Team:Paris/Modeling/More From2Ode Expl

Return to "from molecular reaction to Non-Linear ODEs"

Polymere of Transcription Factor

As it is explained in the introduction for the case of "FlhDC", instead of considering every polymere and complexes of proteins, we simplify those phenomenon by dealing with "abstract" proteins which represents the real transcription factors. It is the case for :

• FlhDC ← FlhD₄C₂
  

It is actually an hexamere [?]

• TetR ← TetR₂
  

TetR dimerizes before acting on pTet [?]

• aTc ← aTc₂
  

aTC dimerizes before binding on TetR [?]

• OmpR^* ← OmpR><P, OmpR><EnvZ
  

OmpR acts on pFlhDC only if phosphorylated (OmpR><P). If we use ompR^*, mutated version of ompR, we consider that the phosphorylation is immediate. If we use envZ, we consider the "basal" presence of OmpR in the cell as an additional parameters, and we suppose that OmpR bound to EnvZ is immediately phosphorylated [?]

Transcription

We suppose that the transcription rate is proportional to the amount of "effecive promoter" in the cell. An effective promoter is either a induced promoter bound to its transcription factor, either a free constitutive promoter. We use the following notations :

Competition Between Transcription Factors

If several transcription factors bind to a given promoter, those several complexations are in competition. To treat this phenomenon, we must know if a reaction is predominant, or if all the steady-states are reached "at the same time", oother rules of order. We made the following hypothesis :

• The contribution of the two inducers FliA and FlhDC on the promoters of "class 2 genes" (pFliL,pFlgA,pFlgB,pFlhB ; U. Alon demonstated it for pFliL in [?] ) are synthetised by a SUM logical gate : that corresponds to the biological hypothesis that the promoter has got two specific sites of binding, one for each TF, and that we can sum the probabilities of binding between the RNAase and the adequat sites.
  
• It seems obvious that the binding sites of OmpR* and of FliA on pFlhDC are completly different. We made the hypothesis that any pFlhDC bound to OmpR is inactivated, whether it is bound to FliA or not.

Dilution/Degradation γ

As we want the cells to remain in "exponential phase of growth", we assume that every protein involved in the system and for the characterisation has a half-life time far superior than the time of cell division. In that way, we consider that every protein in the cell disappears only by dilution, whose value would be estimated in the wet-lab.

Translation : RBS issue

We use the GFPgenerator (E0240) in association with its RBS (B0032), to caracterise the "expression of the gene behind a given promoter". However, the Ribosome Binding Site of the gene influence it's expression. By considering that the translation rate depends near linearly of the Ribosome Binding Site (as we guess ; it gives the affinity between the mRNA and the ribosome !), we introduce a constant factor between the expected value of a protein and its real value.

Moreover, we observe that at the steady state, the "disappearance rate" γ has got a linear effect on the concentration of the protein at equilibrium, too.

For instance, if we consider a protein prot coded by its corresponding gene, put behind one of our "characterized promoters", with its natural RBS, we will have between our construction (expected value, given by GFP) and the real expression of prot the following relation

where

It does not invalid our will of "systematic characterization", because this error is reported once prot is considered as transcription factor for the next step in which it is involved...

↓ why it is not a problem ? ↑

Let's see on our example why it is not a problem. Our wish is to characterize the strength of the promoter pFliL, in function of the transcriptions factors FlhDC and FliA. As it is explained in the characterisation protocol, it is done in two steps : First, we study the activity of the promoters pTet and pBad, in function of the small molecules we introduce in the medium. It gives, with the standard RBS B0034, respectively the functions ƒ1 and ƒ2. Then, we put the genes flhDC and fliA behind pTet and pBad, but with their natural RBS. In this order, we will get [FlhDC] = {coefflhDC} * ƒ1 and [FliA] = {coeffliA} * ƒ2. So,the pFliL added will have a given activity, measured with the RBS B0032 throught the expression of GFP. We have shown in our modeliing that we expect it to have the following analytical expression : that gives, with previous considerations and by dividing numerator and denominator by {coefgene}n : Thus, we can still evaluate the expression by keeping ƒ1 and ƒ2, but we must notice that the dissociation constant which we will estimate is not exactly the dissociation constant of the complexation FT><pFliL. From now on, we understand that the calculated quantities of proteins by our global model are not the one in the "reality". Still, the purpose of our "Characterization Approach" is kept, by considering relative expression of the proteins... We can moreover notice that, by using this method in other circumstances, we can get predictive quantities and real constants, by keeping always the same RBS (for example B0032) in all constructs !