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{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}}
=='''Our Results'''==


 
:During the run of the summer 2006, we had time to study six 2-dimensional Dynamical Systems. Unfortunately we lacked time to carry out a thorough analysis of the 3D model.In order of complexity, the 2D models are:
: We had time to study the following five Dynamical Systems. In order of complexity:
<br><br>
:* '''2D Model 1: Lotka – Volterra'''  
:*<font size="4"> '''2D Model 1: Lotka – Volterra''' </font size="4">
:::[[Image:Model1.PNG]]
:::[[Image:Model1.PNG]]
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1| Detail Analysis for Model 1]]</b>
::*Lotka-Volterra is the first (and most famous) model for prey-predator interactions and is notoriously endowed with some very appealing properties. Lotka-Volterra also was a major inspiration for the design of the molecular predation oscillator.
::*Lotka-Volterra is the first (and most famous) model for prey-predator interactions. We detail  here some of its (very appealing) properties.


 
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model1| Detailed Analysis for Lotka-volterra]]</b>
:* '''2D Model 2: Bounded Prey Growth'''  
<br><br>
:*<font size="4"> '''2D Model 2: Bounded Prey Growth'''</font size="4">
:::[[Image:Model2.PNG]]
:::[[Image:Model2.PNG]]
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model2| Detail Analysis for Model 2]]</b>
::*Lotka-Volterra is far too simple to yield essential results on the complex 2D model.  
::*Lotka-Volterra is far too simple a model to yield any valuable results on the complex 2D model we wish to study. We start to investigate the influence of the various components of the system here by bounding the growth of the preys.
::*We start to investigate the influence of various components of the system by bounding the growth of the preys.
::*The prey (AHL) production is limited by the limited number of promoters. The number of promoter is directly responsible for the production. The rate of production will becomes linear once the number of promoters is saturated. This enzyme activation-site like behaviours can be modelled similarly to the Michaels Menton model
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model2| Detailed Analysis for Model with Bounded Prey Growth]]</b>
 
<br><br>
:* '''2D Model 3: Bounded Predator and Prey Growth'''
:*<font size="4">  '''2D Model 3: Bounded Predator and Prey Growth'''</font size="4">
:::[[Image:Model3.PNG]]
:::[[Image:Model3.PNG]]
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3| Detail Analysis for Model 3]]</b>
::*Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate anymore.  
::*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.
::*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
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3| Detailed Analysis for Model with Bounded Growths]]</b>
 
<br><br>
:* '''2D Model 3a: Bounded Predator and Prey Growth'''
[[Image:blockdiagram.jpg|thumb|600px|center|The path from Lotka-Volterra to the 2D model of the Predation Oscillator ]]
<br>
:*<font size="4">'''2D Model 3bis: Bounded Prey Growth and Prey Killing '''</font size="4"> 
:::[[Image:Model3a.PNG]]
:::[[Image:Model3a.PNG]]
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a| Detail Analysis for Model 3a]]</b>
::*We have studied this model in parallel with Model 3.
::*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.
::*Instead of bounding the production of the predator, we bound the degradation of preys
::*Similarly, the production of the predator is also limited by the number of promoters
::* In both cases the goal was to investigate whether the various terms of the model could balance each other and yield oscillations. 
 
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3a| Detailed Analysis for Model with bounded prey growth and degradation]]</b>
:* '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''
<br><br>
:*<font size="4"> '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''</font size="4">
:::[[Image:Model4.PNG]]
:::[[Image:Model4.PNG]]
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4| Detail Analysis for Model 4]]</b>
::* Bounding growth and killing yielded oscillations; bounding prey and predator growths did not.
::*We have obtained oscillations but unfortunately the killing term for the preys remains unrealistic and need regulating. we show such regulation introduces in the system some undesirable properties.
::* We now combine both previous models and get one step closer to the final system
::*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/Results/2D Model4| Detailed Analysis for Model 4]]</b>
 
<br><br>
:* '''Final 2D Model : 2D Model 5'''
:* <font size="4">'''Final 2D Model : 2D Model 5'''</font size="4">
:::[[Image:Model5.PNG]]
:::[[Image:Model5.PNG]]
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model5| Detail Analysis for Model 5]]</b>
::*Model 4 can be made to oscillate but also exhibits some very unrealistic properties.
::*Model 4 can be made to oscillate. However, it also exhibits some very unwelcome properties. Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
::* 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.
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model5| Detailed Analysis of the complete 2D Model]]</b>


::*We investigate the properties of the 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.
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::*The "-eU" term here is the "natural" decay rate of AHL. However, this is not mainly due to the half-life of the AHL since AHL is quite stable itself. The dominant contribution to this decay rate is the "wash-out" rate in the chemostat.  
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::*AHL is small molecules that are free to move in the cells and medium. Hence it will be "wash-out" when we pump out the medium from the chemostat
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::*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
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Latest revision as of 07:40, 2 November 2006

Analysis of the Model of the Molecular Predation Oscillator


Our Results

During the run of the summer 2006, we had time to study six 2-dimensional 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
  • Lotka-Volterra is the first (and most famous) model for prey-predator interactions and is notoriously endowed with some very appealing properties. Lotka-Volterra also was a major inspiration for the design of the molecular predation oscillator.



  • 2D Model 2: Bounded Prey Growth



  • 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.
  • Detailed Analysis for Model with Bounded Growths



The path from Lotka-Volterra to the 2D model of the Predation Oscillator


  • 2D Model 3bis: Bounded Prey Growth and Prey Killing



  • 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys
  • Bounding growth and killing yielded oscillations; bounding prey and predator growths did not.
  • We now combine both previous models and get one step closer to the final system
  • Detailed Analysis for Model 4



  • Final 2D Model : 2D Model 5
  • Model 4 can be made to oscillate but also exhibits 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.
  • Detailed Analysis of the complete 2D Model

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