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 with MATLAB with equations listed as below:

And parameters as

Parameter	Value	Unit	Description	Source
a_G	2	10^-6M/min	GFP production rate constant
a_C	2	10^-6M/min	CI production rate constant	[3]
a_L1	1	10^-6M/min	LacI production rate constant
a_L2	1	10^-6M/min	LacIM1 production rate constant
b_C	8.x10^-3	10^-6M	Binding strength of CI on LacI operator
b_L	8.x10^-1	10^-6M	Binding strength of LacI or LacIM1 on GFP operator
b_R	1.x10^-2	10^-6M	Binding strength of Luminesensor on corresponding operator
r_G	6.92x10^-2	min^-1	GFP dissociation rate constant	[1]
r_C	6.92x10^-2	min^-1	CI dissociation rate constant	[2]
r_L	2.31x10^-2	min^-1	LacI and LacIM1 dissociation rate constant	Thermal Principle
r_R	2.31x10^-2	min^-1	Luminesensor dissociation rate constant	[2]
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 result is 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 above, we discovered that the activation and decay of Luminesensor 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 Luminesensor does not totally dimerized.

Parameter Analysis

After modeling the prototype Luminesensor, 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 Luminesensor.

Function	Parameter	Description	Remark
Reduce responsing time	k₁	Vivid lighting decay rate constant	Mainly on process from Light to Dark
Reduce responsing time	k₃	rate constant of monomer LexA releasing from specific binding site
Enhance contrast	K₂	Vivid association equilibrium constant	More dimerization provides more binding opportunity
Enhance contrast	K₅	dimered LexA binding equilibrium constant	More binding affinity

Reference

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

@@ 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>vivid decay rate constant</td><td></td>
+     <td>a<sub>G</sub></td><td>2</td><td>10<sup>-6</sup>M/min</td><td>GFP production rate constant</td><td></td>
     </tr><tr>
-     <td>a<sub>C</sub></td><td>2</td><td>10<sup>-6</sup>M/min</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>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="#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>
     </tr><tr>
-     <td>a<sub>L1</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>monomer LexA releasing rate constant from specific binding site</td><td></td>
+     <td>a<sub>L1</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>LacI production rate constant</td><td></td>
     </tr><tr>
-     <td>a<sub>L2</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>binded monomer LexA dissociation rate constant</td><td></td>
+     <td>a<sub>L2</sub></td><td>1</td><td>10<sup>-6</sup>M/min</td><td>LacIM1 production rate constant</td><td></td>
     </tr><tr>
-     <td>b<sub>C</sub></td><td>8.x10<sup>-3</sup></td><td>10<sup>-6</sup>M</td><td>dimered LexA releasing rate constant from specific binding site</td><td></td>
+     <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></td>
     </tr><tr>
-     <td>b<sub>L</sub></td><td>8.x10<sup>-1</sup></td><td>10<sup>-6</sup>M</td><td>equilibrium excitation constant on dark</td><td></td>
+     <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></td>
     </tr><tr>
-     <td>b<sub>R</sub></td><td>1.x10<sup>-2</sup></td><td>10<sup>-6</sup>M</td><td>equilibrium excitation constant on light</td><td></td>
+     <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>
     </tr><tr>
-     <td>r<sub>G</sub></td><td>6.92x10<sup>-2</sup></td><td>min<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>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="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>
     </tr><tr>
-     <td>r<sub>C</sub></td><td>6.92x10<sup>-2</sup></td><td>min<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>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="#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>
     </tr><tr>
-     <td>r<sub>L</sub></td><td>2.31x10<sup>-2</sup></td><td>min<sup>-1</sup></td><td>binded monomer LexA association equilibrium constant</td><td>Thermal Principle</td>
+     <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>Thermal Principle</td>
     </tr><tr>
-     <td>r<sub>R</sub></td><td>2.31x10<sup>-2</sup></td><td>min<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>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><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>
     </tr><tr>
-     <td>I<sub>0</sub></td><td>1000</td><td>AU</td><td>initial concentration of <i>Luminesensor</i> in ground state</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>
     </tr><tr>
-     <td>k</td><td>500</td><td>10<sub>-6</sub>M</td><td>initial concentration of <i>Luminesensor</i> in active 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>K</td><td>10000</td><td>AU</td><td>initial concentration of dimered <i>Luminesensor</i></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>