Difference between revisions of "Team:OUC-China/miniToe"

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     <tr>
 
     <tr>
 
       <th scope="row"> \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\]</th>
 
       <th scope="row"> \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\]</th>
โˆ’
       <td>A</td>
+
       <td align="center">A</td>
โˆ’
       <td>15</td>
+
       <td align="center">15</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘š๐‘…๐‘๐ด_{Csy4}\]</th>
 
       <th scope="row">\[๐‘š๐‘…๐‘๐ด_{Csy4}\]</th>
โˆ’
       <td>B</td>
+
       <td align="center">B</td>
โˆ’
       <td>0</td>
+
       <td align="center">0</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[Protein_{๐ถ๐‘ ๐‘ฆ4}\]</th>
 
       <th scope="row">\[Protein_{๐ถ๐‘ ๐‘ฆ4}\]</th>
โˆ’
       <td>C</td>
+
       <td align="center">C</td>
โˆ’
       <td>0</td>
+
       <td align="center">0</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\]</th>
 
       <th scope="row">\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\]</th>
โˆ’
       <td>D</td>
+
       <td align="center">D</td>
โˆ’
       <td>0</td>
+
       <td align="center">0</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘š๐‘…๐‘๐ด_{gfp}\]</th>
 
       <th scope="row">\[๐‘š๐‘…๐‘๐ด_{gfp}\]</th>
โˆ’
       <td>E</td>
+
       <td align="center">E</td>
โˆ’
       <td>0</td>
+
       <td align="center">0</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด\]</th>
 
       <th scope="row">\[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด\]</th>
โˆ’
       <td>F</td>
+
       <td align="center">F</td>
โˆ’
       <td>0</td>
+
       <td align="center">0</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[Protein_{gfp}\]</th>
 
       <th scope="row">\[Protein_{gfp}\]</th>
โˆ’
       <td>G</td>
+
       <td align="center">G</td>
โˆ’
       <td>0</td>
+
       <td align="center">0</td>
โˆ’
       <td>\[mol/L\]</td>
+
       <td align="center">\[mol/L\]</td>
 
     </tr>
 
     </tr>
 
   </tbody>
 
   </tbody>
Line 293: Line 293:
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{1}\]</th>
 
       <th scope="row">\[๐‘˜_{1}\]</th>
โˆ’
       <td>The transcription rate of\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] </td>
+
       <td align="center">The transcription rate of\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>1.955</td>
+
       <td align="center">1.955</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{d1}\]</th>
 
       <th scope="row">\[๐‘˜_{d1}\]</th>
โˆ’
       <td>The degradation rate of\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] </td>
+
       <td align="center">The degradation rate of\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.002</td>
+
       <td align="center">0.002</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{2}\]</th>
 
       <th scope="row">\[๐‘˜_{2}\]</th>
โˆ’
       <td>The transcription rate of\[๐‘š๐‘…๐‘๐ด_{Csy4}\]  </td>
+
       <td align="center">The transcription rate of\[๐‘š๐‘…๐‘๐ด_{Csy4}\]  </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>1.116</td>
+
       <td align="center">1.116</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{d2}\]</th>
 
       <th scope="row">\[๐‘˜_{d2}\]</th>
โˆ’
       <td>The degradation rate of\[๐‘š๐‘…๐‘๐ด_{Csy4}\] </td>
+
       <td align="center">The degradation rate of\[๐‘š๐‘…๐‘๐ด_{Csy4}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.241</td>
+
       <td align="center">0.241</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{p2}\]</th>
 
       <th scope="row">\[๐‘˜_{p2}\]</th>
โˆ’
       <td>The translation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}\]  </td>
+
       <td align="center">The translation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}\]  </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>1.134</td>
+
       <td align="center">1.134</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{dp2}\]</th>
 
       <th scope="row">\[๐‘˜_{dp2}\]</th>
โˆ’
       <td>The degradation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}\] </td>
+
       <td align="center">The degradation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>6.547</td>
+
       <td align="center">6.547</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{on}\]</th>
 
       <th scope="row">\[๐‘˜_{on}\]</th>
โˆ’
       <td>The binding constant of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] and \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\]</td>
+
       <td align="center">The binding constant of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] and \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\]</td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>23995.469</td>
+
       <td align="center">23995.469</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{off}\]</th>
 
       <th scope="row">\[๐‘˜_{off}\]</th>
โˆ’
       <td>The dissociation constant of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] and \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\]</td>
+
       <td align="center">The dissociation constant of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] and \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\]</td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>2.703</td>
+
       <td align="center">2.703</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{dc1}\]</th>
 
       <th scope="row">\[๐‘˜_{dc1}\]</th>
โˆ’
       <td>The degradation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\] </td>
+
       <td align="center">The degradation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.024</td>
+
       <td align="center">0.024</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{obs}\]</th>
 
       <th scope="row">\[๐‘˜_{obs}\]</th>
โˆ’
       <td>The cleavage rate of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\]  </td>
+
       <td align="center">The cleavage rate of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\]  </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.327</td>
+
       <td align="center">0.327</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{d3}\]</th>
 
       <th scope="row">\[๐‘˜_{d3}\]</th>
โˆ’
       <td>The degradation rate of \[๐‘š๐‘…๐‘๐ด_{gfp}\] </td>
+
       <td align="center">The degradation rate of \[๐‘š๐‘…๐‘๐ด_{gfp}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.472</td>
+
       <td align="center">0.472</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{dc2}\]</th>
 
       <th scope="row">\[๐‘˜_{dc2}\]</th>
โˆ’
       <td>The degradation rate of \[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA\]  </td>
+
       <td align="center">The degradation rate of \[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA\]  </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.024</td>
+
       <td align="center">0.024</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{p1}\]</th>
 
       <th scope="row">\[๐‘˜_{p1}\]</th>
โˆ’
       <td>The translation rate of \[Protein_{gfp}\]</td>
+
       <td align="center">The translation rate of \[Protein_{gfp}\]</td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>1.711</td>
+
       <td align="center">1.711</td>
 
     </tr>
 
     </tr>
 
     <tr>
 
     <tr>
 
       <th scope="row">\[๐‘˜_{dp1}\]</th>
 
       <th scope="row">\[๐‘˜_{dp1}\]</th>
โˆ’
       <td>The degradation rate of \[Protein_{gfp}\] </td>
+
       <td align="center">The degradation rate of \[Protein_{gfp}\] </td>
โˆ’
       <td>\[h^{-1}\]</td>
+
       <td align="center">\[h^{-1}\]</td>
โˆ’
       <td>0.479</td>
+
       <td align="center">0.479</td>
 
     </tr>
 
     </tr>
 
   </tbody>
 
   </tbody>

Revision as of 09:16, 15 October 2018

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miniToe

Reactions


We can describe our miniToe system to be followings:
\[\rightarrow [๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\rightarrow [๐‘š๐‘…๐‘๐ด_{Csy4}]\] \[[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]\rightarrow [๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}] + [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}]\] \[[Protein_{๐ถ๐‘ ๐‘ฆ4}]+[crRNA-RBS-mRNA_{gfp}]\leftrightarrow [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\rightarrow [๐‘š๐‘…๐‘๐ด_{gfp}] + [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด]\] \[[๐‘š๐‘…๐‘๐ด_{gfp}]\rightarrow [๐‘š๐‘…๐‘๐ด_{gfp}] + [๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{gfp}]\] \[[crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\] \[[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]\rightarrow \emptyset\] \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}]\rightarrow \emptyset\] \[[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\rightarrow \emptyset\] \[[๐‘š๐‘…๐‘๐ด_{gfp}]\rightarrow \emptyset\] \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด]\rightarrow \emptyset\] \[[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{gfp}]\rightarrow \emptyset\]

Two equations, describing the functional binding and cleavage of Csy4 protein in biology, and three parameters  kon  koff  kobs describing the same things in mathematics, are the core of our model.

ODEs


To simulate the dynamics of GFP, we use ordinary differential equations to model the reactions above. And ODEs are given as follows:
\[\frac{d[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]}{dt}=๐‘˜_{1}-๐‘˜_{d1}[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[-๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[+๐‘˜_{off}[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\frac{d[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]}{dt}=๐‘˜_{2}-๐‘˜_{d2}[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]\] \[\frac{d[Protein_{๐ถ๐‘ ๐‘ฆ4}]}{dt}=๐‘˜_{p2}[๐‘š๐‘…๐‘๐ด_{๐ถ๐‘ ๐‘ฆ4}]-๐‘˜_{dp2}[Protein_{๐ถ๐‘ ๐‘ฆ4}]\] \[-๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[+๐‘˜_{off}[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\frac{d[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]}{dt}=๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[-๐‘˜_{of}[Protein_{๐ถ๐‘ ๐‘ฆ4}][Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\] \[-๐‘˜_{d1}[Protein_{๐ถ๐‘ ๐‘ฆ4}][Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\] \[-๐‘˜_{obs}[Protein_{๐ถ๐‘ ๐‘ฆ4}][Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{gfp}]\] \[\frac{d[๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]}{dt}=๐‘˜_{obs}[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]-๐‘˜_{d3}[๐‘š๐‘…๐‘๐ด_{gfp}]\] \[\frac{d[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA]}{dt}=๐‘˜_{obs}[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]-๐‘˜_{dc2}[๐‘ƒ๐‘Ÿ๐‘œ๐‘ก๐‘’๐‘–๐‘›_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด]\] \[-๐‘˜_{on}[Protein_{๐ถ๐‘ ๐‘ฆ4}][๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[+๐‘˜_{off}[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]\] \[\frac{d[Protein_{gfp}]}{dt}=๐‘˜_{p1}[๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}]-๐‘˜_{dp1}[Protein_{gfp}]\]

For the readability, the complex symbol is simplified as:
\[\frac{d[A]}{dt}=๐‘˜_{1}-๐‘˜_{d1}[A]-๐‘˜_{on}[C][A]+๐‘˜_{off}[D]\] \[\frac{d[B]}{dt}=๐‘˜_{2}-๐‘˜_{d2}[B]\] \[\frac{d[C]}{dt}=๐‘˜_{p2}[B]-๐‘˜_{dp2}[C]-๐‘˜_{on}[C][A]+๐‘˜_{off}[D]\] \[\frac{d[D]}{dt}=๐‘˜_{on}[C][A]-๐‘˜_{dp2}[C]-๐‘˜_{off}[D]-๐‘˜_{dc1}[D]-๐‘˜_{obs}[D]\] \[\frac{d[E]}{dt}=๐‘˜_{obs}[D]-๐‘˜_{d3}[E]\] \[\frac{d[F]}{dt}=๐‘˜_{obs}[D]-๐‘˜_{d3}[F]\] \[\frac{d[G]}{dt}=๐‘˜_{p1}[E]-๐‘˜_{dp1}[G]\]

Data Processing


The leak in the experiment is a big problem in estimating parameters in our ODEs model, so we processing the data by following formula๏ผ› \[Data(without leak)=Data(+IPTG)-Data(-IPTG)\]
By doing this, we can reduce some factor which may be influenced estimation, not just the leak, but also some background noise. So we can get more precise parameters of the Csy4.

Species, symbols and parameters

Species Symbol Initial value Units
\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] A 15 \[mol/L\]
\[๐‘š๐‘…๐‘๐ด_{Csy4}\] B 0 \[mol/L\]
\[Protein_{๐ถ๐‘ ๐‘ฆ4}\] C 0 \[mol/L\]
\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\] D 0 \[mol/L\]
\[๐‘š๐‘…๐‘๐ด_{gfp}\] E 0 \[mol/L\]
\[Protein_{๐ถ๐‘ ๐‘ฆ4}-๐‘๐‘Ÿ๐‘…๐‘๐ด\] F 0 \[mol/L\]
\[Protein_{gfp}\] G 0 \[mol/L\]

Because the [๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด๐‘”๐‘“๐‘] is under controlled by a constitutive promoter, so we set the initial concentration to 15mol/L .
The other parament we used in the ODEs is listed in the following table:
Parameters Definition Units Value
\[๐‘˜_{1}\] The transcription rate of\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] \[h^{-1}\] 1.955
\[๐‘˜_{d1}\] The degradation rate of\[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] \[h^{-1}\] 0.002
\[๐‘˜_{2}\] The transcription rate of\[๐‘š๐‘…๐‘๐ด_{Csy4}\] \[h^{-1}\] 1.116
\[๐‘˜_{d2}\] The degradation rate of\[๐‘š๐‘…๐‘๐ด_{Csy4}\] \[h^{-1}\] 0.241
\[๐‘˜_{p2}\] The translation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}\] \[h^{-1}\] 1.134
\[๐‘˜_{dp2}\] The degradation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}\] \[h^{-1}\] 6.547
\[๐‘˜_{on}\] The binding constant of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] and \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] \[h^{-1}\] 23995.469
\[๐‘˜_{off}\] The dissociation constant of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] and \[๐‘๐‘Ÿ๐‘…๐‘๐ด โˆ’ ๐‘…๐ต๐‘† โˆ’ ๐‘š๐‘…๐‘๐ด_{๐‘”๐‘“๐‘}\] \[h^{-1}\] 2.703
\[๐‘˜_{dc1}\] The degradation rate of\[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA-RBS-mRNA_{๐‘”๐‘“๐‘}\] \[h^{-1}\] 0.024
\[๐‘˜_{obs}\] The cleavage rate of \[Protein_{๐ถ๐‘ ๐‘ฆ4}\] \[h^{-1}\] 0.327
\[๐‘˜_{d3}\] The degradation rate of \[๐‘š๐‘…๐‘๐ด_{gfp}\] \[h^{-1}\] 0.472
\[๐‘˜_{dc2}\] The degradation rate of \[Protein_{๐ถ๐‘ ๐‘ฆ4}-crRNA\] \[h^{-1}\] 0.024
\[๐‘˜_{p1}\] The translation rate of \[Protein_{gfp}\] \[h^{-1}\] 1.711
\[๐‘˜_{dp1}\] The degradation rate of \[Protein_{gfp}\] \[h^{-1}\] 0.479

Simulation


With the help of Simbiology toolbox in Matlab๏ผŒwe simulate our miniToe system for 30h, the result can be seen in the Fig.1.

Fig.1 The dynamics of GFP by model prediction

We compare the experimental data to the simulation:

Fig.2 The comparison between experimental data and simulation data

Discussion


Combining the biology and math, we now discuss why the dynamics of GFP is like the curve in the Fig.1. In order to explain in detail, we plot the dynamics of all species in the miniToe system in Fig.3.

Fig.3 The dynamics of all species in the miniToe system

As we can see in the Fig.3, the red line refers to the dynamics of GFP, which is increasing in the beginning and then drop down to a stable level. The reason for this is that the cleavage rate of Csy4 is faster than the production rate of[crRNA-RBS-mRNAgfp] , which cause that the [mRNAgfp] is decreasing after 10 hours. Before we add IPTG into our system to induce the Tac promoter, the [crRNA-RBS-mRNAgfp] is accumulated because it is under control by a constitutive promoter. After we add IPTG, the initial concentration of [crRNA-RBS-mRNAgfp] plays an important role in the production of GFP during the early 10 hours. Even the Csy4; cleavage rate is faster than the production rate of [crRNA-RBS-mRNAgfp] , the [crRNA-RBS-mRNAgfp] which is accumulated before make the keep increasing. But after the initial amount of [crRNA-RBS-mRNAgfp] is used out, the [mRNAgfp] then drop down into a stable level with the balance of the production rate and decay rate, which result in the drop down of GFP curve.

Sensitivities Analysis


After building the ODE model, we try to do something more deeply to our miniToe system by analyzing the sensitivity of parameters. Fig.4 shows the numerical integration of sensitivities of parameters in 30 hours.

Fig.4 The numerical integration of sensitivities of parameters in 30h

According to the sensitivity analysis, we can find that three core parameters,kon ,koff , kobs , which is related to the protein Csy4, has different effect in the expression of GFP. The two binding parameters,kon ,koff , will not influence GFP while the cleavage parameter, , will influence the expression in GFP, which indicated if we change the wild-type Csy4 to some mutation then we can achieve the different expression of GFP.
Enlighten by the sensitivity analysis, we give a prediction curve that shows that what will happen in the GFP expression curve if we change the Csy4, and it can be seen in the Fig.5.

Fig.5 The curve of GFP with the changing cleavage rate

And the Fig.6 shows that the relationship between the stable expression level of GFP and the claevage rate,kobs .

Fig.6 The relationship between the stable expression level of GFP and the claevage rate

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