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">
-
   <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>k<sub>1</sub></td><td>3.x10<sup>-4</sup></td><td>s<sup>-1</sup></td><td>vivid decay rate constant</td><td></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>k<sub>2</sub></td><td>5.6x10<sup>-5</sup></td><td>s<sup>-1</sup></td><td>vivid dissociation rate constant</td><td><a href="#ref3" title="Zoltowski, B.D., Vaccaro, B., and Crane, B.R. (2009). Mechanism-based tuning of a LOV domain photoreceptor. Nat. Chem. Biol. 5: 827: 834">[3]</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>k<sub>3</sub></td><td>8.x10<sup>-4</sup></td><td>s<sup>-1</sup></td><td>monomer LexA releasing rate constant from specific binding site</td><td></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>k<sub>4</sub></td><td>1.x10<sup>-3</sup></td><td>s<sup>-1</sup></td><td>binded monomer LexA dissociation rate constant</td><td></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>k<sub>5</sub></td><td>1.x10<sup>-4</sup></td><td>s<sup>-1</sup></td><td>dimered LexA releasing rate constant from specific binding site</td><td></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>K<sub>1</sub>(Dark)</td><td>0</td><td>1</td><td>equilibrium excitation constant on dark</td><td></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>K<sub>1</sub>(Light)</td><td>1.x10<sup>+3</sup></td><td>1</td><td>equilibrium excitation constant on light</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>K<sub>2</sub></td><td>7.7x10<sup>-5</sup></td><td>(n mol/L)<sup>-1</sup></td><td>vivid association equilibrium constant</td><td><a href="#ref1" title="Zoltowski, B.D., Crane, B.R.(2008). Light Activation of the LOV Protein Vivid Generates a Rapidly Exchanging Dimer.Biochemistry, 47: 7012: 7019 ">[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>K<sub>3</sub></td><td>1.x10<sup>-3</sup></td><td>(n mol/L)<sup>-1</sup></td><td>monomer LexA binding equilibrium constant with specific binding site</td><td><a href="#ref2" title="2. Mohana-Borges, R., Pacheco, A.B., Sousa, F.J., Foguel, D., Almeida, D.F., and Silva, J.L. (2000). LexA repressor forms stable dimers in solution. The role of specific DNA in tightening protein-protein interactions. J. Biol. Chem., 275: 4708: 4712">[2]</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>K<sub>4</sub></td><td>K<sub>2</sub>xK<sub>5</sub>/K<sub>3</sub></td><td>(n mol/L)<sup>-1</sup></td><td>binded monomer LexA association equilibrium constant</td><td>Thermal Principle</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>
   </tr><tr>
-
     <td>K<sub>5</sub></td><td>1.</td><td>(n mol/L)<sup>-1</sup></td><td>dimered LexA binding equilibrium constant</td><td><a href="#ref2" title="Mohana-Borges, R., Pacheco, A.B., Sousa, F.J., Foguel, D., Almeida, D.F., and Silva, J.L. (2000). LexA repressor forms stable dimers in solution. The role of specific DNA in tightening protein-protein interactions. J. Biol. Chem., 275: 4708: 4712">[2]</a></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>
   </tr><tr>
   </tr><tr>
-
     <td>[L<sub>G</sub>]<sub>0</sub></td><td>1000</td><td>n mol/L</td><td>initial concentration of <i>Luminesensor</i> in ground state</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>[L<sub>A</sub>]<sub>0</sub></td><td>0</td><td>n mol/L</td><td>initial concentration of <i>Luminesensor</i> in active state</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>
+
-
    <td>[L<sub>A</sub><sup>2</sup>]<sub>0</sub></td><td>0</td><td>n mol/L</td><td>initial concentration of dimered <i>Luminesensor</i></td><td></td>
+
-
  </tr><tr>
+
-
    <td>[D<sub>L</sub>]<sub>0</sub></td><td>100</td><td>n mol/L</td><td>initial concentration of free specific binding site on DNA</td><td>high-copy plasmid</td>
+
-
  </tr><tr>
+
-
    <td>[L<sub>G</sub>D<sub>L</sub>]<sub>0</sub></td><td>0</td><td>n mol/L</td><td>initial concentration of dimered <i>Luminesensor</i> binded <i>Luminesensor</i> in ground state</td><td></td>
+
-
  </tr><tr>
+
-
    <td>[L<sub>A</sub>D<sub>L</sub>]<sub>0</sub></td><td>0</td><td>n mol/L</td><td>initial concentration of dimered <i>Luminesensor</i> binded <i>Luminesensor</i> in active state</td><td></td>
+
-
  </tr><tr>
+
-
    <td>[L<sub>A</sub><sup>2</sup>D<sub>L</sub>]<sub>0</sub></td><td>0</td><td>n mol/L</td><td>initial concentration of binded and dimered <i>Luminesensor</i></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>
 +
<div class="PKU_context floatR">
<div class="PKU_context floatR">
  <h3 id="title2">Parameter Analysis</h3>
  <h3 id="title2">Parameter Analysis</h3>
  <p>
  <p>
-
In order to verify the robustness of <i>Luminesensor</i> function, we simulated this reaction network with a <!--<a href="/Team:Peking/Modeling/Appendix/Stochastic">-->stochastic model<!--</a>-->. By estimating the volume of a cell, we converted the concentration of a component into the number of molecules by 1 n mol/L : 1. The results are shown below:
+
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>
 +
<table style="width:600px;"><tr>
 +
  <td>Parameter</td>
 +
  <td>Function</td>
 +
  <td>Description</td>
 +
  <td>Remark</td>
 +
</tr><tr>
 +
  <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>
 +
</tr><tr>
 +
  <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>
 +
  <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>
 +
</tr><tr>
 +
  <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>
  <p>
  <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">
  <div class="floatC">
-
   <img src="/wiki/images/c/c3/Peking2012_sto_YL.png" alt="Simulation Result" style="width:600px;"/>
+
   <img src="/wiki/images/4/4e/G(1).png" alt="Simulation Result" style="width:500px;"/>
-
  <div>
+
   <p class="description" style="text-align:center;">
   <p class="description" style="text-align:center;">
-
Figure 2. Stochastic Simulation Result of Prototype <i>Luminesensor</i>.
+
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>
   </p>
-
  </div>
 
  </div>
  </div>
  <p>
  <p>
-
According to Figure 2 above, noise does not influence this system. Thus the <i>Luminesensor</i> is expected to work theoretically. Besides, the average value of stochastic simulation is consistent with the result of ODE model, which in turn proves the self-consistency of our ODE model.
+
Then, we tuned &alpha;<sub>C</sub>, the production rate of CI:
  </p>
  </p>
-
</div>
+
<div class="floatC">
-
<div class="PKU_context floatR">
+
  <img src="/wiki/images/5/55/C(0.2).png" alt="Simulation Result" style="width:500px;"/>
-
<h3 id="title3">Simulation for GFP Expression <br />Regulated by the <i>Luminesensor</i></h3>
+
  <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>
  <p>
-
In order to see whether our model is predictive for the downstream gene expression under control of the <i>Luminesensor</i>, transcription and translation process were incorporated into the modeling of DNA binding process. In addition, we considered the delay of translation initiation time and the growth of cell. The simulation below(Figure 3) represents the GFP expression regulated by the <i>Luminesensor</i>. After a long time in light condition, where GFP expression is inhibited, from <i>t=0h</i>, the cells are moved into dark and begin to express GFP. The GFP expression level varying with time was recorded in this simulation.
+
Ultimately, we tuned I<sub>0</sub>/K, the ratio of central light intensity to the sensitivity of <i>Luminesensor</i>:
-
</p>
+
</p>
  <div class="floatC">
  <div class="floatC">
-
   <img src="/wiki/images/0/07/Wild_type.png" alt="Simulation Result" style="width:500px;"/>
+
   <img src="/wiki/images/c/c1/I0(0.01).png" alt="Simulation Result" style="width:500px;"/>
-
   <div>
+
  <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;">
   <p class="description" style="text-align:center;">
-
Figure 3. ODE Simulation Result is correspond to the experiment data of GFP expression level according to time from, which suggests that our model is effective to present the experiment situation.
+
Figure 11. Ring Pattern Simulation for <i>I<sub>0</sub>/K=1x10<sup>0</sup>M/min</i>.
   </p>
   </p>
-
  </div>
 
  </div>
  </div>
</div>
</div>
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  <p></p>
  <p></p>
  <ul class="refer"><li id="ref1">
  <ul class="refer"><li id="ref1">
-
1. Zoltowski, B.D., Crane, B.R.(2008). Light Activation of the LOV Protein Vivid Generates a Rapidly Exchanging Dimer. <i>Biochemistry</i>, 47: 7012: 7019
+
1. Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134
-
  </li><li id = "ref2">
+
-
2. Mohana-Borges, R., Pacheco, A.B., Sousa, F.J., Foguel, D., Almeida, D.F., and Silva, J.L. (2000). LexA repressor forms stable dimers in solution. The role of specific DNA in tightening protein-protein interactions. <i>J. Biol. Chem.</i>, 275: 4708: 4712
+
-
  </li><li id = "ref3">
+
-
3. Zoltowski, B.D., Vaccaro, B., and Crane, B.R. (2009). Mechanism-based tuning of a LOV domain photoreceptor. <i>Nat. Chem. Biol.</i> 5: 827: 834
+
  </li></ul>
  </li></ul>
</div>
</div>
</html>{{Template:Peking2012_Color_Epilogue}}
</html>{{Template:Peking2012_Color_Epilogue}}

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
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