```
import tellurium as te
import roadrunner
import matplotlib.pyplot as plt
r = te.loada("""
J1: $Xo -> S; e1*(k1*Xo - k2*S)
J2: S ->; e2*k3*S
Xo = 1
k1 = 0.5; k2 = 0.2
k3 = 0.45
e1 = 0.01; e2 = 0.99
""")
r.steadyState()
x = []; y = [];
for i in range (49):
r.e1 = r.e1 + 0.02
r.e2 = 1 - r.e1 # Total assumed to be one
r.simulate()
r.steadyState()
x.append (r.e1)
y.append (r.J1)
plt.figure(figsize=(12,10))
plt.grid(b=True, which='major', color='#666666', linestyle='-')
plt.grid(b=True, which='minor', color='#999999', linestyle='-', alpha=0.2)
plt.minorticks_on()
plt.plot (x, y, linewidth = 2)
plt.xlabel('$e_1$', fontsize=36)
plt.ylabel('Flux', fontsize=36)
```

The following graph show the results from the simulation (the font sizes might be on the big size if you have a low res monitor):
You can see the flux reaching a maximum at around $e_1 = 0.6$, meaning also that $e_2 = 0.4$ in order to keep the total fixed. The actual position of the peak wil depend on the rate constants in the rate expressions.
Let's assume we are at the maxium. The slope, $dJ/de_1$, at the maxium is obviously zero. That means if we were to move a small amount of protein from $e_1$ to $e_2$ the flux won't change. We can write this experiment in terms of the two flux control coefficients:
$$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{\delta e_1}{e_1} + C^J_{e_2} \frac{\delta e_2}{e_2} $$
However, we know that in this particlar experiment the change in $e_1$ is the same but oppisite to the change in $e_2$. That is $\delta e_1 + \delta e_2 = 0$ or
$$ \delta e_1 = -\delta e_2$$
Replacing $e_2$ with $-\delta e_1$ gives:
$$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{\delta e_1}{e_1} - C^J_{e_2} \frac{\delta e_1}{e_2} $$
$$ \frac{\delta J}{J} = 0 = C^J_{e_1} \frac{1}{e_1} - C^J_{e_2} \frac{1}{e_2} $$
$$ C^J_{e_1} \frac{1}{e_1} = C^J_{e_2} \frac{1}{e_2} $$
Giving the final result:
$$ \frac{C^J_{e_1}}{C^J_{e_2}} = \frac{e_1}{e_2} $$
That is, when the protein distribution is optimized to maximize the flux, the flux control coefficients are in the same ratio as the ratio of enzyme amounts. This generalizes to any size pathway with multiple enzymes.
This gives us a tantgilizing suggestion that we can obtain the flux control coeficients just by measuring the protein levels.
There is obviously a lot more one can write there and maybe I do that in future blogs but for now you can get further information:

S. Waley, “A note on the kinetics of multi-enzyme systems,” Biochemical Journal, vol. 91, no. 3, p. 514, 1964.

J Burns: “Studies on complex enzyme system.” https://era.ed.ac.uk/handle/1842/13276, 1971 (page 141-) I have a LaTeX version at: https://github.com/hsauro/JumBurnsThesis

Guy Brown, Total cell protein concentration as an evolutionary constraint on the metabolic control distribution in cells,” Journal of theoretical biology, vol. 153, no. 2, pp. 195–203, 1991.

E. Klipp and R. Heinrich, “Competition for enzymes in metabolic pathways:: Implications for optimal distributions of enzyme concentrations and for the distribution of flux control,” Biosystems, vol. 54, no. 1-2, pp. 1–14, 1999

Sauro HM Systems Biology: An Introduction to Metabolic Control Analysis, 2018

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