Control coefficients^[1] are used to describe how much influence (i.e., control) a given reaction step has on the steady-state flux or species concentration level. In practice, this can be accomplished by changing the expression level of a given enzyme and measuring the resulting changes in flux and metabolite levels. Control coefficients form a central component of metabolic control analysis.

There are two primary control coefficients:

• Flux Control Coefficients
• Concentration Control Coefficients.

The simplest way to look at control coefficients is as the scaled derivatives of the steady-state change in an observable with respect to a change in enzyme activity. For example, the flux control coefficients can be written as:

${\displaystyle C_{e_{i}}^{J}={\frac {dJ}{de_{i}}}{\frac {e_{i}}{J}}={\frac {d\ln J}{d\ln e_{i}}}\approx {\frac {J\%}{e_{i}\%}}}$

while the concentration control coefficients can be written as:

${\displaystyle C_{e_{i}}^{s_{j}}={\frac {ds_{j}}{de_{i}}}{\frac {e_{i}}{s_{j}}}={\frac {d\ln s_{j}}{d\ln e_{i}}}\approx {\frac {s_{j}\%}{e_{i}\%}}}$

Control coefficients can have any value that includes negative and positive values. A negative value indicates that the observable in question decreases as a result of the change in enzyme activity.

In theory, other observables, such as growth rate, or even combinations of observables, can be defined using a control coefficient. But flux and concentration control coefficients are by far the most commonly used.

The approximation in terms of percentages makes control coefficients easier to measure and more intuitively understandable.

Control coefficients are useful because they tell us how much influence each enzyme or protein has in a biochemical reaction network.

It is important to note that control coefficients are not fixed values but will change depending on the state of the pathway or organism. If an organism shifts to a new nutritional source, then the control coefficients in the pathway will change.

Formal Definition

One criticism of the concept of the control coefficient as defined above is that it is dependent on being described relative to a change in enzyme activity. Instead, the Berlin school^[2] defined control coefficients in terms of changes to local rates brought about by any suitable parameter, which could include changes to enzyme levels or the action of drugs. Hence a more general definition is given by the following expressions:

${\displaystyle C_{v_{i}}^{J}=\left({\frac {dJ}{dp}}{\frac {p}{J}}\right){\bigg /}\left({\frac {\partial v_{i}}{\partial p}}{\frac {p}{v_{i}}}\right)={\frac {d\ln J}{d\ln v_{i}}}}$

and concentration control coefficients by

${\displaystyle C_{v_{i}}^{s}=\left({\frac {ds}{dp}}{\frac {p}{s}}\right){\bigg /}\left({\frac {\partial v_{i}}{\partial p}}{\frac {p}{v_{i}}}\right)={\frac {d\ln s}{d\ln v_{i}}}}$

In the above expression, ${\displaystyle p}$ could be any convenient parameter. For example, a drug, changes in enzyme expression etc. The advantage is that the control coefficient becomes independent of the applied perturbation. For control coefficients defined in terms of changes in enzyme expression, it is often assumed that the effect on the local rate by changes to the enzyme activity is proportional so that:

${\displaystyle C_{v_{i}}^{X}=C_{e_{i}}^{X}}$

See also

• Elasticity coefficient
• Metabolic control analysis
• Summation theorems (biochemistry)

References