Team:XJTU-China/model-genetic-circuits

Modelling

Genetic Circuits

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

The model of genetic circuits

Background

Based on the model of toggle switch, we designed a genetic circuits as the figure below shows.

The promoter $\lambda P_R$ $\lambda P_{R}$ controls the production of lacI, aroG, trpB and trpA, which catalyse the synthesis of tryptophan. The promoter $P_{lac}$ $P_{lac}$ controls the production of cl857 and pykA, which promote the growth of E. coli.

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

$$\begin{array}{}\text{(3.1)}& \{\begin{array}{ll}\frac{\mathrm{d}[rlacI]}{\mathrm{d}t}& =k_{syn,lacI}\cdot {\alpha }_{\lambda P_R,0}\cdot [dlacI]\\ & +k_{syn,lacI}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dlacI]\\ & -k_{de,lacI}\cdot [rlacI],\\ \frac{\mathrm{d}[raroG]}{\mathrm{d}t}& =k_{syn,aroG}\cdot {\alpha }_{\lambda P_R,0}\cdot [daroG]\\ & +k_{syn,aroG}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [daroG]\\ & -k_{de,aroG}\cdot [raroG],\\ \frac{\mathrm{d}[rtrpB]}{\mathrm{d}t}& =k_{syn,trpB}\cdot {\alpha }_{\lambda P_R,0}\cdot [dtrpB]\\ & +k_{syn,trpB}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dtrpB]\\ & -k_{de,trpB}\cdot [rtrpB],\\ \frac{\mathrm{d}[rtrpA]}{\mathrm{d}t}& =k_{syn,trpA}\cdot {\alpha }_{\lambda P_R,0}\cdot [dtrpA]\\ & +k_{syn,trpA}\cdot {\alpha }_{\lambda P_R}\cdot \frac{1}{[pcI857]}\cdot [dtrpA]\\ & -k_{de,trpA}\cdot [rtrpA].\end{array}\end{array}$$

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

$$\begin{array}{}\text{(3.2)}& \{\begin{array}{ll}\frac{\mathrm{d}[rcI857]}{\mathrm{d}t}& =k_{syn,cI857}\cdot {\alpha }_{P_{lac},0}\cdot [dcI857]\\ & +k_{syn,cI857}\cdot {\alpha }_{P_{lac}}\cdot \frac{[IPTG]}{[IPTG]+[placI]}\cdot [dcI857]\\ & -k_{de,cI857}\cdot [rcI857],\\ \frac{\mathrm{d}[rpykA]}{\mathrm{d}t}& =k_{syn,pykA}\cdot {\alpha }_{P_{lac},0}\cdot [dpykA]\\ & +k_{syn,pykA}\cdot {\alpha }_{P_{lac}}\cdot \frac{[IPTG]}{[IPTG]+[placI]}\cdot [dpykA]\\ & -k_{de,pykA}\cdot [rpykA].\end{array}\end{array}$$

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

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

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)	$678\mathrm{bp}$ $678\mathrm{bp}$	$1$ $1$
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

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

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

• lacI, aroG, trpA and trpB (promoted by $\lambda P_R$ $\lambda P_R$) has the same trend of GFP in the model of toggle switch;
• cl857 and pykA (promoted by $P_{lac}$ $P_{lac}$) has the same trend of RFP in the model of toggle switch.

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 Toggle Switch Go: The Model of Synthesis of Tryptophan Go: The Model of Production

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