4.2. Control Coefficients#

A control coefficient measures the influcence a change in an enzyme has on the steady-state flux or metabolite concentrations.

Imagine a simple metabolite pathway such as the one below is at ateady-state:

\[X_o \stackrel{e_1}{\longrightarrow} S_1 \stackrel{e_2}{\longrightarrow} S_2 \stackrel{e_3}{\longrightarrow} S_3 \stackrel{e_4}{\longrightarrow} X_1\]

We will assume that the boundary species \(X_o\) and \(X_1\) are fixed n order to sustain sa steady-state. At steady-state there will be a pathway flux, \(J\) and steady-state level of metabolites.

Let us now imgine we increase the level of enzyme \(e_2\) by an amount \(\delta e_2\). This wil cause the steady-state to change resulting in changes in \(J\) and metabolites \(S_1, S_2\), and \(S_3\). If we assume these changde by \(\delta J\), \(\delta S_1, \delta S_2\), and \(\delta S_3\).

We can assess the influence by the pertubtion in \(e_2\) had by looking at the ratio. For example the effect on the flux can be written as:

\[\frac{\delta J}{\delta e_2}\]

This ratio has units. For convenince we can eliminate the units by multiplying and dividing by \(e_2\) and \(J\) to give us a unit-less measure of influence:

\[\frac{\delta J}{\delta e_2} \frac{e_2}{J}\]

This can be reexpressed in the following form:

\[\frac{\delta J}{J} / \frac{\delta e_2}{e_2}\]

This can now be seen as a ratio of relative changes. In general, the relationship between changes in enzyme and say a flux is non-linear. This means that the ratio will depend on the size of the perturbation in \(\delta e_2\). To remidie this, we can reduce the size of perturbation such that in the limit, we obtain a ratio of differentials independent of the size of the perturbation. We call this ratio the flux control coefficient, \(C^J_{e_2}\):

\[C^J_{e_2} = \frac{d J}{d e_2} \frac{e_2}{J}\]

A similar kind of measurement can be made with respect to the metabolite levels such that we can define concentraton control coefficients, one for each metabolite:

\[C^{s_1}_{e_2} = \frac{d s_1}{d e_2} \frac{e_2}{s_1},\quad C^{s_2}_{e_2} = \frac{d s_2}{d e_2} \frac{e_2}{s_2},\quad C^{s_2}_{e_2} = \frac{d s_2}{d e_2} \frac{e_2}{s_2}\]

Given that there are four enzymes there will in total be twelve concentraton control coeffiicients.

Since all steps carry the same flux at steady-state, there will only be four flux control coefficients, one for each enzyme.


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