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

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Analysis of the Model of the Biological Oscillator



Generalities on the Model

  • Introduction

The molecular predator-prey system can be modelled by the following 3D Dynamical System

  • Simplified Version of the Model
In order to make the analysis of the dynamical system easier we formalise it as follows
The coordinates U,V and W respectively stand for the concentrations of AHL, aiiA and LuxR.
Their time derivatives stand for their growth rates. The bounded growth is due to gene expression.
Degradation of the prey (AHL) is partly due to an enzymatic reaction (as shows the second term of the first equation). Finally The model comprises dissipative terms of the model (eU,dV,dW) that are due to the washout of the chemostat.
  • General Remarks on the Model of Interest
In this part we only analyse the model presented above. We do not consider its possible (and natural) extensions with an exponent in the model and with leakage terms.
From an experimental point of view, restricting our analysis makes sense. Another part of the Imperial College IGEM project involved building test-objects to measure the growth of AHL,aiiA and LuxR as well as the degradation of AHL by LuxR. Too few measurements were carried out this year to challenge the assumptions of our model.
However, we are very aware of the pitfalls of limiting our analysis and suggest extending in iGEM2007. For the sake of completeness, some preliminary results on these extensions (and their consequences) are discussed in [link]. Anyone who is interested in the finest aspects of the analysis of our model are strongly advised to consult the page.

Overview of Our Approach

Principle 1: Study of Ever More Complex Models

We took an incremental approach to the study of the 3D model. Under a reasonable set of assumptions the 3D model can be turned into a 2D system. We used this property fully as we built up our analysis towards that of the full 3D model:
- First we studied 2D models
- We started with Lotka-Volterra
- We increased the complexity of the model until we obtained the 2D model equivalent to the 3D model (under the set of assumptions)
- See Result section for list of models
- Then time permitting: the whole 3D Model


Breaking up the complexity of the 3D system the way we did it had several notable advantages.
First it allowed us to learn the basics of Dynamical Systems Analysis and practice them on simple models. Second, by modifying the various terms of the model we got a better understanding on how they interact with each other. Finally, it allowed us to learn from the previous calculations and thus to push them in an effective manner.


Principle 2: Normalisation Effort for a System Analysis

In order to make our results easy to read and comprehend, we normalised the various steps of our analyses and the presentation of our results.

  • 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 are 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
  • 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

Our Results

We had time to study the following five Dynamical Systems. In order of complexity:
  • 2D Model 1: Lotka – Volterra
  • Detail Analysis for Model 1
  • 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
  • Detail Analysis for Model 2
  • 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
  • Detail Analysis for Model 3
  • 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 3a: Bounded Predator and Prey Growth
  • Detail Analysis for Model 3a
  • 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
  • Detail Analysis for Model 4
  • We have obtained oscillations but unfortunately the killing term for the preys remains unrealistic and need regulating. we show 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
  • Detail Analysis for 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 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 on the 2D Model
    • 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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