IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model2

Bounded Prey Growth

Introduction

• As we know from previous analysis, the Lotka-Volterra equation is too ideal to achieve.
• e.g. in real natural system, it is impossible to have infinite exponential growth. The growth will be limited by certain mechanism, so does for our system.
• After analysis, we found we could modify our system by Michaelis-Menten Model

Assumption made by using Michaelis-Menten

• The behaviours of promoter and activator are exactly the same as enzyme and substrate except that activator is not used up as substrate. (Hence, the Km should be Kd, the equilibrium constant).
• The LuxR concentration reaches steady state much faster than any others, hence it is assumed to be a constant in our rate equation.
• Although the actual product of prey cell is not AHL (but LuxI), we assume AHL to be our final product, since LuxI is directly responsible for AHL production. Hence AHL can be treated as being reproduced itself exponentially.

Relevance of the Model

• Now the production of prey is limited by Michaelis-Menten mechanism, i.e. the rate of production will be exponential initially and linear thereafter.
• This characteristic also has been verified by our experimental result
• However, the rest of the equation is still too ideal to achieve and hence required further modification.

ODE (Ordinary Differential Equation)

• V: Maximum Speed of prey production
• kd : Equilibrium Constant

Physical interpretation of the equations

• The prey (AHL) production is limited by the limited number of promoters. This is modeled by Michaelis-Menten kinetics. The rate of production becomes linear when number of promoters is saturated.

Basic Results on the Steady Points and Vector Field

Steady Points

• After analysis, there are two stationary points. (0, 0) & (d/c, V*c/b/(k*c+d))
• The existence of both stationary points is independent of the parameters
• After Jacobian analysis, the stationary point (0, 0) is a saddle point, while (d/c, a/b) is stable

Vector Field

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, (d/c, V*c/b/(k*c+d)) – stable

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

Conclusion

Summary of Results

• With the change in the equation, we could see the drastic change in the system behaviour
• Hence it is important to carefully analysis each model to check whether we are able to get the required oscillation

Appendices

Related documents

• Stability analysis