Team:Peking/Modeling/Ring/Simulation

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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:

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:

Parameter	Value	Unit	Description	Source
α_G	2	10^-6M/min	GFP production rate constant	[1]
α_C	2	10^-6M/min	CI production rate constant	[1]
α_L1	1	10^-6M/min	LacI production rate constant	[1]
α_L2	1	10^-6M/min	LacIM1 production rate constant	[1]
θ_C	8.x10^-3	10^-6M	Binding strength of CI on LacI operator	[1]
θ_L	8.x10^-1	10^-6M	Binding strength of LacI or LacIM1 on GFP operator	[1]
θ_R	1.x10^-2	10^-6M	Binding strength of Luminesensor on corresponding operator
γ_G	6.92x10^-2	min^-1	GFP dissociation rate constant	[1]
γ_C	6.92x10^-2	min^-1	CI dissociation rate constant	[1]
γ_L	2.31x10^-2	min^-1	LacI and LacIM1 dissociation rate constant	[1]
I₀	1000	AU	Maximum light intensity in the middle of the plate
k	500	10^-6M	Luminesensor activation rate under light
K	10000	AU	Light sensitivity of Luminesensor activation

The simulation results in static state are shown below:

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

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
α_G/γ_G	If 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 GFP	The production rate of GFP is easily tuned.
α_C/(θ_C*γ_C)	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 CI	The production rate of CI is easily tuned
(kI₀)/(θ_RK)	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 parameters	Tend 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, α_G/γ_G, α_C/(θ_C*γ_C), and (k*I₀)/(θ_R*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 α_G, the production rate of GFP:

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

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

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

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

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

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

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

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

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

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

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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   <h3 id="title1">ODE Model</h3>
   <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>
   <div class="floatC">
@@ Line 17: / Line 17: @@
 <br /><br />
   <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>
   <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>
   <div class="floatC">
@@ Line 27: / Line 27: @@
      <td>Parameter</td><td>Value</td><td>Unit</td><td>Description</td><td>Source</td>
     </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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
    </table>
   </div>
   <p>
-The simulation result is shown below:
+The simulation results in static state are shown below:
   </p>
   <div class="floatC">
@@ Line 81: / Line 69: @@
   </div>
   <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>
 </div>
 <div class="PKU_context floatR">
   <h3 id="title2">Parameter Analysis</h3>
   <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>
+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/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;">
-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>
-  </div>
   </div>
   <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>
-</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>
-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">
-   <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;">
-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>
-  </div>
   </div>
 </div>
@@ Line 119: / Line 150: @@
   <p></p>
   <ul class="refer"><li id="ref1">
-. 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
+. Subhayu Basu et al.(2005), A synthetic multicellular system for programmed pattern formation. <i>Nature</i>, vol.434: 1130: 1134
-  </li><li id = "ref2">
-. 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">
-. 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>
 </div>
 </html>{{Template:Peking2012_Color_Epilogue}}