Goal

To simulate how the oscillator works.

Methods

1. Establish ODE equations based on Mass-action law;
2. Investigate reasonable parameter sets 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 Bacterial strains and plasmids

Parameter	Meaning	Value
D	Ratio of promoters which don't combine with any protein among all promoters	$\dfrac{1}{1+\frac{a^2}{a_0}+\frac{r^4}{r_0}}$