Revision as of 06:42, 17 October 2021

Team:XJTU-China/model-population-dynamics

Model

The model of genetic circuits

Background

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

$\lambda P_{R}$ $P_{lac}$ controls the production of cl857 and pykA, which promote the growth of E. coli.

Theory

$x$ $[dx]$ $[rx]$ $[px]$ $[IPTG]$ be the concentration of IPTG.

$\lambda P_R$ , we have

\begin{matrix} (3.1) & {\begin{cases} \frac{d [r l a c I]}{d t} & = k_{s y n, l a c I} \cdot α_{λ P_{R}, 0} \cdot [d l a c I] \\ + k_{s y n, l a c I} \cdot α_{λ P_{R}} \cdot \frac{1}{[p c I 857]} \cdot [d l a c I] \\ - k_{d e, l a c I} \cdot [r l a c I], \\ \frac{d [r a r o G]}{d t} & = k_{s y n, a r o G} \cdot α_{λ P_{R}, 0} \cdot [d a r o G] \\ + k_{s y n, a r o G} \cdot α_{λ P_{R}} \cdot \frac{1}{[p c I 857]} \cdot [d a r o G] \\ - k_{d e, a r o G} \cdot [r a r o G], \\ \frac{d [r t r p B]}{d t} & = k_{s y n, t r p B} \cdot α_{λ P_{R}, 0} \cdot [d t r p B] \\ + k_{s y n, t r p B} \cdot α_{λ P_{R}} \cdot \frac{1}{[p c I 857]} \cdot [d t r p B] \\ - k_{d e, t r p B} \cdot [r t r p B], \\ \frac{d [r t r p A]}{d t} & = k_{s y n, t r p A} \cdot α_{λ P_{R}, 0} \cdot [d t r p A] \\ + k_{s y n, t r p A} \cdot α_{λ P_{R}} \cdot \frac{1}{[p c I 857]} \cdot [d t r p A] \\ - k_{d e, t r p A} \cdot [r t r p A] . \end{cases} \end{matrix}

$k_{syn,x}$ $k_{de,x}$ $x$ $\alpha_{\lambda P_R,0}$ $\lambda P_R$ $\alpha_{\lambda P_R}$ $\lambda P_R$ .

$P_{lac}$ , we have

\begin{matrix} (3.2) & {\begin{cases} \frac{d [r c I 857]}{d t} & = k_{s y n, c I 857} \cdot α_{P_{l a c}, 0} \cdot [d c I 857] \\ + k_{s y n, c I 857} \cdot α_{P_{l a c}} \cdot \frac{[I P T G]}{[I P T G] + [p l a c I]} \cdot [d c I 857] \\ - k_{d e, c I 857} \cdot [r c I 857], \\ \frac{d [r p y k A]}{d t} & = k_{s y n, p y k A} \cdot α_{P_{l a c}, 0} \cdot [d p y k A] \\ + k_{s y n, p y k A} \cdot α_{P_{l a c}} \cdot \frac{[I P T G]}{[I P T G] + [p l a c I]} \cdot [d p y k A] \\ - k_{d e, p y k A} \cdot [r p y k A] . \end{cases} \end{matrix}

$\alpha_{P_{lac},0}$ $P_{lac}$ $\alpha_{P_{lac}}$ $P_{lac}$ .

For the translation of protein translation, we have

\begin{matrix} (3.3) & {\begin{cases} \frac{d [p l a c I]}{d t} & = k p_{s y n, l a c I} [r l a c I] - k p_{d e, l a c I} [p l a c I], \\ \frac{d [p a r o G]}{d t} & = k p_{s y n, a r o G} [r a r o G] - k p_{d e, a r o G} [p a r o G], \\ \frac{d [p t r p B]}{d t} & = k p_{s y n, t r p B} [r t r p B] - k p_{d e, t r p B} [p t r p B], \\ \frac{d [p t r p A]}{d t} & = k p_{s y n, t r p A} [r t r p A] - k p_{d e, t r p A} [p t r p A], \\ \frac{d [p c l 857]}{d t} & = k p_{s y n, c l 857} [r c l 857] - k p_{d e, c l 857} [p c l 857], \\ \frac{d [p p y k A]}{d t} & = k p_{s y n, p y k A} [r p y k A] - k p_{d e, p y k A} [p p y k A] . \end{cases} \end{matrix}

$kp_{syn,x}$ $kp_{de,x}$ $x$ respectively.

Parameter

The parameters are shown in the table below.

PARAMETER	VALUE	REFERENCE
$k_{syn,x}$	$0.019\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model
$k_{de,x}$	$0.0013\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model
$kp_{syn,x}$	$0.47\mathrm{s^{-1}}$	https://2018.igem.org/Team:NUS_Singapore-A/Model
$kp_{de,x}$	$0.136\mathrm{s^{-1}}$	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)	$678\mathrm{bp}$	$1$
lacI	$1083\mathrm{bp}$	$0.626$
aroG	$1053\mathrm{bp}$	$0.644$
trpB	$1194\mathrm{bp}$	$0.568$
trpA	$807\mathrm{bp}$	$0.84$
cl857	$714\mathrm{bp}$	$0.95$
pykA	$1443\mathrm{bp}$	$0.47$

Result

$t=1000\mathrm{min}$ $t=2000\mathrm{min}$ , 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.

$\lambda P_R$ ) has the same trend of GFP in the model of toggle switch;
$P_{lac}$ ) 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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