Team:XJTU-China/model-production

Modelling

Synthesis of Tryptophan

•  Background
•  Theory
•  Parameter
•  Result and Conclusion

The model of production

Background

Integrate the models we build, the final model of production can be established.

Here, the population of E. coli can influence the production rate in the model of synthesis of tryptophan, while the concentration of pyruvate can influence the growth rate of E. coli. Meanwhile, the production of catalyzers in synthesis of tryptophan is controlled by the genetic circuits.

With these models, we can study the final output (relative concentration) of tryptophan, and determine the optimal production strategy.

Theory

According to the model of population dynamics,

$$\begin{array}{}\text{(5.1)}& \{\begin{array}{ll}\frac{\mathrm{d}N}{\mathrm{d}t}& =rN(1-\frac{N}{K}),\\ r& =r([Pyr]).\end{array}\end{array}$$

According to the model of genetic circuits, for promoters, we have

$$\begin{array}{}\text{(5.2)}& \{\begin{array}{ll}\frac{\mathrm{d}[rlacI]}{\mathrm{d}t}& =k_{syn,lacI}\cdot {\alpha }_{\lambda P_R,0}\cdot [dlacI]\\ & +k_{syn,lacI}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dlacI]\\ & -k_{de,lacI}\cdot [rlacI],\\ \frac{\mathrm{d}[raroG]}{\mathrm{d}t}& =k_{syn,aroG}\cdot {\alpha }_{\lambda P_R,0}\cdot [daroG]\\ & +k_{syn,aroG}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [daroG]\\ & -k_{de,aroG}\cdot [raroG],\\ \frac{\mathrm{d}[rtrpB]}{\mathrm{d}t}& =k_{syn,trpB}\cdot {\alpha }_{\lambda P_R,0}\cdot [dtrpB]\\ & +k_{syn,trpB}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dtrpB]\\ & -k_{de,trpB}\cdot [rtrpB],\\ \frac{\mathrm{d}[rtrpA]}{\mathrm{d}t}& =k_{syn,trpA}\cdot {\alpha }_{\lambda P_R,0}\cdot [dtrpA]\\ & +k_{syn,trpA}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dtrpA]\\ & -k_{de,trpA}\cdot [rtrpA]\\ \frac{\mathrm{d}[rcI857]}{\mathrm{d}t}& =k_{syn,cI857}\cdot {\alpha }_{P_{lac},0}\cdot [dcI857]\\ & +k_{syn,cI857}\cdot {\alpha }_{P_{lac}}\cdot \frac{[IPTG]}{[IPTG]+[placI]}\cdot [dcI857]\\ & -k_{de,cI857}\cdot [rcI857],\\ \frac{\mathrm{d}[rpykA]}{\mathrm{d}t}& =k_{syn,pykA}\cdot {\alpha }_{P_{lac},0}\cdot [dpykA]\\ & +k_{syn,pykA}\cdot {\alpha }_{P_{lac}}\cdot \frac{[IPTG]}{[IPTG]+[placI]}\cdot [dpykA]\\ & -k_{de,pykA}\cdot [rpykA];\end{array}\end{array}$$

and for protein, we have

$$\begin{array}{}\text{(3.3)}& \{\begin{array}{ll}\frac{\mathrm{d}[placI]}{\mathrm{d}t}& =k{p}_{syn,lacI}[rlacI]-k{p}_{de,lacI}[placI],\\ \frac{\mathrm{d}[paroG]}{\mathrm{d}t}& =k{p}_{syn,aroG}[raroG]-k{p}_{de,aroG}[paroG],\\ \frac{\mathrm{d}[ptrpB]}{\mathrm{d}t}& =k{p}_{syn,trpB}[rtrpB]-k{p}_{de,trpB}[ptrpB],\\ \frac{\mathrm{d}[ptrpA]}{\mathrm{d}t}& =k{p}_{syn,trpA}[rtrpA]-k{p}_{de,trpA}[ptrpA],\\ \frac{\mathrm{d}[pcl857]}{\mathrm{d}t}& =k{p}_{syn,cl857}[rcl857]-k{p}_{de,cl857}[pcl857],\\ \frac{\mathrm{d}[ppykA]}{\mathrm{d}t}& =k{p}_{syn,pykA}[rpykA]-k{p}_{de,pykA}[ppykA].\end{array}\end{array}$$

According to the model of synthesis of tryptophan,

$$\begin{array}{}\text{(5.3)}& \{\begin{array}{ll}\frac{1}{N}\cdot \frac{\mathrm{d}[Glc]}{\mathrm{d}t}& =-\frac{v_{max,1}\cdot [Glc]}{[Glc]+k_{m,1}},\\ \frac{1}{N}\cdot \frac{\mathrm{d}[PEP]}{\mathrm{d}t}& =\frac{v_{max,1}\cdot [Glc]}{[Glc]+k_{m,1}}-\frac{k_{cat,aroG}\cdot [aroG]\cdot [PEP]}{[PEP]+k_{m,aroG}}\\ & -\frac{k_{cat,pykA}\cdot [pykA]\cdot [PEP]}{[PEP]+k_{m,pykA}},\\ \frac{1}{N}\cdot \frac{\mathrm{d}[DAHP]}{\mathrm{d}t}& =\frac{k_{cat,aroG}\cdot [aroG]\cdot [PEP]}{[PEP]+k_{m,aroG}}-\frac{v_{max,2}\cdot [DAHP]}{[DAHP]+k_{m,2}},\\ \frac{1}{N}\cdot \frac{\mathrm{d}[Pyr]}{\mathrm{d}t}& =\frac{k_{cat,pykA}\cdot [pykA]\cdot [PEP]}{[PEP]+k_{m,pykA}},\\ \frac{1}{N}\cdot \frac{\mathrm{d}[3IGP]}{\mathrm{d}t}& =\frac{v_{max,2}\cdot [DAHP]}{[DAHP]+k_{m,2}}-\frac{k_{cat,trpAB}\cdot [trpAB]\cdot [3IGP]}{[3IGP]+k_{m,trpAB}},\\ \frac{1}{N}\cdot \frac{\mathrm{d}[Trp]}{\mathrm{d}t}& =\frac{k_{cat,trpAB}\cdot [trpAB]\cdot [3IGP]}{[3IGP]+k_{m,trpAB}}.\end{array}\end{array}$$

Parameter

The parameters are shown in the table below.

Parameter	Value	Reference
$k$ $k$	$6.08\times {10}^9\mathrm{CFU}/\mathrm{ml}$ $6.08×10^9\mathrm{CFU/ml}$	https://2018.igem.org/Team:Lund/Model/GrowthCurves/Results
$k_{syn,x}$ $k_{syn,x}$	$0.019{\mathrm{s}}^{-1}$ $0.019\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model
$k_{de,x}$ $k_{de,x}$	$0.0013{\mathrm{s}}^{-1}$ $0.0013\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model
$k{p}_{syn,x}$ $kp_{syn,x}$	$0.47{\mathrm{s}}^{-1}$ $0.47\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model
$k{p}_{de,x}$ $kp_{de,x}$	$0.136{\mathrm{s}}^{-1}$ $0.136\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model

Meanwhile, the correction is performed according to the CDS length.

NAME	LENGTH	CORRECTION RATE
RFP (Benchmark)	$678\mathrm{bp}$ $678\mathrm{bp}$	$1$ $1$
lacI	$1083\mathrm{bp}$ $1083\mathrm{bp}$	$0.626$ $0.626$
aroG	$1053\mathrm{bp}$ $1053\mathrm{bp}$	$0.644$ $0.644$
trpB	$1194\mathrm{bp}$ $1194\mathrm{bp}$	$0.568$ $0.568$
trpA	$807\mathrm{bp}$ $807\mathrm{bp}$	$0.84$ $0.84$
cl857	$714\mathrm{bp}$ $714\mathrm{bp}$	$0.95$ $0.95$
pykA	$1443\mathrm{bp}$ $1443\mathrm{bp}$	$0.47$ $0.47$

Result and conclusion

Let the initial value of $[Glc]$ $[Glc]$ be $20$ $20$. Add IPTG at the beginning, and change the time of raising the temperature. Let $t_2$ $t_2$ be the time of raising temperature. The final outputs of $[Pyr]$ $[Pyr]$ and $[Trp]$ $[Trp]$ are shown in the figure below.

From the figure, we can see that:

• When $t_2<1170\min$ $t_2<1170\mathrm{min}$, the output of tryptophan increases slowly as $t_2$ $t_2$ increases;
• When $t_2=1170\min$ $t_2=1170\mathrm{min}$, the maximun output of tryptophan is $15.43$ $15.43$ ( $77.15\mathrm{\%}$ $77.15\%$ of $20$ $20$);
• When $1170\min <t_2<2000\min$ $1170\mathrm{min}<t_2<2000\mathrm{min}$, the output of tryptophan drops sharply as $t_2$ $t_2$ increases;
• Finally, when $t_2>2000\min$ $t_2>2000\mathrm{min}$, the output of tryptophan is around $8$ $8$ ( $40\mathrm{\%}$ $40\%$ of $20$ $20$).

Next, let the time of raising temperature be $1170\min$ $1170\mathrm{min}$, and consider the three models we build before. The figure below shows the population density of E. coli.

From the figure, we can see that:

• When $N<\frac{K}{2}$ $N<\dfrac{K}{2}$, the population density grows exponentially;
• When $N>\frac{K}{2}$ $N>\dfrac{K}{2}$, the environmental resources have a restrictive effect on E. coli;
• Finally the population density approaches $K$ $K$;
• The population density reach balance at about $33\mathrm{h}$ $33\mathrm{h}$.

The figure below shows the concentration change of gene product.

From the figure, we can see that:

• There are two stable states during the period of time;
• After raising temperature at $1170\min$ $1170\mathrm{min}$, the concentrations of cl857 and pykA go down, while the concentrations of lacI, aroG, trpA and trpB go up.

The figure below shows the concentration change during the synthesis of tryptophan.

From the figure, we can see that:

• 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 $1900\min$ $1900\mathrm{min}$, and after that it goes down;
• The product of DAHP is 3IGP, and 3IGP immediately converts to Trp;
• The final products of reactions are Pyr and Trp, whose concentrations are stable at $3000\min$ $3000\mathrm{min}$.

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Return to MODELLING Go: The Model of Population Dynamics Go: The Model of Genetic Circuits Go: The Model of Synthesis of Tryptophan Go: The Model of Synthesis of Tryptophan

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