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background-image: url("https://static.igem.org/mediawiki/2016/8/83/Tokyo_Tech_Background.png"); | background-image: url("https://static.igem.org/mediawiki/2016/8/83/Tokyo_Tech_Background.png"); | ||
+ | background-repeat: no-repeat; | ||
+ | background-position: center center; | ||
+ | background-size: cover; | ||
+ | background-attachment: fixed; | ||
} | } | ||
#globalWrapper, #content{ | #globalWrapper, #content{ | ||
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− | #main_contents | + | #main_contents .normal_text{ |
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text-indent: 18px; | text-indent: 18px; | ||
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-webkit-transiton: all 0.6s ease; | -webkit-transiton: all 0.6s ease; | ||
transition: all 0.6s ease; | transition: all 0.6s ease; | ||
+ | } | ||
+ | /************************** this page setting ***************************/ | ||
+ | #modeling_detail_wrapper{ | ||
+ | font-size: 8px; | ||
+ | margin: 0; | ||
+ | } | ||
+ | #modeling_detail_wrapper table{ | ||
+ | font-size: 8px; | ||
+ | } | ||
+ | |||
+ | #modeling_detail_expressions{ | ||
+ | width: 45%; | ||
+ | margin-right: auto; | ||
+ | float: left; | ||
+ | } | ||
+ | #modeling_detail_parameters{ | ||
+ | width: 45%; | ||
+ | margin-left: auto; | ||
+ | float: left; | ||
+ | } | ||
+ | |||
+ | #main_contents p.caption{ | ||
+ | font-size: 16px; | ||
} | } | ||
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<div id="main_contents"> | <div id="main_contents"> | ||
<div id="page_header" class="container container_top"> | <div id="page_header" class="container container_top"> | ||
− | <h1 align="center"> | + | <h1 align="center">Detailed description</h1> |
− | <div id=" | + | </div><!-- page_header --> |
+ | <div id="modeling_development" class="container"> | ||
+ | <div id="modeling_development_header" class="container_header"> | ||
+ | <h2><span>Model development</span></h2> | ||
+ | </div><!-- /modeling_development_header --> | ||
+ | <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 | + | The following segments describe in detail how the equations were developed with the <span style ="font-style : italic">mazEF</span> system. |
− | + | </p> | |
− | + | </div><!-- modeling_development_contents --> | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | </div><!-- | + | |
<div id="modeling_maz_contents" class="container_contents"> | <div id="modeling_maz_contents" class="container_contents"> | ||
+ | <div style="text-align: center;"> | ||
<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> | <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. | + | <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 | + | <h2>Differencial equations</h2> |
<h3>Snow White</h3> | <h3>Snow White</h3> | ||
\begin{equation} | \begin{equation} | ||
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\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[mRNA_{RhlI}]}{dt} = leak_{ | + | \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} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
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\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[mRNA_{MazF}]}{dt} = leak_{ | + | \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] | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | ||
\end{equation} | \end{equation} | ||
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\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{ | + | \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] |
\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[DiMazF]}{dt} = k_{ | + | \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ |
+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] | + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] | ||
\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{ | + | \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE] |
\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[DiMazE]}{dt} = k_{ | + | \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ |
+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] | ||
\end{equation} | \end{equation} | ||
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\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[mRNA_{LasI}]}{dt} = leak_{ | + | \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] | - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] | ||
\end{equation} | \end{equation} | ||
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\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[mRNA_{MazF}]}{dt} = leak_{ | + | \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] | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | ||
\end{equation} | \end{equation} | ||
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\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{ | + | \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] |
\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[DiMazF]}{dt} = k_{ | + | \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ |
+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] | + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] | ||
\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{ | + | \frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE] |
\end{equation} | \end{equation} | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[DiMazE]}{dt} = k_{ | + | \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ |
+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE] | ||
\end{equation} | \end{equation} | ||
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<h3>Prince</h3> | <h3>Prince</h3> | ||
\begin{equation} | \begin{equation} | ||
− | \frac{d[mRNA_{AmiE}]}{dt} = leak_{ | + | \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} | \end{equation} | ||
\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 | + | <h2>Explanation about parameters</h2> |
<table border="1" style="margin: auto;"> | <table border="1" style="margin: auto;"> | ||
<tbody> | <tbody> | ||
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<tr><td>$$P_{max}$$ </td><td> Carrying capacity </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>$$E_{DiMazF}$$ </td><td> Effect of MazF dimer on growth rate</td></tr> | ||
− | <tr><td>$$k$$ </td><td> Transcription rate of | + | <tr><td>$$k$$ </td><td> Transcription rate of downstream of Pcon </td></tr> |
− | <tr><td>$$leak_{ | + | <tr><td>$$leak_{Plux}$$ </td><td> Leakage of Plux </td></tr> |
− | <tr><td>$$leak_{ | + | <tr><td>$$leak_{Prhl}$$ </td><td> Leakage of Prhl </td></tr> |
− | <tr><td>$$\kappa_{Lux}$$ </td><td> Maximum transcription rate of mRNA under | + | <tr><td>$$\kappa_{Lux}$$ </td><td> Maximum transcription rate of mRNA under Plux</td></tr> |
− | <tr><td>$$\kappa_{Rhl}$$ </td><td> Maximum transcription rate of | + | <tr><td>$$\kappa_{Rhl}$$ </td><td> Maximum transcription rate of downstream of Prhl </td></tr> |
− | <tr><td>$$n_{Lux}$$ </td><td> Hill coefficient for | + | <tr><td>$$n_{Lux}$$ </td><td> Hill coefficient for Plux</td></tr> |
− | <tr><td>$$n_{Rhl}$$ </td><td> Hill coefficient for | + | <tr><td>$$n_{Rhl}$$ </td><td> Hill coefficient for Prhl</td></tr> |
<tr><td>$$K_{mLux}$$ </td><td> Lumped paremeter for the Lux System</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>$$K_{mRhl}$$ </td><td> Lumped paremeter for the Rhl System</td></tr> | ||
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<tr><td>$$f_{mRNA_{MazE}}$$ </td><td> The number of ACA sequences in \(mRNA_{MazE}\) </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>$$\alpha$$ </td><td> Translation rate of Protein </td></tr> | ||
− | <tr><td>$$k_{ | + | <tr><td>$$k_{DiMazF}$$ </td><td> Formation rate of MazF dimer </td></tr> |
− | <tr><td>$$k_{- | + | <tr><td>$$k_{-DiMazF}$$ </td><td> Dissociation rate of MazF dimer </td></tr> |
− | <tr><td>$$k_{ | + | <tr><td>$$k_{DiMazE}$$ </td><td> Formation rate of MazE dimer </td></tr> |
− | <tr><td>$$k_{- | + | <tr><td>$$k_{-DiMazE}$$ </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> Formation rate of Maz hexamer </td></tr> | ||
<tr><td>$$k_{-Hexa}$$ </td><td> Dissociation rate of Maz hexamer</td></tr> | <tr><td>$$k_{-Hexa}$$ </td><td> Dissociation rate of Maz hexamer</td></tr> | ||
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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> | + | <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 /> $$ \tag{ | + | <p>$$ \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>$$ |
− | \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$ <br /> $$ \tag{ | + | \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$ <br /> $$ \tag{1-2} $$</p> |
− | <p>$$ | + | <p>$$ |
− | \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{ | + | \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="normal_text">The equations above describe how | + | <p class="caption"><span style="font-weight: bold;">Eq.1. </span> Differential equation of cell population</p> |
− | Equations ( | + | <p class="normal_text">The equations above describe how each cell grows in the culture. |
− | This factor is designed so that its value is small when the concentration of MazF dimers is | + | 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> | ||
</li><!-- /1.1. Cell Population --> | </li><!-- /1.1. Cell Population --> | ||
− | <li><h2> | + | <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> | + | <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 | + | <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 /> | + | <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> | + | <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 | + | <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> | + | <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: | + | <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. | + | <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 | + | <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> |
− | + | ||
<h3>Snow White</h3> | <h3>Snow White</h3> | ||
+ | <p>$$\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} $$</p> | ||
+ | <p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$ <br /> | ||
+ | $$\tag{2-2}$$</p> | ||
+ | <p>$$ \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ | ||
+ | + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$<br />$$ \tag{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{2-4} $$</p> | ||
+ | <p>$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE]$$ | ||
+ | <br />$$\tag{2-5}$$</p> | ||
<p>$$ | <p>$$ | ||
− | \frac{d[ | + | \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ |
− | + | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{2-6} $$</p> | |
− | + | <p>$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$ <br />$$ \tag{2-7}$$</p> | |
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + | ||
− | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{ | + | |
− | <p>$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$ <br />$$ \tag{ | + | |
<h3>Queen</h3> | <h3>Queen</h3> | ||
<p>$$ | <p>$$ | ||
− | \frac{d[mRNA_{MazF}]}{dt} = leak_{ | + | \frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ |
− | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$<br />$$ \tag{ | + | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$<br />$$ \tag{2-8} $$</p> |
− | <p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{ | + | <p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$ <br /> |
− | $$\tag{ | + | $$\tag{2-9}$$</p> |
− | <p>$$ \frac{d[DiMazF]}{dt} = k_{ | + | <p>$$ \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ |
− | + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$<br />$$ \tag{ | + | + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$<br />$$ \tag{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{ | + | <p>$$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$ <br />$$ \tag{2-11} $$</p> |
− | <p>$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{ | + | <p>$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE]$$ |
− | <br />$$\tag{ | + | <br />$$\tag{2-12}$$</p> |
<p>$$ | <p>$$ | ||
− | \frac{d[DiMazE]}{dt} = k_{ | + | \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ |
− | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <br />$$\tag{ | + | + 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{ | + | <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. | + | <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 ( | + | <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> |
− | + | 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 | + | <li><h3>2.2. Cleavage by MazF dimers</h3> |
− | <li><h3> | + | <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">MazF dimers recognize and cleave | + | |
− | + | ||
− | + | ||
<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{ | + | $$ <br />$$ \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_{ | + | <br />$$ \tag{3-2} $$</p> |
− | <br />$$ \tag{ | + | <p>$$\frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ |
− | <p>$$\frac{d[mRNA_{MazF}]}{dt} = leak_{ | + | |
- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | ||
− | $$ <br /> $$\tag{ | + | $$ <br /> $$\tag{3-3}$$</p> |
<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 />$$ \tag{ | + | <br />$$ \tag{3-4} $$</p> |
<h3>Queen</h3> | <h3>Queen</h3> | ||
<p>$$\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 />$$ \tag{ | + | $$<br />$$ \tag{3-5} $$</p> |
<p>$$ | <p>$$ | ||
− | \frac{d[mRNA_{LasI}]}{dt} = leak_{ | + | \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{ | + | - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$ <br />$$ \tag{3-6} $$</p> |
− | <p>$$\frac{d[mRNA_{MazF}]}{dt} = leak_{ | + | <p>$$\frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ |
- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | ||
− | $$ <br /> $$\tag{ | + | $$ <br /> $$\tag{3-7}$$</p> |
<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 />$$ \tag{ | + | <br />$$ \tag{3-8} $$</p> |
− | <p class="caption"><span style="font-weight: bold">Eq. | + | <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 | + | </li><!-- /1.2. the Maz system --> |
− | <li><h2> | + | <li><h2>3. Signaling molecules</h2> |
− | <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.< | + | <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.5-5-3. Reaction of signaling molecules</span></p></div> | |
− | + | <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> | |
− | + | 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> | |
− | + | After translation, C4 and C12 are enzymatically synthesized by LasI and RhlI from some substrates respectively.<br> | |
− | + | 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> | |
− | + | Applying mass action kinetic laws, we obtain the following set of differential equations.</p> | |
− | <p class="normal_text">Equations (1), (4) and (7) describe the concentrations of mRNAs under the AHL | + | <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[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[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \tag{4-5}$$</p> | ||
+ | <p>$$\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[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="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> | ||
+ | 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> | ||
</li> | </li> | ||
</ul><!-- /modeling_maz_list --> | </ul><!-- /modeling_maz_list --> | ||
</div><!-- /modeling_maz_contents --> | </div><!-- /modeling_maz_contents --> | ||
+ | |||
+ | |||
+ | <div class="back_link contents"> | ||
+ | <p class="normal_text"><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#mathematical_model">Back to Model page.</a></p> | ||
+ | </div> | ||
</div><!-- /modeling_maz_system --> | </div><!-- /modeling_maz_system --> | ||
− | <div id=" | + | |
− | <div id=" | + | <!-- /UNTIL HERE the Maz system --> |
− | <h2><span> | + | |
− | </div><!-- / | + | |
− | <div id=" | + | <div id="parameter_discriptions" class="container"> |
− | <p class="normal_text"> | + | <div id="parameter_discription_header" class="container_header"> |
− | <p class="normal_text"> | + | <h2><span>Parameters</span></h2> |
− | <p class="normal_text"> | + | </div><!-- /parameter_discription_header --> |
− | <p class=" | + | |
− | <p class="normal_text"> | + | <!-- この辺から表 --> |
− | + | <div id="parameter_discription_contents" class="container_contents"> | |
− | + | <table border="1" width="800px"> | |
− | </div><!-- / | + | <tbody> |
− | </div><!-- / | + | <tr> |
+ | <th>Parameter</th> | ||
+ | <th>Value</th> | ||
+ | <th>Description</th> | ||
+ | <th>Reference</th> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ g $$</td> | ||
+ | <td>$$ 0.0123 $$</td> | ||
+ | <td>Growth rate of each cells</td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#population">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ P_{max} $$</td> | ||
+ | <td>$$3.3 $$</td> | ||
+ | <td>Carrying capacity</td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#population">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ E_{DiMazF} $$</td> | ||
+ | <td>$$ 0.462234 nM^{-1} min^{-1} $$ </td> | ||
+ | <td>Effect of MazF dimer on growth rate of each cells</td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#population">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k $$</td> | ||
+ | <td>$$5 min^{-1}$$</td> | ||
+ | <td>Transcription rate of downstream of Ptet</td> | ||
+ | <td>Reference<a href ="#references">[1]</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ leak_{Plux} $$</td> | ||
+ | <td>$$ 2.26 min^{-1} $$</td> | ||
+ | <td>Leakage of Plux</td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ leak_{Prhl} $$</td> | ||
+ | <td>$$ 4.654 min^{-1} $$</td> | ||
+ | <td>Leakage of Prhl</td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ κ_{Lux} $$</td> | ||
+ | <td>$$ 6.984 nM^{-1} min^{-1} $$ </td> | ||
+ | <td>Maximum transcription rate of under streams of Plux</td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ κ_{Rhl} $$</td> | ||
+ | <td>$$ 14.95 nM^{-1} min^{-1} $$ </td> | ||
+ | <td>Maximum transcription rate of understreams of Prhl</td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ n_{Lux} $$</td> | ||
+ | <td>$$ 0.76 $$</td> | ||
+ | <td>Hill coefficient for Plux</td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ n_{Rhl} $$</td> | ||
+ | <td>$$ 5 $$</td> | ||
+ | <td>Hill cofficient for Prhl</td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ K_{mLux} $$</td> | ||
+ | <td>$$ 116.24nM $$</td> | ||
+ | <td>Lumped parameter for the Lux system</td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ K_{mRhl} $$</td> | ||
+ | <td>$$ 1000 nM $$</td> | ||
+ | <td>Lumped parameter for the Rhl system</td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #more">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ F_{DiMazF} $$</td> | ||
+ | <td> $$ 5 nM^{-1} min^{-1} $$</td> | ||
+ | <td>Cutting rate at ACA sequences on mRNA by MazF dimers </td> | ||
+ | <td> Assumption </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f $$</td> | ||
+ | <td>$$ 0.299 $$</td> | ||
+ | <td>The probability of distinction of ACA sequences on each mRNA</td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f_{mRNA_{RFP}} $$</td> | ||
+ | <td>$$ 10 $$</td> | ||
+ | <td>The number of ACA sequences on mRNA_{RFP}</td> | ||
+ | <td> Extraction of data </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f_{mRNA_{GFP}} $$</td> | ||
+ | <td>$$ 23 $$</td> | ||
+ | <td>The number of ACA sequences on mRNA_{GFP}</td> | ||
+ | <td> Extraction of data </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f_{mRNA_{RhlI}} $$</td> | ||
+ | <td>$$ 1 $$</td> | ||
+ | <td>The number of ACA sequences on mRNA_{RhlI}</td> | ||
+ | <td> Extraction of data </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f_{mRNA_{LasI}} $$</td> | ||
+ | <td>$$ 10 $$</td> | ||
+ | <td>The number of ACA sequences on mRNA_{LasI}</td> | ||
+ | <td> Extraction of data </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f_{mRNA_{MazF}} $$</td> | ||
+ | <td> $$2$$ </td> | ||
+ | <td>The number of ACA sequences on mRNA_{MazF}</td> | ||
+ | <td> Extraction of data </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ f_{mRNA_{MazE}} $$</td> | ||
+ | <td> $$2$$ </td> | ||
+ | <td>The number of ACA sequences on mRNA_{MazE}</td> | ||
+ | <td> Extraction of data </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ α $$</td> | ||
+ | <td> $$ 0.04 min_{-1} $$ </td> | ||
+ | <td>Translation rate of </td> | ||
+ | <td> Assumption </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k_{DiMazF}$$</td> | ||
+ | <td> $$ 6.82 nM_{-1} min_{-1} $$ </td> | ||
+ | <td>Formation rate of MazF dimer </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k_{-Di_{MazF}}$$</td> | ||
+ | <td> $$ 6.24 nM^{-1} min^{-1} $$ </td> | ||
+ | <td>Formation rate of MazF dimer </td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k_{Di_{MazE}}$$</td> | ||
+ | <td> $$ 3.46 nM^{-1} min^{-1} $$ </td> | ||
+ | <td>Formation rate of MazF dimer </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k_{-Di_{MazE}}$$</td> | ||
+ | <td> $$ 7.25 min^{-1} $$ </td> | ||
+ | <td>Dissociation rate of MazF dimer </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k_{Hexa}$$</td> | ||
+ | <td> $$ 4.51 nM^{-1} min^{-1} $$ </td> | ||
+ | <td>Formation rate of Maz hexamer </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ k_{-Hexa}$$</td> | ||
+ | <td> $$ 4.05 min^{-1} $$ </td> | ||
+ | <td>Dissociation rate of Maz hexamer </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ p_{C4}$$</td> | ||
+ | <td> $$ 0.07 min^{-1} $$ </td> | ||
+ | <td> Production rate of C4HSL by RhlI </td> | ||
+ | <td> Assumption </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ p_{C12}$$</td> | ||
+ | <td> $$ 0.07 min^{-1} $$ </td> | ||
+ | <td> Production rate of 3OC12HSL by LasI </td> | ||
+ | <td> Assumption </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ D $$</td> | ||
+ | <td> $$ 0.1 nM^{-1} min^{-1} $$ </td> | ||
+ | <td> Decomposition rate of 3OC12HSL by AmiE </td> | ||
+ | <td> Assumption </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td>$$ d $$</td> | ||
+ | <td>$$ 0.2773 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of mRNA </td> | ||
+ | <td> Leference<a href ="#references">[2] </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{RFP} $$</td> | ||
+ | <td>$$ 0.005 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of RFP </td> | ||
+ | <td> Assumption </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{GFP} $$</td> | ||
+ | <td>$$ 0.005 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of GFP </td> | ||
+ | <td> Assumption </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{RhlI} $$</td> | ||
+ | <td>$$ 0.0167 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of RhlI </td> | ||
+ | <td> Leference<a href ="#references">[1] </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{LasI} $$</td> | ||
+ | <td>$$ 0.0167 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of LasI </td> | ||
+ | <td> Leference<a href ="#references">[1] </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{MazF} $$</td> | ||
+ | <td>$$ 0.7 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of MazF </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{DiMazF} $$</td> | ||
+ | <td>$$ 0.17 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of MazF dimer </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{MazE} $$</td> | ||
+ | <td>$$ 0.55 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of MazE </td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{DiMazE} $$</td> | ||
+ | <td>$$ 0.416 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of MazE dimer </td> | ||
+ | <td> <a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a> </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{Hexa} $$</td> | ||
+ | <td>$$ 0.511 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of Maz hexameter </td> | ||
+ | <td><a href="https://2016.igem.org/Team:Tokyo_Tech/Model #toxin">Fitted to experimental data</a></td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{C4} $$</td> | ||
+ | <td>$$ 0.000222 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of C4HSL </td> | ||
+ | <td> Literature<a href ="#references">[3] </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{C12} $$</td> | ||
+ | <td>$$ 0.004 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of 3OC12HSL </td> | ||
+ | <td> Literature<a href ="#references">[4] </td> | ||
+ | </td> | ||
+ | <tr> | ||
+ | <td>$$ d_{AmiE} $$</td> | ||
+ | <td>$$ 0.001 min^{-1} $$ </td> | ||
+ | <td> Degradation rate of AmiE </td> | ||
+ | <td> Assumption </td> | ||
+ | </td> | ||
+ | </tbody> | ||
+ | </table> | ||
+ | </div><!-- /parameter_discription_contents --> | ||
+ | |||
+ | <!-- この辺まで表 --> | ||
+ | <div class="back_link contents"> | ||
+ | <p class="normal_text"><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#mathematical_model">Back to Model page.</a></p> | ||
+ | </div> | ||
+ | </div><!-- /parameter_discriptions --> | ||
+ | |||
+ | <div id="references" class="container container_bottom"> | ||
+ | <div id="references_header" class="container_header"> | ||
+ | <h2><span>References</span></h2> | ||
+ | </div><!-- /references_header --> | ||
+ | <div id="references_contents" class="container_contents"> | ||
+ | <p class="normal_text"> [1] <a href="https://2014.igem.org/Team:ETH_Zurich" target="_blank">https://2014.igem.org/Team:ETH_Zurich</a></p> | ||
+ | <p class="normal_text"> [2] <a href="http://www.ncbi.nlm.nih.gov/pubmed/10329160" target="_blank">http://www.ncbi.nlm.nih.gov/pubmed/10329160</a></p> | ||
+ | <p class="normal_text"> [3] <a href="https://www.ncbi.nlm.nih.gov/pubmed/19584835" target="_blank">https://www.ncbi.nlm.nih.gov/pubmed/19584835</a></p> | ||
+ | <p class="normal_text"> [4] <a href="https://2015.igem.org/Team:Technion_HS_Israel" target="_blank">https://2015.igem.org/Team:Technion_HS_Israel</a></p> | ||
+ | </div><!-- /references_contents --> | ||
+ | </div><!-- /references --> | ||
</div><!-- /main_contents --> | </div><!-- /main_contents --> | ||
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Latest revision as of 00:46, 20 October 2016
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Detailed description
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.
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 |
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
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 |