What is a control coefficient?
First we should indicate what we mean by control. The term control has a special meaning in biochemical control analysis. Control refers to the ability of a system parameter to affect a system variable. For example, changes to the external glucose concentration in a microbial culture will most likely change the culture’s growth rate. The concentration of glucose therefore has ‘control’ over the growth rate. Engineering an enzyme in pathway so that its kcat is larger will result in changes to the pathways flux and metabolite concentrations. Changes to the promoter consensus sequence of a particular gene will result in changes to the concentration of the expressed protein and any other variables that depends on that protein. It is possible to quantify this concept of control by either measuring or computing control coefficients.
Control coefficients come in two flavors, flux control and concentration control. First consider flux control:
A flux control coefficient measures the relative steady state change in pathway flux (J) in response to a relative change in enzyme activity, 
, often through changes in enzyme concentration. This definition assumes that the enzyme concentration is under the direct control of the experimenter and as such can be classed as a parameter. This assumes that a change on the level of the enzyme does not change the level of enzymes. This assumption will not always necessarily be true, in which case the control coefficient can be generalized to be independent of any particular parameter. For now however, with loss of generality, that we can change the enzyme concentration without affecting other enzymes. We define the flux control coefficients as follows:
![\[C^J_{e_i} = \frac{dJ}{de_i} \frac{e_i}{J} \right) = \frac{d\ln J}{d\ln e_i}\]](https://web.archive.org/web/20110713122902im_/http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-ee9218b57860adc4739cea3643b416e7_l3.png "Rendered by QuickLaTeX.com")
The more generalized definition in terms of changes to the local reaction rate 
of step 
is given by:
![\[C^J_{v_i} = \left( \frac{dJ}{dp} \frac{p}{J} \right) \bigg/ \left( \frac{dv_i}{dp}\frac{p}{v_i} \right) = \frac{d\ln J}{d\ln v_i}\]](https://web.archive.org/web/20110713122902im_/http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-35aa60f381cf80334428c8df7746ad9e_l3.png "Rendered by QuickLaTeX.com")
so that the definition is now independent of the particular parameter, 
, used to perturb the reaction step. A very important property of all control coefficients is that they can only be measured in the intact system. It is not possible to isolate an enzyme and try to measure its control coefficient. The effect that a particular parameter, for example , has on a flux (or concentration) is a system property and depends on all the enzymes in the pathway. This is why it is not possible, or at least very difficult, to judge the importance of an enzyme by just looking at the enzyme alone.
A concentration control coefficient measures the relative steady state change in a species concentration (S) in response to a relative change in enzyme activity, , often through changes in enzyme concentration. This definition assumes that the enzyme concentration is under the direct control of the experimenter and as such can be classed as a parameter. It is important to note that concentration control coefficients are properties of the intact system and cannot be measured from the isolated reaction step or enzyme. The same comments that were made with respect to the flux control coefficient applies here.
![\[C^S_{e_i} = \frac{dS}{de_i} \frac{e_i}{S} \right) = \frac{d\ln S}{d\ln e_i}\]](https://web.archive.org/web/20110713122902im_/http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-33c41be106711b94f49345310d622ba7_l3.png "Rendered by QuickLaTeX.com")
Or in generalized form:
![\[C^S_{v_i} = \frac{dS}{dv_i} \frac{v_i}{S} \right) = \frac{d\ln S}{d\ln v_i}\]](https://web.archive.org/web/20110713122902im_/http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-a5b83803b73f5028ada49f360eca8a8c_l3.png "Rendered by QuickLaTeX.com")
What do the control coefficients actually mean, and why are they defined the way they are? The first thing to note about the control coefficient is that they are dimensionless. This is because we scale the derivative to eliminate units. The second important point is that as a result of the scaling, a control coefficients is a ratio of relative changes. This means that, roughly speaking, a control coefficient measures the effect a certain percentage change in enzyme concentration has on the percentage change in flux or metabolite concentration. For example a flux control coefficient of 0.4 means, that a 1% increase in the enzyme concentration, results in as 0.4% change in the steady state flux.
![\[C^J_{e_i} = \left. \frac{\delta J}{J} \right/ \frac{\delta e_i}{e_i} \approx \frac{J\%}{ e_i\% }\]](https://web.archive.org/web/20110713122902im_/http://blog.analogmachine.org/wp-content/ql-cache/quicklatex.com-c2599bb01d6a9c22a5f007b869a60f1a_l3.png "Rendered by QuickLaTeX.com")
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