IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results
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: We had time to study the following five Dynamical Systems. In order of complexity:  : We had time to study the following five Dynamical Systems. In order of complexity:  
:* '''2D Model 1: Lotka – Volterra'''  :* '''2D Model 1: Lotka – Volterra''' 
Revision as of 15:53, 29 October 2006
Analysis of the Model of the Molecular Predation Oscillator
 Introduction
 Our Approach
 Model Simplication
 Our Results
 Conclusion
 Appendix
 We had time to study the following five Dynamical Systems. In order of complexity:
 2D Model 1: Lotka – Volterra

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

 Detail Analysis for Model 2
 LotkaVolterra 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.
 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 activationsite like behaviours can be modelled similarly to the Michaels Menton model
 2D Model 3: Bounded Predator and Prey Growth

 Detail Analysis for 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.
 Similarly, the production of the predator is also limited by the number of promoters
 2D Model 3a: Bounded Predator and Prey Growth

 Detail Analysis for Model 3a
 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
 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys

 Detail Analysis for Model 4
 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.
 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.
 Final 2D Model : 2D Model 5

 Detail Analysis for Model 5
 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.
 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.
 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