Proof by perturbation

The perturbation proof by Kacser and Burns^[1] is given as follows.

Given the simple linear pathway catalyzed by two enzymes ${\displaystyle e_{1}}$ and ${\displaystyle e_{2}}$:

${\displaystyle X{\stackrel {e_{1}}{\longrightarrow }}S{\stackrel {e_{2}}{\longrightarrow }}}$

where ${\displaystyle X}$ is the fixed boundary species. Let us increase the concentration of enzyme ${\displaystyle e_{1}}$ by an amount ${\displaystyle \delta e_{1}}$. This will cause the steady state flux and concentration of ${\displaystyle S}$, and all downstream species beyond ${\displaystyle e_{2}}$ to increase. The concentration of ${\displaystyle X}$ is now decreased such that the flux and steady-state concentration of ${\displaystyle S}$ 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:

${\displaystyle {\begin{array}{r}\left.{\dfrac {\delta v_{1}}{v_{1}}}=\varepsilon _{x}^{1}{\dfrac {\delta x}{x}}+\varepsilon _{e_{1}}^{1}{\dfrac {\delta e_{1}}{e_{1}}}=0\right\}{\text{ Local equation }}\\[5pt]\left.{\dfrac {\delta J}{J}}=R_{x}^{J}{\dfrac {\delta x}{x}}+C_{e_{1}}^{J}{\dfrac {\delta e_{1}}{e_{1}}}=0\right\}{\text{ System equation }}\end{array}}}$

There is no ${\displaystyle s}$ term in either equation because the concentration of ${\displaystyle s}$is unchanged. Both right-hand sides of the equations are guaranteed to be zero by construction. The term ${\displaystyle \delta e_{1}/e_{1}}$ can be eliminated by combining both equations. If we also assume that the reaction rate for an enzyme-catalyzed reaction is proportional to the enzyme concentration, then ${\displaystyle \varepsilon _{e_{1}}^{1}=1}$, therefore:

${\displaystyle 0=R_{x}^{J}{\frac {\delta x}{x}}-C_{e_{1}}^{J}\varepsilon _{x}^{1}{\frac {\delta x}{x}}}$

Since ${\displaystyle \delta e_{1}/e_{1}\neq 0}$

this yields:

${\displaystyle R_{x}^{J}=C_{e_{1}}^{J}\varepsilon _{x}^{1}}$.

This proof can be generalized to the case where ${\displaystyle X}$ may act at multiple sites.

Pure algebraic proof

The pure algebraic proof is more complex^[3][4] and requires consideration of the system equation:

${\displaystyle {\bf {N}}{\bf {v}}(s(p),p)=0}$

where ${\displaystyle {\bf {N}}}$ is the stoichiometry matrix and ${\displaystyle {\bf {v}}}$ the rate vector. 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 ${\displaystyle p}$ yields, after rearrangement:

${\displaystyle {\dfrac {ds}{dp}}=\left[-{\bf {N}}{\dfrac {\partial v}{\partial s}}\right]^{-1}{\dfrac {\partial v}{\partial p}}}$

The inverted term is the unscaled control coefficient so that after scaling, it is possible to write:

${\displaystyle R_{p}^{s}=C_{v}^{s}\varepsilon _{p}^{v}}$

To derive the flux response coefficient theorem, we must use the additional equation:

${\displaystyle {\bf {v}}={\bf {v}}({\bf {s}}(p),p)}$