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^[1] 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.^[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.

Statement of 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 ${\displaystyle S_{n}}$ is different from the local species ${\displaystyle S_{m}}$.

${\displaystyle \sum _{i}C_{i}^{J}\varepsilon _{s}^{i}=0}$

${\displaystyle \sum _{i}C_{i}^{s_{n}}\varepsilon _{s_{m}}^{i}=0\quad n\neq m}$

${\displaystyle \sum _{i}C_{i}^{s_{n}}\varepsilon _{s_{m}}^{i}=-1\quad n=m}$

Proof

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. This can be illustrated with the following pathway:

${\displaystyle {\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}S_{2}{\stackrel {v_{3}}{\longrightarrow }}S_{3}{\stackrel {v_{4}}{\longrightarrow }}}$

Let us make a change to the rate through ${\displaystyle v_{2}}$ by increasing the concentration of enzyme ${\displaystyle e_{2}}$. Assume ${\displaystyle e_{2}}$ is increased by an amount, ${\displaystyle \delta e_{2}}$. This will result in a change to the steady-state of the pathway. The concentrations of ${\displaystyle s_{2},s_{3}}$, and the flux, ${\displaystyle J}$ through the pathway will increase, and the concentration of ${\displaystyle s_{1}}$ will decrease because it is upstream of the disturbance.

Impose a second change to the pathway such that the flux, ${\displaystyle J}$ is restored to what it was before the original change. Since the flux increased when ${\displaystyle e_{2}}$ was changed, the flux can be decreased by decreasing one of the other enzyme levels. If the concentration of ${\displaystyle e_{3}}$ is decreased, this will reduce the flux. Decreasing ${\displaystyle e_{3}}$ will also cause the concentration of ${\displaystyle s_{2}}$ to further increase. However, ${\displaystyle s_{1}}$ and ${\displaystyle s_{3}}$ will change in the opposite direction compared to when ${\displaystyle e_{2}}$ was increased.

When ${\displaystyle e_{3}}$ is sufficiently changed so that the flux is restored to its original value, the concentrations of ${\displaystyle s_{1}}$ and ${\displaystyle s_{3}}$ will also be restored to their original values. It is only ${\displaystyle s_{2}}$ that will differ. This is true because the flux through ${\displaystyle v_{1}}$ is now the same as it was originally (since we’ve restored the flux), and ${\displaystyle e_{1}}$ has not been manipulated in anyway. This means that the concentration of ${\displaystyle s_{1}}$ and all species upstream of ${\displaystyle s_{1}}$ must be the same as they were before the modulations occurred. The same arguments apply to ${\displaystyle s_{3}}$ and all species downstream of ${\displaystyle v_{4}}$.

The net result is that ${\displaystyle e_{2}}$ has been increased by ${\displaystyle \delta e_{2}}$ resuling a change in flux of ${\displaystyle \delta J}$. The concentration of ${\displaystyle e_{3}}$ was decreased such that the flux was restored to it original value, ${\displaystyle \delta J=0}$. In the process, ${\displaystyle s_{2}}$ changed by ${\displaystyle \delta s_{2}}$ but neither ${\displaystyle s_{1}}$ or ${\displaystyle s_{3}}$. In fact no other species in the entire system has changed other than ${\displaystyle s_{2}}$.

Perturbations involved in the connectivity theorem. a) First perturbation (orange) at step 2; b) Second counter perturbation at step 3, resulting in no change in flux, J but a change to species S2. No other species changes

This thought experiment can be expressed mathematically as follows. The system equations in terms of the flux control coefficients can be written as:

${\displaystyle {\frac {\delta J}{J}}=0=C_{2}^{J}{\frac {\delta e_{2}}{e_{2}}}+C_{3}^{J}{\frac {\delta e_{3}}{e_{3}}}}$

There are only two terms because only ${\displaystyle e_{2}}$ and ${\displaystyle e_{3}}$ were changed.

The local change at each step can be written for ${\displaystyle v_{2}}$ and ${\displaystyle v_{2}}$ in terms of elasticities:

${\displaystyle 0={\frac {\delta v_{2}}{v_{2}}}={\frac {\delta e_{2}}{e_{2}}}+\varepsilon _{2}^{2}{\frac {\delta s_{2}}{s_{2}}}}$

${\displaystyle 0={\frac {\delta v_{3}}{v_{3}}}={\frac {\delta e_{3}}{e_{3}}}+\varepsilon _{2}^{3}{\frac {\delta s_{2}}{s_{2}}}}$

Note that ${\displaystyle \delta e_{2}/e_{2}}$ won't necessarily equal ${\displaystyle \delta e_{2}/e_{3}}$ and by construction both rates, ${\displaystyle v_{2}}$ and ${\displaystyle v_{3}}$ showed no change. Also by construction only ${\displaystyle s_{2}}$ changed.

The local equation can be rearranged as:

${\displaystyle {\frac {\delta e_{2}}{e_{2}}}=-\varepsilon _{2}^{2}{\frac {\delta s_{2}}{s_{2}}}}$

${\displaystyle {\frac {\delta e_{3}}{e_{3}}}=-\varepsilon _{2}^{3}{\frac {\delta s_{2}}{s_{2}}}}$

The right-hand sides can be inserted into the system equation the change in flux:

${\displaystyle 0={\frac {\delta J}{J}}=-\left(C_{e_{2}}^{J}\varepsilon _{2}^{2}{\frac {\delta s_{2}}{s_{2}}}+C_{e_{3}}^{J}\varepsilon _{2}^{3}{\frac {\delta s_{2}}{s_{2}}}\right)}$

Therefore:

${\displaystyle 0={\frac {\delta s_{2}}{s_{2}}}\left(C_{e_{2}}^{J}\varepsilon _{2}^{2}+C_{e_{3}}^{J}\varepsilon _{2}^{3}\right)}$

However, by construction of the perturbations, ${\displaystyle \delta s_{2}/s_{2}}$ does not equal zero, hence we arrive at the connectivity theorem:

${\displaystyle 0=C_{e_{2}}^{J}\varepsilon _{2}^{2}+C_{e_{3}}^{J}\varepsilon _{2}^{3}}$

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.

Interpretation

The flux control coefficient connectivity theorem is the easiest to understand. Starting with a simple two step pathway:

${\displaystyle X_{o}{\stackrel {v_{1}}{\longrightarrow }}S_{1}{\stackrel {v_{2}}{\longrightarrow }}X_{1}}$

where ${\displaystyle X_{o}}$ and ${\displaystyle X_{1}}$ are fixed species so that the pathway can reach a steady-state. ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$ are the reaction rates for the first and second steps.

We can write the flux connectivity theorem for this simple system as follows:

${\displaystyle C_{1}^{J}\varepsilon _{1}^{1}+C_{2}^{J}\varepsilon _{1}^{2}=0}$

where ${\displaystyle \varepsilon _{1}^{1}}$ is the elasticity of the first step ${\displaystyle v_{1}}$ with respect to the species ${\displaystyle S_{1}}$ and ${\displaystyle \varepsilon _{1}^{2}}$ is the elasticity of the second step ${\displaystyle v_{2}}$ with respect to the species ${\displaystyle S_{1}}$. It is easier to interpret the equation with a slight rearrangement to the following form:

${\displaystyle {\frac {C_{1}^{J}}{C_{2}^{J}}}=-{\frac {\varepsilon _{1}^{2}}{\varepsilon _{1}^{1}}}}$

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 ${\displaystyle \varepsilon _{1}^{1}}$ and vice versa. Likewise a high value for the flux control coefficient on step two is associated with a low elasticity ${\displaystyle \varepsilon _{1}^{2}}$.

This can be explained as follows: If ${\displaystyle \varepsilon _{1}^{1}}$ is high (in absolute terms, since it is negative) then a change at ${\displaystyle v_{1}}$ will be resisted by the elasticity, hence the flux control coefficient on step one will be low.

See also

• Branched pathways
• Control coefficient (biochemistry)
• Elasticity coefficient
• Metabolic control analysis
• Summation theorems (biochemistry)

References