IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results
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(→'''Our Results''') 
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  :  +  :During the run of the summer 2006, we had time to study six 2dimensional Dynamical Systems. Unfortunately we lacked time to carry out a thorough analysis of the 3D model. 
  :  +  :In order of complexity, the 2D models are: 
+  <br><br>  
+  :<font size="4"> '''2D Model 1: Lotka – Volterra''' </font size="4">  
:::[[Image:Model1.PNG]]  :::[[Image:Model1.PNG]]  
  ::*LotkaVolterra is the first (and most famous) model for preypredator interactions  +  ::*LotkaVolterra is the first (and most famous) model for preypredator interactions and is notoriously endowed with some very appealing properties. LotkaVolterra also was a major inspiration for the design of the molecular predation oscillator. 
  +  
  :* '''2D Model 2: Bounded Prey Growth'''  +  ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1 Detail Analysis for Lotkavolterra]]</b> 
+  <br><br>  
+  :<font size="4"> '''2D Model 2: Bounded Prey Growth'''</font size="4">  
:::[[Image:Model2.PNG]]  :::[[Image:Model2.PNG]]  
::*LotkaVolterra is far too simple to yield essential results on the complex 2D model.  ::*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  +  ::*We start to investigate the influence of various components of the system by bounding the growth of the preys. 
  +  
  +  
::*<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>  
  +  <br><br>  
  :  +  :<font size="4"> '''2D Model 3: Bounded Predator and Prey Growth'''</font size="4"> 
:::[[Image:Model3.PNG]]  :::[[Image:Model3.PNG]]  
  ::*Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate.  +  ::*Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate anymore. 
::*We now seek ways to obtain oscillations by bounding the growth terms of both preys and predators.  ::*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  ::*Similarly, the production of the predator is also limited by the number of promoters available 
Revision as of 08:55, 30 October 2006
Analysis of the Model of the Molecular Predation Oscillator
 Introduction
 Our Approach
 Model Simplication
 Our Results
 Conclusion
 Appendix
Our Results
 During the run of the summer 2006, we had time to study six 2dimensional Dynamical Systems. Unfortunately we lacked time to carry out a thorough analysis of the 3D model.
 In order of complexity, the 2D models are:
 2D Model 1: Lotka – Volterra
 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 by bounding the growth of the preys.
 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 anymore.
 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