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.
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.
Goal
In this part, we want to answer to three questions:
What is the sensitivity of our system?
What is the time response?
What steady states will our system always reach?
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:
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.
We first have the complexation of cAMP with CRP to get (CRP-CAMP). It is modeled by this biological reaction:
Thus, we get the evolution speed of (CRP-CAMP), r(CRP-cAMP):
Then, we have the conservation equations:
Thus, we get:
Eventually, we get:
where
We have
Eventually, we have to choose between the two solutions
These two solutions are positive, because:
To chose which solution is the good one, we know that:
Eventually, we get:
(CRP-cAMP) is the transcription factor of the gene arac. When it appears in the network, it activates the production of the protein Arac. This is modeled by a Hill function. In addition, there is some outflow linked to the promoter pAraBAD, which is the promoter regulating arac, thus there is a basal production of Arac. We take into account this basal production, because we need to know if because of them our system will always be turned on, thus useless. Arac is also naturally degraded by the bacterium. Thus, we get as the equation of evolution of Arac
concentration:
Then, the protein Arac complexes with arabinose to create Arac active, written Arac*. It is modeled by the following chemical equation:
We get the evolution of Arac* rArac*:
With the qssa, we get:
In addition we have the conservation equation of Arac:
We assume that we have
Then, Arac* with (CRP-cAMP) are the transcription factors of of the gene ca. When they appear in the network the protein Ca is produced. The product of two hill functions models this. For the same reasons as for Arac we take into account the basal production of the adenylate cyclase. In addition it is degraded by the bacterium.
Thus, we get the equation:
Eventually, Ca catalyzes the production of cAMP with ATP. We have the following enzymatic reaction:
We use the Michaelis Menten model for an enzymatic reaction, thus we get the evolution of cAMP rcAMP:
In addition, we assume that [ATP]>>KM. Eventually we get:
"
What is the sensitivity of our system?
What is the time response?
What steady states will our system always reach?
We have 
Eventually, we have to choose between the two solutions
Eventually, we get:
Then, Arac* with (CRP-cAMP) are the transcription factors of of the gene ca. When they appear in the network the protein Ca is produced. The product of two hill functions models this. For the same reasons as for Arac we take into account the basal production of the adenylate cyclase. In addition it is degraded by the bacterium.
Thus, we get the equation:
