Latest revision as of 21:12, 21 October 2021

Team:XJTU-China/model-population-dynamics

Modelling

The model of synthesis of tryptophan

Background

Before establishing the final model of production, we need to study the synthesis of tryptophan. The reaction are shown in the figure below.

Here, ① and ② represent complex multistep reaction.

We use the Michaelis-Menten equation to establish our model. Michaelis Menten equation is a model that is designed to generally explain the velocity and the gross mechanism of the reaction that is carried out by enzyme catalysts. Michaelis Menten hypothesis is one of the best known models in biochemistry to determine the catalyst kinetics of a reaction.

Theory

$[Glc]$ $[PEP]$ $[DAHP]$ $[3IGP]$ $[Trp]$ $[Pyr]$ be the concentration of Glc, PEP, DAHP, 3IGP, Trp and Pyr respectively. For reaction ①, with Michaelis-Menten equation, we have

\begin{matrix} (4.1) & {\begin{cases} \frac{d [G l c]}{d t} = - \frac{v_{m a x, 1} \cdot [G l c]}{[G l c] + k_{m, 1}}, \\ \frac{d [P E P]}{d t} = \frac{v_{m a x, 1} \cdot [G l c]}{[G l c] + k_{m, 1}} - \frac{v_{m a x, a r o G} \cdot [P E P]}{[P E P] + k_{m, a r o G}} - \frac{v_{m a x, p y k A} \cdot [P E P]}{[P E P] + k_{m, p y k A}}, \end{cases} \end{matrix}

$v_{max,x}$ $k_{m,x}$ $x$ respectively.

For the reation catalyzed by aroG and pykA, we have

\begin{matrix} (4.2) & {\begin{cases} \frac{d [D A H P]}{d t} = \frac{v_{m a x, a r o G} \cdot [P E P]}{[P E P] + k_{m, a r o G}} - \frac{v_{m a x, 2} \cdot [D A H P]}{[D A H P] + k_{m, 2}}, \\ \frac{d [P y r]}{d t} = \frac{v_{m a x, p y k A} \cdot [P E P]}{[P E P] + k_{m, p y k A}}, \end{cases} \end{matrix}

where

\begin{matrix} (4.3) & {\begin{cases} v_{m a x, a r o G} = k_{c a t, a r o G} \cdot [a r o G], \\ v_{m a x, p y k A} = k_{c a t, p y k A} \cdot [p y k A], \end{cases} \end{matrix}

$k_{cat,x}$ $x$ $(4.3)$ $[aroG]$ $[pykA]$ , the synthetic rate of Pyr increases.

For reaction ②, we have

\begin{matrix} (4.4) & \frac{d [3 I G P]}{d t} = \frac{v_{m a x, 2} \cdot [D A H P]}{[D A H P] + k_{m, 2}} - \frac{v_{m a x, t r p A B} \cdot [3 I G P]}{[3 I G P] + k_{m, t r p A B}} . \end{matrix}

For the reation catalyzed by trpA/B, we have

\begin{matrix} (4.5) & \frac{d [T r p]}{d t} = \frac{v_{m a x, t r p A B} \cdot [3 I G P]}{[3 I G P] + k_{m, t r p A B}}, \end{matrix}

where

\begin{matrix} (4.6) & v_{m a x, t r p A B} = k_{c a t, t r p A B} \cdot [t r p A B] . \end{matrix}

Result

$[aroG]=1$ $[pykA]=0.15$ $[trpAB]=2.5$ . The result is shown in the figure below.

Conclusion

The figure shows the concentration change during the multistep reactions.

When reaction starts, Glc begin to convert to PEP, and PEP immediately turns into Pyr and DAHP;
$200\mathrm{min}$ , and after that it goes down;
$600\mathrm{min}$ , and after that it goes down as well;
The final products of reactions are Pyr and Trp, whose concentrations are stable after reactions.

Reference

https://www.vedantu.com/chemistry/michaelis-menten-kinetics

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