrations
of the glycolytic metabolites [cf. Demongeot et al. ( 2007a , 2007b ) and Glade et al.
( 2007 ) for a more general approach of the potential-Hamiltonian decomposition].
Let us suppose now that we measure the outflows J 1 and J 2 (cf. Fig. 4.12 ). Then
from the system (S1) we can calculate the sharing parameter

þ

(which regulates the
pentose pathway and the low glycolysis dispatching) from the steady-state
equations equalizing the in- and outflows at each step. By denoting the stationary
state x *

α

{ x i *} i ¼ 1,4 , we have:
V 1 x 1 n

¼

x 1 n V 2 x 2 =ð x 2 Þ V 2 x 3 =ð x 3 Þ¼

=ð 1 þ Þ¼ J ; 1 þ L α 1 þ J

V 2 x 2 =ð x 2 Þ¼ V 3 x 3 =ð x 3 Þ

ð 1 αÞ 1 þ J 1 ; 1 þ

L 0 V 3 Þ=

¼ J 2 ð V 4 þ V 4

Hence, we can calculate α by using the following formula:

2 J 2 LV 2 ð

L 0 V 3 Þþα L 0 V 3 Þ

α V 4 þ ð JV 4 J 2 LV 2 ð V 4 þ Þ þ J 1 V 4 J

2

K 0 ¼ 0 ;

¼ 0or α αð 1 K Þþ

where we have denoted:

L 0 V 3 Þ and K 0 ¼ð L 0 V 3 Þ:

K ¼ JV 4 = J 2 LV 2 ð V 4 þ J 1 V 4 J Þ= J 2 LV 2 ð V 4 þ

When the flux of the k th step in a metabolic network has reached its stable
stationary value

Φ k , then the notion of control strength C ki exerted by the metabolite

x i on this flux Φ k is defined by (Kaczer and Burns 1973 ; Wolf and Heinrich 2000 ):

C ki ¼ @ Log ΔΦ k =@ Log Δ x i

and we have:

8 k ¼ 1 ; n ; Σ i¼ 1 ;n C ki ¼ 1 :