Team:XJTU-China/model-toggle-switch

Modelling

Toggle Switch

•  Background
•  Theory
•  Parameter
•  Result
•  Conclusion
•  Reference

The model of toggle switch

Background

Next, to verify the effect of toggle switch, we design a genetic circuits as the figure below shows.

The promoter $\lambda P_R$ $\lambda P_{R}$ controls the production of lacI and green fluorescent protein (GFP), and can be promoted by raising temperature. The promoter $P_{lac}$ $P_{lac}$ controls the production of cI857 and red fluorescent protein (RFP), and can be promoted by IPTG. cl857 restricts the promoter $\lambda P_R$ $\lambda P_{R}$, while lacI restricts the promoter $P_{lac}$ $P_{lac}$.

Theory

For $x$ $x$, let $[dx]$ $[dx]$, $[rx]$ $[rx]$ and $[px]$ $[px]$ be the concentration of corresponding DNA, RNA and protein. And let $[IPTG]$ $[IPTG]$ be the concentration of IPTG.

For the transcription of promoter $\lambda P_R$ $\lambda P_R$, we have

$$\begin{array}{}\text{(2.1)}& \{\begin{array}{ll}\frac{\mathrm{d}[rlacI]}{\mathrm{d}t}& =k_{syn,lacI}\cdot {\alpha }_{\lambda P_R,0}\cdot [dlacI]\\ & +k_{syn,lacI}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dlacI]\\ & -k_{de,lacI}\cdot [rlacI],\\ \frac{\mathrm{d}[rGFP]}{\mathrm{d}t}& =k_{syn,GFP}\cdot {\alpha }_{\lambda P_R,0}\cdot [dGFP]\\ & +k_{syn,GFP}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dGFP]\\ & -k_{de,GFP}\cdot [rGFP],\end{array}\end{array}$$

where $k_{syn,x}$ $k_{syn,x}$ and $k_{de,x}$ $k_{de,x}$ are the synthesis rate and the degradation rate of $x$ $x$ respectively, ${\alpha }_{\lambda P_R,0}$ $\alpha_{\lambda P_R,0}$ is the background expression rate of $\lambda P_R$ $\lambda P_R$, and ${\alpha }_{\lambda P_R}$ $\alpha_{\lambda P_R}$ is the induced expression rate of $\lambda P_R$ $\lambda P_R$.

For the transcription of promoter $P_{lac}$ $P_{lac}$, we have

$$\begin{array}{}\text{(2.2)}& \{\begin{array}{ll}\frac{\mathrm{d}[rcI857]}{\mathrm{d}t}& =k_{syn,cI857}\cdot {\alpha }_{P_{lac},0}\cdot [dcI857]\\ & +k_{syn,cI857}\cdot {\alpha }_{P_{lac}}\cdot \frac{[IPTG]}{[IPTG]+[placI]}\cdot [dcI857]\\ & -k_{de,cI857}\cdot [rcI857],\\ \frac{\mathrm{d}[rRFP]}{\mathrm{d}t}& =k_{syn,RFP}\cdot {\alpha }_{P_{lac},0}\cdot [dRFP]\\ & +k_{syn,RFP}\cdot {\alpha }_{P_{lac}}\cdot \frac{[IPTG]}{[IPTG]+[placI]}\cdot [dRFP]\\ & -k_{de,RFP}\cdot [rRFP],\end{array}\end{array}$$

where ${\alpha }_{P_{lac},0}$ $\alpha_{P_{lac},0}$ is the background expression rate of $P_{lac}$ $P_{lac}$, and ${\alpha }_{P_{lac}}$ $\alpha_{P_{lac}}$ is the induced expression rate of $P_{lac}$ $P_{lac}$.

For the translation of protein translation, we have

$$\begin{array}{}\text{(2.3)}& \{\begin{array}{ll}\frac{\mathrm{d}[placI]}{\mathrm{d}t}& =k{p}_{syn,lacI}[rlacI]-k{p}_{de,lacI}[placI],\\ \frac{\mathrm{d}[pGFP]}{\mathrm{d}t}& =k{p}_{syn,GFP}[rGFP]-k{p}_{de,GFP}[pGFP],\\ \frac{\mathrm{d}[pcI857]}{\mathrm{d}t}& =k{p}_{syn,cI857}[rcI857]-k{p}_{de,cI857}[pcI857],\\ \frac{\mathrm{d}[pRFP]}{\mathrm{d}t}& =k{p}_{syn,RFP}[rRFP]-k{p}_{de,RFP}[pRFP],\end{array}\end{array}$$

where $k{p}_{syn,x}$ $kp_{syn,x}$ and $k{p}_{de,x}$ $kp_{de,x}$ are the synthesis rate and the degradation rate of the protein of $x$ $x$ respectively.

Parameter

The parameters are shown in the table below.

PARAMETER	VALUE	REFERENCE
$k_{syn,x}$ $k_{syn,x}$	$0.019{\mathrm{s}}^{-1}$ $0.019\mathrm{s^{-1}}$	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)	$678\mathrm{bp}$ $678\mathrm{bp}$	$1$ $1$
GFP	$773\mathrm{bp}$ $773\mathrm{bp}$	$0.877$ $0.877$
cI857	$714\mathrm{bp}$ $714\mathrm{bp}$	$0.95$ $0.95$
lacI	$1083\mathrm{bp}$ $1083\mathrm{bp}$	$0.626$ $0.626$

Result

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.

Conclusion

Analysis the equations first, and conclusions are as follows.

• In equation $(2.1)$ $(2.1)$, the change rates of $[rlacI]$ $[rlacI]$ and $[rGFP]$ $[rGFP]$ decrease as $[pcI857]$ $[pcI857]$ increases;
• In equation $(2.2)$ $(2.2)$, the change rates of $[rcI857]$ $[rcI857]$ and $[rRFP]$ $[rRFP]$ decrease as $[placI]$ $[placI]$ increases;
• In equation $(2.2)$ $(2.2)$, the factor $\frac{[IPTG]}{[IPTG]+[placI]}$ $\dfrac{[IPTG]}{[IPTG]+[placI]}$ makes sure that $[IPTG]$ $[IPTG]$ restrains the production of RFP when its concentration is low, and promotes the production when its concentration is high.

From the figure, we can see that there are three stable states.

• 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.

Reference

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.

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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 Production

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