Control coefficients measure the response of a biochemical pathway to changes in enzyme activity.
The response coefficient, as originally defined by Kacser and Burns,[1] is a measure of how external factors such as inhibitors, pharmaceutical drugs, or boundary species affect the steady-state fluxes and species concentrations.
The flux response coefficient is defined by:
is the steady-state pathway flux.
Similarly, the concentration response coefficient is defined by the expression:
is the concentration of the external factor.
The response coefficient measures how sensitive a pathway is to changes in external factors other than enzyme activities.
The flux response coefficient is related to control coefficients and elasticities through the following relationship:
Likewise, the concentration response coefficient is related by the following expression:
, can act at multiple sites.
For example, a given drug might act on multiple protein sites.
These results show that the action of an external factor, such as a drug, has two components: When designing drugs for therapeutic action, both aspects must therefore be considered.
[2] There are various ways to prove the response theorems: The perturbation proof by Kacser and Burns[1] is given as follows.
Given the simple linear pathway catalyzed by two enzymes
is the fixed boundary species.
Let us increase the concentration of enzyme
This will cause the steady state flux and concentration of
is now decreased such that the flux and steady-state concentration of
is restored back to their original values.
These changes allow one to write down the following local and systems equations for the changes that occurred:
term in either equation because the concentration of
Both right-hand sides of the equations are guaranteed to be zero by construction.
If we also assume that the reaction rate for an enzyme-catalyzed reaction is proportional to the enzyme concentration, then
This proof can be generalized to the case where
may act at multiple sites.
The pure algebraic proof is more complex[3][4] and requires consideration of the system equation:
In this derivation, we assume there are no conserved moieties in the network, but this doesn't invalidate the proof.
Using the chain rule and differentiating with respect to
The inverted term is the unscaled control coefficient so that after scaling, it is possible to write:
To derive the flux response coefficient theorem, we must use the additional equation: