IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model3

Introduction

• After analysis, the production of predator should be modified by Michaelis-Menten Model

Relevance of the Model

• This is our second step by step approach to our final system.
• As the system model becomes more complicated, the analysis will be more difficult as well
• New methods needed to analyse the system behaviour
• After this analysis, we will include the killing of prey modification

ODE (Ordinary Differential Equation)

Physical interpretation of the equations

• The predator production is limited by the limited number of promoters as well. This is modeled by Michaelis-Menten kinetics.
• However, the LuxR in the predator is the direct product after the promoter is activated. Hence LuxR cannot be treated saturated to a constant.
• The rate of production becomes linear when number of promoters is saturated.

Dimensionless Version of the System

• Now the system is much more complicated, the number of parameters is already 6
• Direct symbolic computation through Matlab seems impossible; the terms will be long and cannot be simplified
• Hence it is necessary to normalized the model to reduce the number of parameters
• The general system behaviours (stable or unstable) will be the same for the dimensionless and original version of the model

• Now the system only have 3 parameters, much easier to analyse.
• We will use this dimension-less version of the Dynamic System for the rest of the study

Basic Results on the Steady Points and Vector Field

Steady Points

• After analysis, there are 3 stationary points, but only 2 in the first quadrant (i.e. X&Y>0)
• The existence of the 2 stationary points is independent of the parameters
• After Jacobian analysis, the stationary point (0, 0) is a saddle point, while the second one is always stable

Limit cycles

• Clearly, the second stable stationary point will ensure the system is stable.
• Trajectories will be remain bounded for all the different parameters

Results of Dynamic Analysis

• 2 stationary points, (0, 0) – saddle, the second point– stable

• Thus we are expected to get a damped oscillation with this model

Conclusion

Summary of Results

• The bounded production will result in stability of the system. This is expected because the rate of production is not longer fast enough; the exponential decay of the prey in the equation will bring the system into equilibrium.

Appendices

Related documents

• Stability analysis