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Branched Pathways
Branched pathways, also known as branch points (not to be confused with the mathematical branch point), are a common pattern found in metabolism. This is where an intermediate species is chemically made or transformed by multiple enzymatic processes. This is contrasted by linear pathways, which only have one enzymatic reaction producing a species and one enzymatic reaction consuming the species.

Branched pathways are present in numerous metabolic reactions, including glycolysis, and the synthesis of lysine, glutamine, and penicillin. Further examples of branched pathways can be found in the production of aromatic amino acids.

One way to investigate the properties of a branched pathway is to carry out computer simulations or to look at the control coefficients for flux and species concentrations using metabolic control analysis.

Simple Branch Pathway
Shown to the right is a simple branched pathway with one input and two outputs. $$J_1$$, $$J_2$$, and $$J_3$$ are steady state fluxes, and conservation of mass dictates:

$$J_1 = J_2 + J_3$$

The fluxes can be controlled by enzyme concentrations $$e_1$$, $$e_2$$, and $$e_3$$ respectively, defining the flux control coefficients.

Following the flux summation theorem and the connectivity theorem the following system of equations can be produced for the simple pathway.

$$C^{J_1}_{e_1} + C^{J_1}_{e_2} + C^{J_1}_{e_3} = 1$$

$$C^{J_2}_{e_1} + C^{J_2}_{e_2} + C^{J_2}_{e_3} = 1$$

$$C^{J_3}_{e_1} + C^{J_3}_{e_2} + C^{J_3}_{e_3} = 1$$

$$C^{J_1}_{e_1} \varepsilon^{v_1}_s + C^{J_1}_{e_2} \varepsilon^{v_2}_s + C^{J_1}_{e_3} \varepsilon^{v_3}_s = 0$$

$$C^{J_2}_{e_1} \varepsilon^{v_1}_s + C^{J_2}_{e_2} \varepsilon^{v_2}_s + C^{J_2}_{e_3} \varepsilon^{v_3}_s = 0$$

$$C^{J_3}_{e_1} \varepsilon^{v_1}_s + C^{J_3}_{e_2} \varepsilon^{v_2}_s + C^{J_3}_{e_3} \varepsilon^{v_3}_s = 0$$

Equations can be paired off, but with three unknown flux control coefficients the system of equations cannot be solved. To produce the third equation the steady state concentration thought experiment is used.

$$C^{J_1}_{e_1} + C^{J_1}_{e_2} + C^{J_1}_{e_3} = 1$$

$$C^{J_1}_{e_1} \varepsilon^{v_1}_s + C^{J_1}_{e_2} \varepsilon^{v_2}_s + C^{J_1}_{e_3} \varepsilon^{v_3}_s = 0$$

Steady State Concentration Thought Experiment
The steady-state concentration thought experiment follows a series of steps.


 * 1) Define the fractional flux through $$J_2$$ and $$J_3$$ as $$\alpha = J_2/J_1$$ and $$1 - \alpha = J_3/J_1$$ respectively.
 * 2) Increase $$e_2$$ by $$\delta e_2$$. This will decrease $$S_1$$ and increase $$J_1$$ through relief of product inhibition.
 * 3) Make a compensatory change in $$J_1$$ by decreasing $$e_1$$ such that $$S_1$$ is restored to its original concentration ($$\delta S_1 = 0$$).
 * 4) Since $$e_1$$ and $$S_1$$ have not changed, $$\delta J_1 = 0$$.

Following these assumptions two sets of equations are produced. The flux branch point theorems and the concentration branch point theorems.

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

$$C^{J1}_{e_2} \frac{\delta e_2}{e_2} + C^{J1}_{e_3}  \frac{\delta e_3}{e_3} =  \frac{\delta J_1}{J_1} = 0  $$

Because $$\delta S_1 = 0$$ and, assuming that the flux rates are directly related to the enzyme concentration thus, the elasticities, $$\varepsilon^{v}_{e_i}$$, equal one, the local equations are:

$$\frac{\delta v_2}{v_2} = \frac{\delta e_2}{e_2}  $$

$$\frac{\delta v_3}{v_3} = \frac{\delta e_3}{e_3}  $$

Substituting $$\frac{\delta v_i}{v_i} $$ for $$\frac{\delta e_i}{e_i}  $$ in the system equation results in:

$$C^{J1}_{e_2} \frac{\delta v_2}{v_2} + C^{J1}_{e_3}  \frac{\delta v_3}{v_3} = 0  $$

Conservation of mass dictates $$\delta J_1 = \delta J_2 + \delta J_3 $$ since $$\delta J_1 = 0  $$ then $$ \delta v_2 = - \delta v_3  $$. Substitution eliminates the $$ \delta v_3 $$ term from the system equation:

$$C^{J1}_{e_2} \frac{\delta v_2}{v_2} - C^{J1}_{e_3}  \frac{\delta v_2}{v_3} = 0  $$

Dividing out $$\frac{\delta v_2}{v_2}  $$ results in:

$$C^{J1}_{e_2} - C^{J1}_{e_3} \frac{v_2}{v_3} = 0  $$

$$v_2 $$ and $$v_3  $$ can be substituted by the fractional rates giving:

$$C^{J1}_{e_2} - C^{J1}_{e_3} \frac{\alpha}{1-\alpha} = 0  $$

Rearrangement yields the final form of the first flux branch point theorem :

$$C^{J1}_{e_2}(1-\alpha) - C^{J1}_{e_3} {\alpha} = 0 $$

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

Flux Branch Point Theorems
$$C^{J_1}_{e_2} (1-\alpha) - C^{J_1}_{e_3} (\alpha) = 0$$

$$C^{J_2}_{e_1} (1-\alpha) + C^{J_2}_{e_3} (\alpha) = 0$$

$$C^{J_3}_{e_1} (\alpha) + C^{J_3}_{e_2} = 0$$

Concentration Branch Point Theorems
$$C^{S1}_{e_2} (1-\alpha) + C^{S1}_{e_3} (\alpha) = 0$$

$$C^{S1}_{e_1} (1-\alpha) + C^{S1}_{e_3} = 0$$

$$C^{S1}_{e_1} (\alpha) + C^{S1}_{e_2} = 0$$

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

$$C^{J2}_{e_1} = \frac{\varepsilon_2}{\varepsilon_2 \alpha + \varepsilon_3 (1-\alpha) - \varepsilon_1}  $$

$$C^{J2}_{e_2} = \frac{\varepsilon_3 (1-\alpha) - \varepsilon_1}{\varepsilon_2 \alpha + \varepsilon_3 (1-\alpha) - \varepsilon_1}  $$

$$C^{J2}_{e_3} = \frac{- \varepsilon_2 (1-\alpha)}{\varepsilon_2 \alpha + \varepsilon_3 (1-\alpha) - \varepsilon_1}  $$

$$C^{S_1}_{e_1} = \frac{1}{\varepsilon_2 \alpha + \varepsilon_3 (1-\alpha) - \varepsilon_1}  $$

$$C^{S_1}_{e_1} = \frac{- \alpha}{\varepsilon_2 \alpha + \varepsilon_3 (1-\alpha) - \varepsilon_1}  $$

$$C^{S_1}_{e_3} = \frac{-(1-\alpha)}{\varepsilon_2 \alpha + \varepsilon_3 (1-\alpha) - \varepsilon_1}  $$