Revision as of 11:08, 18 October 2016

Model

1. Overview

To reproduce the story of ”Snow White”, we have designed the cell-cell communication system with improved or characterized parts and collected data from comprehensive experiments. Furthermore, we constructed the mathematical model in order to simulate the behavior of the whole system and to confirm the feasibility of our story. This simulation successfully contributed to give the suggestions to wet lab experiments and confirm the feasibility of our system. In addition, in order to utilize TA (Toxin-Antitoxin) system, we developed a new software in Java for adjusting the number of ACA sequences, which MazF dimer recognizes and cleaves in mRNAs.

2. Story Simulation

2-1. Mathematical Model

To simulate our gene circuits, we developed an ordinary differential equation model.

[Detail Description for Modeling]

ODE and parameters

Differencial Equations

Snow White

\begin{equation} \frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{RhlI}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] \end{equation} \begin{equation} \frac{d[RFP]}{dt} = \alpha [mRNA_{RFP}] - d_{RFP}[RFP] \end{equation} \begin{equation} \frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \end{equation} \begin{equation} \frac{d[C4]}{dt} = p_{C4}[RhlI]P_{Snow White} - d_{C4}[C4] \end{equation} \begin{equation} \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] \end{equation} \begin{equation} \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] \end{equation} \begin{equation} \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{equation} \begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE] \end{equation} \begin{equation} \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] \end{equation} \begin{equation} \frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa] \end{equation} \begin{equation} \frac{dP_{Snow White}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}} \right) P_{Snow White} \end{equation}

Queen

\begin{equation} \frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{LasI}]}{dt} = leak_{P_{rhl}} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\ 　　　　　　　- d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] \end{equation} \begin{equation} \frac{d[GFP]}{dt} = \alpha [mRNA_{GFP}] - d_{GFP}[GFP] \end{equation} \begin{equation} \frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \end{equation} \begin{equation} \frac{d[C12]}{dt} = p_{C12}[LasI]P_{Queen} - d_{C12}[C12] - D[C12][AmiE] \end{equation} \begin{equation} \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] \end{equation} \begin{equation} \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] \end{equation} \begin{equation} \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{equation} \begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE] \end{equation} \begin{equation} \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] \end{equation} \begin{equation} \frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa] \end{equation} \begin{equation} \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}\\ \end{equation}

Prince

\begin{equation} \frac{d[mRNA_{AmiE}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^n}{K_{mLux}^n + [C12]^n} - d[mRNA_{AmiE}] \end{equation} \begin{equation} \frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \end{equation} \begin{equation} \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \end{equation}

Explanation about Parameters

Parameter	Description
$$g$$	Growth rate of each cells
$$P_{max}$$	Carrying capacity
$$E_{DiMazF}$$	Effect of MazF dimer on growth rate
$$k$$	Transcription rate of mRNA under $P_{tet}$
$$leak_{P_{lux}}$$	Leakage of $P_{lux}$
$$leak_{P_{rhl}}$$	Leakage of $P_{rhl}$
$$\kappa_{Lux}$$	Maximum transcription rate of mRNA under $P_{lux}$
$$\kappa_{Rhl}$$	Maximum transcription rate of mRNA under $P_{rhl}$
$$n_{Lux}$$	Hill coefficient for $P_{lux}$
$$n_{Rhl}$$	Hill coefficient for $P_{rhl}$
$$K_{mLux}$$	Lumped paremeter for the Lux System
$$K_{mRhl}$$	Lumped paremeter for the Rhl System
$$F_{DiMazF}$$	Cutting rate at ACA sequences on mRNA by MazF dimer
$$f$$	The probability of distinction of ACA sequencess in each mRNA
$$f_{mRNA_{RFP}}$$	The number of ACA sequences in $mRNA_{RFP}$
$$f_{mRNA_{GFP}}$$	The number of ACA sequences in $mRNA_{GFP}$
$$f_{mRNA_{RhlI}}$$	The number of ACA sequences in $mRNA_{RhlI}$
$$f_{mRNA_{LasI}}$$	The number of ACA sequences in $mRNA_{LasI}$
$$f_{mRNA_{MazF}}$$	The number of ACA sequences in $mRNA_{MazF}$
$$f_{mRNA_{MazE}}$$	The number of ACA sequences in $mRNA_{MazE}$
$$\alpha$$	Translation rate of Protein
$$k_{Di_{MazF}}$$	Formation rate of MazF dimer
$$k_{-Di_{MazF}}$$	Dissociation rate of MazF dimer
$$k_{Di_{MazE}}$$	Formation rate of MazE dimer
$$k_{-Di_{MazE}}$$	Dissociation rate of MazE dimer
$$k_{Hexa}$$	Formation rate of Maz hexamer
$$k_{-Hexa}$$	Dissociation rate of Maz hexamer
$$p_{C4}$$	Production rate of C4HSL by RhlI
$$p_{C12}$$	Production rate of 3OC12HSL by LuxI
$$D$$	Decomposition rate of 3OC12HSL by AmiE
$$d$$	Degradation rate of mRNA
$$d_{RFP}$$	Degradation rate of RFP
$$d_{GFP}$$	Degradation rate of GFP
$$d_{RhlI}$$	Degradation rate of RhlI
$$d_{LasI}$$	Degradation rate of LasI
$$d_{MazF}$$	Degradation rate of MazF
$$d_{DiMazF}$$	Degradation rate of MazF dimer
$$d_{MazE}$$	Degradation rate of MazE
$$d_{DiMazE}$$	Degradation rate of MazE dimer
$$d_{Hexa}$$	Degradation rate of Maz Hexamer
$$d_{C4}$$	Degradation rate of C4HSL
$$d_{C12}$$	Degradation rate of 3OC12HSL
$$d_{AmiE}$$	Degradation rate of AmiE

2-2. Results

As a result, we obtained and confirmed the desirable behavior of the whole system by modifying and improving parts. As described below, our simulation showed appropriate transition of fluorescence for the story.

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

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

In the blue area of Fig, 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, C12 is being synthesized thanks to the C4’s appearance. 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, 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, 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. Fitting

TEST

4. Sensitivity Analysis

We performed the sensitivity analysis descried in this section in order to examine which parameter dominates the story.

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.

Parameter	Range	Step size
$$D$$	$$0.0001 < D < 0.001$$	$$0.0001$$
$$p_{C4}$$	$$0.0001 < p_{C4} < 1$$	$$0.001$$
$$p_{C12}$$	$$0.0001 < p_{C12} < 1$$	$$0.001$$
$$α$$	$$0.01 < α < 0.2$$	$$0.01$$
$$d_{AmiE}$$	$$0.001 < d_{AmiE} < 1$$	$$0.001$$

As a result, we obtained the following parameter ranges.

Parameter	Value
$$D$$	$$0.0056 < D < 0.001$$
$$p_{C4}$$	$$0.0029 < p_{C4} < 0.778$$
$$p_{C12}$$	$$0.001 < p_{C12} < 0.217$$
$$α$$	$$0.01 < α < 0.16$$
$$d_{AmiE}$$	$$0.001 < d_{AmiE} < 1$$

4-2. the Prince coli should be put in in the middle

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. ??.Number of individuals when the Prince coli is introduced from the beginning

Fig. ??.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) 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).

Fig. ??.Number of individuals when the Prince coli is introduced at t = 700

Fig. ??.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 ), so C12 can exist until t=700 and then decreases thanks to the augment of AmiE (Fig ).
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.

Fig. 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 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 on Prhl so its expression level increases.

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.

Fig. Concentrations of GFP and RFP dependencies of production rate of C4 by RhlI

Each line corresponds to the transition of the concentration of RFP/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 reproduce 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. 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 reproduce the story.

4-5. Translation rate of protein

Translation rate affects the production of protein.

Fig. Concentrations of GFP and RFP dependencies of translation rate of proteins

Each line corresponds to the transition of the concentration of RFP/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 reproduce the story.

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.

Fig. Concentrations of GFP and RFP dependencies of decomposition rate of C12 by AmiE

Each line corresponds to the transition of the concentration of RFP/GFP with a certain value of D. 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 reproduce the story.

4-7. Degradation rate of AmiE

Fig. Concentration of GFP and RFP dependencies of degradation rate of AmiE

Each line corresponds to the transition of the concentration of RFP/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.

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.

Fig. Relation of degradation rate of GFP and RFP

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

5. Software

5-1. Abstract

We developed a new software named ACADwarfs. This software helps to control the sensitivity of the protein to MazF by regulating the number of ACA sequences in the mRNA sequence. ACADwarfs can increase or decrease the number of ACA sequences on mRNA without changing the amino acid sequences that the mRNA specifies or frameshifts resulted from insertion of bases without considering.
Therewith we improve the practicality of the characteristic of the mazEF system. For example you can let protein A express constantly by eliminating ACA sequences of the sequence, while letting protein B suspend to express, at the desirable timing, by expression of MazF.
This software also evades using rare codons, so you don’t have to worry about them.

5-2. Key achievement

・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

5-3. Work flow

1) Start ACA Dwarfs
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 bottom window. Modified codons will be written in capital letters and the rest in small letters so you can easily locate the modified parts.
And under it you can see the number of ACA sequences on the sequence before been modified following “previous number” and the number of ACA sequences on the sequence after been modified 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. Time-dependent change of the concentrations of fluorescence proteins with DNA before regulation

Fig. Time-dependent change of the concentrations of fluorescence proteins with DNA after regulation

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.

@@ Line 108: / Line 108: @@
 				<h3 class="link"><a href="#translation"><font size="2.7">&nbsp;&nbsp;&nbsp;4-5. Translation rate of protein</font></a></h3>
 				<h3 class="link"><a href="#decomposition"><font size="2.7">&nbsp;&nbsp;&nbsp;4-6. Decomposition rate of O3C12HSL by AmiE</font></a></h3>
-				<h3 class="link"><a href="#degradation_amie"><font size="2.7">&nbsp;&nbsp;&nbsp;4-7. Degradation rate of AmiE</font></a></h3>
+				<h3 class="link"><a href="#degradation_amie"><font size="2.7">&nbsp;&nbsp;&nbsp;4-6. Degradation rate of AmiE</font></a></h3>
-				<h3 class="link"><a href="#degradation_protein"><font size="2.7">&nbsp;&nbsp;&nbsp;4-8. Degradation rate of RFP and GFP</font></a></h3>
+				<h3 class="link"><a href="#degradation_protein"><font size="2.7">&nbsp;&nbsp;&nbsp;4-7. Degradation rate of RFP and GFP</font></a></h3>
 				<h3 class="link"><a href="#software">5. Software</a></h3>
 				<h3 class="link"><a href="#abstract"><font size="2.7">&nbsp;&nbsp;&nbsp;5-1. Abstract</font></a></h3>
@@ Line 354: / Line 354: @@
 								<tbody>
 									<tr><td>Parameter </td><td> Range </td><td> Step size </td></tr>
-									<tr><td> $$ D $$ </td><td> $$ 0.0001 < D < 0.001 $$ </td><td> $$ 0.0001 $$ </td></tr>
+									<tr><td>$$D$$ </td><td> $$0.0001 < D < 0.001$$</td><td>$$0.0001$$</td></tr>
-									<tr><td> $$ p_{C4} $$ </td><td> $$ 0.0001 < p_{C4} < 1 $$</td><td> $$ 0.001 $$ </td></tr>
+									<tr><td>$$p_{C4}$$ </td><td> $$0.0001 < p_{C4} < 1$$</td><td>$$0.001$$</td></tr>
-									<tr><td> $$ p_{C12} $$ </td><td> $$ 0.0001 < p_{C12} < 1 $$</td><td> $$ 0.001 $$ </td></tr>
+									<tr><td>$$p_{C12}$$ </td><td> $$0.0001 < p_{C12} < 1$$</td><td>$$0.001$$</td></tr>
-									<tr><td> $$ α $$ </td><td> $$ 0.01 < α < 0.2 $$</td><td> $$ 0.01 $$ </td></tr>
+									<tr><td>$$α$$ </td><td> $$0.01 < α < 0.2$$</td><td>$$0.01$$</td></tr>
-									<tr><td> $$ d_{AmiE} $$ </td><td> $$ 0.001 < d_{AmiE} < 1 $$ </td><td> $$ 0.001 $$ </td></tr>
+									<tr><td>$$d_{AmiE}$$ </td><td> $$0.001 < d_{AmiE} < 1$$</td><td>$$0.001$$</td></tr>
 							    </tbody>
 					    </table>
@@ Line 366: / Line 366: @@
 								<tbody>
 									<tr><td>Parameter </td><td> Value </td></tr>
-									<tr><td> $$ D $$ </td><td> $$ 0.0056 < D < 0.001 $$ </td></tr>
+									<tr><td>$$D$$ </td><td> $$0.0056 < D < 0.001$$</td></tr>
-									<tr><td> $$ p_{C4} $$ </td><td> $$ 0.0029 < p_{C4} < 0.778 $$ </td></tr>
+									<tr><td>$$p_{C4}$$ </td><td> $$0.0029 < p_{C4} < 0.778$$</td></tr>
-									<tr><td> $$ p_{C12} $$ </td><td> $$ 0.001 < p_{C12} < 0.217 $$ </td></tr>
+									<tr><td>$$p_{C12}$$ </td><td> $$0.001 < p_{C12} < 0.217$$</td></tr>
-									<tr><td> $$ α $$ </td><td> $$ 0.01 < α < 0.16 $$ </td></tr>
+									<tr><td>$$α$$ </td><td> $$0.01 < α < 0.16$$</td></tr>
-									<tr><td> $$ d_{AmiE} $$ </td><td> $$ 0.001 < d_{AmiE} < 1 $$ </td></tr>
+									<tr><td>$$d_{AmiE}$$ </td><td> $$0.001 < d_{AmiE} < 1$$</td></tr>
 							    </tbody>
 					    </table>
 </p>
+</div> </div> <!-- requirements-->
 <br>
 			<div id="prince_coli">
@@ Line 397: / Line 397: @@
                     <p class="normal_text">On the other hand, if we introduce the Prince coli at t=700, the number of Prince coli does not increment much (Fig ), so C12 can exist until t=700 and then decreases thanks to the augment of AmiE (Fig ).<br>In conclusion, if we introduce the Prince coli at t=700, the circuit will behave accordingly.</p>
 </p>
+</div></div> <!-- header -->
 <br>
 			<div id="prhl">
@@ Line 410: / Line 410: @@
 </p>
 </div>
+</div></div> <!-- 4-3-->
 <br>
 			<div id="production_ahl">
@@ Line 433: / Line 434: @@
 </p>
 </div>
+</div></div> <!--4-4-->
 <br>
 			<div id="translation">
@@ Line 448: / Line 451: @@
 </p>
 </div>
+</div></div> <!-- 4-5-->
 <br>
 			<div id="decomposition">
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 </p>
 </div>
+</div></div> <!--4-6-->
 <br>
 			<div id="degradation _amie">
@@ Line 477: / Line 482: @@
 </p>
 </div>
+</div></div> <!--4-7-->
 <br>
 			<div id="degradation_protein">
@@ Line 489: / Line 495: @@
 					<div align="center"><p class="caption" style="font-size: 16px; text-align: center;"><span style="font-weight: bold;">Fig. Relation of degradation rate of GFP and RFP</span>
-                    <p class="normal_text">The story is reproduced only if the degradation rate of RFP and GFP are the same order of magnitude.</p>
+                    <p class="normal_text">The story is reproduced only if the degradation rate of RFP and GFP are the same.</p>
 </div>
+</div></div> <!--4-8-->
 		</div><!-- /analysis_contents -->
 		</div><!-- /analysis -->
-<br>
 		<div id="software" class="container">
 			<div id="software_header" class="container_header">

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