IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results

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('''Our Results''')
Current revision (09:40, 2 November 2006) (view source)
('''Our Results''')
 
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{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}}
{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}}
=='''Our Results'''==
=='''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:
: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:
<br><br>
<br><br>
-
:<font size="4"> '''2D Model 1: Lotka – Volterra''' </font size="4">
+
:*<font size="4"> '''2D Model 1: Lotka – Volterra''' </font size="4">
:::[[Image:Model1.PNG]]
:::[[Image:Model1.PNG]]
::*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 and is notoriously endowed with some very appealing properties. Lotka-Volterra also was a major inspiration for the design of the molecular predation oscillator.
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1| Detail Analysis for Lotka-volterra]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model1| Detailed Analysis for Lotka-volterra]]</b>
<br><br>
<br><br>
-
:<font size="4"> '''2D Model 2: Bounded Prey Growth'''</font size="4">  
+
:*<font size="4"> '''2D Model 2: Bounded Prey Growth'''</font size="4">  
:::[[Image:Model2.PNG]]
:::[[Image:Model2.PNG]]
::*Lotka-Volterra is far too simple to yield essential results on the complex 2D model.  
::*Lotka-Volterra 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.
::*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 with Bounded Prey Growth]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model2| Detailed Analysis for Model with Bounded Prey Growth]]</b>
<br><br>
<br><br>
-
:<font size="4">  '''2D Model 3: Bounded Predator and Prey Growth'''</font size="4">  
+
:*<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 anymore.  
::*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.
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3| Detail Analysis for Model with Bounded Growths]]</b>
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3| Detailed Analysis for Model with Bounded Growths]]</b>
<br><br>
<br><br>
-
:<font size="4">   '''2D Model 3a: Bounded Predator and Prey Growth'''</font size="4">   
+
[[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]]
-
::*We have studied this model in parallel with the previous model
+
::*We have studied this model in parallel with Model 3.
-
::*Instead of bounding the production of the predator, we bound the degradation of prey
+
::*Instead of bounding the production of the predator, we bound the degradation of preys
-
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a| Detail Analysis for Model 3a]]</b>
+
::* 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]]
-
::*Combining the previous two models
+
::* Bounding growth and killing yielded oscillations; bounding prey and predator growths did not.
-
::*Such combination introduces some undesirable properties in the system.
+
::* We now combine both previous models and get one step closer to the final 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/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]]
-
::*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 but also exhibits some very unrealistic properties.
-
::*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.
+
::* Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
-
::*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.
+
::*We investigate the properties of this final 2D model and prove that the new dissipative term confers it some very interesting characteristics.
-
::*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
+
::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model5| Detailed Analysis of the complete 2D Model]]</b>
-
::*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>
+
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Current revision

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
Image:Model1.PNG
  • 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
Image:Model2.PNG



  • 2D Model 3: Bounded Predator and Prey Growth
Image:Model3.PNG
  • 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
The path from Lotka-Volterra to the 2D model of the Predation Oscillator


  • 2D Model 3bis: Bounded Prey Growth and Prey Killing
Image:Model3a.PNG



  • 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys
Image:Model4.PNG
  • 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
Image:Model5.PNG
  • 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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