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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}} |
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| | <font size="6"><center>'''Welcome to the Analysis Page '''</center></font size="6"> |
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| | =='''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 |
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| =='''Generalities on the Model'''==
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| * '''Introduction'''
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| The molecular predator-prey system can be modelled by the following 3D Dynamical System
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| :::[[Image:biological model.png]]
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| * '''Simplified Version of the Model'''
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| :In order to make the analysis of the dynamical system easier we formalise it as follows
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| :::[[Image:3Dmodel.png]]
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| :The coordinates U,V and W respectively stand for the concentrations of AHL, aiiA and LuxR.
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| :Their time derivatives stand for their growth rates. The bounded growth is due to gene expression.
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| :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.
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| * '''General Remarks on the Model of Interest'''
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| : 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.
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| : 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.
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| :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.
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| =='''Overview of Our Approach '''==
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| <font size="4"> Principle 1: Study of Ever More Complex Models</font size>
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| :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:
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| ::- First we studied 2D models
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| :::- We started with Lotka-Volterra
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| :::- We increased the complexity of the model until we obtained the 2D model equivalent to the 3D model (under the set of assumptions)
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| :::- See [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical_Analyses#Our_Results| Result]] section for list of models
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| ::- Then time permitting: the whole 3D Model
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| :Breaking up the complexity of the 3D system the way we did it had several notable advantages.
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| :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.
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| <font size="4"> Principle 2: Normalisation Effort for a System Analysis </font size>
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| 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.
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| * '''First page of the analysis: most important results'''
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| : See the following [[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Template Results| template]] for the way we normalised
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| * '''Following Pages: Display of important simulations (and comments)'''
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| :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
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| * '''More detailed analyses are made available for downloading'''
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| ** Detailed analyses = proofs of results, additional simulations...
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| ** pdf and Doc suggested formats for attachments
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| == '''Can We Learn Anything About the 3D Model from 2D Models?''' ==
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| * '''From 3D to 2D'''
| | =='''Presentation of our Dynamical System'''== |
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| :* Similarity of Derivatives of V and W in Model
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| ::*Only their dissipative terms (-d1V and -d2W ) vary
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| ::*consequence: Simple hypotheses lead to a very big simplification
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| ::*2D analysis is much simpler, and still will give us valid prediction on whether the system will oscillate.
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| :* Required Hypotheses for Simplification
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| ::* Hypothesis 1: d1=d2
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| ::* Hypothesis 2: [aiiA] = [LuxR] initially ( that is time t=0)
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| :::*The assumption of d1=d2 is feasible because aiiA and LuxR within the cells will be washed out at the same rate in chemostat
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| :::*As long as we can ensure the washing out rate is much more dominant than their natural half-life. (Easily achieved)
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| :*Under previous 2 Hypothesis
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| ::*Both aiiA and LuxR will start at the same concentration, and same rate of production, and same rate of degradation
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| ::*Hence they will be at the same concentration thoughout
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| :* System then simplifies to
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| ::[[Image:3Dmodel-simple.png]]
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| :* '''NB''': Hypothesis 2 is not really needed
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| ::* If d1=d2 W-V decays to 0 exponentially (with a time constant 1/d1)
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| ::* Therefore after a little time we can assume V=W
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| ::* The larger d1 is the faster the assumption becomes valid
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| ::* Hence the larger the difference between initial value of V &W, the longer the settling time of reaching V=W only
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| ::* 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.
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| * '''Problem : There is a Huge Difference Between 2D and 3D'''
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| :* Poincare- Bendixson Theorem works for 1D and 2D only, not 3D!!!
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| :** We have simple requirements for a limit cycle in 2D
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| :** In 3D it is more complex - much more complex
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| :* Can we really afford to make the hypotheses and reduce the system to 2D?
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| ::* If the hypotheses are exactly met: Yes!
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| ::* In practice : there will be slight errors
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| :::* Slight error on Hypothesis 2: not important
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| :::* Slight error on hypothesis 1:
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| ::::* [aiiA] and [LuxR] get more and more out sync
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| ::::* However, if the hypotheses are almost met we can hope to have a few cycles
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| ::::* 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
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| :*Studying 2D model will also help us understand 3D model more
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| * '''Conclusion'''
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| :* Yes there is a lot to learn from the 2D model
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| :* A word of caution:
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| ::[[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer]]
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| <br style="clear:both;"/>
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| :*The simulation above shows individual cycles of [aiiA] and [LuxR]
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| ::* Frequencies are equal
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| ::* Profiles very similar
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| ::* Peak amplitudes different
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| :*Clearly for such cycles d1=d2 was not met. We therefore have to study the 3D case in its entirety at some point
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| :*However for our current interest of whether the system can result in generation, 2D case of d1=d2 should be enough
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| =='''Our Results'''==
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| : We had time to study the following five Dynamical Systems. In order of complexity:
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| :* '''2D Model 1: Lotka – Volterra'''
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| :::[[Image:Model1.PNG]]
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| ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model1| Detail Analysis for Model 1]]</b>
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| ::*Lotka-Volterra is the first (and most famous) model for prey-predator interactions. We detail here some of its (very appealing) properties.
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| :* '''2D Model 2: Bounded Prey Growth'''
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| :::[[Image:Model2.PNG]]
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| ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model2| Detail Analysis for Model 2]]</b>
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| ::*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.
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| ::*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
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| :* '''2D Model 3: Bounded Predator and Prey Growth'''
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| :::[[Image:Model3.PNG]]
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| ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3| Detail Analysis for Model 3]]</b>
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| ::*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.
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| ::*Similarly, the production of the predator is also limited by the number of promoters
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| :* '''2D Model 3a: Bounded Predator and Prey Growth'''
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| :::[[Image:Model3a.PNG]]
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| ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model3a| Detail Analysis for Model 3a]]</b>
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| ::*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.
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| ::*Similarly, the production of the predator is also limited by the number of promoters
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| :* '''2D Model 4: Bounded Predator and Prey Growth with Controlled Killing of Preys'''
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| :::[[Image:Model4.PNG]]
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| ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model4| Detail Analysis for Model 4]]</b>
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| ::*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.
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| ::*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.
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| :* '''Final 2D Model : 2D Model 5'''
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| :::[[Image:Model5.PNG]]
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| ::*<b>[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/2D Model5| Detail Analysis for Model 5]]</b>
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| ::*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.
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| ::*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.
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| ::*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.
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| ::*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
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| ::*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
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| =='''Conclusion'''==
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| I suggest organising the conclusion as follows
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| * '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Conclusion| General Conclusion of our Studies]] '''
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| ** Existence limit cycles
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| ** Influence of parameters
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| ** What was learnt from modifying the model
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| * '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Experimental Recommendations| Recommendations for the Experiments]] '''
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| ** Translate the obtained results into recommendations for the experiments
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| ** Focus on Sensitivity Issues (speed of convergence)
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| ** Recommendations for Measurements (link them to shape of limit-cycle)
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| * '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Future Works| Future Works on the 2D Model]]''' | | :*The molecular predator-prey system has been modelled by the following 3D Dynamical System <br> |
| ** Is there something left to do in 2D?
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| ** What should be done in 3D?
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| ** Focus in 3D :How to get a cycle
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| ** Any Good Theoretical Criteria?
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| * '''[[IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Future Works3D| Future Works on the 3D Model]]''' | | [[Image:biological model.png|center]] |
| | <br> |
| | :* 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 |
| | <center><font size="3">'''Formalised Model''' </font size="3">[[Image:3Dmodel.png]]</center> |
| | <br><br> |
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| == '''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. |
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| | <br> |
| | [[Image:block diagram.jpg|thumb|600px|center|the block diagram for the 3D model]] |
| | <br><br> |
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| * '''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]
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| ** Tutorial 2 : Application of Poincare - bendixson [http://www.openwetware.org/images/6/66/WIKI_Document_Number_2_-_Poincare.pdf Poincare Bendixson]
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| ** 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]
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| ** 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]
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| * '''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. |
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| *'''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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