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	3.x10^-4	s^-1	vivid decay rate constant
a_C	5.6x10^-5	s^-1	vivid dissociation rate constant	[3]
a_L1	8.x10^-4	s^-1	monomer LexA releasing rate constant from specific binding site
a_L2	1.x10^-3	s^-1	binded monomer LexA dissociation rate constant
b_C	1.x10^-4	s^-1	dimered LexA releasing rate constant from specific binding site
b_L(Dark)	0	1	equilibrium excitation constant on dark
b_R(Light)	1.x10⁺³	1	equilibrium excitation constant on light
r_G	7.7x10^-5	(n mol/L)^-1	vivid association equilibrium constant	[1]
r_C	1.x10^-3	(n mol/L)^-1	monomer LexA binding equilibrium constant with specific binding site	[2]
r_L	K₂xK₅/K₃	(n mol/L)^-1	binded monomer LexA association equilibrium constant	Thermal Principle
r_R	1.	(n mol/L)^-1	dimered LexA binding equilibrium constant	[2]
I₀	1000	n mol/L	initial concentration of Luminesensor in ground state
k	0	n mol/L	initial concentration of Luminesensor in active state
K	0	n mol/L	initial concentration of dimered Luminesensor

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. Zoltowski, B.D., Crane, B.R.(2008). Light Activation of the LOV Protein Vivid Generates a Rapidly Exchanging Dimer. Biochemistry, 47: 7012: 7019
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
3. Zoltowski, B.D., Vaccaro, B., and Crane, B.R. (2009). Mechanism-based tuning of a LOV domain photoreceptor. Nat. Chem. Biol. 5: 827: 834

@@ Line 27: / Line 27: @@
      <td>Parameter</td><td>Value</td><td>Unit</td><td>Description</td><td>Source</td>
     </tr><tr>
-     <td>{\alpha}<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>a<sub>G</sub></td><td>3.x10<sup>-4</sup></td><td>s<sup>-1</sup></td><td>vivid decay rate constant</td><td></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>a<sub>C</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>
     </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>a<sub>L1</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>
     </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>a<sub>L2</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>
     </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>b<sub>C</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>
     </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>b<sub>L</sub>(Dark)</td><td>0</td><td>1</td><td>equilibrium excitation constant on dark</td><td></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>b<sub>R</sub>(Light)</td><td>1.x10<sup>+3</sup></td><td>1</td><td>equilibrium excitation constant on light</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>r<sub>G</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>
     </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>r<sub>C</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>
     </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>r<sub>L</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>
     </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>r<sub>R</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>
     </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>I<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>
     </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>0</td><td>n mol/L</td><td>initial concentration of <i>Luminesensor</i> in active state</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>
+     <td>K</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>