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

 
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<div id="main_contents">
 
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<div id="page_header" class="container container_top">
 
<div id="page_header" class="container container_top">
<h1 align="center">Detailed Description</h1>
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<h1 align="center">Detailed description</h1>
 
</div><!-- page_header -->
 
</div><!-- page_header -->
 
<div id="modeling_development" class="container">
 
<div id="modeling_development" class="container">
 
<div id="modeling_development_header" class="container_header">
 
<div id="modeling_development_header" class="container_header">
<h2><span>Modeling Development</span></h2>
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<h2><span>Model development</span></h2>
 
</div><!-- /modeling_development_header -->
 
</div><!-- /modeling_development_header -->
 
<div id="modeling_development_contents" class="container_contents">
 
<div id="modeling_development_contents" class="container_contents">
 
<p class="normal_text">To simulate the cell-cell communication system, we developed an ordinary differential equation model.
 
<p class="normal_text">To simulate the cell-cell communication system, we developed an ordinary differential equation model.
The following sentences describe how the equations were developed with Maz system.
+
The following segments describe in detail how the equations were developed with the <span style ="font-style : italic">mazEF</span> system.
 
</p>
 
</p>
 
</div><!-- modeling_development_contents -->
 
</div><!-- modeling_development_contents -->
 
<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"><div align="center"><img src="https://static.igem.org/mediawiki/2016/5/52/T--Tokyo_Tech--Model_Details_1.png" /></div></a>
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<div style="text-align: center;">
<p class="caption"><span style="font-weight: bold;">Fig. 4-2-1. </span> Maz System Gene Circuit</p>
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<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.5-5-1. The <span style ="font-style : italic">mazEF</span> system gene circuit</span></p></div>
 
<div id="modeling_detail" class="off">
 
<div id="modeling_detail" class="off">
 
<div id="modeling_detail_wrapper">
 
<div id="modeling_detail_wrapper">
 
<div id="modeling_detail_expressions">
 
<div id="modeling_detail_expressions">
<h2>Differencial Equations</h2>
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<h2>Differencial equations</h2>
 
<h3>Snow White</h3>
 
<h3>Snow White</h3>
 
\begin{equation}
 
\begin{equation}
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</div><!-- /modeling_detail_expressions -->
 
</div><!-- /modeling_detail_expressions -->
 
<div id="modeling_detail_parameter">
 
<div id="modeling_detail_parameter">
<h2>Explanation about Parameters</h2>
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<h2>Explanation about parameters</h2>
 
<table border="1" style="margin: auto;">
 
<table border="1" style="margin: auto;">
 
<tbody>
 
<tbody>
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<p class="normal_text" style="text-align:center;"><a href="javascript:void(0);" onClick="show('modeling_detail');" class="showHidden">Expressions</a></p>
 
<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. Cell Population</h2>
+
<li><h2>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 /> $$  \tag{1-1} $$</p>
 
<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 /> $$  \tag{1-1} $$</p>
 
<p>$$
 
<p>$$
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<p>$$
 
<p>$$
 
\frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{1-3} $$ </p>
 
\frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{1-3} $$ </p>
<p class="caption"><span style="font-weight: bold;">Eq.1. </span> Differential Equation of Cell Population</p>
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<p class="caption"><span style="font-weight: bold;">Eq.1. </span> Differential equation of cell population</p>
<p class="normal_text">The equations above describe how cells grow in the culture.
+
<p class="normal_text">The equations above describe how each cell grows in the culture.
Equations (1-1), (1-2) and (1-3) describe the populations of Snow White, the Queen and the Prince. (1-3) is described by the logistic growth equation, but (1-1) and (1-2) are represented by the growth inhibition by MazF dimers.
+
Equations (1-1), (1-2) and (1-3) describe the populations of Snow White <span style ="font-style : italic">coli</span>, the Queen <span style ="font-style : italic">coli</span> and the Prince <span style ="font-style : italic">coli</span>. (1-3) is described by the logistic growth equation, but (1-1) and (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 high, and its value converges to 1 when the concentration of MazF dimers is low.</p>
 
This factor is designed so that its value is small when the concentration of MazF dimers is high, and its value converges to 1 when the concentration of MazF dimers is low.</p>
 
</li><!-- /1.1. Cell Population -->
 
</li><!-- /1.1. Cell Population -->
<li><h2>2. Maz System</h2>
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<li><h2>2. The <span style ="font-style : italic">mazEF</span> system</h2>
 
<ul id="modeling_maz_system" class="non_dotted_list">
 
<ul id="modeling_maz_system" class="non_dotted_list">
<li><h3>2.1. Expression of Maz System</h3>
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<li><h3>2.1. Expression of the <span style ="font-style : italic">mazEF</span> system</h3>
<p class="normal_text">After translation, MazE and MazF each form an  dimer which can be activated to exert its function.</p>
+
<p class="normal_text">After translation, MazE and MazF each form a dimer which can be activated to exert its function.<div style="text-align: center;"></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 />
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<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"  style="width: 600px;"/></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>
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<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"  style="width: 600px;"/></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 MazF dimers sandwich a MazE 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>
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<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"  style="width: 700px;"/></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>
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<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: 800px;" /></a>
<p class="caption"><span style="font-weight: bold">Fig. 4-2-2. </span> Reaction of Maz System</p>
+
<p class="caption"><span style="font-weight: bold">Fig.5-5-2. Reaction of the <span style ="font-style : italic">mazEF</span> system</span></p></div>
<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 <span style ="font-style : italic">coli</span> and the Queen <span style ="font-style : italic">coli</span> decrease because of their original degradation and the cleavage at ACA sequences by MazF dimers.<br>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>
+
 
<h3>Snow White</h3>
 
<h3>Snow White</h3>
<p>$$
+
<p>$$\frac{d[mRNA_{MazF}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\
\frac{d[mRNA_{MazF}]}{dt} =  leak_{Plux} + \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{2-1} $$</p>
 
       - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$<br />$$ \tag{2-1} $$</p>
 
<p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$ <br />
 
<p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$ <br />
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       + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{2-13} $$</p>
 
       + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{2-13} $$</p>
 
<p>$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$ <br />$$ \tag{2-14}$$</p>
 
<p>$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$ <br />$$ \tag{2-14}$$</p>
<p class="caption"><span style="font-weight: bold">Eq. 2. </span>Differential Equations of Maz System</p>
+
<p class="caption"><span style="font-weight: bold">Eq. 2. </span>Differential equations of the <span style ="font-style : italic">mazEF</span> system</p>
<p class="normal_text">Equations (2-1) and (2-8) describe the concentration of mRNAs under the AHL inducing promoters.
+
<p class="normal_text">Equations (2-1) and (2-8) describe the concentration of mRNAs under AHL-inducible promoters. Thus, they comprise terms of production by leaky expression of promoters, terms of production by Hill function dependent on the concentration of C4HSL (C4) and 3OC12HSL (C12), terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.<br>
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 (2-2), (2-3), (2-5), (2-6), (2-7), (2-9), (2-10), (2-12), (2-13) and (2-14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.</p>
Since Equations (2-2), (2-3), (2-5), (2-6), (2-7), (2-9), (2-10), (2-12), (2-13) and (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 the <span style ="font-style : italic">mazEF</span> system -->
</li><!-- /1.2.1. Expression of Maz System -->
+
 
<li><h3>2.2. Cleavage by MazF dimers</h3>
 
<li><h3>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 ACA sequences in mRNAs, thus acting as a toxin.We estimated the rate of recognitions of ACA sequences by MazF dimers at $$ 1-(1-f)^n $$ where n is the number of ACA sequences in mRNA and  f is the probability of distinction of ACA sequences on each 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.</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>
+
 
<h3>Snow White</h3>
 
<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]
 
<p>$$\frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF]
 
$$ <br />$$ \tag{3-1} $$</p>
 
$$ <br />$$ \tag{3-1} $$</p>
<p>$$
+
<p>$$\frac{d[mRNA_{RhlI}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
\frac{d[mRNA_{RhlI}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
+
 
<br />$$ \tag{3-2} $$</p>
 
<br />$$ \tag{3-2} $$</p>
 
<p>$$\frac{d[mRNA_{MazF}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\
 
<p>$$\frac{d[mRNA_{MazF}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\
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<h3>Queen</h3>
 
<h3>Queen</h3>
 
<p>$$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF]
 
<p>$$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF]
$$<br />$$ \tag{3-5} %%</p>
+
$$<br />$$ \tag{3-5} $$</p>
 
<p>$$
 
<p>$$
 
\frac{d[mRNA_{LasI}]}{dt} =  leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\
 
\frac{d[mRNA_{LasI}]}{dt} =  leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\
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<p>$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
 
<p>$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
 
<br />$$ \tag{3-8} $$</p>
 
<br />$$ \tag{3-8} $$</p>
<p class="caption"><span style="font-weight: bold">Eq. 3. </span>Differential Equations of mRNAs</p>
+
<p class="caption"><span style="font-weight: bold">Eq. 3. </span>Differential equations of mRNA concentrations</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 only original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
 
</p>
 
</p>
 
</li><!-- /1.2.2. Cleavage by MazF dimers -->
 
</li><!-- /1.2.2. Cleavage by MazF dimers -->
 
</ul><!-- /modeling_maz_system -->
 
</ul><!-- /modeling_maz_system -->
</li><!-- /1.2. Maz System -->
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</li><!-- /1.2. the Maz system -->
<li><h2>3. Signal Molecules</h2>
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<li><h2>3. Signaling molecules</h2>
<a href="https://static.igem.org/mediawiki/2016/c/c4/T--Tokyo_Tech--Model_Details_6.png"><img src="https://static.igem.org/mediawiki/2016/c/c4/T--Tokyo_Tech--Model_Details_6.png" style="width: 300px;" /></a>
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<div style="text-align: center;"><a href="https://static.igem.org/mediawiki/2016/c/c4/T--Tokyo_Tech--Model_Details_6.png"><img src="https://static.igem.org/mediawiki/2016/c/c4/T--Tokyo_Tech--Model_Details_6.png" style="width: 800px;" /></a>
<p class="caption"><span style="font-weight:bold;">Fig. 4-2-3. </span> Reaction of Signal Molecules</p>
+
<p class="caption"><span style="font-weight:bold;">Fig.5-5-3. Reaction of signaling molecules</span></p></div>
<p class="normal_text">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.</p>
+
<p class="normal_text">Snow White <span style ="font-style : italic">coli</span> expresses RhlI under Plux induced by C12, the Queen <span style ="font-style : italic">coli</span> expresses LasI under Prhl induced by C4 and the Prince <span style ="font-style : italic">coli</span>  expresses AmiE under Plux induced by C12.<br>
<p class="normal_text">The mRNAs of Snow White and the Queen decrease from original degradation and the cleavage at ACA sequences by MazF dimers.
+
The mRNAs of Snow White <span style ="font-style : italic">coli</span> and the Queen <span style ="font-style : italic">coli</span> decrease from original degradation and the cleavage at ACA sequences by MazF dimers. On the other hand, those of the Prince <span style ="font-style : italic">coli</span> don’t have any MazF genes so they decrease from original degradation only.<br>
On the other hand, those of the Prince don’t have any MazF gene so they decrease from only original degradation.</p>
+
After translation, C4 and C12 are enzymatically synthesized by LasI and RhlI from some substrates respectively.<br>
<p class="normal_text">After translation, C12 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 C4 and C12 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.C4 decreases from original degradation only meanwhile C12 decreases from both original degradation and degradation by AmiE, which the Prince <span style ="font-style : italic">coli</span> produces.<br>
For simplicity, we assumed that the amount of substrates is sufficient so that the C12 / C4 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.</p>
+
Applying mass action kinetic laws, we obtain the following set of differential equations.</p>
<p class="normal_text">C4 decreases from original degradation meanwhile C12 decreases from both original degradation and degradation by AmiE, which Prince products.</p>
+
<p class="normal_text">Applying mass action kinetic laws, we obtain the following set of differential equations.</p>
+
 
<p>$$ \frac{d[mRNA_{RhlI}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] $$<br />$$\tag{4-1}$$</p>
 
<p>$$ \frac{d[mRNA_{RhlI}]}{dt} =  leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] $$<br />$$\tag{4-1}$$</p>
 
<p>$$\frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \tag{4-2}$$</p>
 
<p>$$\frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \tag{4-2}$$</p>
<p>$$ \frac{d[Rhl AHL]}{dt} = p_{Rhl}[RhlI]P_{Snowwhite} - d_{RhlAHL}[RhlAHL] \tag{4-3} $$</p>
+
<p>$$ \frac{d[C4]}{dt} = p_{Rhl}[RhlI]P_{Snowwhite} - d_{C4}[C4] \tag{4-3} $$</p>
 
<p>$$ \frac{d[mRNA_{LasI}]}{dt} =  leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$<br />$$\tag{4-4}$$</p>
 
<p>$$ \frac{d[mRNA_{LasI}]}{dt} =  leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$<br />$$\tag{4-4}$$</p>
 
<p>$$\frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \tag{4-5}$$</p>
 
<p>$$\frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \tag{4-5}$$</p>
<p>$$\frac{d[LasAHL]}{dt} = p_{Las}[LasI]P_{Stepmother} - d_{LasAHL}[LasAHL] - D[LasAHL][AmiE]$$ <br /> $$\tag{4-6}$$</p>
+
<p>$$\frac{d[C12]}{dt} = p_{C12}[LasI]P_{Stepmother} - d_{C12}[C12] - D[C12][AmiE]$$ <br /> $$\tag{4-6}$$</p>
 
<p>$$\frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{AmiE}]$$ <br />$$\tag{4-7}$$</p>
 
<p>$$\frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{AmiE}]$$ <br />$$\tag{4-7}$$</p>
 
<p>$$\frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \tag{4-8} $$</p>
 
<p>$$\frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \tag{4-8} $$</p>
  
<p class="caption"><span style="font-weight: bold;">Eq. 4. </span> Differential Equations of Signaling Molecules</p>
+
<p class="caption"><span style="font-weight: bold;">Eq. 4. </span> Differential equations of signaling molecules</p>
  
<p class="normal_text">Equations (4-1), (4-4) and (4-7) describe the concentrations of mRNAs under the AHL inducing promoters.
+
<p class="normal_text">Equations (4-1), (4-4) and (4-7) describe the concentrations of mRNAs under the AHL-inducible promoters.Thus, they comprise terms of production by leaky expression of promoters, terms of production by Hill function depending on the concentration of C4 and C12, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.<br>
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.</p>
+
The other ODEs describe how the concentrations of materials change in individuals, on the other hand (4-3), (4-6) describe the concentrations of C4 and C12 in the whole culture medium.</p>
<p class="normal_text">The other ODEs describe how the concentrations of materials change in individuals, on the other hand (4-3), (4-6) describe the concentrations of C4 C12 in the whole culture medium.</p>
+
 
</li>
 
</li>
 
</ul><!-- /modeling_maz_list -->
 
</ul><!-- /modeling_maz_list -->
Line 411: Line 403:
 
</div><!-- /modeling_maz_system -->
 
</div><!-- /modeling_maz_system -->
  
<!-- /UNTIL HERE MAZ SYSTEM -->
+
<!-- /UNTIL HERE the Maz system -->
  
  
Line 433: Line 425:
 
<td>$$ 0.0123 $$</td>
 
<td>$$ 0.0123 $$</td>
 
<td>Growth rate of each cells</td>
 
<td>Growth rate of each cells</td>
<td>Fitted to experimental data</td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#population">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 439: Line 431:
 
<td>$$3.3 $$</td>
 
<td>$$3.3 $$</td>
 
<td>Carrying capacity</td>
 
<td>Carrying capacity</td>
<td>Fitted to experimental data</td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#population">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 445: Line 437:
 
<td>$$ 0.462234 nM^{-1} min^{-1} $$ </td>
 
<td>$$ 0.462234 nM^{-1} min^{-1} $$ </td>
 
<td>Effect of MazF dimer on growth rate of each cells</td>
 
<td>Effect of MazF dimer on growth rate of each cells</td>
<td>Fitted to experimental data</td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#population">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 457: Line 449:
 
<td>$$ 2.26 min^{-1} $$</td>
 
<td>$$ 2.26 min^{-1} $$</td>
 
<td>Leakage of Plux</td>
 
<td>Leakage of Plux</td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 463: Line 455:
 
<td>$$ 4.654 min^{-1} $$</td>
 
<td>$$ 4.654 min^{-1} $$</td>
 
<td>Leakage of Prhl</td>
 
<td>Leakage of Prhl</td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 469: Line 461:
 
<td>$$ 6.984 nM^{-1} min^{-1} $$ </td>
 
<td>$$ 6.984 nM^{-1} min^{-1} $$ </td>
 
<td>Maximum transcription rate of under streams of Plux</td>
 
<td>Maximum transcription rate of under streams of Plux</td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 475: Line 467:
 
<td>$$ 14.95 nM^{-1} min^{-1} $$ </td>
 
<td>$$ 14.95 nM^{-1} min^{-1} $$ </td>
 
<td>Maximum transcription rate of understreams of Prhl</td>
 
<td>Maximum transcription rate of understreams of Prhl</td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 481: Line 473:
 
<td>$$ 0.76 $$</td>
 
<td>$$ 0.76 $$</td>
 
<td>Hill coefficient for Plux</td>
 
<td>Hill coefficient for Plux</td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 487: Line 479:
 
<td>$$ 5 $$</td>
 
<td>$$ 5 $$</td>
 
<td>Hill cofficient for Prhl</td>
 
<td>Hill cofficient for Prhl</td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 493: Line 485:
 
<td>$$ 116.24nM $$</td>
 
<td>$$ 116.24nM $$</td>
 
<td>Lumped parameter for the Lux system</td>
 
<td>Lumped parameter for the Lux system</td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 499: Line 491:
 
<td>$$ 1000 nM $$</td>
 
<td>$$ 1000 nM $$</td>
 
<td>Lumped parameter for the Rhl system</td>
 
<td>Lumped parameter for the Rhl system</td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 511: Line 503:
 
<td>$$ 0.299 $$</td>
 
<td>$$ 0.299 $$</td>
 
<td>The probability of distinction of ACA sequences on each mRNA</td>
 
<td>The probability of distinction of ACA sequences on each mRNA</td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
 
<td>$$ f_{mRNA_{RFP}} $$</td>
 
<td>$$ f_{mRNA_{RFP}} $$</td>
 
<td>$$ 10 $$</td>
 
<td>$$ 10 $$</td>
<td>$$ The number of ACA sequences on mRNA_{RFP} $$</td>
+
<td>The number of ACA sequences on mRNA_{RFP}</td>
 
<td> Extraction of data </td>
 
<td> Extraction of data </td>
 
</tr>
 
</tr>
Line 522: Line 514:
 
<td>$$ f_{mRNA_{GFP}} $$</td>
 
<td>$$ f_{mRNA_{GFP}} $$</td>
 
<td>$$ 23 $$</td>
 
<td>$$ 23 $$</td>
<td>$$ The number of ACA sequences on mRNA_{GFP} $$</td>
+
<td>The number of ACA sequences on mRNA_{GFP}</td>
 
<td> Extraction of data </td>
 
<td> Extraction of data </td>
 
</tr>
 
</tr>
Line 528: Line 520:
 
<td>$$ f_{mRNA_{RhlI}} $$</td>
 
<td>$$ f_{mRNA_{RhlI}} $$</td>
 
<td>$$ 1 $$</td>
 
<td>$$ 1 $$</td>
<td>$$ The number of ACA sequences on mRNA_{RhlI} $$</td>
+
<td>The number of ACA sequences on mRNA_{RhlI}</td>
 
<td> Extraction of data </td>
 
<td> Extraction of data </td>
 
</tr>
 
</tr>
Line 534: Line 526:
 
<td>$$ f_{mRNA_{LasI}} $$</td>
 
<td>$$ f_{mRNA_{LasI}} $$</td>
 
<td>$$ 10 $$</td>
 
<td>$$ 10 $$</td>
<td>$$ The number of ACA sequences on mRNA_{LasI} $$</td>
+
<td>The number of ACA sequences on mRNA_{LasI}</td>
 
<td> Extraction of data </td>
 
<td> Extraction of data </td>
 
</tr>
 
</tr>
Line 540: Line 532:
 
<td>$$ f_{mRNA_{MazF}} $$</td>
 
<td>$$ f_{mRNA_{MazF}} $$</td>
 
<td> $$2$$ </td>
 
<td> $$2$$ </td>
<td>$$ The number of ACA sequences on mRNA_{MazF} $$</td>
+
<td>The number of ACA sequences on mRNA_{MazF}</td>
 
<td> Extraction of data </td>
 
<td> Extraction of data </td>
 
</tr>
 
</tr>
Line 546: Line 538:
 
<td>$$ f_{mRNA_{MazE}} $$</td>
 
<td>$$ f_{mRNA_{MazE}} $$</td>
 
<td> $$2$$ </td>
 
<td> $$2$$ </td>
<td>$$ The number of ACA sequences on mRNA_{MazE} $$</td>
+
<td>The number of ACA sequences on mRNA_{MazE}</td>
 
<td> Extraction of data </td>
 
<td> Extraction of data </td>
 
</tr>
 
</tr>
Line 559: Line 551:
 
<td> $$ 6.82 nM_{-1} min_{-1} $$ </td>
 
<td> $$ 6.82 nM_{-1} min_{-1} $$ </td>
 
<td>Formation rate of MazF dimer </td>
 
<td>Formation rate of MazF dimer </td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 565: Line 557:
 
<td> $$ 6.24 nM^{-1} min^{-1} $$ </td>
 
<td> $$ 6.24 nM^{-1} min^{-1} $$ </td>
 
<td>Formation rate of MazF dimer </td>
 
<td>Formation rate of MazF dimer </td>
<td> Fitted to experimental data </td>
+
<td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 571: Line 563:
 
<td> $$ 3.46 nM^{-1} min^{-1} $$ </td>
 
<td> $$ 3.46 nM^{-1} min^{-1} $$ </td>
 
<td>Formation rate of MazF dimer </td>
 
<td>Formation rate of MazF dimer </td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 577: Line 569:
 
<td> $$ 7.25 min^{-1} $$ </td>
 
<td> $$ 7.25 min^{-1} $$ </td>
 
<td>Dissociation rate of MazF dimer </td>
 
<td>Dissociation rate of MazF dimer </td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 583: Line 575:
 
<td> $$ 4.51 nM^{-1} min^{-1} $$ </td>
 
<td> $$ 4.51 nM^{-1} min^{-1} $$ </td>
 
<td>Formation rate of Maz hexamer </td>
 
<td>Formation rate of Maz hexamer </td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 589: Line 581:
 
<td> $$ 4.05 min^{-1} $$ </td>
 
<td> $$ 4.05 min^{-1} $$ </td>
 
<td>Dissociation rate of Maz hexamer </td>
 
<td>Dissociation rate of Maz hexamer </td>
<td> Fitted to experimental data </td>
+
<td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</tr>
 
</tr>
 
<tr>
 
<tr>
Line 643: Line 635:
 
    <td>$$ 0.7 min^{-1} $$ </td>
 
    <td>$$ 0.7 min^{-1} $$ </td>
 
    <td> Degradation rate of MazF </td>
 
    <td> Degradation rate of MazF </td>
    <td> Fitted to experimental data </td>
+
    <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</td>
 
</td>
 
<tr>
 
<tr>
Line 649: Line 641:
 
    <td>$$ 0.17 min^{-1} $$ </td>
 
    <td>$$ 0.17 min^{-1} $$ </td>
 
    <td> Degradation rate of MazF dimer </td>
 
    <td> Degradation rate of MazF dimer </td>
    <td> Fitted to experimental data </td>
+
    <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td>
 
</td>
 
</td>
 
<tr>
 
<tr>
Line 655: Line 647:
 
    <td>$$ 0.55 min^{-1} $$ </td>
 
    <td>$$ 0.55 min^{-1} $$ </td>
 
    <td> Degradation rate of MazE </td>
 
    <td> Degradation rate of MazE </td>
    <td> Fitted to experimental data </td>
+
    <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td>
 
</td>
 
</td>
 
<tr>
 
<tr>
Line 661: Line 653:
 
    <td>$$ 0.416 min^{-1} $$ </td>
 
    <td>$$ 0.416 min^{-1} $$ </td>
 
    <td> Degradation rate of MazE dimer </td>
 
    <td> Degradation rate of MazE dimer </td>
    <td> Fitted to experimental data </td>
+
    <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td>
 
</td>
 
</td>
 
<tr>
 
<tr>
Line 667: Line 659:
 
    <td>$$ 0.511 min^{-1} $$ </td>
 
    <td>$$ 0.511 min^{-1} $$ </td>
 
    <td> Degradation rate of Maz hexameter </td>
 
    <td> Degradation rate of Maz hexameter </td>
    <td> Fitted to experimental data </td>
+
    <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td>
 
</td>
 
</td>
 
<tr>
 
<tr>

Latest revision as of 00:46, 20 October 2016

Model development

To simulate the cell-cell communication system, we developed an ordinary differential equation model. The following segments describe in detail how the equations were developed with the mazEF system.

Fig.5-5-1. The mazEF 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_{Plux} + \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_{Plux} + \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_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] \end{equation} \begin{equation} \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\        + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{equation} \begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE] \end{equation} \begin{equation} \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[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_{Prhl} + \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_{Plux} + \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_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] \end{equation} \begin{equation} \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\        + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] \end{equation} \begin{equation} \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE] \end{equation} \begin{equation} \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[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_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - 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 downstream of Pcon
$$leak_{Plux}$$ Leakage of Plux
$$leak_{Prhl}$$ Leakage of Prhl
$$\kappa_{Lux}$$ Maximum transcription rate of mRNA under Plux
$$\kappa_{Rhl}$$ Maximum transcription rate of downstream of Prhl
$$n_{Lux}$$ Hill coefficient for Plux
$$n_{Rhl}$$ Hill coefficient for Prhl
$$K_{mLux}$$ 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
$$k_{DiMazF}$$ Formation rate of MazF dimer
$$k_{-DiMazF}$$ Dissociation rate of MazF dimer
$$k_{DiMazE}$$ Formation rate of MazE dimer
$$k_{-DiMazE}$$ 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. Cell population

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

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

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

    Eq.1. Differential equation of cell population

    The equations above describe how each cell grows in the culture. Equations (1-1), (1-2) and (1-3) describe the populations of Snow White coli, the Queen coli and the Prince coli. (1-3) is described by the logistic growth equation, but (1-1) and (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 high, and its value converges to 1 when the concentration of MazF dimers is low.

  • 2. The mazEF system

    • 2.1. Expression of the mazEF system

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


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

      Fig.5-5-2. Reaction of the mazEF system

      The mRNAs of Snow White coli and the Queen coli decrease because of their original degradation and the cleavage at ACA sequences by MazF dimers.
      Applying mass action kinetic laws, we obtain the following set of differential equations.

      Snow White

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

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

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

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

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

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

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

      Queen

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

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

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

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

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

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

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

      Eq. 2. Differential equations of the mazEF system

      Equations (2-1) and (2-8) describe the concentration of mRNAs under AHL-inducible promoters. Thus, they comprise terms of production by leaky expression of promoters, terms of production by Hill function dependent on the concentration of C4HSL (C4) and 3OC12HSL (C12), terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
      Since equations (2-2), (2-3), (2-5), (2-6), (2-7), (2-9), (2-10), (2-12), (2-13) and (2-14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.

    • 2.2. Cleavage by MazF dimers

      MazF dimers recognize and cleave ACA sequences in mRNAs, thus acting as a toxin.We estimated the rate of recognitions of ACA sequences by MazF dimers at $$ 1-(1-f)^n $$ where n is the number of ACA sequences in mRNA and f is the probability of distinction of ACA sequences on each 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

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

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

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

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

      Queen

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

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

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

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

      Eq. 3. Differential equations of mRNA concentrations

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

  • 3. Signaling molecules

    Fig.5-5-3. Reaction of signaling molecules

    Snow White coli expresses RhlI under Plux induced by C12, the Queen coli expresses LasI under Prhl induced by C4 and the Prince coli expresses AmiE under Plux induced by C12.
    The mRNAs of Snow White coli and the Queen coli decrease from original degradation and the cleavage at ACA sequences by MazF dimers. On the other hand, those of the Prince coli don’t have any MazF genes so they decrease from original degradation only.
    After translation, C4 and C12 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 C4 and C12 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.C4 decreases from original degradation only meanwhile C12 decreases from both original degradation and degradation by AmiE, which the Prince coli produces.
    Applying mass action kinetic laws, we obtain the following set of differential equations.

    $$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] $$
    $$\tag{4-1}$$

    $$\frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \tag{4-2}$$

    $$ \frac{d[C4]}{dt} = p_{Rhl}[RhlI]P_{Snowwhite} - d_{C4}[C4] \tag{4-3} $$

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

    $$\frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \tag{4-5}$$

    $$\frac{d[C12]}{dt} = p_{C12}[LasI]P_{Stepmother} - d_{C12}[C12] - D[C12][AmiE]$$
    $$\tag{4-6}$$

    $$\frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{AmiE}]$$
    $$\tag{4-7}$$

    $$\frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \tag{4-8} $$

    Eq. 4. Differential equations of signaling molecules

    Equations (4-1), (4-4) and (4-7) describe the concentrations of mRNAs under the AHL-inducible promoters.Thus, they comprise terms of production by leaky expression of promoters, terms of production by Hill function depending on the concentration of C4 and C12, 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 (4-3), (4-6) describe the concentrations of C4 and C12 in the whole culture medium.

Parameters

Parameter Value Description Reference
$$ g $$ $$ 0.0123 $$ Growth rate of each cells Fitted to experimental data
$$ P_{max} $$ $$3.3 $$ Carrying capacity Fitted to experimental data
$$ E_{DiMazF} $$ $$ 0.462234 nM^{-1} min^{-1} $$ Effect of MazF dimer on growth rate of each cells Fitted to experimental data
$$ k $$ $$5 min^{-1}$$ Transcription rate of downstream of Ptet Reference[1]
$$ leak_{Plux} $$ $$ 2.26 min^{-1} $$ Leakage of Plux Fitted to experimental data
$$ leak_{Prhl} $$ $$ 4.654 min^{-1} $$ Leakage of Prhl Fitted to experimental data
$$ κ_{Lux} $$ $$ 6.984 nM^{-1} min^{-1} $$ Maximum transcription rate of under streams of Plux Fitted to experimental data
$$ κ_{Rhl} $$ $$ 14.95 nM^{-1} min^{-1} $$ Maximum transcription rate of understreams of Prhl Fitted to experimental data
$$ n_{Lux} $$ $$ 0.76 $$ Hill coefficient for Plux Fitted to experimental data
$$ n_{Rhl} $$ $$ 5 $$ Hill cofficient for Prhl Fitted to experimental data
$$ K_{mLux} $$ $$ 116.24nM $$ Lumped parameter for the Lux system Fitted to experimental data
$$ K_{mRhl} $$ $$ 1000 nM $$ Lumped parameter for the Rhl system Fitted to experimental data
$$ F_{DiMazF} $$ $$ 5 nM^{-1} min^{-1} $$ Cutting rate at ACA sequences on mRNA by MazF dimers Assumption
$$ f $$ $$ 0.299 $$ The probability of distinction of ACA sequences on each mRNA Fitted to experimental data
$$ f_{mRNA_{RFP}} $$ $$ 10 $$ The number of ACA sequences on mRNA_{RFP} Extraction of data
$$ f_{mRNA_{GFP}} $$ $$ 23 $$ The number of ACA sequences on mRNA_{GFP} Extraction of data
$$ f_{mRNA_{RhlI}} $$ $$ 1 $$ The number of ACA sequences on mRNA_{RhlI} Extraction of data
$$ f_{mRNA_{LasI}} $$ $$ 10 $$ The number of ACA sequences on mRNA_{LasI} Extraction of data
$$ f_{mRNA_{MazF}} $$ $$2$$ The number of ACA sequences on mRNA_{MazF} Extraction of data
$$ f_{mRNA_{MazE}} $$ $$2$$ The number of ACA sequences on mRNA_{MazE} Extraction of data
$$ α $$ $$ 0.04 min_{-1} $$ Translation rate of Assumption
$$ k_{DiMazF}$$ $$ 6.82 nM_{-1} min_{-1} $$ Formation rate of MazF dimer Fitted to experimental data
$$ k_{-Di_{MazF}}$$ $$ 6.24 nM^{-1} min^{-1} $$ Formation rate of MazF dimer Fitted to experimental data
$$ k_{Di_{MazE}}$$ $$ 3.46 nM^{-1} min^{-1} $$ Formation rate of MazF dimer Fitted to experimental data
$$ k_{-Di_{MazE}}$$ $$ 7.25 min^{-1} $$ Dissociation rate of MazF dimer Fitted to experimental data
$$ k_{Hexa}$$ $$ 4.51 nM^{-1} min^{-1} $$ Formation rate of Maz hexamer Fitted to experimental data
$$ k_{-Hexa}$$ $$ 4.05 min^{-1} $$ Dissociation rate of Maz hexamer Fitted to experimental data
$$ p_{C4}$$ $$ 0.07 min^{-1} $$ Production rate of C4HSL by RhlI Assumption
$$ p_{C12}$$ $$ 0.07 min^{-1} $$ Production rate of 3OC12HSL by LasI Assumption
$$ D $$ $$ 0.1 nM^{-1} min^{-1} $$ Decomposition rate of 3OC12HSL by AmiE Assumption
$$ d $$ $$ 0.2773 min^{-1} $$ Degradation rate of mRNA Leference[2]
$$ d_{RFP} $$ $$ 0.005 min^{-1} $$ Degradation rate of RFP Assumption
$$ d_{GFP} $$ $$ 0.005 min^{-1} $$ Degradation rate of GFP Assumption
$$ d_{RhlI} $$ $$ 0.0167 min^{-1} $$ Degradation rate of RhlI Leference[1]
$$ d_{LasI} $$ $$ 0.0167 min^{-1} $$ Degradation rate of LasI Leference[1]
$$ d_{MazF} $$ $$ 0.7 min^{-1} $$ Degradation rate of MazF Fitted to experimental data
$$ d_{DiMazF} $$ $$ 0.17 min^{-1} $$ Degradation rate of MazF dimer Fitted to experimental data
$$ d_{MazE} $$ $$ 0.55 min^{-1} $$ Degradation rate of MazE Fitted to experimental data
$$ d_{DiMazE} $$ $$ 0.416 min^{-1} $$ Degradation rate of MazE dimer Fitted to experimental data
$$ d_{Hexa} $$ $$ 0.511 min^{-1} $$ Degradation rate of Maz hexameter Fitted to experimental data
$$ d_{C4} $$ $$ 0.000222 min^{-1} $$ Degradation rate of C4HSL Literature[3]
$$ d_{C12} $$ $$ 0.004 min^{-1} $$ Degradation rate of 3OC12HSL Literature[4]
$$ d_{AmiE} $$ $$ 0.001 min^{-1} $$ Degradation rate of AmiE Assumption