The Control Analysis of Signal Transduction - Systems Biology of Metabolic and Signaling Networks - page 41

“ i ”. p is a parameter that affects process i selectively. As indicated by the
parentheses, the derivatives are both partial but in a different sense. The one in
the numerator is that where only one of all process activities in the system is
modulated, but all metabolite concentrations and hence process rates may vary as
a consequence of that modulation; it is a systems property. With the proviso that all
other process activities are not modulated, it can be seen as a total derivative. The
one in the denominator modulates the process activity at constant values of all other
metabolite concentrations that affect the process; like the elasticity coefficient (see
below) it is a component property even though that component property may itself
vary with the system's state.
The most common control coefficients quantify the control of fluxes or metabo-
lite concentrations. However, any variable of the system will have a control
coefficient defined by equations analogous to Eq. ( 3.1 ) (Westerhoff and Dam
1987 ). There is no need for the system to be at steady state (Westerhoff 2008 ); if
it is not then x is a function of time, but until now the perturbation in process activity
has been considered to be time independent. As many enzyme-catalysed reaction
rates are proportional to the enzyme concentration (at least in a certain range of
enzyme concentrations), control coefficients can often be written using total
enzyme concentration E i as parameter (Kacser and Burn 1973 ):

d x
d E i

E i
J ¼

dln x
dln E i :

C i ¼ (3.2)

The definition of concentration control coefficient pertains equally to gene
expression, signal transduction, as well as metabolic pathways, as well as to their
integration (Kahn and Westerhoff 1991 ). In metabolic pathways fluxes are defined
as the amount of a chemical element flowing down the pathway per unit time and
unit biomass. In gene expression pathways, fluxes may be defined as the rate of
synthesis of the mRNA, the rate of synthesis of the protein, and the chemical flux
catalysed by that protein. In a signal transduction pathway it may make sense to
define the ultimate flux as the steady rate of the process that is activated by the final
signalling protein, which could well correspond to the concentration of the signal-
ling state of that protein multiplied by a rate constant.
Figure 3.1 shows flux control coefficients of various enzymes in one, two, and
three reactions' metabolic pathways. The rate-limiting enzymes in a metabolic
pathway must have a flux control coefficient equal to 1. In this example, this only
occurs in the one-enzyme pathway. Moreover, a rate-limiting enzyme does not
remain rate-limiting upon overexpression. Indeed, most overexpression studies of
enzymes have revealed that large increases of enzyme concentrations are not
accompanied by equivalent increases in pathway flux. Usually, whilst one is
increasing the amount of the hypothetically rate-limiting enzyme, its control over
the pathway flux decreases until it approaches 0 eventually. Concentration control
coefficients of various enzymes in three reactions' metabolic pathway are shown in
Table 3.1 .