Branched pathways

A simple branched pathway has one key property related to the conservation of mass. In general, the rate of change of the branch species based on the above figure is given by:

${\displaystyle {\frac {ds_{1}}{dt}}=v_{1}-(v_{2}+v_{3})}$ 

At steady-state the rate of change of ${\displaystyle S_{1}}$  is zero. This gives rise to a steady-state constraint among the branch reaction rates:

${\displaystyle v_{1}=v_{2}+v_{3}}$ 

Such constraints are key to computational methods such as flux balance analysis.

Branched pathways have unique control properties compared to simple linear chain or cyclic pathways. These properties can be investigated using metabolic control analysis. The fluxes can be controlled by enzyme concentrations ${\displaystyle e_{1}}$ , ${\displaystyle e_{2}}$ , and ${\displaystyle e_{3}}$  respectively, described by the corresponding flux control coefficients. To do this the flux control coefficients with respect to one of the branch fluxes can be derived. The derivation is shown in a subsequent section. The flux control coefficient with respect to the upper branch flux, ${\displaystyle J_{2}}$  are given by:

${\displaystyle C_{e_{1}}^{J_{2}}={\frac {\varepsilon _{2}}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{2}}^{J_{2}}={\frac {\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{3}}^{J_{2}}={\frac {-\varepsilon _{2}(1-\alpha )}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

where ${\displaystyle \alpha }$  is the fraction of flux going through the upper arm, ${\displaystyle J_{2}}$ , and ${\displaystyle 1-\alpha }$  the fraction going through the lower arm, ${\displaystyle J_{3}}$ . ${\displaystyle \varepsilon _{1},\varepsilon _{2},}$  and ${\displaystyle \varepsilon _{3}}$  are the elasticities for ${\displaystyle s_{1}}$  with respect to ${\displaystyle v_{1},v_{2},}$  and ${\displaystyle v_{3}}$  respectively.

For the following analysis, the flux ${\displaystyle J_{2}}$  will be the observed variable in response to changes in enzyme concentrations.

There are two possible extremes to consider, either most of the flux goes through the upper branch ${\displaystyle J_{2}}$  or most of the flux goes through the lower branch, ${\displaystyle J_{3}}$ . The former, depicted in panel a), is the least interesting as it converts the branch in to a simple linear pathway. Of more interest is when most of the flux goes through ${\displaystyle J_{3}}$ 

If most of the flux goes through ${\displaystyle J_{3}}$ , then ${\displaystyle \alpha \rightarrow 0}$  and ${\displaystyle 1-\alpha \rightarrow 1}$  (condition (b) in the figure), the flux control coefficients for ${\displaystyle J_{2}}$  with respect to ${\displaystyle e_{2}}$  and ${\displaystyle e_{3}}$  can be written:

${\displaystyle C_{e_{2}}^{J_{2}}\rightarrow 1}$ 

${\displaystyle C_{e_{3}}^{J_{2}}\rightarrow {\frac {\varepsilon _{2}}{\varepsilon _{1}-\varepsilon _{3}}}}$ 

That is, ${\displaystyle e_{2}}$  acquires proportional influence over its own flux, ${\displaystyle J_{2}}$ . Since ${\displaystyle J_{2}}$  only carries a very small amount of flux, any changes in ${\displaystyle e_{2}}$  will have little effect on ${\displaystyle S}$ . Hence the flux through ${\displaystyle e_{2}}$  is almost entirely governed by the activity of ${\displaystyle e_{2}}$ . Because of the flux summation theorem and the fact that ${\displaystyle C_{e_{2}}^{J_{2}}=1}$ , it means that the remaining two coefficients must be equal and opposite in value. Since ${\displaystyle C_{e_{1}}^{J_{2}}}$  is positive, ${\displaystyle C_{e_{3}}^{J_{2}}}$  must be negative. This also means that in this situation, there can be more than one Rate-limiting step (biochemistry) in a pathway.

Unlike a linear pathway, values for ${\displaystyle C_{e_{3}}^{J_{2}}}$ and ${\displaystyle C_{e_{1}}^{J_{2}}}$  are not bounded between zero and one. Depending on the values of the elasticities, it is possible for the control coefficients in a branched system to greatly exceed one.^[3] This has been termed the branchpoint effect by some in the literature.^[4]

The following branch pathway model (in antimony format) illustrates the case ${\displaystyle J_{1}}$  and ${\displaystyle J_{3}}$  have very high flux control and step J2 has proportional control.

   J1: $Xo -> S1; e1*k1*Xo
   J2:  S1 ->; e2*k3*S1/(Km1 + S1)
   J3:  S1 ->; e3*k4*S1/(Km2 + S1)
     
   k1 = 2.5;
   k3 = 5.9; k4 = 20.75
   Km1 = 4; Km2 = 0.02
   Xo =5; 
   e1 = 1; e2 = 1; e3 = 1

A simulation of this model yields the following values for the flux control coefficients with respect to flux ${\displaystyle J_{2}}$ 

In a linear pathway, only two sets of theorems exist, the summation and connectivity theorems. Branched pathways have an additional set of branch centric summation theorems. When combined with the connectivity theorems and the summation theorem, it is possible to derive the control equations shown in the previous section. The deviation of the branch point theorems is as follows.

1. Define the fractional flux through ${\displaystyle J_{2}}$  and ${\displaystyle J_{3}}$  as ${\displaystyle \alpha =J_{2}/J_{1}}$  and ${\displaystyle 1-\alpha =J_{3}/J_{1}}$  respectively.
2. Increase ${\displaystyle e_{2}}$  by ${\displaystyle \delta e_{2}}$ . This will decrease ${\displaystyle S_{1}}$  and increase ${\displaystyle J_{1}}$  through relief of product inhibition.^[5]
3. Make a compensatory change in ${\displaystyle J_{1}}$  by decreasing ${\displaystyle e_{1}}$  such that ${\displaystyle S_{1}}$  is restored to its original concentration (hence ${\displaystyle \delta S_{1}=0}$ ).
4. Since ${\displaystyle e_{1}}$  and ${\displaystyle S_{1}}$  have not changed, ${\displaystyle \delta J_{1}=0}$ .

Following these assumptions two sets of equations can be derived: the flux branch point theorems and the concentration branch point theorems.^[6]

From these assumptions, the following system equation can be produced:

${\displaystyle C_{e_{2}}^{J_{1}}{\frac {\delta e_{2}}{e_{2}}}+C_{e_{3}}^{J_{1}}{\frac {\delta e_{3}}{e_{3}}}={\frac {\delta J_{1}}{J_{1}}}=0}$ 

Because ${\displaystyle \delta S_{1}=0}$  and, assuming that the flux rates are directly related to the enzyme concentration thus, the elasticities, ${\displaystyle \varepsilon _{e_{i}}^{v}}$ , equal one, the local equations are:

${\displaystyle {\frac {\delta v_{2}}{v_{2}}}={\frac {\delta e_{2}}{e_{2}}}}$ 

${\displaystyle {\frac {\delta v_{3}}{v_{3}}}={\frac {\delta e_{3}}{e_{3}}}}$ 

Substituting ${\displaystyle {\frac {\delta v_{i}}{v_{i}}}}$  for ${\displaystyle {\frac {\delta e_{i}}{e_{i}}}}$  in the system equation results in:

${\displaystyle C_{e_{2}}^{J_{1}}{\frac {\delta v_{2}}{v_{2}}}+C_{e_{3}}^{J_{1}}{\frac {\delta v_{3}}{v_{3}}}=0}$ 

Conservation of mass dictates ${\displaystyle \delta J_{1}=\delta J_{2}+\delta J_{3}}$  since ${\displaystyle \delta J_{1}=0}$  then ${\displaystyle \delta v_{2}=-\delta v_{3}}$ . Substitution eliminates the ${\displaystyle \delta v_{3}}$  term from the system equation:

${\displaystyle C_{e_{2}}^{J_{1}}{\frac {\delta v_{2}}{v_{2}}}-C_{e_{3}}^{J_{1}}{\frac {\delta v_{2}}{v_{3}}}=0}$ 

Dividing out ${\displaystyle {\frac {\delta v_{2}}{v_{2}}}}$  results in:

${\displaystyle C_{e_{2}}^{J_{1}}-C_{e_{3}}^{J_{1}}{\frac {v_{2}}{v_{3}}}=0}$ 

${\displaystyle v_{2}}$  and ${\displaystyle v_{3}}$  can be substituted by the fractional rates giving:

${\displaystyle C_{e_{2}}^{J_{1}}-C_{e_{3}}^{J_{1}}{\frac {\alpha }{1-\alpha }}=0}$ 

Rearrangement yields the final form of the first flux branch point theorem:^[6]

${\displaystyle C_{e_{2}}^{J_{1}}(1-\alpha )-C_{e_{3}}^{J_{1}}{\alpha }=0}$ 

Similar derivations result in two more flux branch point theorems and the three concentration branch point theorems.

${\displaystyle C_{e_{2}}^{J_{1}}(1-\alpha )-C_{e_{3}}^{J_{1}}(\alpha )=0}$ 

${\displaystyle C_{e_{1}}^{J_{2}}(1-\alpha )+C_{e_{3}}^{J_{2}}(\alpha )=0}$ 

${\displaystyle C_{e_{1}}^{J_{3}}(\alpha )+C_{e_{2}}^{J_{3}}=0}$ 

${\displaystyle C_{e_{2}}^{S_{1}}(1-\alpha )+C_{e_{3}}^{S_{1}}(\alpha )=0}$ 

${\displaystyle C_{e_{1}}^{S_{1}}(1-\alpha )+C_{e_{3}}^{S_{1}}=0}$ 

${\displaystyle C_{e_{1}}^{S_{1}}(\alpha )+C_{e_{2}}^{S_{1}}=0}$ 

Following the flux summation theorem^[7] and the connectivity theorem^[8] the following system of equations can be produced for the simple pathway.^[9]

${\displaystyle C_{e_{1}}^{J_{1}}+C_{e_{2}}^{J_{1}}+C_{e_{3}}^{J_{1}}=1}$ 

${\displaystyle C_{e_{1}}^{J_{2}}+C_{e_{2}}^{J_{2}}+C_{e_{3}}^{J_{2}}=1}$ 

${\displaystyle C_{e_{1}}^{J_{3}}+C_{e_{2}}^{J_{3}}+C_{e_{3}}^{J_{3}}=1}$ 

${\displaystyle C_{e_{1}}^{J_{1}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{1}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{1}}\varepsilon _{s}^{v_{3}}=0}$ 

${\displaystyle C_{e_{1}}^{J_{2}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{2}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{2}}\varepsilon _{s}^{v_{3}}=0}$ 

${\displaystyle C_{e_{1}}^{J_{3}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{3}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{3}}\varepsilon _{s}^{v_{3}}=0}$ 

Using these theorems plus flux summation and connectivity theorems values for the concentration and flux control coefficients can be determined using linear algebra.^[6]

${\displaystyle C_{e_{1}}^{J_{2}}={\frac {\varepsilon _{2}}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{2}}^{J_{2}}={\frac {\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{3}}^{J_{2}}={\frac {-\varepsilon _{2}(1-\alpha )}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{1}}^{S_{1}}={\frac {1}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{1}}^{S_{1}}={\frac {-\alpha }{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

${\displaystyle C_{e_{3}}^{S_{1}}={\frac {-(1-\alpha )}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}}$ 

• Control coefficient (biochemistry)
• Elasticity coefficient
• Metabolic control analysis