Summation theorems (biochemistry)

The summation of the flux and concentration control coefficients were discovered independently by the Kacser/Burns group and the Heinrich/Rapoport group in the early 1970s and late 1960s.

If we define the control coefficients using enzyme concentration, then the summation theorems are written as:

${\displaystyle \sum _i{C}_{e_i}^J=1}$

${\displaystyle \sum _i{C}_{e_i}^s=0}$

However these theorems depend on the assumption that reaction rates are proportional to enzyme concentration. An alternative way to write the theorems is to use control coefficients that are defined with respect to the local rates which is therefore independent of how rates respond to changes in enzyme concentration:

${\displaystyle \sum _i{C}_{v_i}^J=1}$

${\displaystyle \sum _i{C}_{v_i}^s=0}$

Although originally derived for simple linear chains of enzyme catalyzed reactions, it became apparent that the theorems applied to pathways of any structure including pathways with complex regulation involving feedback control.

Derivation

There are different ways to derive the summation theorems. One is analytical and rigorous using a combination of linear algebra and calculus. The other is less rigorous, but more operational and intuitive. The latter derivation is shown here.

Consider the two-step pathway:

${\displaystyle {\text{X}}_o\stackrel{v_1}{⟶}\text{S}\stackrel{v_2}{⟶}{\text{X}}_1}$

where ${\displaystyle X_o}$ and ${\displaystyle X_1}$ are fixed species so that the system can achieve a steady-state.

Let the pathway be at steady-state and imagine increasing the concentration of enzyme, ${\displaystyle e_1}$, catalyzing the first step, ${\displaystyle v_1}$, by an amount, ${\displaystyle \delta e_1}$. The effect of this is to increase the steady-state levels of S and flux, J. Let us now increase the level of ${\displaystyle e_2}$ by ${\displaystyle \delta e_2}$ such that the change in S is restored to the original value it had at steady-state.

The net effect of these two changes is by definition, ${\displaystyle \delta s=0}$.

There are two ways to look at this thought experiment, from the perspective of the system and from the perspective of local changes. For the system we can compute the overall change in flux or species concentration by adding the two control coefficient terms, thus:

${\displaystyle \frac{\delta J}{J}=C_{e_1}^J\frac{\delta e_1}{e_1}+C_{e_2}^J\frac{\delta e_2}{e_2}}$

${\displaystyle \frac{\delta s}{s}=C_{e_1}^s\frac{\delta e_1}{e_1}+C_{e_2}^s\frac{\delta e_2}{e_2}=0}$

We can also look at what is happening locally at every reaction step for which there will be two: one for ${\displaystyle v_1}$, and another for ${\displaystyle v_2}$. Since the thought experiment guarantees that ${\displaystyle \delta s=0}$, the local equations are quite simple:

${\displaystyle \frac{\delta v_1}{v_1}={\epsilon }_{e_1}^{v_1}\frac{\delta e_1}{e_1}}$

${\displaystyle \frac{\delta v_2}{v_2}={\epsilon }_{e_1}^{v_1}\frac{\delta e_2}{e_2}}$

where the ${\displaystyle \epsilon }$ terms are the elasticities. However, because the enzyme elasticity is equal to one, these reduce to:

${\displaystyle \frac{\delta v_1}{v_1}=\frac{\delta e_1}{e_1}}$

${\displaystyle \frac{\delta v_2}{v_2}=\frac{\delta e_2}{e_2}}$

Because the pathway is linear, at steady-state, ${\displaystyle v_1=v_2=J}$. We can substitute these expressions into the system equations to give:

${\displaystyle \frac{\delta J}{J}=C_{e_1}^J\frac{\delta v_1}{v_1}+C_{e_2}^J\frac{\delta v_2}{v_2}}$

${\displaystyle \frac{\delta s}{s}=C_{e_1}^s\frac{\delta v_1}{v_1}+C_{e_2}^s\frac{\delta v_2}{v_2}=0}$

Note that at steady state the change in ${\displaystyle v_1}$ and ${\displaystyle v_2}$ must be the same, therefore ${\displaystyle \delta v_1/v_1=\delta v_2/v_2}$.

Setting ${\displaystyle \alpha =\delta J/J=\delta v_1/v_1=\delta v_2/v_2}$, we can rewrite the above equations as:

${\displaystyle \alpha =C_{e_1}^J\alpha +C_{e_2}^J\alpha =\alpha (C_{e_1}^J+C_{e_2}^J)}$

${\displaystyle 0=C_{e_1}^s\alpha +C_{e_2}^s\alpha =\alpha (C_{e_1}^s+C_{e_2}^s)}$

We then conclude through cancelation of ${\displaystyle \alpha }$ since ${\displaystyle \alpha \ne 0}$, that:

${\displaystyle 1=C_{e_1}^J+C_{e_2}^J}$

${\displaystyle 0=C_{e_1}^s+C_{e_2}^s}$