Connectivity theorems
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.
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.
[1] This is an important observation that highlights the orthogonal nature of these two sets of theorem.
[2] As with the summation theorems, the connectivity theorems can also be proved using more rigorous mathematical approaches involving calculus and linear algebra.
[3][4][5] Here the more intuitive and operational proofs will be used to prove the connectivity theorems.
Two basic sets of theorems exists, one for flux and another for concentrations.
The concentration connectivity theorems are divided again depending on whether the system species
The operational proof for the flux connectivity theorem[1] relies on making perturbations to enzyme levels such that the pathway flux is unchanged but a single metabolite level is changed.
will change in the opposite direction compared to when
is sufficiently changed so that the flux is restored to its original value, the concentrations of
was decreased such that the flux was restored to it original value,
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:
won't necessarily equal
The right-hand sides can be inserted into the system equation the change in flux:
The operational method can also be used for systems where a given metabolite can influence multiple steps.
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.
The flux control coefficient connectivity theorem is the easiest to understand.
Starting with a simple two step pathway:
are fixed species so that the pathway can reach a steady-state.
We can write the flux connectivity theorem for this simple system as follows:
It is easier to interpret the equation with a slight rearrangement to the following form:
The equation indicates that the ratio of the flux control coefficients is inversely proportional to the elasticities.
That is, a high flux control coefficient on step one is associated with a low elasticity
Likewise a high value for the flux control coefficient on step two is associated with a low elasticity
is high (in absolute terms, since it is negative) then a change at
will be resisted by the elasticity, hence the flux control coefficient on step one will be low.
