Team:HUST-China/Modelling/DDE Model

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DDE MODEL

Goal

To simulate how the oscillator works.

Methods

1. Establish DDEs based on mass action law;
2. Investigate reasonable parameters set from previous researches;
3. Simulation;

Results

We solved these DDEs with R language. We also went one step further. We simulated the situation in which lag obeys a specific gaussian distribution, and the lag $\tau$ changes in every certain interval. We hoped by running a random test, we could get closer to real life situation. The results are below.

(a) A numeric solve of AraC

(b) 5 random tests numeric solve of AraC


Fig 1.(a)A numeric solve of AraC when lag $\tau = 2$min, Arabinose concentration is 5%, IPTG concentration is 1mM, time interval is 0.1min. (b)numeric solve of AraC concentration versus time of 5 random tests, when Arabinose concentration is 0.7%, IPTG concentration is 10mM, and $\tau \sim (2.0,0.3^2)$.

The period of this particular solve is 49.0minutes. The numeric solve of DDEs shows that the supposed oscillator is feasible. On the other hand, interval between every adjacent peak is different in a single random test, thus period is calculated by average intervals. Even so, the average period of each random test is different from each other: $T_1$= 43.95min, $T2$= 47.65min, $T_3$= 40.625min, $T_4$ = 39.375min, $T_5$ = 45.975min. Also, the amplitude of each curve is different. The random solve suggests that extern factors might be introduced to force the period to be the same.

Backgrounds

Fig 2.The pathway of genetic oscillator


The arabinose Operon and the lac Operon is the core to the functioning of the oscillator. With the presence of Arabinose, dimeric AraC can induce the expression of downstream gene; On the other hand, with minor presence of IPTG, tetrameric LacI may suppress the expression of downstream gene.
According to law of mass action, we had: $$\frac{dR_a}{dt} = copy_a(k_3D_1+k_4D_2+k_5D)-d_{a/r}R_a$$ $$\frac{dR_r}{dt} = copy_r(k_3D_1+k_4D_2+k_5D)-d_{a/r}R_r$$ $$\frac{da_{uf}}{dt} = t_aR_a-k_{fa}a_{uf} - \lambda f(x)a_{uf}$$ $$\frac{dr_{uf}}{dt} = t_rR_r-k_{fr}r_{uf} - f(x)r_{uf}$$ $$\frac{da}{dt} = k_{fa}a_{uf} - \lambda f(x)a$$ $$\frac{dr}{dt} = k_{fr}r_{uf} - f(x)r$$
Table 1-1 Parameters and variables used in DDEs
Parameters and Variables Meaning Value
$a_0$ Dissociation rate constant of AraC binding and unbinding with promoters $\dfrac{(6.25+ara^2)(1+\frac{IPTG^2}{3.24})}{101ara^2}$
$r_0$ Dissociation rate constant of LacI binding and unbinding with promoters $\dfrac{1}{2000000\frac{0.19}{1.0+(\frac{IPTG}{0.035})^2+0.01}}$
$D$ Ratio of promoters which don't combine with any protein among all promoters $\dfrac{1}{1+\frac{a^2}{a_0}+\frac{r^4}{r_0}}$
$D_1$ Ratio of operons combined with AraC dimer $\dfrac{a^2}{a_0(1+\frac{a^2}{a_0}+\frac{r^4}{r_0})}$
$D_2$ Ratio of operons combined with tetrameric LacI $\dfrac{r^4}{r_0(1+\frac{a^2}{a_0}+\frac{r^4}{r_0})}$
$a$ AraC protein(activator)
$r$ LacI protein(repressor)
$R_a$ mRNA of AraC
$R_r$ mRNA of LacI
$a_{uf}$ Unfolded AraC
$r_{uf}$ Unfolded LacI
$copy_a$ plasmid containing araC copies that are transfected into E.coli 50
$copy_r$ plasmid containing lacI copies that are transfected into E.coli 25
$k_3$ Transcriptional reaction rates constants of $D_1$ 196/min
$k_4$ Transcriptional reaction rates constants of $D_2$ 0/min
$k_5$ Transcriptional reaction rates constants of D 5.6/min
$d_{a/r}$ Degradation rate constants 10.54/min
$t_a$ Translational reaction rate constants of araC 90/min
$t_r$ Translational reaction rate constants of lacI 90/min
$k_{fa}$ Folding rate constants of AraC 0.9/min
$k_{fr}$ Folding rate constants of LacI 0.9/min
$f(x)$ Degradation rate constants $\frac{1080}{0.1+X}$
$\lambda f(x)$ Degradation rate constants $\frac{2887.92}{0.1+X}$
$\tau$ Time delay because of the transcription and translation time 2min

References

Team:NTU-Singapore/Modelling/Parameter
Jesse Stricker et al., 2008, Supplementary Information From A fast, robust and tunable synthetic gene oscillator, Nature 456, 516-519