text: Control equations can also be derived by considering the effect of perturbations on the system. Consider that reaction rates  and  are determined by two enzymes  and  respectively. Changing either enzyme will result in a change to the steady state level of  and the steady state reaction rates . Consider a small change in  of magnitude . This will have a number of effects, it will increase  which in turn will increase  which in turn will increase . Eventually the system will settle to a new steady state. We can describe these changes by focusing on the change in  and . The change in , which we designate , came about as a result of the change . Because we are only considering small changes we can express the change  in terms of  using the relation
  where the derivative  measures how responsive  is to changes in . The derivative can be computed if we know the rate law for . For example, if we assume that the rate law is  then the derivative is . We can also use a similar strategy to compute the change in  as a result of the change . This time the change in  is a result of two changes, the change in  itself and the change in . We can express these changes by summing the two individual contributions:
  We have two equations, one describing the change in  and the other in . Because we allowed the system to settle to a new steady state we can also state that the change in reaction rates must be the same (otherwise it wouldn't be at steady state). That is we can assert that . With this in mind we equate the two equations and write
  Solving for the ratio  we obtain:
  In the limit, as we make the change  smaller and smaller, the left-hand side converges to the derivative :
  We can go one step further and scale the derivatives to eliminate units. Multiplying both sides by  and dividing both sides by  yields the scaled derivatives:
  The scaled derivatives on the right-hand side are the elasticities,  and the scaled left-hand term is the scaled sensitivity coefficient or concentration control coefficient, 
  We can simplify this expression further. The reaction rate  is usually a linear function of . For example, in the Briggs–Haldane equation, the reaction rate is given by . Differentiating this rate law with respect to  and scaling yields .
  Using this result gives:
  A similar analysis can be done where  is perturbed. In this case we obtain the sensitivity of  with respect to :
  The above expressions measure how much enzymes  and  control the steady state concentration of intermediate . We can also consider how the steady state reaction rates  and  are affected by perturbations in  and . This is often of importance to metabolic engineers who are interested in increasing rates of production. At steady state the reaction rates are often called the fluxes and abbreviated to  and . For a linear pathway such as this example, both fluxes are equal at steady-state so that the flux through the pathway is simply referred to as . Expressing the change in flux as a result of a perturbation in  and taking the limit as before we obtain
  The above expressions tell us how much enzymes  and  control the steady state flux. The key point here is that changes in enzyme concentration, or equivalently the enzyme activity, must be brought about by an external action.