IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/3Dto2D: Difference between revisions
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<big>'''From 3D to 2D'''</big> | <big>'''From 3D to 2D'''</big> | ||
* | *Simplification is possible because of the similarity of the growth rates of V and W in the full 3D Model | ||
** | **Their complex production terms are identical | ||
** Only their dissipative terms (- | **Only their dissipative terms (-d1*V and -d2*W ) varies | ||
* | **A simple hypotheses could lead to a very big simplification in our analysis | ||
*2D analysis is much simpler, and still will give us valid prediction on whether the system will oscillate. | **A 2D analysis is much simpler, and still will give us valid prediction on whether the system will oscillate. | ||
* Required Hypotheses for Simplification | *Required Hypotheses for Simplification | ||
** Hypothesis 1: d1=d2 | ** Hypothesis 1: d1=d2 | ||
** Hypothesis 2: [aiiA] = [LuxR] initially ( that is time t=0) | ** Hypothesis 2: [aiiA] = [LuxR] initially ( that is at 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 | ***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) | ***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 | *Under previous 2 Hypothesis | ||
**Both aiiA and LuxR will start at the same concentration, and same rate of production | **Both aiiA and LuxR will start at the same concentration, and the same rate of production and degradation | ||
**Hence they will be at the same concentration thoughout | **Hence they will be at the same concentration thoughout | ||
* System then | * System then can be simplified to | ||
::[[Image:3Dmodel-simple.png]] | ::[[Image:3Dmodel-simple.png]] | ||
* '''NB''': Hypothesis 2 is not really | * '''NB''': Hypothesis 2 is not really essential | ||
** If d1=d2 W | ** If d1=d2, the difference between W and V will decay to 0 exponentially (with a time constant 1/d1) | ||
** Therefore after a little time we can assume V=W | ** Therefore after a little time we can assume V=W | ||
** The larger d1 | ** The larger d1, the faster the assumption becomes valid | ||
** | ** The larger the difference between initial values 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. | ** In particular we are sure that the condition on the parameters for obtaining a limit cycle will still be identical in 2D and 3D despite of the initial concentrations of U V W. | ||
<br> | |||
<big>'''Problem : There is a Huge Difference Between 2D and 3D'''</big> | <big>'''Problem : There is a Huge Difference Between 2D and 3D'''</big> | ||
* Poincare-Bendixson Theorem works for 1D and 2D only, but not 3D!!! | |||
** We only need simple requirements for a limit cycle in 2D | |||
** In 3D the requirement is more complex - or much more complex | |||
* Can we really afford to assume the hypotheses and reduce the system to 2D? | |||
**If our hypotheses are exactly met: Yes! | |||
**In practice: there might be slight errors | |||
***Slight error on Hypothesis 2: not important | |||
* Can we really afford to | ***Slight error on hypothesis 1: | ||
** If | ****[aiiA] and [LuxR] get more and more out of synchronisation | ||
** In practice : there | ****However, if the hypotheses are almost met, we can hope to have a few synchronised cycles | ||
*** Slight error on Hypothesis 2: not important | ****We hope that the conditions on the parameter to generate limit cycle will be the same, the only differences are the amplitude, frequency and phase difference of the oscillations | ||
*** Slight error on hypothesis 1: | |||
**** [aiiA] and [LuxR] get more and more out | |||
**** 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 | |||
*However, studying the 2D model will also help us understand the 3D model more | |||
<br> | |||
<big> '''Conclusion'''</big> | <big> '''Conclusion'''</big> | ||
* | *There is a lot to learn from the 2D model | ||
* A word of caution: | *A word of caution: | ||
::[[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer]] | ::[[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer]] | ||
<br style="clear:both;"/> | <br style="clear:both;"/> |
Revision as of 02:53, 30 October 2006
Analysis of the Model of the Molecular Predation Oscillator
Model Simplification: Can We Learn Anything from 2D Models?
From 3D to 2D
- Simplification is possible because of the similarity of the growth rates of V and W in the full 3D Model
- Their complex production terms are identical
- Only their dissipative terms (-d1*V and -d2*W ) varies
- A simple hypotheses could lead to a very big simplification in our analysis
- A 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 at 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 the same rate of production and degradation
- Hence they will be at the same concentration thoughout
- System then can be simplified to
- NB: Hypothesis 2 is not really essential
- If d1=d2, the difference between W and V will decay to 0 exponentially (with a time constant 1/d1)
- Therefore after a little time we can assume V=W
- The larger d1, the faster the assumption becomes valid
- The larger the difference between initial values 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 still 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, but not 3D!!!
- We only need simple requirements for a limit cycle in 2D
- In 3D the requirement is more complex - or much more complex
- Can we really afford to assume the hypotheses and reduce the system to 2D?
- If our hypotheses are exactly met: Yes!
- In practice: there might be slight errors
- Slight error on Hypothesis 2: not important
- Slight error on hypothesis 1:
- [aiiA] and [LuxR] get more and more out of synchronisation
- However, if the hypotheses are almost met, we can hope to have a few synchronised cycles
- We hope that the conditions on the parameter to generate limit cycle will be the same, the only differences are the amplitude, frequency and phase difference of the oscillations
- However, studying the 2D model will also help us understand the 3D model more
Conclusion
- There is a lot to learn from the 2D model
- A word of caution:
- 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
- The simulation above shows individual cycles of [aiiA] and [LuxR]