The stoichiometric structure and mass-conservation properties of biochemical pathways gives rise to a series of theorems or relationships between the control coefficients and the control coefficients and elasticities. There are a large number of such relationships depending on the pathway configuration (e.g. linear, branched or cyclic) which have been documented and discovered by various authors. The term theorem has been used to describe these relationships because they can be proved in terms of more elementary concepts. The operational proofs in particular are of this nature.
The most well known of these theorems are the summation theorems for the control coefficients and the connectivity theorems which relate control coefficients to the elasticities.  The focus of this page are the connectivity theorems.
When deriving the summation theorems, a thought experiment was conducted that involved manipulating enzyme activities such that concentrations were unaffected but fluxes changed. The connectivity theorems use the opposite thought experiment, that is enzyme activities are changed such that concentrations change but fluxes are unchanged. This is an important observation that highlights the orthogonal nature of these two sets of theorem.
As with the summation theorems, the connectivity theorems can also be proved using more rigorous mathematical approaches involving calculus and linear algebra. Here the more intuitive and operational proofs will be used to prove the connectivity theorems.