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>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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
-     <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>
      <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>
@@ Line 53: / Line 51: @@
      <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>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>
    </table>
   </div>
   <p>
-The simulation result is shown below:
+The simulation results in static state are shown below:
   </p>
   <div class="floatC">
@@ Line 71: / 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>
@@ Line 78: / Line 76: @@
   <h3 id="title2">Parameter Analysis</h3>
   <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>
   <table style="width:600px;"><tr>
-  <td>Function</td>
    <td>Parameter</td>
+  <td>Function</td>
    <td>Description</td>
    <td>Remark</td>
   </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>
-   <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>
-   <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>
-   <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>
+ <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 class="PKU_context floatR">
   <h3 id="title4">Reference</h3>