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

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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.
  • Detail Analysis for Model with Bounded Growths

2D Model 3bis: Bounded Prey Growth and Prey Killing
  • We have studied this model in parallel with Model 3.
  • Instead of bounding the production of the predator, we bound the degradation of preys
  • In both cases the goal was to investigate whether the various terms of the model could balance each other and yield oscillations.
  • Detail Analysis for Model with bounded prey growth and degradation

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
  • 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 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.
  • 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
  • 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
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