IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results
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:* '''2D Model 1: Lotka – Volterra'''  :* '''2D Model 1: Lotka – Volterra'''  
:::[[Image:Model1.PNG]]  :::[[Image:Model1.PNG]]  
+  ::*LotkaVolterra is the first (and most famous) model for preypredator interactions. There are some very appealing properties.  
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1 Detail Analysis for Model 1]]</b>  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1 Detail Analysis for Model 1]]</b>  
  
  
:* '''2D Model 2: Bounded Prey Growth'''  :* '''2D Model 2: Bounded Prey Growth'''  
:::[[Image:Model2.PNG]]  :::[[Image:Model2.PNG]]  
+  ::*LotkaVolterra is far too simple to yield essential results on the complex 2D model.  
+  ::*We start to investigate the influence of various components of the system, e.g. bounding of the growth of the preys.  
+  ::*The prey (AHL) production is bounded by the limited number of promoters. The number of activated promoter is directly responsible for the rate of production.  
+  ::*The rate of production will become linear once the number of promoters is saturated. This enzyme activationsitelimited like behaviours can be modelled similarly to the Michaels Menton model  
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model2 Detail Analysis for Model 2]]</b>  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model2 Detail Analysis for Model 2]]</b>  
  
  
:* '''2D Model 3: Bounded Predator and Prey Growth'''  :* '''2D Model 3: Bounded Predator and Prey Growth'''  
:::[[Image:Model3.PNG]]  :::[[Image:Model3.PNG]]  
+  ::*Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate.  
+  ::*We now seek ways to obtain oscillations by bounding the growth terms of both preys and predators.  
+  ::*Similarly, the production of the predator is also limited by the number of promoters available  
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3 Detail Analysis for Model 3]]</b>  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3 Detail Analysis for Model 3]]</b>  
  
  
:* '''2D Model 3a: Bounded Predator and Prey Growth'''  :* '''2D Model 3a: Bounded Predator and Prey Growth'''  
:::[[Image:Model3a.PNG]]  :::[[Image:Model3a.PNG]]  
+  ::*We have studied this model in parallel with the previous model  
+  ::*Instead of bounding the production of the predator, we bound the degradation of prey  
+  ::*The degradation of prey(AHL) by predator(aiiA) is truely an enzyme reaction, hence the killing of prey can be modelled by Michaelis Menton directly  
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a Detail Analysis for Model 3a]]</b>  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a Detail Analysis for Model 3a]]</b>  
  
  
:* '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''  :* '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''  
:::[[Image:Model4.PNG]]  :::[[Image:Model4.PNG]]  
+  ::*Combining the previous two models  
+  ::*Such combination introduces some undesirable properties in the system.  
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4 Detail Analysis for Model 4]]</b>  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4 Detail Analysis for Model 4]]</b>  
  
  
:* '''Final 2D Model : 2D Model 5'''  :* '''Final 2D Model : 2D Model 5'''  
:::[[Image:Model5.PNG]]  :::[[Image:Model5.PNG]]  
  +  ::*Model 4 can be made to oscillate but exhibit some very unrealistic properties. Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.  
  ::*Model 4 can be made to oscillate  +  ::*We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics –among other things it prevents all the problems that may be encountered with Model 4. 
  +  
  ::*We investigate the properties of  +  
  +  
::*The "eU" term here is the "natural" decay rate of AHL. However, this is not mainly due to the halflife of the AHL since AHL is quite stable itself. The dominant contribution to this decay rate is the "washout" rate in the chemostat.  ::*The "eU" term here is the "natural" decay rate of AHL. However, this is not mainly due to the halflife of the AHL since AHL is quite stable itself. The dominant contribution to this decay rate is the "washout" rate in the chemostat.  
::*AHL is small molecules that are free to move in the cells and medium. Hence it will be "washout" when we pump out the medium from the chemostat  ::*AHL is small molecules that are free to move in the cells and medium. Hence it will be "washout" when we pump out the medium from the chemostat  
::*This will allow us to have a extra feature to change the magnitude of the parameter "e" and maybe give us a better control of the system  ::*This will allow us to have a extra feature to change the magnitude of the parameter "e" and maybe give us a better control of the system  
+  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model5 Detail Analysis for Model 5]]</b> 
Revision as of 05:11, 30 October 2006
Analysis of the Model of the Molecular Predation Oscillator
 Introduction
 Our Approach
 Model Simplication
 Our Results
 Conclusion
 Appendix
Our Results
 We had time to study the following five Dynamical Systems. In order of complexity:
 2D Model 1: Lotka – Volterra

 LotkaVolterra is the first (and most famous) model for preypredator interactions. There are some very appealing properties.
 Detail Analysis for Model 1
 2D Model 2: Bounded Prey Growth

 LotkaVolterra is far too simple to yield essential results on the complex 2D model.
 We start to investigate the influence of various components of the system, e.g. bounding of the growth of the preys.
 The prey (AHL) production is bounded by the limited number of promoters. The number of activated promoter is directly responsible for the rate of production.
 The rate of production will become linear once the number of promoters is saturated. This enzyme activationsitelimited like behaviours can be modelled similarly to the Michaels Menton model
 Detail Analysis for Model 2
 2D Model 3: Bounded Predator and Prey Growth

 Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate.
 We now seek ways to obtain oscillations by bounding the growth terms of both preys and predators.
 Similarly, the production of the predator is also limited by the number of promoters available
 Detail Analysis for Model 3
 2D Model 3a: Bounded Predator and Prey Growth

 We have studied this model in parallel with the previous model
 Instead of bounding the production of the predator, we bound the degradation of prey
 The degradation of prey(AHL) by predator(aiiA) is truely an enzyme reaction, hence the killing of prey can be modelled by Michaelis Menton directly
 Detail Analysis for Model 3a
 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys

 Combining the previous two models
 Such combination introduces some undesirable properties in the system.
 Detail Analysis for Model 4
 Final 2D Model : 2D Model 5

 Model 4 can be made to oscillate but exhibit some very unrealistic properties. Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
 We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics –among other things it prevents all the problems that may be encountered with Model 4.
 The "eU" term here is the "natural" decay rate of AHL. However, this is not mainly due to the halflife of the AHL since AHL is quite stable itself. The dominant contribution to this decay rate is the "washout" rate in the chemostat.
 AHL is small molecules that are free to move in the cells and medium. Hence it will be "washout" when we pump out the medium from the chemostat
 This will allow us to have a extra feature to change the magnitude of the parameter "e" and maybe give us a better control of the system
 Detail Analysis for Model 5