Team:Peking/Modeling/Ring/Simulation

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  <h3 id="title1">ODE Model</h3>
  <h3 id="title1">ODE Model</h3>
  <p>
  <p>
-
According to the previous circuit and ODE model, we listed all the differential equations <!--(<a href="/Team:Peking/Modeling/Appendix/ODE">detail here</a>)--> and simulated this system with MATLAB with equations listed as below:
+
According to the previous circuit and ODE model, we listed all the differential equations <!--(<a href="/Team:Peking/Modeling/Appendix/ODE">detail here</a>)--> and simulated this system in MATLAB with equations listed as below:
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
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<br /><br />
<br /><br />
  <div class="floatC">
  <div class="floatC">
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   <img src="/wiki/images/c/c6/ODE_formula.jpg" alt="Formulae" style="width:400px;"/>
+
   <img src="/wiki/images/e/ee/Peking2012_Formula015.png" alt="Formulae" style="width:400px;"/>
  </div>
  </div>
  <p>
  <p>
-
And parameters as
+
We applied the inverse square law to describe the light intensity distribution on the plate according to different radius, with a central intensity <i>I<sub>0</sub></i> in a region of <i>r=1mm</i>. Here, parameters are:
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
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     <td>Parameter</td><td>Value</td><td>Unit</td><td>Description</td><td>Source</td>
     <td>Parameter</td><td>Value</td><td>Unit</td><td>Description</td><td>Source</td>
   </tr><tr>
   </tr><tr>
-
     <td>a<sub>G</sub></td><td>2</td><td>10<sup>-6</sup>M/min</td><td>GFP production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&alpha;<sub>G</sub></td><td>2</td><td>10<sup>-6</sup>M/min</td><td>GFP production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>a<sub>C</sub></td><td>2</td><td>10<sup>-6</sup>M/min</td><td>CI production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&alpha;<sub>C</sub></td><td>2</td><td>10<sup>-6</sup>M/min</td><td>CI production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>a<sub>L1</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>LacI production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&alpha;<sub>L1</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>LacI production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>a<sub>L2</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>LacIM1 production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&alpha;<sub>L2</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>LacIM1 production rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>b<sub>C</sub></td><td>8.x10<sup>-3</sup></td><td>10<sup>-6</sup>M</td><td>Binding strength of CI on LacI operator</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&theta;<sub>C</sub></td><td>8.x10<sup>-3</sup></td><td>10<sup>-6</sup>M</td><td>Binding strength of CI on LacI operator</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>b<sub>L</sub></td><td>8.x10<sup>-1</sup></td><td>10<sup>-6</sup>M</td><td>Binding strength of LacI or LacIM1 on GFP operator</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&theta;<sub>L</sub></td><td>8.x10<sup>-1</sup></td><td>10<sup>-6</sup>M</td><td>Binding strength of LacI or LacIM1 on GFP operator</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>b<sub>R</sub></td><td>1.x10<sup>-2</sup></td><td>10<sup>-6</sup>M</td><td>Binding strength of <i>Luminesensor</i> on corresponding operator</td><td></td>
+
     <td>&theta;<sub>R</sub></td><td>1.x10<sup>-2</sup></td><td>10<sup>-6</sup>M</td><td>Binding strength of <i>Luminesensor</i> on corresponding operator</td><td></td>
   </tr><tr>
   </tr><tr>
-
     <td>r<sub>G</sub></td><td>6.92x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>GFP dissociation rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&gamma;<sub>G</sub></td><td>6.92x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>GFP dissociation rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>r<sub>C</sub></td><td>6.92x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>CI dissociation rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&gamma;<sub>C</sub></td><td>6.92x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>CI dissociation rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
   </tr><tr>
   </tr><tr>
-
     <td>r<sub>L</sub></td><td>2.31x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>LacI and LacIM1 dissociation rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
+
     <td>&gamma;<sub>L</sub></td><td>2.31x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>LacI and LacIM1 dissociation rate constant</td><td><a href="#ref1" title="Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134">[1]</a></td>
-
  </tr><tr>
+
-
    <td>r<sub>R</sub></td><td>2.31x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td><i>Luminesensor</i> dissociation rate constant</td><td></td>
+
   </tr><tr>
   </tr><tr>
     <td>I<sub>0</sub></td><td>1000</td><td>AU</td><td>Maximum light intensity in the middle of the plate </td><td></td>
     <td>I<sub>0</sub></td><td>1000</td><td>AU</td><td>Maximum light intensity in the middle of the plate </td><td></td>
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     <td>k</td><td>500</td><td>10<sup>-6</sup>M</td><td><i>Luminesensor</i> activation rate under light</td><td></td>
     <td>k</td><td>500</td><td>10<sup>-6</sup>M</td><td><i>Luminesensor</i> activation rate under light</td><td></td>
   </tr><tr>
   </tr><tr>
-
     <td>K</td><td>10000</td><td>AU</td><td>light sensitivity of <i>Luminesensor</i> activation </td><td></td>
+
     <td>K</td><td>10000</td><td>AU</td><td>Light sensitivity of <i>Luminesensor</i> activation </td><td></td>
   </tr>
   </tr>
   </table>
   </table>
  </div>
  </div>
  <p>
  <p>
-
The simulation result is shown below:
+
The simulation results in static state are shown below:
  </p>
  </p>
  <div class="floatC">
  <div class="floatC">
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  </div>
  </div>
  <p>
  <p>
-
From the Figure 1 above, we discovered that the activation and decay of <i>Luminesensor</i> are the key points of progress, and the activating rate is the most sensitive to light intensity. The promoter will be repressed even though the <i>Luminesensor</i> does not totally dimerized.
+
From the Figure 1 & 2 above, we discover that, with wildtype parameters, ring-like pattern is formed based on sender-receiver communication through bio-luminescence.
  </p>
  </p>
</div>
</div>
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  <h3 id="title2">Parameter Analysis</h3>
  <h3 id="title2">Parameter Analysis</h3>
  <p>
  <p>
-
After modeling the prototype <i>Luminesensor</i>, we attempted to optimize it in a rational way. We have tuned the parameters both up and down, one by one, and finally discovered four parameters which predominantly influence the performance of the <i>Luminesensor</i>.
+
After modeling the ring-like pattern formation with wildtype parameters, we attempted to optimize it in a rational way. We have tuned the parameters both up and down, one by one, and finally discovered several parameters which predominantly influence the expression intensity, ring radius, and band width of pattern formation.
  </p>
  </p>
  <table style="width:600px;"><tr>
  <table style="width:600px;"><tr>
-
  <td>Function</td>
 
   <td>Parameter</td>
   <td>Parameter</td>
 +
  <td>Function</td>
   <td>Description</td>
   <td>Description</td>
   <td>Remark</td>
   <td>Remark</td>
  </tr><tr>
  </tr><tr>
-
   <td rowspan="2">Reduce responsing time</td>
+
   <td>&alpha;<sub>G</sub>/&gamma;<sub>G</sub></td><td>If increasing, the expression intensity will be amplified, but the ring radius and the band width will not change. </td><td>Related to the production and dissociation of GFP</td><td>The production rate of GFP is easily tuned.</td>
-
  <td>k<sub>1</sub></td><td>Vivid lighting decay rate constant</td><td>Mainly on process from Light to Dark</td>
+
  </tr><tr>
  </tr><tr>
-
   <td>k<sub>3</sub></td><td>rate constant of monomer LexA releasing from specific binding site</td><td></td>
+
   <td>&alpha;<sub>C</sub>/(&theta;<sub>C</sub>*&gamma;<sub>C</sub>)</td><td>If increasing, the expression intensity will increase, but the ring radius will decease and the band width will not change. </td><td>Related to the production and dissociation of CI</td><td>The production rate of CI is easily tuned</td>
  </tr><tr>
  </tr><tr>
-
   <td rowspan="2">Enhance contrast</td>
+
   <td>(k*I<sub>0</sub>)/(&theta;<sub>R</sub>*K)</td><td>If increasing, the ring radius and the band width will increase, leaving the expression amplitude unchanged. </td><td>Related to the light intensity emitted by sender cells and the activation rate, light sensitivity, and binding efficiency of <i>Luminesensor</i>. </td><td>Light intensity could be tuned, although the effect may noe be obvious experimentally.</td>
-
  <td>K<sub>2</sub></td><td>Vivid association equilibrium constant</td><td>More dimerization provides more binding opportunity</td>
+
  </tr><tr>
  </tr><tr>
-
   <td>K<sub>5</sub></td><td>dimered LexA binding equilibrium constant</td><td>More binding affinity</td>
+
   <td>LacI and LacIM1 related parameters</td><td>Tend to influence all three criteria.</td><td>Related to the production and dissociation rate and binding efficiency of LacI and LaciM1. </td><td>Tuning is not useful to make a better pattern.</td>
  </tr></table>
  </tr></table>
-
 
+
<p>
 +
As we can see, &alpha;<sub>G</sub>/&gamma;<sub>G</sub>, &alpha;<sub>C</sub>/(&theta;<sub>C</sub>*&gamma;<sub>C</sub>), and (k*I<sub>0</sub>)/(&theta;<sub>R</sub>*K) are the most important and accessible parameters for pattern formation. To make it clear, we tuned several of the parameters each in one of the three groups to see the effect on pattern formation, while holding other parameters unchanged.
 +
<br /><br />
 +
Firstly, we tuned &alpha;<sub>G</sub>, the production rate of GFP:
 +
</p>
 +
<div class="floatC">
 +
  <img src="/wiki/images/4/4e/G(1).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 3. Ring Pattern Simulation for <i>&alpha;<sub>G</sub>=1x10<sup>-6</sup>M/min</i>.
 +
  </p>
 +
  <img src="/wiki/images/3/31/G(2).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 4. Ring Pattern Simulation for <i>&alpha;<sub>G</sub>=2x10<sup>-6</sup>M/min</i>.
 +
  </p>
 +
  <img src="/wiki/images/5/59/G(4).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 5. Ring Pattern Simulation for <i>&alpha;<sub>G</sub>=4x10<sup>-6</sup>M/min</i>.
 +
  </p>
 +
</div>
 +
<p>
 +
Then, we tuned &alpha;<sub>C</sub>, the production rate of CI:
 +
</p>
 +
<div class="floatC">
 +
  <img src="/wiki/images/5/55/C(0.2).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 6. Ring Pattern Simulation for <i>&alpha;<sub>C</sub>=0.2x10<sup>-6</sup>M/min</i>.
 +
  </p>
 +
  <img src="/wiki/images/8/83/C(2).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 7. Ring Pattern Simulation for <i>&alpha;<sub>C</sub>=2x10<sup>-6</sup>M/min</i>.
 +
  </p>
 +
  <img src="/wiki/images/b/bd/C(20).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 8. Ring Pattern Simulation for <i>&alpha;<sub>C</sub>=20x10<sup>-6</sup>M/min</i>.
 +
  </p>
 +
</div>
 +
<p>
 +
Ultimately, we tuned I<sub>0</sub>/K, the ratio of central light intensity to the sensitivity of <i>Luminesensor</i>:
 +
</p>
 +
<div class="floatC">
 +
  <img src="/wiki/images/c/c1/I0(0.01).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 9. Ring Pattern Simulation for <i>I<sub>0</sub>/K=1x10<sup>-2</sup>M/min</i>.
 +
  </p>
 +
  <img src="/wiki/images/5/5a/I0(0.1).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 10. Ring Pattern Simulation for <i>I<sub>0</sub>/K=1x10<sup>-1</sup>M/min</i>.
 +
  </p>
 +
  <img src="/wiki/images/b/ba/I0(1).png" alt="Simulation Result" style="width:500px;"/>
 +
  <p class="description" style="text-align:center;">
 +
Figure 11. Ring Pattern Simulation for <i>I<sub>0</sub>/K=1x10<sup>0</sup>M/min</i>.
 +
  </p>
 +
</div>
</div>
</div>
-
 
-
 
<div class="PKU_context floatR">
<div class="PKU_context floatR">
  <h3 id="title4">Reference</h3>
  <h3 id="title4">Reference</h3>

Latest revision as of 12:23, 26 October 2012

ODE Model

According to the previous circuit and ODE model, we listed all the differential equations and simulated this system in MATLAB with equations listed as below:

Formulae


Formulae

We applied the inverse square law to describe the light intensity distribution on the plate according to different radius, with a central intensity I0 in a region of r=1mm. Here, parameters are:

ParameterValueUnitDescriptionSource
αG210-6M/minGFP production rate constant[1]
αC210-6M/minCI production rate constant[1]
αL1110-6M/minLacI production rate constant[1]
αL2110-6M/minLacIM1 production rate constant[1]
θC8.x10-310-6MBinding strength of CI on LacI operator[1]
θL8.x10-110-6MBinding strength of LacI or LacIM1 on GFP operator[1]
θR1.x10-210-6MBinding strength of Luminesensor on corresponding operator
γG6.92x10-2min-1GFP dissociation rate constant[1]
γC6.92x10-2min-1CI dissociation rate constant[1]
γL2.31x10-2min-1LacI and LacIM1 dissociation rate constant[1]
I01000AUMaximum light intensity in the middle of the plate
k50010-6MLuminesensor activation rate under light
K10000AULight sensitivity of Luminesensor activation

The simulation results in static state are shown below:

Simulation Result

Figure 1. ODE Simulation in a plate of the ring-like pattern formation.

Simulation Result

Figure 2. ODE Simulation for the radial expression amplitude of the ring-like pattern formation.

From the Figure 1 & 2 above, we discover that, with wildtype parameters, ring-like pattern is formed based on sender-receiver communication through bio-luminescence.

Parameter Analysis

After modeling the ring-like pattern formation with wildtype parameters, we attempted to optimize it in a rational way. We have tuned the parameters both up and down, one by one, and finally discovered several parameters which predominantly influence the expression intensity, ring radius, and band width of pattern formation.

Parameter Function Description Remark
αGGIf increasing, the expression intensity will be amplified, but the ring radius and the band width will not change. Related to the production and dissociation of GFPThe production rate of GFP is easily tuned.
αC/(θCC)If increasing, the expression intensity will increase, but the ring radius will decease and the band width will not change. Related to the production and dissociation of CIThe production rate of CI is easily tuned
(k*I0)/(θR*K)If increasing, the ring radius and the band width will increase, leaving the expression amplitude unchanged. Related to the light intensity emitted by sender cells and the activation rate, light sensitivity, and binding efficiency of Luminesensor. Light intensity could be tuned, although the effect may noe be obvious experimentally.
LacI and LacIM1 related parametersTend to influence all three criteria.Related to the production and dissociation rate and binding efficiency of LacI and LaciM1. Tuning is not useful to make a better pattern.

As we can see, αGG, αC/(θCC), and (k*I0)/(θR*K) are the most important and accessible parameters for pattern formation. To make it clear, we tuned several of the parameters each in one of the three groups to see the effect on pattern formation, while holding other parameters unchanged.

Firstly, we tuned αG, the production rate of GFP:

Simulation Result

Figure 3. Ring Pattern Simulation for αG=1x10-6M/min.

Simulation Result

Figure 4. Ring Pattern Simulation for αG=2x10-6M/min.

Simulation Result

Figure 5. Ring Pattern Simulation for αG=4x10-6M/min.

Then, we tuned αC, the production rate of CI:

Simulation Result

Figure 6. Ring Pattern Simulation for αC=0.2x10-6M/min.

Simulation Result

Figure 7. Ring Pattern Simulation for αC=2x10-6M/min.

Simulation Result

Figure 8. Ring Pattern Simulation for αC=20x10-6M/min.

Ultimately, we tuned I0/K, the ratio of central light intensity to the sensitivity of Luminesensor:

Simulation Result

Figure 9. Ring Pattern Simulation for I0/K=1x10-2M/min.

Simulation Result

Figure 10. Ring Pattern Simulation for I0/K=1x10-1M/min.

Simulation Result

Figure 11. Ring Pattern Simulation for I0/K=1x100M/min.

Reference

  • 1. Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. Nature, vol.434: 1130: 1134
  • Totop Totop