IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/3Dto2D: Difference between revisions

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== '''Model Simplification: Can We Learn Anything from 2D Models?''' ==
== '''Model Simplification''' ==
<br><br>
<font size="4">'''From 3D to 2D'''</font size="4">
*<font size="4">'''Why we can simplify the 3d Model into a 2D Model'''</font size="4">
:*Simplification is possible because of the similarity of the growth rates of the predator terms (V and W) in the 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.
<br><br>
<br><br>
*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'''
*<font size="4">'''Required Hypotheses for Simplification'''</font size="4">
** Hypothesis 1: d1=d2
:* Hypothesis 1: to ensure V and W have same growth rates
** Hypothesis 2: [aiiA] = [LuxR] initially ( that is at time t=0)
::* '''Hypothesis 1: d1=d2''')
***The assumption of d1=d2 is feasible because aiiA and LuxR within the cells will be washed out at the same rate in chemostat
:* Hypothesis 2:To have equality of the initial conditions
***As long as we can ensure the washing out rate is much more dominant than their natural half-life. (Easily achieved)
::* '''Hypothesis 2: [aiiA] = [LuxR]''' at time t=0
*Under previous 2 Hypothesis
:* Under previous 2 Hypotheses
**Both aiiA and LuxR will start at the same concentration, and the same rate of production and degradation
::* aiiA and LuxR  start at the same concentration
**Hence they will be at the same concentration thoughout
::* they have the same rate of production and degradation
::* hence they have at the same concentration throughout
:* System then can be simplified to
[[Image:3Dmodel-simple.png|center]]
<br><br>
[[Image:simplification.jpg|thumb|600px|center|Summary of our approach]]
<br>
*<font size="4"> '''Validity of the hypotheses'''</font size="4">
:* Hypothesis 1 : d1=d2
::* 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) the assumption should hold
:* Hypothesis 2 is not really essential
::* it is fortunate as it was hard to ensure
::* 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.
<br><br>


* System then can be simplified to
*<font size="4">'''Problem : in Theory , there is a Huge Difference Between 2D and 3D'''</font size="4>
::[[Image:3Dmodel-simple.png]]
:* 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
:* So are our results in 2D worth anything ?
::*If our hypotheses are exactly met: Yes!
::*In practice hypotheses not exactly met, but we have a margin of error
::*A slight error on Hypothesis 2 is not important
::*Slight error on hypothesis 1 (d1 not strictly equal to d2): 2 Scenarii
:::* Scenario 1: (the kind one)
::::* For d1=d2 and a range of parameters well chosen we have oscillations
::::* Because the system is well behaved , we still have oscillations at the vicinity of these parameters (hence for d1 slightly different from d2)
:::* Scenario 2: (the not so nice one)
::::*[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
<br><br>
 
*<font size="4">'''Conclusion'''</font size="4">
:*There is a lot to learn from the 2D model
:*A word of caution:
:::[[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer|center]]
<br style="clear:both;"/>
::*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 and yet we have oscillations. 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


* '''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.
<br>
<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?
<html>
**If our hypotheses are exactly met: Yes!
<!-- Start of StatCounter Code -->
**In practice: there might be slight errors
<script type="text/javascript" language="javascript">
***Slight error on Hypothesis 2: not important
var sc_project=1999441;
***Slight error on hypothesis 1:
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****[aiiA] and [LuxR] get more and more out of synchronisation
var sc_partition=18;
****However, if the hypotheses are almost met, we can hope to have a few synchronised 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 and phase difference of the oscillations
</script>


*However, studying the 2D model will also help us understand the 3D model more
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<br>
<!-- End of StatCounter Code -->
<big> '''Conclusion'''</big>
</html>
*There is a lot to learn from the 2D model
*A word of caution:  
::[[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer]]
<br style="clear:both;"/>
:*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

Latest revision as of 06:12, 1 November 2006

Analysis of the Model of the Molecular Predation Oscillator


Model Simplification



  • Why we can simplify the 3d Model into a 2D Model
  • Simplification is possible because of the similarity of the growth rates of the predator terms (V and W) in the 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: to ensure V and W have same growth rates
  • Hypothesis 1: d1=d2)
  • Hypothesis 2:To have equality of the initial conditions
  • Hypothesis 2: [aiiA] = [LuxR] at time t=0
  • Under previous 2 Hypotheses
  • aiiA and LuxR start at the same concentration
  • they have the same rate of production and degradation
  • hence they have at the same concentration throughout
  • System then can be simplified to



Summary of our approach


  • Validity of the hypotheses
  • Hypothesis 1 : d1=d2
  • 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) the assumption should hold
  • Hypothesis 2 is not really essential
  • it is fortunate as it was hard to ensure
  • 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 : in Theory , 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
  • So are our results in 2D worth anything ?
  • If our hypotheses are exactly met: Yes!
  • In practice hypotheses not exactly met, but we have a margin of error
  • A slight error on Hypothesis 2 is not important
  • Slight error on hypothesis 1 (d1 not strictly equal to d2): 2 Scenarii
  • Scenario 1: (the kind one)
  • For d1=d2 and a range of parameters well chosen we have oscillations
  • Because the system is well behaved , we still have oscillations at the vicinity of these parameters (hence for d1 slightly different from d2)
  • Scenario 2: (the not so nice one)
  • [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



  • Conclusion
  • 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 and yet we have oscillations. 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


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