2-1. Mathematical model

2-2. Results

Fig. 5-2-2-1.Time-dependent change of the concentrations of fluorescence proteins

Fig. 5-2-2-1.The story of "Snow White"

In the blue area of Fig.5-2-2-1, the fluorescence intensity starts to increase. Snow White coli’s fluorescence intensity exceeds that of the Queen coli’s. It represents Snow White got fairer more and more.
In the pink area of Fig.5-2-2-1, C12 is being synthesized thanks to the appearance of C4. And the concentration of C12 increases. As a result, the toxin MazF augments inside of Snow White coli and the Queen coli, suppressing the increment of fluorescent proteins. It is as if the Mirror’s answer transformed the Queen into a Witch, so she can give Snow White a poisoned apple.
In the green area of Fig.5-2-2-1, C12 increases and the MazF inside Snow White coli induced by it increases even more. So the GFP exceeds the RFP. It looks as if Snow White bit the apple, sinking into unconsciousness promptly.
In the yellow area of Fig.5-2-2-1, the AmiE synthesized by the introduced Prince coli decomposes the C12 molecules so the amount of MazF inside Snow White diminishes and the amount of C4 increases. It looks as if Prince lifted Snow White and she opened her eyes.

3-1. Population growth

First, we tried to model the growth curve of the system. When the number of E.coli approaches a certain value, the growth will stop. We defined the value in the culture as P_max. Then the population growth equation for our system is described as follows:
$$ \frac{dP}{dt} = g(1 - \frac{P}{P_{max}})P$$
where g is the population growth rate.
This equation can be analytically solved as:
$$ P = \frac{P_{0} P_{max} e^{gt}}{P_{max} - P_{0} + P_{0} e^{gt}}$$
where P ₀ is the population at t = 0. We used this equation to fit the experimental data.

We got the experimental data from {Toxin assay} for this fitting. We estimated the parameters
g = 0.0123
and
P_max =3.6
,respectively.
These parameters can be used for Snow White, Queen and Prince in the same way.

3-2. Toxin-Antitoxin system

Since there are few researches 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 estimation. 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 algorithm to fit the parameters to the experimental data. We used ODs at 7 points and fit them to experimental ODs.

As a result, we obtained the following parameters.

3-3. Promoters

We measured the relation between the AHL inputted and the fluorescence intensity with our data.
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.
$$ \frac{d[GFP]}{dt} = leak_{Prhl} + \frac{κ_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} $$
The same holds true for Lux system.
$$ \frac{d[GFP]}{dt} = leak_{Plux} + \frac{κ_{Lux}[C12]^n{_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^n{_{Lux}}} $$ 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

3-4. More realistic model with mRNA

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.

4-1. Requirements

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 reproduces 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.

4-2. the Prince coli should be put in during the process

We run simulations in order to determine whether we would get a better behavior let we introduce the Prince coli at the beginning or halfway of the story.

Fig.5-4-2-1.Number of individuals when the Prince coli is introduced from the beginning

Fig.5-4-2-2.AHL concentrations when the Prince coli is introduced from the beginning

As a result, if we introduce the Prince coli from the beginning, the number of Prince coli increases too much (Fig.5-4-2-1) so the AmiE the Prince coli produces augments and the decomposition of C12 also overly occurs. So C12 is almost inexistent in the medium (Fig.5-4-2-2).

Fig.5-4-2-3.Number of individuals when the Prince coli is introduced at t = 700

Fig.5-4-2-4.AHL concentrations when the Prince coli is introduced at t = 700

On the other hand, if we introduce the Prince coli at t=700, the number of Prince coli does not increment much (Fig.5-4-2-3), so C12 can exist until t=700 and then decreases thanks to the augment of AmiE (Fig.5-4-2-4).
In conclusion, if we introduce the Prince coli at t=700, the circuit will behave accordingly.

4-3. Prhl should be changed

In order to confirm whether by using a combination of the current promoters we would be able to reproduce the story with our gene circuit, we performed some simulations based on the results of our assays.

4-4. Production rate of C4HSL and 3OC12HSL by RhlI and LasI

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.

4-5. Translation rate of protein

Translation rate affects the production of protein.

4-6. Decomposition rate of C12 by AmiE

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.

4-7. Degradation rate of AmiE

4-8. Degradation rate of RFP and GFP

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 reproduced.

5-1. Abstract

・Provided the tool regulating the number of ACA sequences
・Released under open-source license so everyone can use it
・Able to correspond to any base arrangements
・Rare codons are evaded
・Extend the application field of mazEF system