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: $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$
$$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
$$ \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE] $$
$$ \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] $$
$$ \frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa] $$
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 ‘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.5-4-3. 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.

Fig.5-4-4-1. Concentrations of GFP and RFP dependencies of production rate of C4 by RhlI

Each line corresponds to the transition of the concentration of RFP and GFP with a certain production rate of C12. Red and green lines correspond to RFP and GFP, respectively. Blighter one indicates higher production rate of C12.
If the production rate of C4 is between 0.029 and 0.778, our system can recreate the story. If this parameter is too small, the production of LasI by the Queen coli is insufficiently inhibited by MazF so C12 increases greatly. As a result, The concentration of GFP overcomes the concentration of RFP and the story does not develop correctly. And if this parameter is too big, the production of LasI by the Queen coli is overly inhibited by MazF so C12 does not increase.
As a result, the concentration of RFP is always greater than the GFP concentration and the story does not develop either.

Fig.5-4-4-2. Concentrations of GFP and RFP dependencies of production rate of C12 by LasI

Each line corresponds to the transition of the concentration of RFP/GFP with a certain production rate of C4. Red and green lines correspond to RFP and GFP, respectively. Blighter one indicates higher production rate of C4.
If production rate of C12 is between 0.001 and 0.217, our system can recreate the story.

Fig.5-4-5. Concentrations of GFP and RFP dependencies of translation rate of proteins

Each line corresponds to the transition of the concentration of RFP and GFP with a certain translation rate. Red and green lines correspond to RFP and GFP, respectively. Blighter one indicates higher translation rate.
If translation rate of protein is between 0.01 and 0.16, our system can recreate the story.

Fig.5-4-6. Concentrations of GFP and RFP dependencies of decomposition rate of C12 by AmiE

Each line corresponds to the transition of the concentration of RFP and GFP with a certain decomposition rate of C12. Red and green lines correspond to RFP and GFP, respectively. Blighter one indicates higher value of decomposition rate of C12.
If degradation rate of C12 is higher than 0.00056, our system can recreate the story.

Fig.5-4-7. Concentration of GFP and RFP dependencies of degradation rate of AmiE

Each line corresponds to the transition of the concentration of RFP and GFP with a certain degradation rate of AmiE. Red and green lines correspond to RFP and GFP, respectively. Blighter one indicates higher degradation rate of AmiE.
Even if we modify the values of the parameters inside the defined range, the concentration of RFP overcomes the concentration of GFP. And even if the degradation rate of AmiE is small, the decomposition rate of C12 by AmiE is high enough so C12 decreases sufficiently.

Fig.5-4-8 Relation of degradation rate of GFP and RFP

The story is recreated only if the degradation rate of RFP and GFP are the same.

5-2. Key achievements

1. Start ACADwarfs
   

   Download the zip file and open it.
   

   

2. Put in the part sequence
   

   Put the part sequence you want to regulate the effect of the mazEF system on in the upper window.
   

   

3. Choose operation
   

   If you want to magnify the effect of mazEF system, choose “increase ACA”.
   

   If you want to lessen the effect of mazEF system, choose “decrease ACA”.
   

   

4. Get sequences
   

   You can get the modified part sequences in the lower window. Modified codons will be written in capital letters and while the rest will be in small letters so you can easily locate the modified parts.
   And under this window you can see the number of ACA sequences on the sequence before the modification of the following "previous number" and the number of ACA sequences on the sequence after the modification of the following "new number".
   

   

5-4. Demonstration

We created a demo to present the features of this software. Using this, we regulated the number of ACA sequences and control the sensitivity of the protein to MazF.

Gene	Pre number of ACA sequences	Post number of ACA sequences
RFP	10	30
GFP	23	39
MazF	2	1
MazE	2	1

We increased the numbers of ACA sequences of RFP and GFP decreased the numbers of ACA sequences of MazF and MazE .

Fig.5-4. Comparison between the results of simulations using original sequences and modified sequences

We can see that the concentration of expressed MazF reacts more keenly after adjusting the ACA sequences than before doing so.

5-5. Download

To download click here.

The code is available on github.