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I'm looking for a rigorous proof of the summation theorem of metabolic control theory. The only sources I find are the original papers by Kacser and Burns 1973 and Heinrich and Rapoport 1974, both of which are unclear to me.

The summation theorem is the following. Consider a linear metabolic pathway

$X_0 \xrightarrow{E_1} X_1 \xrightarrow{E_2} X_2 \cdots \xrightarrow{E_n} X_n$

Let $J$ be the steady-state flux through the pathway, and let $e_i$ be the concentrations of the enzyme $E_i$. The flux control coefficients $C_i$ is defined as the partial derivatives

$C_i = {\partial (\ln J)} / {\partial (\ln e_i)}$

The summation theorem states that for any such system, it holds that $\sum_{i=1}^n C_i = 1$. Intuitively this means that if the concentration of some enzyme has a large impact on flux, then the other must have a relatively smaller impact. It seems reasonable, but is far from obvious I think. So a proof is needed.

The proof (sketch) given by Kacser and Burns seem to rest on the assumption that $\partial (\ln J) = \sum_{i=1}^n \partial (\ln e_i)$, and given this, the proof is trivial; but this assumption is never explained further. Heinrich and Rapaport states the summation theorem in passing (p.94, "it is worth mentioning that the following relation holds"), but I don't see a proof. (But their paper is rather dense so I may have missed it.)

Anyone know of a proof of the general case?

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