IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/3Dto2D: Difference between revisions
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== '''Model Simplification''' == | == '''Model Simplification''' == | ||
<br><br> | <br><br> | ||
<font size="4">'''Why we can simplify the 3d Model into a 2D Model'''</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> | ||
*<font size="4">'''Required Hypotheses for Simplification'''</font size="4"> | |||
:* 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 | |||
<font size="4">'''Required Hypotheses for Simplification'''</font size="4"> | :* Under previous 2 Hypotheses | ||
::* aiiA and LuxR start at the same concentration | |||
* Hypothesis 1: to ensure V and W have same growth rates | ::* they have the same rate of production and degradation | ||
:* '''Hypothesis 1: d1=d2''') | ::* hence they have at the same concentration throughout | ||
* Hypothesis 2:To have equality of the initial conditions | :* System then can be simplified to | ||
:* '''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 | |||
[[Image:3Dmodel-simple.png|center]] | [[Image:3Dmodel-simple.png|center]] | ||
<br><br> | <br><br> | ||
* Hypothesis 1 : d1=d2 | [[Image:simplification.jpg|thumb|600px|center|Summary of our approach]] | ||
:* The assumption of d1=d2 is feasible because aiiA and LuxR within the cells will be washed out at the same rate in chemostat. | <br> | ||
:* 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 | *<font size="4"> '''Validity of the hypotheses'''</font size="4"> | ||
* Hypothesis 2 is not really essential | :* Hypothesis 1 : d1=d2 | ||
:* it is fortunate as it was hard to ensure | ::* The assumption of d1=d2 is feasible because aiiA and LuxR within the cells will be washed out at the same rate in chemostat. | ||
:* if d1=d2, the difference between W and V will decay to 0 exponentially (with a time constant 1/d1) | ::* 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 | ||
:* therefore after a little time we can assume V=W | :* Hypothesis 2 is not really essential | ||
:* the larger d1, the faster the assumption becomes valid | ::* it is fortunate as it was hard to ensure | ||
:* the larger the difference between initial values of V & W, the longer the settling time of reaching V=W only | ::* if d1=d2, the difference between W and V will decay to 0 exponentially (with a time constant 1/d1) | ||
:* 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. | ::* 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> | <br><br> | ||
<font size="4">'''Problem : in Theory , there is a Huge Difference Between 2D and 3D'''</font size="4 | *<font size="4">'''Problem : in Theory , there is a Huge Difference Between 2D and 3D'''</font size="4> | ||
:* Poincare-Bendixson Theorem works for 1D and 2D only, but not 3D | |||
* Poincare-Bendixson Theorem works for 1D and 2D only, but not 3D | ::* We only need simple requirements for a limit cycle in 2D | ||
:* We only need simple requirements for a limit cycle in 2D | ::* In 3D the requirement is more complex - or much more complex | ||
:* In 3D the requirement is more complex - or much more complex | :* So are our results in 2D worth anything ? | ||
* So are our results in 2D worth anything ? | ::*If our hypotheses are exactly met: Yes! | ||
:*If our hypotheses are exactly met: Yes! | ::*In practice hypotheses not exactly met, but we have a margin of error | ||
:*In practice hypotheses not exactly met, but we have a margin of error | ::*A slight error on Hypothesis 2 is not important | ||
:*A slight error on Hypothesis 2 is not important | ::*Slight error on hypothesis 1 (d1 not strictly equal to d2): 2 Scenarii | ||
:*Slight error on hypothesis 1 (d1 not strictly equal to d2): 2 Scenarii | :::* Scenario 1: (the kind one) | ||
::* Scenario 1: (the kind one) | ::::* For d1=d2 and a range of parameters well chosen we have oscillations | ||
:::* 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) | ||
:::* 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) | ||
::* Scenario 2: (the not so nice one) | ::::*[aiiA] and [LuxR] get more and more out of synchronisation | ||
:::*[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 | ||
:::*However, if the hypotheses are almost met, we can hope to have a few synchronised cycles | |||
<br><br> | <br><br> | ||
<font size="4">'''Conclusion'''</font size="4"> | *<font size="4">'''Conclusion'''</font size="4"> | ||
:*There is a lot to learn from the 2D model | |||
*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|center]] | ||
::[[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer|center]] | |||
<br style="clear:both;"/> | <br style="clear:both;"/> | ||
:*The simulation above shows individual cycles of [aiiA] and [LuxR] | ::*The simulation above shows individual cycles of [aiiA] and [LuxR] | ||
:** Frequencies are equal | ::** Frequencies are equal | ||
:** Profiles very similar | ::** Profiles very similar | ||
:** Peak amplitudes different | ::** 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 | ::**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 | ::*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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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
- 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:
- 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
- The simulation above shows individual cycles of [aiiA] and [LuxR]
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