IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses: Difference between revisions

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::<font size="6">Analysis of the Model of the Biological Oscillator </font size>
{{Template:IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analysis}}
<br><br>
<font size="6"><center>'''Welcome to the Analysis Page '''</center></font size="6">
<br><br><br>


=='''Introduction'''==
:*The present part of the i-coli Reporter deals with the analysis of a complex 3 dimensional dynamical system that model the dynamic of the Molecular Predation System
:* Since the analysis was long and complex, we have split its results in different sections that are accessible with the blue tabs above.
:* These different sections present the logical progression in our analysis
:* We suggest you browse them in the order shown above
:* We hope you enjoy your reading


=='''Generalities on the Model'''==
* '''Introduction'''
*After a careful analyis, the system can be modelled as following
:::[[Image:2d model 0a.PNG]]
*Click [http://openwetware.org/wiki/IGEM:IMPERIAL/2006/project/Oscillator/Modelling here] for detail analysis
* '''Our Version of the Model'''
:We formalise the model as follows
:::[[Image:3Dmodel.png]]
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!!
=='''Overview of Our Approach '''==
<font size="4"> Principle 1: Study of Ever More Complex Models</font size>
* '''Our Roadmap'''
** We did not Tackle 3D Model Outright
** First We Studied 2D models
*** From Lotka-Volterra to the full 2D model
*** See [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical_Analyses#Our_Results| 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)
<font size="4"> Principle 2: Normalisation Effort for a System Analysis </font size>
* '''First page of the analysis: most important results'''
: See the following [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Template Results| template]] for the way we normalised
* '''Following Pages: Display of Important Simulations (and Comments)'''
:see the following [http://www.openwetware.org/images/1/18/WIKI_Document_Number_4_-_Presentation_of_Results.pdf  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
   
   
:* Required Hypotheses for Simplification
=='''Presentation of our Dynamical System'''==
::* 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.
 
:* System then simplifies to
::[[Image:3Dmodel-simple.png]]
 
:* '''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.
 
 
* '''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
 
* '''Conclusion'''
:* Yes there is a lot to learn from the 2D model
:* A word of caution:
::[[Image:2d model 0b.PNG]]
: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
 
=='''Our Results'''==
 
 
: We had time to study the following five Dynamical Systems. In order of complexity:
:* '''2D Model 1: Lotka – Volterra'''  
:::[[Image:Model1.PNG]]
 
:Lotka-Volterra is the first (and most famous) model for prey-predator interactions. We detail [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1| here]] some of its (very appealing) properties.
 
 
:* '''2D Model 2: Bounded Prey Growth'''
:::[[Image:Model2.PNG]]
 
: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 [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model2| 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'''
:::[[Image:Model3.PNG]]
 
:Bounding the growth of the preys only stabilises the system to the extent we cannot make it oscillate. We [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3| 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'''
:::[[Image:Model4.PNG]]
 
:We have obtained oscillations but unfortunately the killing term for the preys remains unrealistic and need regulating. we show [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4| 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'''
:::[[Image:Model5.PNG]]
 
: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 [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model5| 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
* '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Conclusion| General Conclusion of our Studies]] '''
** Existence limit cycles
** Influence of parameters
** What was learnt from modifying the model
 
 
* '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Experimental Recommendations| 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)


:*The molecular predator-prey system has been modelled by the following 3D Dynamical System <br>


* '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Future Works| Future Works]]'''
[[Image:biological model.png|center]]
** Is there something left to do in 2D?
<br>
** What should be done in 3D?
:* As is often the case in Mathematics, we renamed the parameters in order to make the system more symmetric and easier to interprete. We formalised the dynamical system  as follows
** Focus in 3D :How to get a cycle
<center><font size="3">'''Formalised Model''' </font size="3">[[Image:3Dmodel.png]]</center>
** Any Good Theoretical Criteria?
<br><br>


== '''Available Resources'''==
== '''How to read the model (the basics)'''==
:*The variables U,V and W respectively stand for the concentrations of AHL, aiiA and LuxR.
:*The time derivatives of U,V  and W stand for their growth rates.
:*Their bounded growth terms is due to gene expression.
:*Degradation of the prey (AHL) is partly due to an enzymatic reaction
:*Finally The model comprises dissipative terms of the model (eU,d1V,d2W) due to the washout in the chemostat.
:* '''Different types of parameters'''
::- ao,bo and co are assumed constant (that is '''without our control''')
::- The washout parameters (d1,d2 and e) are assumed '''within our control'''. However, experimental contraints prevent them from getting very small or very large
::- a, b and c are '''within our control'''. Out of design (2 cell system) they vary accordingly to the concentrations of cells.
   
<br>
[[Image:block diagram.jpg|thumb|600px|center|the block diagram for the 3D model]]
<br><br>


* '''Tutorials on the Analysis of a Dynamic System'''
=='''General Remarks on the Model of Interest'''==
** Tutorial 1 : Basic Principles of Dynamic Analysis [http://www.openwetware.org/images/9/98/WIKI_Document_Number_1_-_Dynamic_Analysis.pdf  Dynamic Analysis]
** Tutorial 2 : Application of Poincare - bendixson [http://www.openwetware.org/images/6/66/WIKI_Document_Number_2_-_Poincare.pdf  Poincare Bendixson]
** Tutorial 3 : Template for the Analysis of a Dynamic System [http://www.openwetware.org/images/e/ea/WIKI_Document_Number_3_-_Template_for_Analysis.pdf  Template for Analysis]
** Tutorial 4 : Suggestions for the Presentation of Simulation Results [http://www.openwetware.org/images/1/18/WIKI_Document_Number_4_-_Presentation_of_Results.pdf  Presentation]


* '''Powerpoint Presentations'''
:*In the analysis part we only deal with the model presented above.
:We will post here the various Powerpoint presentation(s) we will give on the Analysis of the 2D model.
:*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. We have built test constructs to measure the growth of AHL, aiiA and LuxR as well as the degradation of AHL by aiiA. Our experimental results generally showed that our model was valid.
:*However, we are very aware of the pitfalls of limiting our analysis, but due to the time constrain, we will not tackle them this year.


*'''Computer Code'''
<html>
:Download these two files [http://openwetware.org/images/8/84/SimulationE.m 1] & [http://openwetware.org/images/6/64/Lotkavolterra2.m 2] to view our matlab programming code to generate simulations
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* '''References'''
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*Lecture 9 & 10 in Modeling Gene Expression and Cell Signaling by [http://www.duke.edu/~you/courses/BME265-05/ Lingchong You]
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Latest revision as of 06:40, 1 November 2006

Analysis of the Model of the Molecular Predation Oscillator




Welcome to the Analysis Page




Introduction

  • The present part of the i-coli Reporter deals with the analysis of a complex 3 dimensional dynamical system that model the dynamic of the Molecular Predation System
  • Since the analysis was long and complex, we have split its results in different sections that are accessible with the blue tabs above.
  • These different sections present the logical progression in our analysis
  • We suggest you browse them in the order shown above
  • We hope you enjoy your reading


Presentation of our Dynamical System

  • The molecular predator-prey system has been modelled by the following 3D Dynamical System


  • As is often the case in Mathematics, we renamed the parameters in order to make the system more symmetric and easier to interprete. We formalised the dynamical system as follows
Formalised Model



How to read the model (the basics)

  • The variables U,V and W respectively stand for the concentrations of AHL, aiiA and LuxR.
  • The time derivatives of U,V and W stand for their growth rates.
  • Their bounded growth terms is due to gene expression.
  • Degradation of the prey (AHL) is partly due to an enzymatic reaction
  • Finally The model comprises dissipative terms of the model (eU,d1V,d2W) due to the washout in the chemostat.
  • Different types of parameters
- ao,bo and co are assumed constant (that is without our control)
- The washout parameters (d1,d2 and e) are assumed within our control. However, experimental contraints prevent them from getting very small or very large
- a, b and c are within our control. Out of design (2 cell system) they vary accordingly to the concentrations of cells.


the block diagram for the 3D model



General Remarks on the Model of Interest

  • In the analysis part we only deal with 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. We have built test constructs to measure the growth of AHL, aiiA and LuxR as well as the degradation of AHL by aiiA. Our experimental results generally showed that our model was valid.
  • However, we are very aware of the pitfalls of limiting our analysis, but due to the time constrain, we will not tackle them this year.

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