Summary

Our modeling includes five steps:

• Establish the model of population dynamics, which displays the population change of E. coli;
• Establish the model of toggle switch, where the production of red fluorescent protein (RFP) and green fluorescent protein (GFP) shows the effect of toggle switch;
• Establish the model of genetic circuits based on the model of toggle switch;
• Establish the model of synthesis of tryptophan based on Michaelis-Menten equation;
• Finally, integrate the above models to establish the model of production.

Establishment of model

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Result and conclusion

The population density of E. coli

Here, $K=6.08\times {10}^9\mathrm{CFU}/\mathrm{ml}$ $K=6.08×10^9\mathrm{CFU/ml}$. The figure below shows the population density of E. coli.

• When $N<{\displaystyle \frac{K}{2}}$ $N<\dfrac{K}{2}$, the population density grows exponentially. And when $N>{\displaystyle \frac{K}{2}}$ $N>\dfrac{K}{2}$, 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 effect of toggle switch

When $t=1000\min$ $t=1000\mathrm{min}$, add IPTG. When $t=2000\min$ $t=2000\mathrm{min}$, remove IPTG and raise temperature. The results are shown in the figure below.

• The change rates of $[rlacI]$ $[rlacI]$ and $[rGFP]$ $[rGFP]$ decrease as $[pcI857]$ $[pcI857]$ increases. The change rates of $[rcI857]$ $[rcI857]$ and $[rRFP]$ $[rRFP]$ decrease as $[placI]$ $[placI]$ increases. $[IPTG]$ $[IPTG]$ restrains the production of RFP when its concentration is low, and promotes the production when its concentration is high;
• At first, the concentration of GFP is more than the concentration of RFP, and green fluorescence appears. After adding IPTG, the concentration of RFP outnumbered GFP, and red fluorescence appears. After removing IPTG and raising temperature, the rank of RFP and GFP exchanged again, and green fluorescence appears.

The product of genetic circuits

Add IPTG at the beginning, and when $t=1170\min$ $t=1170\mathrm{min}$, remove IPTG and raise temperature. The figure below shows the concentration change of gene product.

• 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 output of tryptophan

Add IPTG at the beginning, and when $t=1170\min$ $t=1170\mathrm{min}$, remove IPTG and raise temperature. The figure below shows the concentration change during the synthesis of tryptophan.

• 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 $3000\min$ $3000\mathrm{min}$.

The best production strategy

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.

• 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 $20$ $20$).

Reference

[1] Verhulst, P.-F. "Recherches mathématiques sur la loi d'accroissement de la population." Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1-41, 1845.

[2] Verhulst, P.-F. "Deuxième mémoire sur la loi d'accroissement de la population." Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20, 1-32, 1847.

[3] XIAN YIN, HYUN-DONG SHIN, et al. 2017. P gas, a Low-pH-Induced Promoter, as a Tool for Dynamic Control of Gene Expression for Metabolic Engineering of Aspergillus niger. Appl Environ Microbiol. [J/OL], 2;83(6):e03222-16.

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

The model of population dynamics

Background

First, we establish the model of population dynamics to study the variation of E. coli population density. Here, we use the Logistic equation to build our model.

The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation known as the logistic map is also widely used.

Theory

Let $N$ $N$ be the population density of E. coli. With the Logistic equation, we know that

where $r$ $r$ and $K$ $K$ are the growth rate and the environmental capacity of E. coli respectively.

Parameter

The parameters are shown in the table below.

Result

Let the initial value of population density be $0.01\mathrm{\%}$ $0.01\%$ of the environmental capacity. The result is shown in the figure below.

Conclusion

Equation $(1.1)$ $(1.1)$ and result show that:

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

Reference

Verhulst, P.-F. "Recherches mathématiques sur la loi d'accroissement de la population." Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1-41, 1845.

Verhulst, P.-F. "Deuxième mémoire sur la loi d'accroissement de la population." Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20, 1-32, 1847.