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

• The simplification is made possible by the similarity of the growth rates of V and W in thefull 3D Model
  • Complex production term is identical
  • Only their dissipative terms (-d1V and -d2W ) varies and does so by a constant
• Consequence: Simple hypotheses lead to a very big simplification
• 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 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 same rate of production, and same rate of degradation
  • Hence they will be at the same concentration thoughout

• System then simplifies to

• NB: Hypothesis 2 is not really needed
  • If d1=d2 W-V decays to 0 exponentially (with a time constant 1/d1)
  • Therefore after a little time we can assume V=W
  • The larger d1 is the faster the assumption becomes valid
  • Hence the larger the difference between initial value 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 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, not 3D!!!
  • We have simple requirements for a limit cycle in 2D
  • In 3D it is more complex - much more complex

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

• Studying 2D model will also help us understand 3D model more

Conclusion

• Yes 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