Difference between revisions of "Team:Tokyo Tech/Modeling Details"

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</div><!-- /modeling_maz_header -->
 
</div><!-- /modeling_maz_header -->
 
<div id="modeling_maz_contents" class="container_contents">
 
<div id="modeling_maz_contents" class="container_contents">
 +
<a href="https://static.igem.org/mediawiki/2016/5/52/T--Tokyo_Tech--Model_Details_1.png"><img src="https://static.igem.org/mediawiki/2016/5/52/T--Tokyo_Tech--Model_Details_1.png" /></a>
 +
<p class="caption"><span style="font-weight: bold;">Fig. 4-2-1. </span> Maz System Gene Circuit</p>
 +
<div id="modeling_detail" class="off">
 +
<div id="modeling_detail_wrapper">
 +
<div id="modeling_detail_expressions">
 +
<h2>Differencial Equations</h2>
 +
<h3>Snow White</h3>
 +
\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}
 +
 +
<h3>Queen</h3>
 +
\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}
 +
 +
<h3>Prince</h3>
 +
\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}
 +
</div><!-- /modeling_detail_expressions -->
 +
<div id="modeling_detail_parameter">
 +
<h2>Explanation about Parameters</h2>
 +
<table border="1" style="margin: auto;">
 +
<tbody>
 +
<tr><td>Parameter </td><td> Description </td></tr>
 +
<tr><td>$$g$$ </td><td> Growth rate of each cells</td></tr>
 +
<tr><td>$$P_{max}$$ </td><td> Carrying capacity </td></tr>
 +
<tr><td>$$E_{DiMazF}$$ </td><td> Effect of MazF dimer on growth rate</td></tr>
 +
<tr><td>$$k$$ </td><td> Transcription rate of mRNA under \(P_{tet}\) </td></tr>
 +
<tr><td>$$leak_{P_{lux}}$$ </td><td> Leakage of \(P_{lux}\) </td></tr>
 +
<tr><td>$$leak_{P_{rhl}}$$ </td><td> Leakage of \(P_{rhl}\) </td></tr>
 +
<tr><td>$$\kappa_{Lux}$$ </td><td> Maximum transcription rate of mRNA under \(P_{lux}\)</td></tr>
 +
<tr><td>$$\kappa_{Rhl}$$ </td><td> Maximum transcription rate of mRNA under \(P_{rhl}\) </td></tr>
 +
<tr><td>$$n_{Lux}$$ </td><td> Hill coefficient for \(P_{lux}\)</td></tr>
 +
<tr><td>$$n_{Rhl}$$ </td><td> Hill coefficient for \(P_{rhl}\)</td></tr>
 +
<tr><td>$$K_{mLux}$$ </td><td> Lumped paremeter for the Lux System</td></tr>
 +
<tr><td>$$K_{mRhl}$$ </td><td> Lumped paremeter for the Rhl System</td></tr>
 +
<tr><td>$$F_{DiMazF}$$ </td><td> Cutting rate at ACA sequences on mRNA by MazF dimer</td></tr>
 +
<tr><td>$$f$$ </td><td> The probability of distinction of ACA sequencess in each mRNA</td></tr>
 +
<tr><td>$$f_{mRNA_{RFP}}$$ </td><td> The number of ACA sequences in \(mRNA_{RFP}\)</td></tr>
 +
<tr><td>$$f_{mRNA_{GFP}}$$ </td><td> The number of ACA sequences in \(mRNA_{GFP}\)</td></tr>
 +
<tr><td>$$f_{mRNA_{RhlI}}$$ </td><td> The number of ACA sequences in \(mRNA_{RhlI}\) </td></tr>
 +
<tr><td>$$f_{mRNA_{LasI}}$$ </td><td> The number of ACA sequences in \(mRNA_{LasI}\)</td></tr>
 +
<tr><td>$$f_{mRNA_{MazF}}$$ </td><td> The number of ACA sequences in \(mRNA_{MazF}\) </td></tr>
 +
<tr><td>$$f_{mRNA_{MazE}}$$ </td><td> The number of ACA sequences in \(mRNA_{MazE}\) </td></tr>
 +
<tr><td>$$\alpha$$ </td><td> Translation rate of Protein </td></tr>
 +
<tr><td>$$k_{Di_{MazF}}$$ </td><td> Formation rate of MazF dimer </td></tr>
 +
 +
<tr><td>$$k_{-Di_{MazF}}$$ </td><td> Dissociation rate of MazF dimer </td></tr>
 +
<tr><td>$$k_{Di_{MazE}}$$ </td><td> Formation rate of MazE dimer </td></tr>
 +
<tr><td>$$k_{-Di_{MazE}}$$ </td><td> Dissociation rate of MazE dimer </td></tr>
 +
<tr><td>$$k_{Hexa}$$ </td><td> Formation rate of Maz hexamer </td></tr>
 +
<tr><td>$$k_{-Hexa}$$ </td><td> Dissociation rate of Maz hexamer</td></tr>
 +
<tr><td>$$p_{C4}$$ </td><td> Production rate of C4HSL by RhlI</td></tr>
 +
<tr><td>$$p_{C12}$$ </td><td> Production rate of 3OC12HSL by LuxI </td></tr>
 +
<tr><td>$$D$$ </td><td> Decomposition rate of 3OC12HSL by AmiE </td></tr>
 +
<tr><td>$$d$$ </td><td> Degradation rate of mRNA </td></tr>
 +
<tr><td>$$d_{RFP}$$ </td><td> Degradation rate of RFP</td></tr>
 +
<tr><td>$$d_{GFP}$$ </td><td> Degradation rate of GFP </td></tr>
 +
<tr><td>$$d_{RhlI}$$ </td><td> Degradation rate of RhlI</td></tr>
 +
<tr><td>$$d_{LasI}$$ </td><td> Degradation rate of LasI</td></tr>
 +
<tr><td>$$d_{MazF}$$ </td><td> Degradation rate of MazF</td></tr>
 +
<tr><td>$$d_{DiMazF}$$ </td><td> Degradation rate of MazF dimer</td></tr>
 +
<tr><td>$$d_{MazE}$$ </td><td> Degradation rate of MazE </td></tr>
 +
<tr><td>$$d_{DiMazE}$$ </td><td> Degradation rate of MazE dimer </td></tr>
 +
<tr><td>$$d_{Hexa}$$ </td><td> Degradation rate of Maz Hexamer </td></tr>
 +
<tr><td>$$d_{C4}$$ </td><td> Degradation rate of C4HSL </td></tr>
 +
<tr><td>$$d_{C12}$$ </td><td> Degradation rate of 3OC12HSL </td></tr>
 +
<tr><td>$$d_{AmiE}$$ </td><td> Degradation rate of AmiE </td></tr>
 +
 +
</tbody>
 +
</table>
 +
</div><!-- /modeling_detail_parameter -->
 +
</div><!-- /modeling_detail_wrapper -->
 +
</div><!-- /modeling_detail -->
 +
<p class="normal_text" style="text-align:center;"><a href="javascript:void(0);" onClick="show('modeling_detail');" class="showHidden">Expressions</a></p>
 
<ul id="modeling_maz_list" class="non_dotted_list">
 
<ul id="modeling_maz_list" class="non_dotted_list">
 
<li><h2>1.1. Cell Population</h2>
 
<li><h2>1.1. Cell Population</h2>
 +
<p>$$ \frac{dP_{Snow White}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}} \right) P_{Snow White} $$ <br /> $$  \tag{4-2-1-1} $$</p>
 +
<p>$$
 +
\frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$ <br /> $$ \tag{4-2-1-2} $$</p>
 +
<p>$$
 +
\frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{4-2-1-3} $$ </p>
 
<p class="normal_text">The equations above describe how cells grow in the culture.
 
<p class="normal_text">The equations above describe how cells grow in the culture.
Equations (1), (2) and (3) describe the populations of Snow White, the Queen and the Prince. (3) is described by the logistic growth equation, but (1) and (2) are represented by the growth inhibition by MazF dimers.
+
Equations (4-2-1-1), (4-2-1-2) and (4-2-1-3) describe the populations of Snow White, the Queen and the Prince. (4-2-1-3) is described by the logistic growth equation, but (4-2-1-1) and (4-2-1-2) are represented by the growth inhibition by MazF dimers.
 
This factor is designed so that its value is small when the concentration of MazF dimers is low, and its value converges to 1 when the concentration of MazF dimers is high.</p>
 
This factor is designed so that its value is small when the concentration of MazF dimers is low, and its value converges to 1 when the concentration of MazF dimers is high.</p>
 
</li><!-- /1.1. Cell Population -->
 
</li><!-- /1.1. Cell Population -->
Line 80: Line 276:
 
<li><h3>1.2.1. Expression of Maz System</h3>
 
<li><h3>1.2.1. Expression of Maz System</h3>
 
<p class="normal_text">After translation, MazE and MazF each form  an stable dimer which can be activated to exert its function.</p>
 
<p class="normal_text">After translation, MazE and MazF each form  an stable dimer which can be activated to exert its function.</p>
 +
<a href="https://static.igem.org/mediawiki/2016/8/88/T--Tokyo_Tech--Model_Details_2.png"><img src="https://static.igem.org/mediawiki/2016/8/88/T--Tokyo_Tech--Model_Details_2.png" /></a><br />
 +
<a href="https://static.igem.org/mediawiki/2016/c/c3/T--Tokyo_Tech--Model_Details_3.png"><img src="https://static.igem.org/mediawiki/2016/c/c3/T--Tokyo_Tech--Model_Details_3.png" /></a>
 
<p class="normal_text">Two MazE dimers sandwich the MazF dimer, forming MazF2-MazE2-MazF2 heterohexamers and suppressing the toxicity of the MazF dimers.</p>
 
<p class="normal_text">Two MazE dimers sandwich the MazF dimer, forming MazF2-MazE2-MazF2 heterohexamers and suppressing the toxicity of the MazF dimers.</p>
 +
<a href="https://static.igem.org/mediawiki/2016/e/ea/T--Tokyo_Tech--Model_Details_4.png"><img src="https://static.igem.org/mediawiki/2016/e/ea/T--Tokyo_Tech--Model_Details_4.png" /></a>
 +
<a href="https://static.igem.org/mediawiki/2016/3/32/T--Tokyo_Tech--Model_Details_5.png"><img src="https://static.igem.org/mediawiki/2016/3/32/T--Tokyo_Tech--Model_Details_5.png" style="width: 500px;" /></a>
 +
<p class="caption"><span style="font-weight: bold">Fig. 4-2-2. </span> Reaction of Maz System</p>
 
<p class="normal_text">The mRNAs of Snow White and the Queen decrease by their original degradation and by the cleavage at ACA sequences by MazF dimers.</p>
 
<p class="normal_text">The mRNAs of Snow White and the Queen decrease by their original degradation and by the cleavage at ACA sequences by MazF dimers.</p>
 
<p class="normal_text">Applying mass action kinetic laws, we obtain the following set of differential equations.</p>
 
<p class="normal_text">Applying mass action kinetic laws, we obtain the following set of differential equations.</p>
<p class="normal_text">Equations (1) and (8) describe the concentration of mRNAs under the AHL inducing promoters.
+
<h3>Snow White</h3>
 +
<p>$$
 +
\frac{d[mRNA_{MazF}]}{dt} =  leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\
 +
       - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$<br />$$ \tag{4-2-2-1} $$</p>
 +
<p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$ <br />
 +
$$\tag{4-2-2-2}$$</p>
 +
<p>$$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\
 +
       + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$<br />$$ \tag{4-2-2-3} $$</p>
 +
<p>$$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$ <br />$$ \tag{4-2-2-4} $$</p>
 +
<p>$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE]$$
 +
<br />$$\tag{4-2-2-5}$$</p>
 +
<p>$$
 +
\frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\
 +
       + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{4-2-2-6} $$</p>
 +
<p>$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$ <br />$$ \tag{4-2-2-7}$$</p>
 +
<h3>Queen</h3>
 +
<p>$$
 +
\frac{d[mRNA_{MazF}]}{dt} =  leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\
 +
       - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$<br />$$ \tag{4-2-2-8} $$</p>
 +
<p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$ <br />
 +
$$\tag{4-2-2-9}$$</p>
 +
<p>$$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\
 +
       + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$<br />$$ \tag{4-2-2-10} $$</p>
 +
<p>$$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$ <br />$$ \tag{4-2-2-11} $$</p>
 +
<p>$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE]$$
 +
<br />$$\tag{4-2-2-12}$$</p>
 +
<p>$$
 +
\frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\
 +
       + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{4-2-2-13} $$</p>
 +
<p>$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$ <br />$$ \tag{4-2-2-14}$$</p>
 +
<p class="caption"><span style="font-weight: bold">Eq. 4-2-2. </span>Differential Equations of Maz System</p>
 +
<p class="normal_text">Equations (4-2-2-1) and (4-2-2-8) describe the concentration of mRNAs under the AHL inducing promoters.
 
Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
 
Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
Since Equations (2), (3), (5), (6), (7), (9), (10), (12), (13) and (14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.</p>
+
Since Equations (4-2-2-2), (4-2-2-3), (4-2-2-5), (4-2-2-6), (4-2-2-7), (4-2-2-9), (4-2-2-10), (4-2-2-12), (4-2-2-13) and (4-2-2-14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.</p>
 
</li><!-- /1.2.1. Expression of Maz System -->
 
</li><!-- /1.2.1. Expression of Maz System -->
 
<li><h3>1.2.2. Cleavage by MazF dimers</h3>
 
<li><h3>1.2.2. Cleavage by MazF dimers</h3>
 
<p class="normal_text">MazF dimers recognize and cleave ACAs in mRNAs, thus acting as Toxin.</p>
 
<p class="normal_text">MazF dimers recognize and cleave ACAs in mRNAs, thus acting as Toxin.</p>
<p class="normal_text">We estimated the rate of recognitions of ACA sequences by MazF dimers at 1-(1-f)^n, where the number of ACA sequences in mRNA.</p>
+
<p class="normal_text">We estimated the rate of recognitions of ACA sequences by MazF dimers at \(1-(1-f)^n\), where the number of ACA sequences in mRNA.</p>
<p class="normal_text">Then, we expressed the rate of degradation by MazF dimers in F(1-(1-f)^{f_{mRNA}}) and obtain the following set of differential equations.</p>
+
<p class="normal_text">Then, we expressed the rate of degradation by MazF dimers in \(F(1-(1-f)^{f_{mRNA}})\) and obtain the following set of differential equations.</p>
 +
<h3>Snow White</h3>
 +
<p>$$\frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF]
 +
$$ <br />$$ \tag{4-2-3-1} $$</p>
 +
<p>$$
 +
\frac{d[mRNA_{RhlI}]}{dt} =  leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
 +
<br />$$ \tag{4-2-3-2} $$</p>
 +
<p>$$\frac{d[mRNA_{MazF}]}{dt} =  leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\
 +
       - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF]
 +
$$ <br /> $$\tag{4-2-3-3}$$</p>
 +
<p>$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
 +
<br />$$ \tag{4-2-3-4} $$</p>
 +
<h3>Queen</h3>
 +
<p>$$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF]
 +
$$<br />$$ \tag{4-2-3-5} %%</p>
 +
<p>$$
 +
\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] $$ <br />$$ \tag{4-2-3-6} $$</p>
 +
<p>$$\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]
 +
$$ <br /> $$\tag{4-2-3-7}$$</p>
 +
<p>$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
 +
<br />$$ \tag{4-2-3-8} $$</p>
 +
<p class="caption"><span style="font-weight: bold">Eq. 4-2-3. </span>Differential Equations of mRNAs</p>
 
<p class="normal_text">The equations above comprise terms of production, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
 
<p class="normal_text">The equations above comprise terms of production, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
 
</p>
 
</p>
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Revision as of 03:08, 13 October 2016

1. Maz System

Fig. 4-2-1. Maz System Gene Circuit

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

Expressions

  • 1.1. Cell Population

    $$ \frac{dP_{Snow White}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}} \right) P_{Snow White} $$
    $$ \tag{4-2-1-1} $$

    $$ \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$
    $$ \tag{4-2-1-2} $$

    $$ \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{4-2-1-3} $$

    The equations above describe how cells grow in the culture. Equations (4-2-1-1), (4-2-1-2) and (4-2-1-3) describe the populations of Snow White, the Queen and the Prince. (4-2-1-3) is described by the logistic growth equation, but (4-2-1-1) and (4-2-1-2) are represented by the growth inhibition by MazF dimers. This factor is designed so that its value is small when the concentration of MazF dimers is low, and its value converges to 1 when the concentration of MazF dimers is high.

  • 1.2. Maz System

    • 1.2.1. Expression of Maz System

      After translation, MazE and MazF each form an stable dimer which can be activated to exert its function.


      Two MazE dimers sandwich the MazF dimer, forming MazF2-MazE2-MazF2 heterohexamers and suppressing the toxicity of the MazF dimers.

      Fig. 4-2-2. Reaction of Maz System

      The mRNAs of Snow White and the Queen decrease by their original degradation and by the cleavage at ACA sequences by MazF dimers.

      Applying mass action kinetic laws, we obtain the following set of differential equations.

      Snow White

      $$ \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\        - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
      $$ \tag{4-2-2-1} $$

      $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$
      $$\tag{4-2-2-2}$$

      $$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\        + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
      $$ \tag{4-2-2-3} $$

      $$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$
      $$ \tag{4-2-2-4} $$

      $$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE]$$
      $$\tag{4-2-2-5}$$

      $$ \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\        + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
      $$\tag{4-2-2-6} $$

      $$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
      $$ \tag{4-2-2-7}$$

      Queen

      $$ \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\        - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
      $$ \tag{4-2-2-8} $$

      $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$
      $$\tag{4-2-2-9}$$

      $$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\        + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
      $$ \tag{4-2-2-10} $$

      $$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$
      $$ \tag{4-2-2-11} $$

      $$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE]$$
      $$\tag{4-2-2-12}$$

      $$ \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\        + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
      $$\tag{4-2-2-13} $$

      $$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
      $$ \tag{4-2-2-14}$$

      Eq. 4-2-2. Differential Equations of Maz System

      Equations (4-2-2-1) and (4-2-2-8) describe the concentration of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers. Since Equations (4-2-2-2), (4-2-2-3), (4-2-2-5), (4-2-2-6), (4-2-2-7), (4-2-2-9), (4-2-2-10), (4-2-2-12), (4-2-2-13) and (4-2-2-14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.

    • 1.2.2. Cleavage by MazF dimers

      MazF dimers recognize and cleave ACAs in mRNAs, thus acting as Toxin.

      We estimated the rate of recognitions of ACA sequences by MazF dimers at \(1-(1-f)^n\), where the number of ACA sequences in mRNA.

      Then, we expressed the rate of degradation by MazF dimers in \(F(1-(1-f)^{f_{mRNA}})\) and obtain the following set of differential equations.

      Snow White

      $$\frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] $$
      $$ \tag{4-2-3-1} $$

      $$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
      $$ \tag{4-2-3-2} $$

      $$\frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\        - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
      $$\tag{4-2-3-3}$$

      $$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
      $$ \tag{4-2-3-4} $$

      Queen

      $$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] $$
      $$ \tag{4-2-3-5} %%

      $$ \frac{d[mRNA_{LasI}]}{dt} = leak_{P_{rhl}} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\        - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$
      $$ \tag{4-2-3-6} $$

      $$\frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\        - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
      $$\tag{4-2-3-7}$$

      $$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
      $$ \tag{4-2-3-8} $$

      Eq. 4-2-3. Differential Equations of mRNAs

      The equations above comprise terms of production, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.

  • 1.3. Signal Molecules

    Snow White expresses RhlI under Plux induced by C12, the Queen expresses LasI under Prhl induced by C4 and the Prince expresses AmiE under Plux induced by C12.

    The mRNAs of Snow White and the Queen decrease from original degradation and the cleavage at ACA sequences by MazF dimers. On the other hand, those of the Prince don’t have any MazF gene so they decrease from only original degradation.

    After translation, C12AHL and C4 are enzymatically synthesized by LasI and RhlI from some substrates respectively. For simplicity, we assumed that the amount of substrates is sufficient so that the C12AHL / C4 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.

    C4 decreases from original degradation meanwhile C12AHL decreases from both original degradation and degradation by AmiE, which Prince products.

    Applying mass action kinetic laws, we obtain the following set of differential equations.

    Equations (1), (4) and (7) describe the concentrations of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.

    The other ODEs describe how the concentrations of materials change in individuals, on the other hand (3), (6) describe the concentrations of C4 C12AHL in the whole culture medium.

2. Yaf System (Alternative Design)

We also designed alternative design with Yaf system as our plan B.

After translation, YafO and YafN exert their function alone.

YafO forms an heterodimer with YafO and suppress the toxicity of YafO.

Applying mass action kinetic laws, we obtain the following set of differential equations.

Equations (1) and (6) describe the concentration of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation by YafO. Since Equations (5) and (10) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding dissociation.