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

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Analysis of the Model of the Molecular Predation Oscillator


Conclusion

Overview of our Results on the Molecular Predation System

  • We haven't carried out the full study of the 3D model of Molecular Predation System
  • Instead we have used some biologically justifiable hypotheses to simplify the model to a 2D model
  • We have also carried out a complete theoretical study of the 2D system and are now able to predict for every combination of parameters how the Molecular Predation System will behave.
  • In particular we have proved that the Molecular Predation System can operate in two modes:
- It can work as an oscillator (oscillating around unique limit cycle)
- It can work in stable regime (both prey and predator populations converge to limit value)


What was learnt from modifying the model

  • Oscillations are the results of a subtle balance between the growth terms and degradation terms of the system.
  • Oscillations appeared to be mainly due to balancing the growth and degradation terms of the preys
  • The washout terms in the model were crucial to control the system


Examples of Oscillations with the Molecular Predation System

Phase Diagram - Oscillation Mode
Phase Diagram - Oscillation Mode
Corresponding Time Signals - Oscillation Mode
Corresponding Time Signals - Oscillation Mode



Control of the Oscillations of the Molecular Predation Oscillator
Control of the oscillations is not as simple as with Lotka-Volterra, but simulations show that we have good control over the amplitude and total control over the frequency as shown below.

Control of Amplitude & Frequency
Control of Amplitude & Frequency



Future Works

Future Works on the 2D Model

  • The normalised 2D model of the molecular predation oscillator has been extensively studied.
  • Some more work on the oscillator still needs doing however
  • The most important work remaining on the normalised 2D model concerns the characterisation of the output of the oscillator with regards to the model parameters
- Amplitude
- Frequency
- Profile (measures of shape in phase diagrams or time diagrams)
  • To be complete the characterisation should also be dome with the experimental constraints in mind (e.g: washout cannot be infinitely small)


Extensions to the 2D Model

  • The model of the molecular predation oscillator overlooks the leakage terms and does not comprise any exponent in the growth terms.
  • With these new terms the dimensionless model would look like
Generalised 2D Model Image:Model-2Dgeneralised.png
  • These assumptions are reasonable a first sight. However, some preliminary results (available in Future Works on the 2D Model) suggest that the oscillator is very sensitive to the assumption on the exponent.
  • We therefore suggest carrying out a thorough analysis of the extended model next year


Future Works on the 3D Model

  • We only studied the case when d1=d2 and thus simplified the 3D model into 2D
  • At the very least the study should be extended to the vicinity of d1=d2
  • Ideally the whole 3D system would be studied
  • NB: the complexity of the study will increase dramatically
- Making sure steady points are unstable is not enough
- We need a solid criterion on how to get a cycle


Stochastic Analysis

  • What is the influence of the distributions of gene-expression parameters such as ao,bo,co?
  • We may in practice need to drop the Dynamical System approach and go fully stochastic
  • An entirely new level of complexity!!!!!

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