The Control Analysis of Signal Transduction - Systems Biology of Metabolic and Signaling Networks - page 42

Fig. 3.1 One, two, and three reactions' metabolic pathways and exemplary flux control
coefficients. A, P and enzymes E 1 ,E 2 , and E 3 are present at fixed concentrations that may be
modulated, whereas steady-state flux J and concentrations of B and C are dependent variables

Table 3.1 Concentration
control coefficients in the
three reactions' metabolic
pathway shown in Fig. 3.1

Enzymes
E 1

Metabolites E 2 E 3

B 2.65 1.98 0.67

C 0.18 0.29 0.47

3.4 The Summation Laws

Control analysis has led to the identification of important properties of pathways
at steady state pertaining to the summation of all the control coefficients of a
pathway. For a steady state the sum of flux control coefficients of all the reactions
in any pathway is equal to unity, using various procedures (Kacser and Burn
1973 ; Heinrich and Rapoport 1974 ; Westerhoff and Dam 1987 ; Giersch 1988 ;
Westerhoff 2008 ) for the generalisation beyond metabolic pathways and steady
state). This law can be derived by making a simultaneous small relative increase
(

x / x ) in all process rates of a metabolic system. Because the relative rates
of production of each metabolite then increase by the same amount as the relative
rates of its consumption, that metabolite's concentration remains unchanged and
hence the steady-state condition is maintained. The metabolic flux through the
pathway then increases exactly by the same relative amount “

δ r x δ

δ r ”, whereas a

transient time decreases by the fraction “ δ r ”. Mathematically this principle says

that the flux is a homogeneous function of first degree, the metabolite
concentrations are homogeneous functions of zero degree, and transient times
are a function of degree minus 1, of all process activities: The summation laws
are corollary of the Euler theorem for homogeneous functions which is also used
in the derivation of the Gibbs-Duhem law (Westerhoff and Dam 1987 ; Giersch
1988 ; Westerhoff 2008 ). For flux control coefficients:
X

C i ¼

1 (3.3)

i

for concentration control coefficients:
X

C X i
i

¼

0 (3.4)

i