The Control Analysis of Signal Transduction - Systems Biology of Metabolic and Signaling Networks - page 45

(Groen et al. 1982 ). In signal transduction, the situation can become almost bizarre,
with all kinases exerting a control of 1 on the steady metabolic flux and all
phosphatases a control of 1 and the k cat of the enzyme exerting the missing

1 (Kahn and Westerhoff 1991 ). For concentration they show that the concept that
a single step limits a concentration is always wrong: if any step limits a concentra-
tion then there must be other steps that limit that concentration in the opposite
direction. And for transient times, it reflects the opposite of limitation by a process
activity: a step limits the magnitude of a transient time, e.g. in signal transduction,
because it is too fast rather than too slow. Control of concentrations must be, and
control of flux and transient time may well be, distributed over multiple steps
hereby, but what determines this distribution? Enter the component properties.
But which component properties?

3.5 The Elasticity Coefficients

The elasticity coefficients of an enzyme measure the most important component
property, which is its immediate response to changes in its immediate environment.
Said changes include changes in the concentrations of substrates, products, or alloste-
ric effector, the concentration ratio of phosphorylated to non-phosphorylated form of
signal transduction proteins, and the concentration of a growth factor such as EGF,
pH, and temperature. Mathematically the elasticity coefficient is defined by ( 3.6 ):

local

@ v i S

J ¼ @

ln v i

J
S

ε ¼

ln S ;

(3.6)

@ S @

where v i denotes the reaction rate and S denotes any metabolite concentration or
other environmental factor.

3.6 Connections Between System-Level Control and
Molecular Properties

The key systems biology property of control analysis is that it relates the kinetic
properties of the component processes to the functional properties of the system.
The connectivity (Kacser and Burn 1973 ; Westerhoff and Chen 1984 ) laws are such
relationships. They relate the control coefficients to elasticity coefficients. The
connectivity theorem for flux control coefficients (Kacser and Burn 1973 ) is valid
for any free variable X (such as the concentration ratio of the phosphorylated to the
non-phosphorylated form of a signal transduction protein) and any flux J in the
network. It refers to the (multiplicative) products of the flux control coefficient of
all steps by their elasticity coefficients towards X . It states that the sum of all of
these is zero: