Fig.5-3-1. Modeled growth curve of E. coli fitted to experiment data

Since there are only few researches that actually discussed on the kinetics of the TA system, we estimated the parameters for our TA system. We got the experimental data from Toxin assay for this fitting. The differential equations representing the TA system are described as follows: \begin{equation} \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] \nonumber \end{equation}
\begin{eqnarray} \frac{d[DiMazF]}{dt} &=& k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \nonumber \\ && + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{eqnarray}
\begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE] \nonumber \end{equation}
\begin{eqnarray} \frac{d[DiMazE]}{dt} &=& k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \nonumber \\ && + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] \end{eqnarray}
\begin{equation} \frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa] \nonumber \end{equation}
We used genetic algorithms to fit this data. We used ODs at 7 points and fit them to experimental ODs.

As a result, we obtained the following parameters.

We measured the relation between the AHL inputted and the fluorescence intensity.
Changing the AHL concentration from 10^-4 to 10⁴ by one power of 10 at a time, we measured the fluorescence intensity 4 hours after the insertion. We then performed the fitting with these data.
In our experimental data we only measured the fluorescence intensity of GFP. However, we can only do the modulation for the concentration of GFP. Therefore, we needed the relationship between the concentration of GFP by modeling and the fluorescence intensity by experiment. We assumed that the fluorescence intensity is proportional to the concentration of GFP. We determined the parameter from experimental data.

Fluorescence intensity = 31.8 [GFP]

From this parameter, we determined the parameters of the promoters.

We defined the following set of differential equations.
$$ \frac{d[GFP]}{dt} = leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} $$
The same holds true for Lux system.
$$ \frac{d[GFP]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} $$ We used MATLAB to fit the parameters to the experimental data and we fit them to the experimental concentrations.

We obtained these parameters as follows:

Leak_Prhl = 0.86
κ_Rhl &=& 1.326
n_Rhl &=& 5
K_mRhl &=& 1000

Leak_Plux = 0.86
κ_Lux = 1.326
n_Lux = 5
K_mLux = 1000

We defined the differential equations and estimated the parameters in the AHL - GFP system as described in 3-3. However, the translation from mRNA to protein was not considered in the model. To simulate the story of "the Snow White," the translation is essential because the whole system includes the toxin-antitoxin system involving the cleavage of mRNA. We also had to take in account in our fitting that the AHL inputted at the beginning decreases with time.
Then, we redefined the following set of differential equations.
$$ \frac{d[mRNA_{GFP}]}{dt} = leak_{Prhl} + \frac{κ_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} - d[mRNA_{GFP}]$$
$$ \frac{d[GFP]}{dt} = α[mRNA_{GFP}] - d_{GFP}[GFP] $$
$$ \frac{d[C4]}{dt} = - d_{C4}[C4] $$
The same holds true for Lux system.
$$ \frac{d[mRNA_{GFP}]}{dt} = leak_{Plux} + \frac{κ_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{GFP}]$$
$$ \frac{d[GFP]}{dt} = α[mRNA_{GFP}] - d_{GFP}[GFP] $$
$$ \frac{d[C12]}{dt} = - d_{C12}[C12] $$
As a result, we obtained these parameters as follows:

Leak_Prhl = 4.65
κ_Rhl = 14.95
n_Rhl = 5
K_mRhl = 1000
and
Leak_Plux = 2.26
κ_Lux = 6.98
n_Lux = 0.76
K_mLux = 116.24

Using these parameters we simulated the reproduction of the story of Snow White.

Fig.5-4-2. The intensity of Plux and Prhl promoters

The diagram above shows that the intensity of the two promoters should be in the red region of the figure.The combination of promoters which we were originally going to use is shown in the graph by the green point. To move this point into the red region, we had to improve Prhl to raise its expression level.

We performed the sensitivity analysis descried in this section in order to examine which parameter dominates the story. We defined these requirements as the “successful Snow White story.”
1) At t = 150
     concentration of RFP > concentration of GFP
2) At t = 700
      concentration of RFP< concentration of GFP
3) At t = 1500
     concentration of RFP > concentration of GFP

We the began to analyze the graph that satisfies these requirements that we could say that recreates the story correctly. In the table bellow we show the range in which we modified each parameter. They were modified one step size at a time.

As a result, we obtained the following parameter ranges.

The signaling molecule production rates by RhlI and LasI can be changed by modifying RhlI and LasI to make more or less signaling molecule in silico.

Translation rate affects the production of protein.

AmiE decomposes C12. The concentration of C12 after input of the Prince coli is changed by AmiE. If the decomposition rate of C12 is too small, C12 does not decrease enough so the MazF inside Snow White coli continues being expressed and the concentration of RFP decreases.

The degradation rate of RFP and GFP is key to the success the story of ‘"Snow White".
These parameters are closely related to the concentrations of GFP and RFP, so we conjectured that if they do not take appropriate values the story can not be correctly recreated.

5-2. Key achievements