Understanding Negative Feedback Systems: Equations, Dynamics, and Activation

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Homework Statement

A protein X is produced at a constant rate β. Its degradation rate γ is controlled by its own concentration x, meaning that: if x < K then γ(x) = γ[SUB]low[/SUB], otherwise γ = γ[SUB]high[/SUB].
K is the repression threshold.

We have to follow an example given in class, which is very similar, that I attached.
Then answer the following questions:

1. Find the steady-state for different parameters.

2. For the case where feedback is activated in the steady-state, describe the dynamics of the system. Calculate the speedup of the dynamics here compared to the open loop case with the same steady-state?

3. Using the interval method, calculate the dynamics for a delay in the feedback in the extreme case where γ[SUB]low[/SUB] = 0. What is the pulse height as a function of the delay time?

4. Can you suggest a simple synthetic system where degradation based feedback can be studied?

Relevant Equations

See the attached file.

We express such equations by $$\frac{dx}{dt} = \beta - \gamma \cdot x$$, t denotes the time.
In this case, γ depends on x, thus the dynamic equation should probably be:
$$\frac{dx}{dt} = \beta - \gamma(x) \cdot x$$

1. Setting the equation to 0 leads to two different cases:
If x < K: x_ST = γ_low / β
If x ≥ K: x_ST = γ_high / β

I can also say (like in the example), that it is dependent on β, so:
If β > γ_low / k: x_ST = γ_low / β
If β ≤ γ_high / k: x_ST = γ_high / β

I don't understand why in the example in this step the divided by K drops and why the ≤ becomes a <.

2. Now my biggest concern is, how do I see, in which case the feedback is activated and in which not?
Then I would take this case and try to solve the equation with initial condition x(t=0)=0.
Probably the outcome is very similar to the example.
How would I proceed?

Any help is very appreciated!

 

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No one any suggestions or ideas/thoughts? :)