<section>

Preliminary

We will use the quasi steady state approximation (QSSA) then. The idea is that there are quick reactions, such as enzymatic ones, complexations, etc… And there are slow reactions such as protein production. We assume that the evolution speed of an element that is created only by quick reaction is null. </br> </br>

<img src="" alt="" />

</br> </br> Indeed, we have these types of evolution for the biological elements. The ones involved only in quick reactions are most of the time in a steady state, and there jump from one steady state to an other has an infinite speed, which doesn’t interest us. </section> <section>

Goal

In this part, we want to answer to three questions: </br> </br>

<img src="" alt="" /> What is the sensitivity of our system?

<img src="" alt="" /> What is the time response?

<img src="" alt="" /> What steady states will our system always reach?

</br> </section> <section>

The system

Here is the schema of the real system, in orange are the reactions which didn’t appear in the simplified system of the overview: </br> </br>

<img src="" alt="" />

</br> cAMP is the quorum sensing molecule. When we put some cAMP out of the system, it enters into the system. Then, it complexes with CRP to create (CRP-cAMP), which is the transcription factor of the gene arac. When some Arac is created, it will complex with arabinose to create Arac*. Arac* is the active form of Arac. Arac* with (CRP-cAMP) are the transcription factors of the gene cya. Then when some protein of adenylate cyclase is produced, it will catalyze the production of cAMP. </br> </br> We first have the complexation of cAMP with CRP to get (CRP-CAMP). It is modeled by this biological reaction: </br> </br>

<img src="" alt="" />

</br> </br> Thus, we get the evolution speed of (CRP-CAMP), r<INF>(CRP-cAMP)<INF>: </section>