Team:Grenoble/Modeling/Amplification/Sensitivity

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Classic Odes Parameters


Parameter Name Value Unit Source
AraC synthesis rate vmAraC 12 10-8 mol.L-1.min-1 Yes [1] and explanation 1
Ca synthesis rate vmAraC 2 10-9 mol.L-1.min-1 No, explanation 2
AraC threshold KAraC 0.310-6 mol.L-1 No, explanation 3
Ca threshold KCa (for (CRP-cAMP) activation) 0.6 10-6 mol.L-1 No, explanation 3
Ca threshold K'Ca (for AraC* activation) 0.6 10-6 mol.L-1 No, explanation 3
Dissociation constant of cAMP and CRP Kc 10-5 mol.L-1 Yes [2]
Dissociation of arabinose with AraC Kc1 2 10-13 mol.L-1 Yes [3]
Degradation rate of AraC αAraC 6 10-3 min-1 Yes [1]
Degradation rate of Ca αCa 6 10-3 min-1 No, explanation 4
Degradation/exports rate of cAMP αcAMP 2.1 min-1 Yes [4]
Arabinose concentration 10-6 mol.L-1 Chosen by the team
CRP concentration 10-6 mol.L-1 Yes [5] and explanation 1
cAMP import constant η 6 10-1 min-1 No, explanation 5
Catalysis constant kcat 810 min-1 No, explanation 6
Hill cooperativity n, η1, η2 1.5 / No, explanation 7
AraC basal value pAraC 12 10-11 mol.L-1.min-1 No, explanation 1
Ca basal value pAraC 2 10-12 mol.L-1.min-1 No, explanation 1


Explanation 1


  • AraC and Ca:

Once the production of AraC has been activated, we have:

In addition, we assume that p_AraC ≪v_mAraC thus, we get:

In addition we know that the maximum number of copies per cell of AraC is 40, thuswhereAvogadro number andis E. coli volume. Thus, we have . Then, we get the value of .

For the adenylate cyclase, but we assumed that there is less adenylate cyclase in the cell than AraC. Indeed, when there is too much cAMP is inhibits the production of the adenylate cyclase [6].

  • CRP:

We have 1000 copies per cell of CRP, and it’s a constant. Then by the same computation, we make the conversion into mol.L-1, and we get the value of CRP.

Explanation 2


We do not know the value of the synthesis rate of adenylate cyclase, but we assume that there is less adenylate cyclase, because if there is a huge amont of adenylate cyclase, there will too much cAMP, and then cAMP will repress the adenylate cyclase production [6]. For the basal production, we make the same assumption as for AraC.

Explanation 3


  • Activations by (CRP-cAMP):

To set the value of this parameter, we plotted the concentration of (CRP-cAMP) at t=2000 min as we varied the initial concentration of cAMP:


Evolution of (CRP-cAMP) concentration at t=2000 minutes in function of the initial concentration of cAMP.


We could notice that the maximum value of (CRP-cAMP) varied between 0 and 10-6 mol.L-1. The value of the threshold had to be in this range of concentration. Not too low, else it would have meant that the proteins are always produced, and not to high, else the genes would never be expressed. In addition we assumed that AraC had to be activated first and without amplification of cAMP, seing that it is only when arac is expressed the adenylate cyclase is produced.

Remark: when we have [cAMP]>> 10-6 mol/L=[CRP], we have by the same reasoning as for AraC:

and we havewhen [cAMP] increases. Thus, in this case we could be sure that the concentration of (CRP-cAMP) wouldn’t exceed 10-6 mol.L-1. In addition, when [cAMP]≤10-6 mol/L, we also know that because of the value of [CRP], [(CRP-cAMP)] wouldn’t exceed this threshold.

  • Activation by AraC*:

Same reasoning, we give the same type of graph:


We could set the value of K'Ca.

Explanation 4


Inspired from AraC. By discussing with the biologists we concluded that Ca is also a stable protein.

Explanation 5


By discussing with the biologists, we assumed that there is around 10 times more cAMP out of the cell than inside the cell.

Explanation 6


In the reference [6], we have kcat=100 min-1 in vitro. However, by discussing with the biologists, we assumed that in vivo this value was higher.

Quorum sensing parameters


Parameter Name Value Unit Source
cAMP diffusion coefficient Ddiff 2.66 10-4 cm2.min-1 Yes, [8]
E. coli volume vc 10-12 cm3 Yes, [9]
E. coli number by cm3 ρ 0.5 10-12 cm-3 No, explanation 1
lenght and width lx = ly 2 cm chosen
Number of points Nx = Ny 20 / chosen
Time Δt 1000 min chosen
Number of time Nt 1000 / chosen
Initial concentration of cAMP N 10-4 mol.L-1 chosen, explanation 2


Explanation 1


We assumed that they would not be glued together, but not too far at the same time.

Explanation 2


We work above the threshold, because we want to know the speed of the diffusion when we have a detection.

References

  • [2] Purification of and properties of the cyclic adenosine 2' ,5'-monophosphate receptor which mediates cyclic adenosine 3',5'-monophosphate dependent gene transcription in E. Coli.
    W.B. Aderson, A. B. Schneider, M. Emmer, R.L. Perlman, and I. Pasta
  • [3] AraC protein, regulation of the L-arabinose operon inEscherichiacoli, and the light switch mechanism of AraC action.
    Robert Schleif Biology Department, Johns Hopkins University, Baltimore, MD, USA.
  • [4] Epstein et Hesse 1975.
  • [6] Regulation of adenylate cyclase in E. coli.
    Edith Gstrein-Reider and Manfred Schweiger, Institut fur Biochemie (nat. Fak.), UniversitAt Innsbruck, A-6020 Innsbruck, Austria.
  • [7] Purification and characterization of adenylate cyclase from E. coli K12.
    Yang and Epstein 1983.
  • [8] Solubility and diffusion coefficient of adenosine 3’ :5’ – monophosphate.
    Martin Dworkin and Kenneth H. Keller, 1976.

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