Difference between revisions of "Team:Tokyo Tech"
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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"); | ||
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| + | // background-image: url("https://static.igem.org/mediawiki/2016/8/83/Tokyo_Tech_Background.png"); | ||
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| + | /**********************************************************************/ | ||
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| + | /************************** this page setting ***************************/ | ||
| + | #modeling_detail_wrapper{ | ||
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</head> | </head> | ||
| + | <body> | ||
| + | <div id="main_contents"> | ||
| + | <div id="page_header" class="container container_top"> | ||
| + | <h1 align="center">Detail Description</h1> | ||
| + | </div><!-- page_header --> | ||
| + | <div id="modeling_development" class="container"> | ||
| + | <div id="modeling_development_header" class="container_header"> | ||
| + | <h2><span>Modeling 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. | ||
| + | The following sentences describe how the equations were developed. | ||
| + | And in this page we expound not only on the model with the Maz system, which we selected as the best TA system for our project, but also on the one with the Yaf system, which we chose as an alternative.</p> | ||
| + | </div><!-- modeling_development_contents --> | ||
| + | <div id="modeling_maz_contents" class="container_contents"> | ||
| + | <a href="https://static.igem.org/mediawiki/2016/5/52/T--Tokyo_Tech--Model_Details_1.png"><img src="https://static.igem.org/mediawiki/2016/5/52/T--Tokyo_Tech--Model_Details_1.png" /></a> | ||
| + | <p class="caption"><span style="font-weight: bold;">Fig. 4-2-1. </span> Maz System Gene Circuit</p> | ||
| + | <div id="modeling_detail" class="off"> | ||
| + | <div id="modeling_detail_wrapper"> | ||
| + | <div id="modeling_detail_expressions"> | ||
| + | <h2>Differencial Equations</h2> | ||
| + | <h3>Snow White</h3> | ||
| + | \begin{equation} | ||
| + | \frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] | ||
| + | \end{equation} | ||
| + | \begin{equation} | ||
| + | \frac{d[mRNA_{RhlI}]}{dt} = leak_{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} | ||
| + | <h3>Queen</h3> | ||
| + | \begin{equation} | ||
| + | \frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] | ||
| + | \end{equation} | ||
| + | \begin{equation} | ||
| + | \frac{d[mRNA_{LasI}]}{dt} = leak_{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} | ||
| + | <h3>Prince</h3> | ||
| + | \begin{equation} | ||
| + | \frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^n}{K_{mLux}^n + [C12]^n} - d[mRNA_{AmiE}] | ||
| + | \end{equation} | ||
| + | \begin{equation} | ||
| + | \frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] | ||
| + | \end{equation} | ||
| + | \begin{equation} | ||
| + | \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} | ||
| + | \end{equation} | ||
| + | </div><!-- /modeling_detail_expressions --> | ||
| + | <div id="modeling_detail_parameter"> | ||
| + | <h2>Explanation about Parameters</h2> | ||
| + | <table border="1" style="margin: auto;"> | ||
| + | <tbody> | ||
| + | <tr><td>Parameter </td><td> Description </td></tr> | ||
| + | <tr><td>$$g$$ </td><td> Growth rate of each cells</td></tr> | ||
| + | <tr><td>$$P_{max}$$ </td><td> Carrying capacity </td></tr> | ||
| + | <tr><td>$$E_{DiMazF}$$ </td><td> Effect of MazF dimer on growth rate</td></tr> | ||
| + | <tr><td>$$k$$ </td><td> Transcription rate of downstream of Pcon </td></tr> | ||
| + | <tr><td>$$leak_{Plux}$$ </td><td> Leakage of Plux </td></tr> | ||
| + | <tr><td>$$leak_{Prhl}$$ </td><td> Leakage of Prhl </td></tr> | ||
| + | <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 downstream of Prhl </td></tr> | ||
| + | <tr><td>$$n_{Lux}$$ </td><td> Hill coefficient for Plux</td></tr> | ||
| + | <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>$$K_{mRhl}$$ </td><td> Lumped paremeter for the Rhl System</td></tr> | ||
| + | <tr><td>$$F_{DiMazF}$$ </td><td> Cutting rate at ACA sequences on mRNA by MazF dimer</td></tr> | ||
| + | <tr><td>$$f$$ </td><td> The probability of distinction of ACA sequencess in each mRNA</td></tr> | ||
| + | <tr><td>$$f_{mRNA_{RFP}}$$ </td><td> The number of ACA sequences in \(mRNA_{RFP}\)</td></tr> | ||
| + | <tr><td>$$f_{mRNA_{GFP}}$$ </td><td> The number of ACA sequences in \(mRNA_{GFP}\)</td></tr> | ||
| + | <tr><td>$$f_{mRNA_{RhlI}}$$ </td><td> The number of ACA sequences in \(mRNA_{RhlI}\) </td></tr> | ||
| + | <tr><td>$$f_{mRNA_{LasI}}$$ </td><td> The number of ACA sequences in \(mRNA_{LasI}\)</td></tr> | ||
| + | <tr><td>$$f_{mRNA_{MazF}}$$ </td><td> The number of ACA sequences in \(mRNA_{MazF}\) </td></tr> | ||
| + | <tr><td>$$f_{mRNA_{MazE}}$$ </td><td> The number of ACA sequences in \(mRNA_{MazE}\) </td></tr> | ||
| + | <tr><td>$$\alpha$$ </td><td> Translation rate of Protein </td></tr> | ||
| + | <tr><td>$$k_{DiMazF}$$ </td><td> Formation rate of MazF dimer </td></tr> | ||
| + | <tr><td>$$k_{-DiMazF}$$ </td><td> Dissociation rate of MazF dimer </td></tr> | ||
| + | <tr><td>$$k_{DiMazE}$$ </td><td> Formation rate of MazE dimer </td></tr> | ||
| + | <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> Dissociation rate of Maz hexamer</td></tr> | ||
| + | <tr><td>$$p_{C4}$$ </td><td> Production rate of C4HSL by RhlI</td></tr> | ||
| + | <tr><td>$$p_{C12}$$ </td><td> Production rate of 3OC12HSL by LuxI </td></tr> | ||
| + | <tr><td>$$D$$ </td><td> Decomposition rate of 3OC12HSL by AmiE </td></tr> | ||
| + | <tr><td>$$d$$ </td><td> Degradation rate of mRNA </td></tr> | ||
| + | <tr><td>$$d_{RFP}$$ </td><td> Degradation rate of RFP</td></tr> | ||
| + | <tr><td>$$d_{GFP}$$ </td><td> Degradation rate of GFP </td></tr> | ||
| + | <tr><td>$$d_{RhlI}$$ </td><td> Degradation rate of RhlI</td></tr> | ||
| + | <tr><td>$$d_{LasI}$$ </td><td> Degradation rate of LasI</td></tr> | ||
| + | <tr><td>$$d_{MazF}$$ </td><td> Degradation rate of MazF</td></tr> | ||
| + | <tr><td>$$d_{DiMazF}$$ </td><td> Degradation rate of MazF dimer</td></tr> | ||
| + | <tr><td>$$d_{MazE}$$ </td><td> Degradation rate of MazE </td></tr> | ||
| + | <tr><td>$$d_{DiMazE}$$ </td><td> Degradation rate of MazE dimer </td></tr> | ||
| + | <tr><td>$$d_{Hexa}$$ </td><td> Degradation rate of Maz Hexamer </td></tr> | ||
| + | <tr><td>$$d_{C4}$$ </td><td> Degradation rate of C4HSL </td></tr> | ||
| + | <tr><td>$$d_{C12}$$ </td><td> Degradation rate of 3OC12HSL </td></tr> | ||
| + | <tr><td>$$d_{AmiE}$$ </td><td> Degradation rate of AmiE </td></tr> | ||
| − | <!-- | + | </tbody> |
| − | <div class=" | + | </table> |
| + | </div><!-- /modeling_detail_parameter --> | ||
| + | </div><!-- /modeling_detail_wrapper --> | ||
| + | </div><!-- /modeling_detail --> | ||
| + | <p class="normal_text" style="text-align:center;"><a href="javascript:void(0);" onClick="show('modeling_detail');" class="showHidden">Expressions</a></p> | ||
| + | <ul id="modeling_maz_list" class="non_dotted_list"> | ||
| + | <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>$$ | ||
| + | \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$ <br /> $$ \tag{1-2} $$</p> | ||
| + | <p>$$ | ||
| + | \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{1-3} $$ </p> | ||
| + | <p class="caption"><span style="font-weight: bold;">Eq.1. </span> Differential Equation of Cell Population</p> | ||
| + | <p class="normal_text">The equations above describe how cells grow 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. | ||
| + | This factor is designed so that its value is small when the concentration of MazF dimers is low, and its value converges to 1 when the concentration of MazF dimers is high.</p> | ||
| + | </li><!-- /1.1. Cell Population --> | ||
| + | <li><h2>2. Maz System</h2> | ||
| + | <ul id="modeling_maz_system" class="non_dotted_list"> | ||
| + | <li><h3>2.1. Expression of Maz System</h3> | ||
| + | <p class="normal_text">After translation, MazE and MazF each form an stable dimer which can be activated to exert its function.</p> | ||
| + | <a href="https://static.igem.org/mediawiki/2016/8/88/T--Tokyo_Tech--Model_Details_2.png"><img src="https://static.igem.org/mediawiki/2016/8/88/T--Tokyo_Tech--Model_Details_2.png" /></a><br /> | ||
| + | <a href="https://static.igem.org/mediawiki/2016/c/c3/T--Tokyo_Tech--Model_Details_3.png"><img src="https://static.igem.org/mediawiki/2016/c/c3/T--Tokyo_Tech--Model_Details_3.png" /></a> | ||
| + | <p class="normal_text">Two MazE dimers sandwich the MazF dimer, forming MazF2-MazE2-MazF2 heterohexamers and suppressing the toxicity of the MazF dimers.</p> | ||
| + | <a href="https://static.igem.org/mediawiki/2016/e/ea/T--Tokyo_Tech--Model_Details_4.png"><img src="https://static.igem.org/mediawiki/2016/e/ea/T--Tokyo_Tech--Model_Details_4.png" /></a> | ||
| + | <a href="https://static.igem.org/mediawiki/2016/3/32/T--Tokyo_Tech--Model_Details_5.png"><img src="https://static.igem.org/mediawiki/2016/3/32/T--Tokyo_Tech--Model_Details_5.png" style="width: 500px;" /></a> | ||
| + | <p class="caption"><span style="font-weight: bold">Fig. 4-2-2. </span> Reaction of Maz System</p> | ||
| + | <p class="normal_text">The mRNAs of Snow White and the Queen decrease by their original degradation and by the cleavage at ACA sequences by MazF dimers.</p> | ||
| + | <p class="normal_text">Applying mass action kinetic laws, we obtain the following set of differential equations.</p> | ||
| + | <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>$$ | ||
| + | \frac{d[DiMazE]}{dt} = k_{DiMazE}}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ | ||
| + | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <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> | ||
| + | <h3>Queen</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 />$$ \tag{2-8} $$</p> | ||
| + | <p>$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$ <br /> | ||
| + | $$\tag{2-9}$$</p> | ||
| + | <p>$$ \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ | ||
| + | + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$<br />$$ \tag{2-10} $$</p> | ||
| + | <p>$$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$ <br />$$ \tag{2-11} $$</p> | ||
| + | <p>$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE]$$ | ||
| + | <br />$$\tag{2-12}$$</p> | ||
| + | <p>$$ | ||
| + | \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ | ||
| + | + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$ <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 class="caption"><span style="font-weight: bold">Eq. 2. </span>Differential Equations of Maz System</p> | ||
| + | <p class="normal_text">Equations (2-1) and (2-8) describe the concentration of mRNAs under the AHL inducing promoters. | ||
| + | Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers. | ||
| + | Since Equations (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 Maz System --> | ||
| + | <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">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> | ||
| + | <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> | ||
| + | <p>$$ | ||
| + | \frac{d[mRNA_{RhlI}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$ | ||
| + | <br />$$ \tag{3-2} $$</p> | ||
| + | <p>$$\frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ | ||
| + | - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] | ||
| + | $$ <br /> $$\tag{3-3}$$</p> | ||
| + | <p>$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$ | ||
| + | <br />$$ \tag{3-4} $$</p> | ||
| + | <h3>Queen</h3> | ||
| + | <p>$$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] | ||
| + | $$<br />$$ \tag{3-5} %%</p> | ||
| + | <p>$$ | ||
| + | \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{3-6} $$</p> | ||
| + | <p>$$\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{3-7}$$</p> | ||
| + | <p>$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$ | ||
| + | <br />$$ \tag{3-8} $$</p> | ||
| + | <p class="caption"><span style="font-weight: bold">Eq. 3. </span>Differential Equations of mRNAs</p> | ||
| + | <p class="normal_text">The equations above comprise terms of production, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers. | ||
| + | </p> | ||
| + | </li><!-- /1.2.2. Cleavage by MazF dimers --> | ||
| + | </ul><!-- /modeling_maz_system --> | ||
| + | </li><!-- /1.2. Maz System --> | ||
| + | <li><h2>3. Signal 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> | ||
| + | <p class="caption"><span style="font-weight:bold;">Fig. 4-2-3. </span> Reaction of Signal Molecules</p> | ||
| + | <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">The mRNAs of Snow White and the Queen decrease from original degradation and the cleavage at ACA sequences by MazF dimers. | ||
| + | On the other hand, those of the Prince don’t have any MazF gene so they decrease from only original degradation.</p> | ||
| + | <p class="normal_text">After translation, C12AHL and C4 are enzymatically synthesized by LasI and RhlI from some substrates respectively. | ||
| + | For simplicity, we assumed that the amount of substrates is sufficient so that the C12AHL / C4 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.</p> | ||
| + | <p class="normal_text">C4 decreases from original degradation meanwhile C12AHL 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[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[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[LasAHL]}{dt} = p_{Las}[LasI]P_{Stepmother} - d_{LasAHL}[LasAHL] - D[LasAHL][AmiE]$$ <br /> $$\tag{4-6}$$</p> | ||
| + | <p>$$\frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^n}{K_{mLux}^n + [C12]^n} - d[mRNA_{AmiE}]$$ <br />$$\tag{4-7}$$</p> | ||
| + | <p>$$\frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \tag{4-8} $$</p> | ||
| + | |||
| + | <p class="caption"><span style="font-weight: bold;">Eq. 4. </span> Differential Equations of Signaling Molecules</p> | ||
| + | |||
| + | <p class="normal_text">Equations (4-1), (4-4) and (4-7) describe the concentrations of mRNAs under the AHL inducing promoters. | ||
| + | Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.</p> | ||
| + | <p class="normal_text">The other ODEs describe how the concentrations of materials change in individuals, on the other hand (4-3), (4-6) describe the concentrations of C4 C12AHL in the whole culture medium.</p> | ||
| + | </li> | ||
| + | </ul><!-- /modeling_maz_list --> | ||
| + | </div><!-- /modeling_maz_contents --> | ||
| − | + | <div class="back_link contents"> | |
| − | + | <p class="normal_text"><a href="https://2016.igem.org/Team:Tokyo_Tech/Model#mathematical_model">Back to Model page.</a></p> | |
| − | + | </div> | |
| − | + | </div><!-- /modeling_maz_system --> | |
| − | + | ||
| − | + | <!-- /UNTIL HERE MAZ SYSTEM --> | |
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| − | + | <div id="parameter_discriptions" class="container"> | |
| − | + | <div id="parameter_discription_header" class="container_header"> | |
| + | <h1><span>Parameters</span></h1> | ||
| + | </div><!-- /parameter_discription_header --> | ||
| + | |||
| + | <!-- この辺から表 --> | ||
| + | <div id="parameter_discription_contents" class="container_contents"> | ||
| + | <table border="1" width="800px"> | ||
| + | <tbody> | ||
| + | <tr> | ||
| + | <th>Parameter</th> | ||
| + | <th>Value</th> | ||
| + | <th>Description</th> | ||
| + | <th>Reference</th> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ g $$</td> | ||
| + | <td>0.0123</td> | ||
| + | <td>Growth rate of each cells</td> | ||
| + | <td>Fitted to experimental data</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ P_{max} $$</td> | ||
| + | <td>3.6</td> | ||
| + | <td>Carrying capacity</td> | ||
| + | <td>Fitted to experimental data</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ E_{DiMazF} $$</td> | ||
| + | <td>$$ 0.462234 nM^{-1} min^{-1} $$ </td> | ||
| + | <td>Effect of MazF dimer on growth rate of each cells</td> | ||
| + | <td>Fitted to experimental data</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k $$</td> | ||
| + | <td>$$5 min^{-1}$$</td> | ||
| + | <td>Transcription rate of downstream of Ptet</td> | ||
| + | <td>Reference</td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ leak_{Plux} $$</td> | ||
| + | <td>$$ 2.26 min^{-1} $$</td> | ||
| + | <td>Leakage of Plux</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ leak_{Prhl} $$</td> | ||
| + | <td>$$ 4.654 min^{-1} $$</td> | ||
| + | <td>Leakage of Prhl</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ κ_{Lux} $$</td> | ||
| + | <td>$$ 6.984 nM^{-1} min^{-1} $$ </td> | ||
| + | <td>Maximum transcription rate of under streams of Plux</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ κ_{Rhl} $$</td> | ||
| + | <td>$$ 14.95 nM^{-1} min^{-1} $$ </td> | ||
| + | <td>Maximum transcription rate of understreams of Prhl</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ n_{Lux} $$</td> | ||
| + | <td> 0.76 </td> | ||
| + | <td>Hill coefficient for Plux</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ n_{Rhl} $$</td> | ||
| + | <td> 5 </td> | ||
| + | <td>Hill cofficient for Prhl</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ K_{mLux} $$</td> | ||
| + | <td>$$ 116.24nM $$</td> | ||
| + | <td>Lumped parameter for the Lux system</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ K_{mRhl} $$</td> | ||
| + | <td>$$ 1000 nM $$</td> | ||
| + | <td>Lumped parameter for the Rhl system</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ F_{DiMazF} $$</td> | ||
| + | <td> $$ 5 nM^{-1} min^{-1} $$</td> | ||
| + | <td>Cutting rate at ACA sequences on mRNA by MazF dimers </td> | ||
| + | <td> Estimated </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ f $$</td> | ||
| + | <td> 0.299 </td> | ||
| + | <td>The probability of distinction of ACA sequences on each mRNA</td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ 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} $$ </td> | ||
| + | <td>Translation rate of </td> | ||
| + | <td> Estimated </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k_{DiMazF}$$</td> | ||
| + | <td> $$ 6.82 nM_{-1} min_{-1} $$ </td> | ||
| + | <td>Formation rate of MazF dimer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k_{-Di_{MazF}}$$</td> | ||
| + | <td> $$ 6.24 nM^{-1} min^{-1} $$ </td> | ||
| + | <td>Formation rate of MazF dimer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k_{Di_{MazE}}$$</td> | ||
| + | <td> $$ 3.46 nM^{-1} min^{-1} $$ </td> | ||
| + | <td>Formation rate of MazF dimer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k_{-Di_{MazE}}$$</td> | ||
| + | <td> $$ 7.25 min^{-1} $$ </td> | ||
| + | <td>Dissociation rate of MazF dimer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k_{Hexa}$$</td> | ||
| + | <td> $$ 4.51 nM^{-1} min^{-1} $$ </td> | ||
| + | <td>Formation rate of Maz hexamer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ k_{-Hexa}$$</td> | ||
| + | <td> $$ 4.05 min^{-1} $$ </td> | ||
| + | <td>Dissociation rate of Maz hexamer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ p_{C4}$$</td> | ||
| + | <td> $$ 0.07 min^{-1} $$ </td> | ||
| + | <td> Production rate of C4HSL by RhlI </td> | ||
| + | <td> Estimated </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ p_{C12}$$</td> | ||
| + | <td> $$ 0.07 min^{-1} $$ </td> | ||
| + | <td> Production rate of 3OC12HSL by LasI </td> | ||
| + | <td> Estimated </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ D $$</td> | ||
| + | <td> $$ 5 nM^{-1} min^{-1} $$ </td> | ||
| + | <td> Decomposition rate of 3OC12HSL by AmiE </td> | ||
| + | <td> Estimated </td> | ||
| + | </tr> | ||
| + | <tr> | ||
| + | <td>$$ d $$</td> | ||
| + | <td>$$ 0.2773 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of mRNA </td> | ||
| + | <td> Leference </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{RFP} $$</td> | ||
| + | <td>$$ 0.018 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of RFP </td> | ||
| + | <td> Leference </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{GFP} $$</td> | ||
| + | <td>$$ 0.04 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of GFP </td> | ||
| + | <td> Leference </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{RhlI} $$</td> | ||
| + | <td>$$ 0.0167 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of RhlI </td> | ||
| + | <td> Leference </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{LasI} $$</td> | ||
| + | <td>$$ 0.0167 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of LasI </td> | ||
| + | <td> Leference </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{MazF} $$</td> | ||
| + | <td>$$ 0.7 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of MazF </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{DiMazF} $$</td> | ||
| + | <td>$$ 0.17 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of MazF dimer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{MazE} $$</td> | ||
| + | <td>$$ 0.55 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of MazE </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{DiMazE} $$</td> | ||
| + | <td>$$ 0.416 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of MazE dimer </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{Hexa} $$</td> | ||
| + | <td>$$ 0.511 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of Maz hexameter </td> | ||
| + | <td> Fitted to experimental data </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{C4} $$</td> | ||
| + | <td>$$ 0.000222 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of C4HSL </td> | ||
| + | <td> Literature </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ d_{C12} $$</td> | ||
| + | <td>$$ 0.004 min^{-1} $$ </td> | ||
| + | <td> Degradation rate of 3OC12HSL </td> | ||
| + | <td> Literature </td> | ||
| + | </td> | ||
| + | <tr> | ||
| + | <td>$$ 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 --> | ||
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| + | |||
| + | <div id="references" class="container container_bottom"> | ||
| + | <div id="references_header" class="container_header"> | ||
| + | <h2><span>References</span></h2> | ||
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| + | <div id="references_contents" class="container_contents"> | ||
| + | <p class="normal_text">[1] <a href="ここにリンク"></a></p> | ||
| + | <p class="normal_text"> [2] <a href="ここにリンク"></a>.</p> | ||
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Revision as of 03:57, 17 October 2016
Detail Description
Modeling Development
To simulate the cell-cell communication system, we developed an ordinary differential equation model. The following sentences describe how the equations were developed. And in this page we expound not only on the model with the Maz system, which we selected as the best TA system for our project, but also on the one with the Yaf system, which we chose as an alternative.
Fig. 4-2-1. Maz System Gene Circuit
Differencial Equations
Snow White
\begin{equation} \frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] \end{equation} \begin{equation} \frac{d[mRNA_{RhlI}]}{dt} = leak_{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}{K_{mLux}^n + [C12]^n} - d[mRNA_{AmiE}] \end{equation} \begin{equation} \frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \end{equation} \begin{equation} \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \end{equation}Explanation about Parameters
| Parameter | Description |
| $$g$$ | Growth rate of each cells |
| $$P_{max}$$ | Carrying capacity |
| $$E_{DiMazF}$$ | Effect of MazF dimer on growth rate |
| $$k$$ | Transcription rate of downstream of Pcon |
| $$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 cells grow 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. This factor is designed so that its value is small when the concentration of MazF dimers is low, and its value converges to 1 when the concentration of MazF dimers is high.
2. Maz System
2.1. Expression of Maz System
After translation, MazE and MazF each form an stable dimer which can be activated to exert its function.
Two MazE dimers sandwich the MazF dimer, forming MazF2-MazE2-MazF2 heterohexamers and suppressing the toxicity of the MazF dimers.
Fig. 4-2-2. Reaction of Maz System
The mRNAs of Snow White and the Queen decrease by their original degradation and by the cleavage at ACA sequences by MazF dimers.
Applying mass action kinetic laws, we obtain the following set of differential equations.
Snow White
$$ \frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
$$ \tag{2-1} $$$$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$
$$\tag{2-2}$$$$ \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ + 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
$$ \tag{2-3} $$$$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$
$$ \tag{2-4} $$$$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE]$$
$$\tag{2-5}$$$$ \frac{d[DiMazE]}{dt} = k_{DiMazE}}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ + k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
$$\tag{2-6} $$$$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
$$ \tag{2-7}$$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 Maz System
Equations (2-1) and (2-8) describe the concentration of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers. Since Equations (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 ACAs in mRNAs, thus acting as Toxin.
We estimated the rate of recognitions of ACA sequences by MazF dimers at \(1-(1-f)^n\), where the number of ACA sequences in mRNA.
Then, we expressed the rate of degradation by MazF dimers in \(F(1-(1-f)^{f_{mRNA}})\) and obtain the following set of differential equations.
Snow White
$$\frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] $$
$$ \tag{3-1} $$$$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
$$ \tag{3-2} $$$$\frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ - d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
$$\tag{3-3}$$$$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
$$ \tag{3-4} $$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 mRNAs
The equations above comprise terms of production, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
3. Signal Molecules
Fig. 4-2-3. Reaction of Signal Molecules
Snow White expresses RhlI under Plux induced by C12, the Queen expresses LasI under Prhl induced by C4 and the Prince expresses AmiE under Plux induced by C12.
The mRNAs of Snow White and the Queen decrease from original degradation and the cleavage at ACA sequences by MazF dimers. On the other hand, those of the Prince don’t have any MazF gene so they decrease from only original degradation.
After translation, C12AHL and C4 are enzymatically synthesized by LasI and RhlI from some substrates respectively. For simplicity, we assumed that the amount of substrates is sufficient so that the C12AHL / C4 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.
C4 decreases from original degradation meanwhile C12AHL decreases from both original degradation and degradation by AmiE, which Prince products.
Applying mass action kinetic laws, we obtain the following set of differential equations.
$$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] $$
$$\tag{4-1}$$$$\frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \tag{4-2}$$
$$ \frac{d[Rhl AHL]}{dt} = p_{Rhl}[RhlI]P_{Snowwhite} - d_{RhlAHL}[RhlAHL] \tag{4-3} $$
$$ \frac{d[mRNA_{LasI}]}{dt} = leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$
$$\tag{4-4}$$$$\frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \tag{4-5}$$
$$\frac{d[LasAHL]}{dt} = p_{Las}[LasI]P_{Stepmother} - d_{LasAHL}[LasAHL] - D[LasAHL][AmiE]$$
$$\tag{4-6}$$$$\frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^n}{K_{mLux}^n + [C12]^n} - d[mRNA_{AmiE}]$$
$$\tag{4-7}$$$$\frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \tag{4-8} $$
Eq. 4. Differential Equations of Signaling Molecules
Equations (4-1), (4-4) and (4-7) describe the concentrations of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.
The other ODEs describe how the concentrations of materials change in individuals, on the other hand (4-3), (4-6) describe the concentrations of C4 C12AHL in the whole culture medium.
Parameters
| Parameter | Value | Description | Reference |
|---|---|---|---|
| $$ g $$ | 0.0123 | Growth rate of each cells | Fitted to experimental data |
| $$ P_{max} $$ | 3.6 | 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 |
| $$ leak_{Plux} $$ | $$ 2.26 min^{-1} $$ | 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 | Estimated |
| $$ f $$ | 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 | Estimated |
| $$ k_{DiMazF}$$ | $$ 6.82 nM_{-1} min_{-1} $$ | 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 | Estimated |
| $$ p_{C12}$$ | $$ 0.07 min^{-1} $$ | Production rate of 3OC12HSL by LasI | Estimated |
| $$ D $$ | $$ 5 nM^{-1} min^{-1} $$ | Decomposition rate of 3OC12HSL by AmiE | Estimated |
| $$ d $$ | $$ 0.2773 min^{-1} $$ | Degradation rate of mRNA | Leference |
| $$ d_{RFP} $$ | $$ 0.018 min^{-1} $$ | Degradation rate of RFP | Leference |
| $$ d_{GFP} $$ | $$ 0.04 min^{-1} $$ | Degradation rate of GFP | Leference |
| $$ d_{RhlI} $$ | $$ 0.0167 min^{-1} $$ | Degradation rate of RhlI | Leference |
| $$ d_{LasI} $$ | $$ 0.0167 min^{-1} $$ | Degradation rate of LasI | Leference |
| $$ d_{MazF} $$ | $$ 0.7 min^{-1} $$ | 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 |
| $$ d_{C12} $$ | $$ 0.004 min^{-1} $$ | Degradation rate of 3OC12HSL | Literature |
| $$ d_{AmiE} $$ | $$ 0.001 min^{-1} $$ | Degradation rate of AmiE | Assumption |
