Team:XJTU-China/model-synthesis-of-tryptophan

Modelling

Synthesis of Tryptophan

•  Background
•  Theory
•  Result
•  Conclusion
•  Reference

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

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

$$\begin{array}{}\text{(4.1)}& \{\begin{array}{l}\frac{\mathrm{d}[Glc]}{\mathrm{d}t}=-\frac{v_{max,1}\cdot [Glc]}{[Glc]+k_{m,1}},\\ \frac{\mathrm{d}[PEP]}{\mathrm{d}t}=\frac{v_{max,1}\cdot [Glc]}{[Glc]+k_{m,1}}-\frac{v_{max,aroG}\cdot [PEP]}{[PEP]+k_{m,aroG}}-\frac{v_{max,pykA}\cdot [PEP]}{[PEP]+k_{m,pykA}},\end{array}\end{array}$$

where $v_{max,x}$ $v_{max,x}$ and $k_{m,x}$ $k_{m,x}$ are the maximum reaction rate and Michaelis constant of reaction $x$ $x$ respectively.

For the reation catalyzed by aroG and pykA, we have

$$\begin{array}{}\text{(4.2)}& \{\begin{array}{l}\frac{\mathrm{d}[DAHP]}{\mathrm{d}t}=\frac{v_{max,aroG}\cdot [PEP]}{[PEP]+k_{m,aroG}}-\frac{v_{max,2}\cdot [DAHP]}{[DAHP]+k_{m,2}},\\ \frac{\mathrm{d}[Pyr]}{\mathrm{d}t}=\frac{v_{max,pykA}\cdot [PEP]}{[PEP]+k_{m,pykA}},\end{array}\end{array}$$

where

$$\begin{array}{}\text{(4.3)}& \{\begin{array}{l}{v}_{max,aroG}=k_{cat,aroG}\cdot [aroG],\\ v_{max,pykA}=k_{cat,pykA}\cdot [pykA],\end{array}\end{array}$$

and $k_{cat,x}$ $k_{cat,x}$ is the catalytic efficiency of $x$ $x$. Equation $(4.3)$ $(4.3)$ shows that with the increase of $[aroG]$ $[aroG]$, the synthetic rate of DAHP increases. And with the increase of $[pykA]$ $[pykA]$, the synthetic rate of Pyr increases.

For reaction ②, we have

$$\begin{array}{}\text{(4.4)}& \frac{\mathrm{d}[3IGP]}{\mathrm{d}t}=\frac{v_{max,2}\cdot [DAHP]}{[DAHP]+k_{m,2}}-\frac{v_{max,trpAB}\cdot [3IGP]}{[3IGP]+k_{m,trpAB}}.\end{array}$$

For the reation catalyzed by trpA/B, we have

$$\begin{array}{}\text{(4.5)}& \frac{\mathrm{d}[Trp]}{\mathrm{d}t}=\frac{v_{max,trpAB}\cdot [3IGP]}{[3IGP]+k_{m,trpAB}},\end{array}$$

where

$$\begin{array}{}\text{(4.6)}& v_{max,trpAB}=k_{cat,trpAB}\cdot [trpAB].\end{array}$$

Result

Set $[aroG]=1$ $[aroG]=1$, $[pykA]=0.15$ $[pykA]=0.15$ and $[trpAB]=2.5$ $[trpAB]=2.5$. 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;
• The concentration of DAHP reaches maximum at about $200\min$ $200\mathrm{min}$, and after that it goes down;
• Next, the concentration of 3IGP reaches maximum at about $600\min$ $600\mathrm{min}$, 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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