IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/3Dto2D

Analysis of the Model of the Molecular Predation Oscillator

• Introduction
• Our Approach
• Model Simplication
• Our Results
• Conclusion
• Appendix

Model Simplification: Can We Learn Anything from 2D Models?

From 3D to 2D

• Simplification is possible because of the similarity of the growth rates of V and W in the full 3D Model
  • Their complex production terms are identical
  • Only their dissipative terms (-d1*V and -d2*W ) varies
  • A simple hypotheses could lead to a very big simplification in our analysis
  • A 2D analysis is much simpler, and still will give us valid prediction on whether the system will oscillate.

• Required Hypotheses for Simplification
  • Hypothesis 1: d1=d2
  • Hypothesis 2: [aiiA] = [LuxR] initially ( that is at time t=0)
    • The assumption of d1=d2 is feasible because aiiA and LuxR within the cells will be washed out at the same rate in chemostat
    • As long as we can ensure the washing out rate is much more dominant than their natural half-life. (Easily achieved)
• Under previous 2 Hypothesis
  • Both aiiA and LuxR will start at the same concentration, and the same rate of production and degradation
  • Hence they will be at the same concentration thoughout

• System then can be simplified to

• NB: Hypothesis 2 is not really essential
  • If d1=d2, the difference between W and V will decay to 0 exponentially (with a time constant 1/d1)
  • Therefore after a little time we can assume V=W
  • The larger d1, the faster the assumption becomes valid
  • The larger the difference between initial values of V & W, the longer the settling time of reaching V=W only
  • In particular we are sure that the condition on the parameters for obtaining a limit cycle will still be identical in 2D and 3D despite of the initial concentrations of U V W.

Problem : There is a Huge Difference Between 2D and 3D

• Poincare-Bendixson Theorem works for 1D and 2D only, but not 3D!!!
  • We only need simple requirements for a limit cycle in 2D
  • In 3D the requirement is more complex - or much more complex

• Can we really afford to assume the hypotheses and reduce the system to 2D?
  • If our hypotheses are exactly met: Yes!
  • In practice: there might be slight errors
    • Slight error on Hypothesis 2: not important
    • Slight error on hypothesis 1:
      • [aiiA] and [LuxR] get more and more out of synchronisation
      • However, if the hypotheses are almost met, we can hope to have a few synchronised cycles
      • We hope that the conditions on the parameter to generate limit cycle will be the same, the only differences are the amplitude, frequency and phase difference of the oscillations

• However, studying the 2D model will also help us understand the 3D model more

Conclusion

• There is a lot to learn from the 2D model
• A word of caution:

Simulation of Full 3D model done by Cell Designer

• The simulation above shows individual cycles of [aiiA] and [LuxR]
  • Frequencies are equal
  • Profiles very similar
  • Peak amplitudes different
  • Clearly for such cycles d1=d2 was not met. We therefore have to study the 3D case in its entirety at some point
• However for our current interest of whether the system can result in generation, 2D case of d1=d2 should be enough