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Overview

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.
If you need further details about the receptor design, you can check the network section here.

This approach arises a few questions we cannot answer without computer models:

1- How sensitive is the signaling module? And what 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 rapidity of the whole system (ie the signaling module, the amplification module and the quorum sensing).

To answer these questions, we need to set the ordinary equations that govern Ca evolution. Thus we can plot the evolution of Ca concentration versus initial dipeptide concentration (sensitivity) as well as the temporal evolution of Ca concentration (rapidity).

ODEs

Let’s begin by considering the cya gene activation by the transcription factor ompR*.
As it is a gene activation, the 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 in the literature 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 Goldbetegivenr-Koshland biochemical switch model.[1]
The resulting equation governs [ompR*] temporal evolution (the square brackets denote concentration) and highlights the fact that the phosphorylation is activated by dipeptide.

where K and K’ are the dimensionless Michaelis-Menten coefficients.
Since the process involved in the production of the new protein Ca proceed at much slower timescale than the phosphorylation process that aims at chemically modifying the existing protein OmpR, 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 float roots x and y. 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.

Parameters

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

Constants	Value	Derivation
Total quantity [omprR]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 Cya	KCya = 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.

[2] http://ecoliwiki.net/colipedia/index.php/ompR:Expression

[3] http://bionumbers.hms.harvard.edu/bionumber.aspx?&id=100004&ver=15&trm=E%20coli%20volume

[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

First of all, we plotted the evolution of the ratio ( [ompR])/( [ompR]_tot ) versus the initial concentration of dipeptide to check if the phenomenological extension of Goldbeter-Koshland model is convenient and gives us the expected results.

We notice that the receptor has a maximal 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 we can say that the phosphorylation cascade is efficient as it enables as to have a maximal response for the expected conditions of use.

Next we need to plot the evolution of [Ca] versus initial dipeptide concentration to assess sensitivity.

The sensitivity of the receptor 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 have an idea of the rapidity of the detection, we plotted the temporal evolution of Ca for two different initial dipeptide concentrations: 10⁸ mol.L-1 and 10^-6 mol.L-1.

As we expected, we notice that we have the expected steady state values (see the graph above): 0.55 10-7 mol.L-1 for an initial dipeptide concentration of 10^-8 mol/L-1 and 1.1 10^-7 mol.L for an initial dipeptide concentration of 10^-6 mol.L-1.

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.

But, Above all we are waiting for biological results to check the estimated parameters values and validate them.

@@ Line 14: / Line 14: @@
 </br>
 </br>
-Once the dipeptide molecule is fixed to the TAP receptor, it activates the phosphorylation of OmpR.</br> OmpR* (the phosphorylated form of OmpR) is the transcription factor that activates the gene expression of cya.
+Once the dipeptide molecule is fixed to the TAP receptor, it activates the phosphorylation of ompR.</br> ompR* (the phosphorylated form of ompR) is the transcription factor that activates the gene expression of cyaA.
 </br>
 If you need further details about the receptor design, you can check the network section <a href="https://2012.igem.org/Team:Grenoble/Biology/Network#10"> here</a>.
@@ Line 31: / Line 31: @@
 </br>
 </br>
-To answer these questions, we need to set the ordinary equations that govern AC evolution.   Thus we can plot the evolution of AC concentration versus initial dipeptide concentration (sensitivity) as well as the temporal evolution of AC concentration (rapidity).
+To answer these questions, we need to set the ordinary equations that govern Ca evolution.   Thus we can plot the evolution of Ca concentration versus initial dipeptide concentration (sensitivity) as well as the temporal evolution of Ca concentration (rapidity).
 </section>
 <section>
 <h1>ODEs</h1>
-Let’s begin by considering the cya gene activation by the transcription factor OmpR*.
+Let’s begin by considering the cya gene activation by the transcription factor ompR*.
 </br>
 As it is a gene activation, the transcription rate is usually modelized by a hill function:
@@ Line 46: / Line 46: @@
 Where Vm is the maximal transcription rate, k is the activation coefficient, p is the basal production coefficient and α the degradation coefficient.
 </br>
-For OmpR phosphorylation, we considered in the literature a model that takes into account the enzymatic mechanism of the Histidine Kinase EnvZ as well as the phosphotransfer and the phosphatase.
+For ompR phosphorylation, we considered in the literature a model that takes into account the enzymatic mechanism of the Histidine Kinase EnvZ as well as the phosphotransfer and the phosphatase.
 </br>
 The model we use is a phenomenological extension of the Goldbetegivenr-Koshland biochemical switch model.<a href="#ref">[1]</a></br>
-  The resulting equation governs [OmpR*] temporal evolution (the square brackets denote concentration) and highlights the fact that the phosphorylation is activated by dipeptide.
+  The resulting equation governs [ompR*] temporal evolution (the square brackets denote concentration) and highlights the fact that the phosphorylation is activated by dipeptide.
 </br>
 </br>
@@ Line 57: / Line 57: @@
 where K and K’ are the dimensionless Michaelis-Menten coefficients.
 </br>
-Since the process involved in the production of the new protein AC proceed at much slower timescale than the phosphorylation process that aims at chemically modifying the existing protein OmpR, time derivative of [OmpR*] is null (click here for more explanation) and [OmpR] and [OmpR*] are linked by a conservation law:
+Since the process involved in the production of the new protein Ca proceed at much slower timescale than the phosphorylation process that aims at chemically modifying the existing protein OmpR, time derivative of [ompR*] is null (click here for more explanation) and [ompR] and [ompR*] are linked by a conservation law:
 </br>
-<center>[OmpR*]+[OmpR]=[OmpR]<SUB>tot</SUB></center>
+<center>[ompR*]+[ompR]=[ompR]<SUB>tot</SUB></center>
 </br>
-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*]:
+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*]:
 </br>
 </br>
@@ Line 73: / Line 73: @@
 </br>
 </br>
-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.
+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.
 </br>
 For the reasons given above the determinant is positive:
@@ Line 107: / Line 107: @@
      <tbody>
 	<tr>
-		<td class="column1">Total quantity [OmprR]<span class="indice">tot</span></td>
+		<td class="column1">Total quantity [omprR]<span class="indice">tot</span></td>
 		<td class="column2">6.8 10<span class="exposant">-8</span> mol.L<span class="exposant">-1</span></td>
-		<td class="column3">The average number of OmpR molecules per cell is 80.769 ± 0.719 <a href="#ref">[2]</a>. Knowing the cell volume (<i>v<span class="indice">c</span> = 1.1 10<span class="exposant">-15</span> L</i><a href="#ref">[3]</a>)<br/> and the Avogadro number <i>N<span class="indice">A</span> = 6.02 10<span class="exposant">-23</span> mol.L<span class="exposant">-1<span></i>, we deduce <br/>[OmpR]<span class="indice">tot</span> = 80/(N<span class="indice">A</span>*v<span class="indice">c</span>) = 6.8 10<span class="exposant">-8</span> mol.L<span class="exposant">-1</span></td>
+		<td class="column3">The average number of ompR molecules per cell is 80.769 ± 0.719 <a href="#ref">[2]</a>. Knowing the cell volume (<i>v<span class="indice">c</span> = 1.1 10<span class="exposant">-15</span> L</i><a href="#ref">[3]</a>)<br/> and the Avogadro number <i>N<span class="indice">A</span> = 6.02 10<span class="exposant">-23</span> mol.L<span class="exposant">-1<span></i>, we deduce <br/>[ompR]<span class="indice">tot</span> = 80/(N<span class="indice">A</span>*v<span class="indice">c</span>) = 6.8 10<span class="exposant">-8</span> mol.L<span class="exposant">-1</span></td>
 	</tr>
 	<tr>
@@ Line 120: / Line 120: @@
 The receptor should at least be sensitive to [dipeptide]=10<span class="exposant">-5</span> mol.L<span class="exposant">-1</span> (it represents the maximum concentration expected <a href="#ref">[4]</a>). We consider V = v[dipeptide], the equation is given by :<br/>
 <center><img src="https://static.igem.org/mediawiki/2012/1/16/DOmpR.png" alt="" /></center>
-First of all, the value of K (resp K') should be in the same range of concentration as [OmpR]<span class="indice">tot</span>. Indeed, if K&raquo;[OmpR]<span class="indice">tot</span> the phosphorylation term becomes negligible and the curve has not the desired evolution. Else if K&laquo;[OmpR]<span class="indice">tot</span>, <img src="https://static.igem.org/mediawiki/2012/6/60/OmpR-K.png" alt="" /><br/>
+First of all, the value of K (resp K') should be in the same range of concentration as [ompR]<span class="indice">tot</span>. Indeed, if K&raquo;[ompR]<span class="indice">tot</span> the phosphorylation term becomes negligible and the curve has not the desired evolution. Else if K&laquo;[ompR]<span class="indice">tot</span>, <img src="https://static.igem.org/mediawiki/2012/6/60/OmpR-K.png" alt="" /><br/>
-and we have a high phosphorylation rate even if almost all OmpR has been phosphorylated. We chose K<span class="indice">Cya</span> = 7 10<span class="exposant">-7</span> mol.L<span class="exposant">-1</span> and K'<span class="indice">Cya</span> = 9 10<span class="exposant">-8</span> mol.L<span class="exposant">-1</span><br/>
+and we have a high phosphorylation rate even if almost all ompR has been phosphorylated. We chose K<span class="indice">cyaA</span> = 7 10<span class="exposant">-7</span> mol.L<span class="exposant">-1</span> and K'<span class="indice">cyaA</span> = 9 10<span class="exposant">-8</span> mol.L<span class="exposant">-1</span><br/>
-Given that <img src="https://static.igem.org/mediawiki/2012/0/05/DOmpR-0.png" alt="" />,<br/> if we consider [OmpR]~[OmprR]<span class="indice">tot</span>&cong;6.8 10<span class="exposant">-8</span> and [OmpR*]&cong;10<span class="exposant">-11</span>&raquo;[OmprR]<span class="indice">tot</span> we find that V&ge;10<span class="exposant">4</span> V'.
+Given that <img src="https://static.igem.org/mediawiki/2012/0/05/DOmpR-0.png" alt="" />,<br/> if we consider [ompR]~[omprR]<span class="indice">tot</span>&cong;6.8 10<span class="exposant">-8</span> and [ompR*]&cong;10<span class="exposant">-11</span>&raquo;[omprR]<span class="indice">tot</span> we find that V&ge;10<span class="exposant">4</span> V'.
 </td>
 	</tr>
 	<tr>
-		<td class="column1">Maximal transcription rate of Cya</td>
+		<td class="column1">Maximal transcription rate of cyaA</td>
-		<td class="column2">V<span class="indice">mCya</span> = 2 10<span class="exposant">-9</span> mol.L<span class="exposant">-1</span>.min<span class="exposant">-1</span></td>
+		<td class="column2">V<span class="indice">mcyaA</span> = 2 10<span class="exposant">-9</span> mol.L<span class="exposant">-1</span>.min<span class="exposant">-1</span></td>
 		<td class="column3">The value of this constant should be understood in the continuity of the network. For full details, consider the <a href="https://2012.igem.org/Team:Grenoble/Modeling/Amplification/Sensitivity#exp2">amplification section, parameters, explanation2</a></td>
 	</tr
 	<tr>
-		<td class="column1">Basal production of AC</td>
+		<td class="column1">Basal production of Ca</td>
-		<td class="column2">p<span class="indice">AC</span> = 2*10<span class="exposant">-12</span> mol.L<span class="exposant">-1</span>.min<span class="exposant">-1</span></td>
+		<td class="column2">p<span class="indice">Ca</span> = 2*10<span class="exposant">-12</span> mol.L<span class="exposant">-1</span>.min<span class="exposant">-1</span></td>
 		<td class="column3">The value of this constant should be understood in the continuity of the network. For full details, consider the <a href="https://2012.igem.org/Team:Grenoble/Modeling/Amplification/Sensitivity#exp1">amplification section, parameters, explanation1</a></td>
 	</tr>
 	<tr>
-		<td class="column1">Degradation rate of AC</td>
+		<td class="column1">Degradation rate of Ca</td>
-		<td class="column2">α<span class="indice">AC</span> = 6 10<span class="exposant">-3</span> min<span class="exposant">-1</span></td>
+		<td class="column2">α<span class="indice">Ca</span> = 6 10<span class="exposant">-3</span> min<span class="exposant">-1</span></td>
 		<td class="column3">The value of this constant should be understood in the continuity of the network. For full details, consider the <a href="https://2012.igem.org/Team:Grenoble/Modeling/Amplification/Sensitivity#exp4">amplification section, parameters, explanation4</a></td>
 	</tr>
@@ Line 143: / Line 143: @@
 		<td class="column1">Activation coefficient of Cya</td>
 		<td class="column2">K<span class="indice">Cya</span> = 10<span class="exposant">-7</span> mol.L<span class="exposant">-1</span></td>
-		<td class="column3">The value K<span class="indice">AC</span> was set considering the maximum value of [AC]. Indeed if we consider the steady state and assume that <i>p<span class="indice">AC</span></i> is negligible compared to the other terms we have : <img src="https://static.igem.org/mediawiki/2012/f/f8/AC.png" alt="" /> where h stands for the Hill function, 0&lt;h&lt;1.<br/>
+		<td class="column3">The value K<span class="indice">Ca</span> was set considering the maximum value of [Ca]. Indeed if we consider the steady state and assume that <i>p<span class="indice">Ca</span></i> is negligible compared to the other terms we have : <img src="https://static.igem.org/mediawiki/2012/f/f8/AC.png" alt="" /> where h stands for the Hill function, 0&lt;h&lt;1.<br/>
-We have then : <img src="https://static.igem.org/mediawiki/2012/a/a4/ACmax.png" alt="" /><br>K<span class="indice">AC</span> should be in the same range as [AC]<span class="indice">max</span> not too high otherwise the gene would never be expressed and not too low otherwise the protein is always produced. We chose K<span class="indice">Cya</span> = 10<span class="exposant">-7</span> mol.L<span class="exposant">-1</span></td>
+We have then : <img src="https://static.igem.org/mediawiki/2012/a/a4/ACmax.png" alt="" /><br>K<span class="indice">Ca</span> should be in the same range as [Ca]<span class="indice">max</span> not too high otherwise the gene would never be expressed and not too low otherwise the protein is always produced. We chose K<span class="indice">cyaA</span> = 10<span class="exposant">-7</span> mol.L<span class="exposant">-1</span></td>
 	</tr>
 	<tr>
@@ Line 171: / Line 171: @@
 <section>
 <h1>Sensitivity</h1>
-First of all, we plotted the evolution of the ratio ( [OmpR])/( [OmpR]<SUB>tot</SUB> ) versus the initial concentration of dipeptide to check if the phenomenological extension of Goldbeter-Koshland model is convenient and gives us the expected results.
+First of all, we plotted the evolution of the ratio ( [ompR])/( [ompR]<SUB>tot</SUB> ) versus the initial concentration of dipeptide to check if the phenomenological extension of Goldbeter-Koshland model is convenient and gives us the expected results.
 </br>
 </br>
@@ Line 180: / Line 180: @@
 </br>
 </br>
-Next we need to plot the evolution of [AC] versus initial dipeptide concentration to assess sensitivity.
+Next we need to plot the evolution of [Ca] versus initial dipeptide concentration to assess sensitivity.
 </br>
 </br>
@@ Line 193: / Line 193: @@
 <section>
 <h1>Time response</h1>
-In order to have an idea of the rapidity of the detection, we plotted the temporal evolution of AC for two different initial dipeptide concentrations: 10<SUP>8</SUP> mol.L<span class="exposant">-1</span> and 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>.
+In order to have an idea of the rapidity of the detection, we plotted the temporal evolution of Ca for two different initial dipeptide concentrations: 10<SUP>8</SUP> mol.L<span class="exposant">-1</span> and 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>.
 </br>
 </br>
@@ Line 204: / Line 204: @@
 </br>
 </br>
-As we expected, we notice that we have the expected steady state values (see the graph above): 0.55 10-7 mol.L<span class="exposant">-1</span> for an initial dipeptide concentration of 10<SUP>-8</SUP> Mol/L and 1.1 10<SUP>-7</SUP> mol.L<span class="exposant"></span> for an initial dipeptide concentration of 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>.
+As we expected, we notice that we have the expected steady state values (see the graph above): 0.55 10-7 mol.L<span class="exposant">-1</span> for an initial dipeptide concentration of 10<SUP>-8</SUP> mol/L<span class="exposant">-1</span> and 1.1 10<SUP>-7</SUP> mol.L<span class="exposant"></span> for an initial dipeptide concentration of 10<SUP>-6</SUP> mol.L<span class="exposant">-1</span>.
 </br>
 </br>
-Moreover, we see that to reach an AC production of 10<SUP>-7</SUP> mol.L<span class="exposan">-1</span>  we need approximately 600 min.
+Moreover, we see that to reach an Ca production of 10<SUP>-7</SUP> mol.L<span class="exposan">-1</span>  we need approximately 600 min.
 <a href="https://static.igem.org/mediawiki/2012/9/9e/Cya_vs_time.zip">Click here</a> to download the commented matlab code that gave us the temporal evolution.

Team:Grenoble/Modeling/Signaling