IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/3Dto2D

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:
 * [[Image:2d model 0b.PNG|thumb|400px|left|Simulation of Full 3D model done by Cell Designer|center]]


 * 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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