Team:Grenoble/Modeling/Signaling

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Signaling module



The design of signaling module is given by the figure below:



Once the dipeptide molecule is fixed to the TAP receptor, it activates the phosphorylation of ompR. ompR* (the phosphorylated form of ompR) is the transcription factor that activates the gene expression of cyaA.
Further details about the receptor design, are available here.

A computer model was developped to answer 2 questions:

1- How sensitive is the signaling module? Wich dipeptide concentration should be present in order to trigger a response?

2- How long does it take for the dipeptide to trigger a response?

Answering these two questions will help us assess the sensitivity and the response time of the whole system (ie the signaling module, the amplification module and the quorum sensing).

First, we define the ordinary differential equations that govern adenylate cyclase (Ca) evolution. Thus we plot the evolution of Ca concentration versus initial dipeptide concentration (sensitivity) as well as the temporal evolution of Ca concentration (time response).

Ordinary Differential Equations

Let’s begin by considering the cya gene activation by the transcription factor ompR*.
As it is a gene activation, the regulated transcription rate is usually modelized by a Hill function:



Where Vm is the maximal transcription rate, k is the activation coefficient, p is the basal production coefficient and α the degradation coefficient.
For ompR phosphorylation, we considered a model that takes into account the enzymatic mechanism of the Histidine Kinase EnvZ as well as the phosphotransfer and the phosphatase.
The model we use is a phenomenological extension of the Goldbeter-Koshland biochemical switch model.[1]
The resulting equation governs the temporal evolution of [ompR*] and highlights the fact that the phosphorylation is activated by a dipeptide.



where K and K’ are the dimensionless Michaelis-Menten coefficients.
Since the process involved in the production of the new protein proceed at much slower timescale than the phosphorylation process that aims at chemically modifying the existing protein OmpR, the time derivative of [ompR*] is null (click here for more explanation) and [ompR] and [ompR*] are linked by a conservation law:
[ompR*]+[ompR]=[ompR]tot

Once we have set the derivative equation of ompR* equals to zero and replaced the value of [ompR] and [ompR*] by their expressions involving the total quantity of ompR, we get a second order polynomial equation of [ompR*]:



If we define the coefficients



We notice that the coefficient “a” is always negative because the dephosphorylation rate of ompR* is lower in value than the phosphorylation rate .As a product of kinetic parameters, the coefficient “c” is positive.
For the reasons given above the determinant is positive:

Δ=b²-4 a c>0


We have then two roots x and y for [ompR*]. Their product is given by x.y = c/a. As “a” and “c” have opposite signs, x and y have opposite signs: We choose the positive root, because [ompR*] is always positive.

Parameters



Here is the link to the parameters of the amplification module we sometimes refer to.

Constants Value Derivation
Total quantity [ompR]tot 6.8 10-8 mol.L-1 The average number of ompR molecules per cell is 80.769 ± 0.719 [2]. Knowing the cell volume (vc = 1.1 10-15 L[3])
and the Avogadro number NA = 6.02 10-23 mol.L-1, we deduce
[ompR]tot = 80/(NA*vc) = 6.8 10-8 mol.L-1
Goldbeter-Koshland model constants v = 80 L-1.min-1
V' = 7 10-8 mol.L-1.min-1
K = 7 10-7 mol.L-1
K' = 9 10-8 mol.L-1		 We could not find these parameters in literature and we hope we will be able to conduct the necessary experiments to set them. Nevertheless, we could use a simple approach to estimate them :
The receptor should at least be sensitive to [dipeptide]=10-5 mol.L-1 (it represents the maximum concentration expected [4]). We consider V = v[dipeptide], the equation is given by :
First of all, the value of K (resp K') should be in the same range of concentration as [ompR]tot. Indeed, if K»[ompR]tot the phosphorylation term becomes negligible and the curve has not the desired evolution. Else if K«[ompR]tot,
and we have a high phosphorylation rate even if almost all ompR has been phosphorylated. We chose KcyaA = 7 10-7 mol.L-1 and K'cyaA = 9 10-8 mol.L-1
Given that ,
if we consider [ompR]~[omprR]tot≅6.8 10-8 and [ompR*]≅10-11»[omprR]tot we find that V≥104 V'.
Maximal transcription rate of cyaA VmcyaA = 2 10-9 mol.L-1.min-1 The value of this constant should be understood in the continuity of the network. For full details, consider the amplification section, parameters, explanation2
Basal production of Ca pCa = 2*10-12 mol.L-1.min-1 The value of this constant should be understood in the continuity of the network. For full details, consider the amplification section, parameters, explanation1
Degradation rate of Ca αCa = 6 10-3 min-1 The value of this constant should be understood in the continuity of the network. For full details, consider the amplification section, parameters, explanation4
Activation coefficient of cyaA KcyaA = 10-7 mol.L-1 The value KCa was set considering the maximum value of [Ca]. Indeed if we consider the steady state and assume that pCa is negligible compared to the other terms we have : where h stands for the Hill function, 0<h<1.
We have then :
KCa should be in the same range as [Ca]max not too high otherwise the gene would never be expressed and not too low otherwise the protein is always produced. We chose KcyaA = 10-7 mol.L-1
Hill Coefficient n = 2 We took a number greater than one to indicate positive cooperativity.


References



  • [1] Alejandra C.Ventura, Jacques-A. Sepulchre, Sofia D.Merajver.
    A Hidden Feedback in Signaling Cascades Is Revealed. PLOS Computational Biology, 2008, 4, 3, e1000041.
  • [4] Michael D.Manson, Volker BlanK and Gabriele Brade.
    Peptide chemotaxis in E.Coli involves the Tap signal transducer and the dipeptide permease.Nature,15 May 1986,321,253-256.
  • [5] Edith Gstrein-Reider and Manfred Schweiger, Institut fur Biochemie (nat. Fak.),UniversitAt Innsbruck, A-6020 Innsbruck, Austria.
    Regulation of adenylate cyclase in E. coli.

Sensitivity

( [ompR])/( [ompR]tot ) is plotted as a function of the initial concentration of dipeptide.



We notice that the receptor has a saturated response at an initial dipeptide concentration of 10-6 mol.L-1 . This concentration represents the expected concentration of dipeptide out of the cell [4]. Thus the sensitivity of the phosphorylation cascade is sufficient.

In the next figure [Ca] is a represented as a function of initial dipeptide concentration.



The sensitivity of the sensor is 10-8 mol.L-1 of initial dipeptide concentration. However, at this level we are not able to assess the whole system sensitivity. We have got to wait for the amplification modeling results and link the two models to get an answer to the sensitivity question.

Click here to download the commented matlab code that gave us the figures above.

Time response

In order to determine the time response, we plotted the temporal evolution of Ca for two different initial dipeptide concentrations: 108 mol.L-1 and 10-6 mol.L-1.







In both cases, steady states is reached after 10h.

Moreover, we see that to reach an Ca production of 10-7 mol.L-1 we need approximately 600 min. Click here to download the commented matlab code that gave us the temporal evolution.

Conclusion

Now we have an idea of the sensitivity and the rapidity of the receptor. However, to make sense, these values should be linked to the modeling results concerning the amplification loop.

Since the amplification loop will both increase and accelerate the response, we expect that the sensitivity and response time determinated without the loop are the minimum compared to the whole system.

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