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

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Analysis of the Model of the Biological Oscillator



Generalities on the Model

  • Introduction
  • After a careful analyis, the system can be modelled as following

--SxE00 07:11, 18 October 2006 (EDT) Please Use single Format for Equations!!! same color scheme/ font/...


  • Click here for detail analysis


  • Our Version of the Model
We formalise the model as follows

Where the coordinates U,V and W respectively stand for the concentrations of AHL, aiiA and LuxR.


  • Please Note
    • We do not consider possible exponents in the models
    • Part of the IGEM project involves modelling the growth and killing terms of AHL, aiiA and LuxR
    • Their results will yield new models most certainly
    • We leave this for next year!!

--SxE00 07:13, 18 October 2006 (EDT) Expand maybe

Overview of Our Approach

Principle 1: Study of Ever More Complex Models

  • Our Roadmap
    • We did not Tackle 3D Model Outright
    • First We Studied 2D models
      • From Lotka-Volterra to the full 2D model
      • See Result section for list of models
    • Then Time Permitting: the 3D Model
  • Ideas Behind our Step-by-step Approach
    • Learn the Basics of Dynamical Systems Analysis on Simple Models
    • Learn of the way the various terms of the model interact with each other
    • Make calculations easier with the final model (learning from previous calculations)


Principle 2: Normalisation Effort for a System Analysis

  • First page of the analysis: most important results
See the following template for the way we normalised
  • Following Pages: Display of Important Simulations (and Comments)
see the following tutorial for a possible way to present simulation results
  • More Detailed Analyses to be Made Available for Downloading
    • Detailed analyses = proofs of results, additional simulations...
    • pdf and Doc suggested formats for attachments

Can We Learn Anything About the 3D Model from 2D Models?

  • From 3D to 2D
  • Similarity of Derivatives of V and W in Model
  • Only their dissipative terms (-d1V and -d2W ) vary
  • consequence: Simple hypotheses lead to a very big simplification

--SxE00 07:23, 18 October 2006 (EDT) Expand


  • Required Hypotheses for Simplification
  • Hypothesis 1: d1=d2
  • Hypothesis 2: [aiiA] = [LuxR] initially ( that is time t=0)
  • d1 can be assumed to be equal to d2 because aiiA and LuxR will be washed out at the same rate in chemostat, and this washing rate will be more dominant than their natural degradation rate.

--SxE00 07:23, 18 October 2006 (EDT) Expand: it is a crucial step in our work!!!


  • 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
  • In particular we are sure that the condition on the parameters for obtaining a limit cycle will be identical in 2D and 3D.

--SxE00 07:23, 18 October 2006 (EDT) Expand maybe


  • 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

--SxE00 07:23, 18 October 2006 (EDT) Expand: it is a delicate point

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

--SxE00 07:23, 18 October 2006 (EDT) If you want to display the simulaton, please edit the picture !!! who cares about the left hand side????

The above simulation 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

--SxE00 07:23, 18 October 2006 (EDT) Expand: we indeed have to study the 3d case but maybe all we need is to study it at the vicinty of d1=d2!!!

Our Results

We had time to study the following five Dynamical Systems. In order of complexity:
  • 2D Model 1: Lotka – Volterra
Lotka-Volterra is the first (and most famous) model for prey-predator interactions. We detail here some of its (very appealing) properties.


  • 2D Model 2: Bounded Prey Growth
Lotka-Volterra is far too simple a model to yield any valuable results on the complex 2D model we wish to study. We start to investigate the influence of the various components of the system here by bounding the growth of the preys.
  • The prey (AHL) production is limited by the limited number of promoters. The number of promoter is directly responsible for the production. The rate of production will becomes linear once the number of promoters is saturated. This enzyme activation-site like behaviours can be modelled similarly to the Michaels Menton model
  • 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. 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


  • 2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys
We have obtained oscillations but unfortunately the killing term for the preys remains unrealistic and need regulating. we show here that such regulation introduces in the system some undesirable properties.
  • 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.


  • Final 2D Model : 2D Model 5
Model 4 can be made to oscillate. However, it also exhibits some very unwelcome properties. Fortunately experimental conditions lead us to introduce a final dissipative term –eU to the derivative of the prey population.
We investigate here the properties of the 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

Conclusion

I suggest organising the conclusion as follows


  • Recommendations for the Experiments
    • Translate the obtained results into recommendations for the experiments
    • Focus on Sensitivity Issues (speed of convergence)
    • Recommendations for Measurements (link them to shape of limit-cycle)


  • Future Works
    • Is there something left to do in 2D?
    • What should be done in 3D?
    • Focus in 3D :How to get a cycle
    • Any Good Theoretical Criteria?

Available Resources

  • Tutorials on the Analysis of a Dynamic System
  • Powerpoint Presentations
We will post here the various Powerpoint presentation(s) we will give on the Analysis of the 2D model.
  • Computer Code
Download these two files 1 & 2 to view our matlab programming code to generate simulations
  • References
  • Lecture 9 & 10 in Modeling Gene Expression and Cell Signaling by Lingchong You
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