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

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 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.

• Detail Analysis for Lotka-volterra

2D Model 2: Bounded Prey Growth

• 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.
• 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 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