Guided example: lac operon

We now translate the lac operon model into the ordinary differential equation (ODE) framework. The components and their regulatory relationships are the same as in the difference equation (DE) model. What changes is how we represent the dynamics: we now express how each component’s value changes instantaneously over time.

As before, we treat external glucose and internal allolactose concentrations as constant inputs, not governed by ODEs.

Model Equations

Let the model components be:

• cAMP(t) – internal cAMP level
  
• CAP(t) – active CAP level
  
• repressor(t) – active Lac repressor
  
• mRNA(t) – polycistronic lac operon transcript
  
• lacProtein(t) – lac enzymes
  
• lacMetabolism(t) – lactose metabolism readout
  

1. cAMP (production, as negatively regulated by glucose, is high when glucose is low)

2. CAP (activation positively regulated by cAMP)

3. Lac Repressor ( inactivation positively regulated by allolactose)

4. mRNA (transcription inhibited by Repressor, enhanced by CAP)

5. Lac Enzymes (translated from mRNA)

6. Lactose Metabolism (abstract readout, proportional to enzyme levels)

lacMetabolism(t) = k_conv​ ⋅ lacEnzymes(t)

Notes

• Each equation describes the instantaneous rate of change of a model component.
  
• The functional forms such as [1 / (1 + glucose)] or [CAP / (CAP + K_CAPtx)] are identical to the DE version.
  

Exploring model behaviour

We can now explore how the model components evolve over time. Figure 16 shows 2 scenarios, differing in the initial values of the model components glucose an allolactose simulated over a time span of 50 minutes.

Figure 17 illustrates how we can simulate the effect of suddenly removing lactose. In this simulation, allolactose is present for the first 100 minutes, after which its value is set to zero, and the simulation continues for another 100 minutes.