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
\alpha₁	3.x10^-4	s^-1	vivid decay rate constant
k₂	5.6x10^-5	s^-1	vivid dissociation rate constant	[3]
k₃	8.x10^-4	s^-1	monomer LexA releasing rate constant from specific binding site
k₄	1.x10^-3	s^-1	binded monomer LexA dissociation rate constant
k₅	1.x10^-4	s^-1	dimered LexA releasing rate constant from specific binding site
K₁(Dark)	0	1	equilibrium excitation constant on dark
K₁(Light)	1.x10⁺³	1	equilibrium excitation constant on light
K₂	7.7x10^-5	(n mol/L)^-1	vivid association equilibrium constant	[1]
K₃	1.x10^-3	(n mol/L)^-1	monomer LexA binding equilibrium constant with specific binding site	[2]
K₄	K₂xK₅/K₃	(n mol/L)^-1	binded monomer LexA association equilibrium constant	Thermal Principle
K₅	1.	(n mol/L)^-1	dimered LexA binding equilibrium constant	[2]
[L_G]₀	1000	n mol/L	initial concentration of Luminesensor in ground state
[L_A]₀	0	n mol/L	initial concentration of Luminesensor in active state
[L_A²]₀	0	n mol/L	initial concentration of dimered Luminesensor
[D_L]₀	100	n mol/L	initial concentration of free specific binding site on DNA	high-copy plasmid
[L_GD_L]₀	0	n mol/L	initial concentration of dimered Luminesensor binded Luminesensor in ground state
[L_AD_L]₀	0	n mol/L	initial concentration of dimered Luminesensor binded Luminesensor in active state
[L_A²D_L]₀	0	n mol/L	initial concentration of binded and 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>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>1</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>