text: The operational proof for the flux connectivity theorem relies on making perturbations to enzyme levels such that the pathway flux is unchanged but a single metabolite level is changed. This can be illustrated with the following pathway:
  Let us make a change to the rate through  by increasing the concentration of enzyme . Assume  is increased by an amount, . This will result in a change to the steady-state of the pathway. The concentrations of , and the flux,  through the pathway will increase, and the concentration of  will decrease because it is upstream of the disturbance.
  Impose a second change to the pathway such that the flux,  is restored to what it was before the original change. Since the flux increased when  was changed, the flux can be decreased by decreasing one of the other enzyme levels. If the concentration of  is decreased, this will reduce the flux. Decreasing  will also cause the concentration of  to further increase. However,  and  will change in the opposite direction compared to when  was increased.
  When  is sufficiently changed so that the flux is restored to its original value, the concentrations of  and  will also be restored to their original values. It is only  that will differ. This is true because the flux through  is now the same as it was originally (since we’ve restored the flux), and  has not been manipulated in anyway. This means that the concentration of  and all
  species upstream of  must be the same as they were before the modulations occurred. The same arguments apply to  and all species downstream of .
  The net result is that  has been increased by  resuling a change in flux of . The concentration of  was decreased such that the flux was restored to it original value, . In the process,   changed by  but neither  or . In fact no other species in the entire system has changed other than .
  This thought experiment can be expressed mathematically as follows. The system equations in terms of the flux control coefficients can be written as:
  There are only two terms because only  and  were changed.
  The local change at each step can be written for  and  in terms of elasticities:
  Note that  won't necessarily equal  and by construction both rates,   and  showed no change. Also by construction only  changed.
  The local equation can be rearranged as:
  The right-hand sides can be inserted into the system equation the change in flux:
  Therefore:
  However, by construction of the perturbations,  does not equal zero, hence we arrive at the connectivity theorem:
  The operational method can also be used for systems where a given metabolite can influence multiple steps. This would apply to cases such as branched systems or systems with negative feedback loops.
  The same approach can be used to derive the concentration connectivity theorems except one can consider either the case that focuses on a single species or a second case where the system equation is written to consider the effect on a distance species.