Team:UPO-Sevilla/Modeling/Signaling

From 2010.igem.org

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<tr class="oddRow"><td>kFecRDeactivation</td><td>k<sub>10</sub></td></tr>
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The differential equation considering the quantities involved is given by:
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<img src="https://static.igem.org/mediawiki/2010/e/e9/UPOModel-React1.png" width="300" alt="Simbiology model"/>
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Knowing that $FecR+FecRa=FecR_{0}$, the initial amount of $FecR$, and some simple calculations, then:
Knowing that $FecR+FecRa=FecR_{0}$, the initial amount of $FecR$, and some simple calculations, then:
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<img src="https://static.igem.org/mediawiki/2010/a/a0/UPOModel-React2.png" width="300" alt="Simbiology model"/>
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If we assume that $bind$ is constant, then the equation is just a first order differential equation, and the evolution of the system is:
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<img src="https://static.igem.org/mediawiki/2010/8/8c/UPOModel-React3.png" width="350" alt="Simbiology model"/>
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Revision as of 14:57, 27 October 2010

The signaling circuit

The signaling circuit 3 described in the Circuit Section has been modeled using Matlab Simbiology desktop. The following diagram shows the different parts of the model we have simulated:

Simbiology model

For the level of detail considered, the main parts simulated are the following (the number correspond to the equations listed in the table in the next section):

  1. Generation of L_aspartate induced by AAL
  2. Diffusion of L_aspartate through the cell wall
  3. Transcription of the aspA, promoted by FecI_a (active)
  4. Translation of aspA
  5. Activation of FecI, induced by the activation of FecR
  6. Activation of FecR induced by FecA-PrhA
  7. Plant cell wall lingand, FecA-PrhA binding

Reactions

The reaction equations for the previous parts, and the reactions rates associated, are summarized in the following table:

#ReactionReactionRateActive
1ecoli.ammonia + ecoli.fumarate + ecoli.AAL <-> ecoli.L_aspartate + ecoli.AALk1*ecoli.ammonia*ecoli.fumarate*ecoli.AAL - k2*ecoli.L_aspartate*ecoli.AALtrue
2ecoli.L_aspartate <-> medium.L_aspartatekWallDiffusion*ecoli.L_aspartate - kWallDiffusionBack*medium.L_aspartatetrue
3ecoli.DNAaspA + ecoli.FecI_a -> ecoli.ARNm_aspA + ecoli.DNAaspA + ecoli.FecI_akTranscript*ecoli.DNAaspA*ecoli.FecI_atrue
4ecoli.ARNm_aspA -> ecoli.AAL + ecoli.ARNm_aspAkTranslation*ecoli.ARNm_aspAtrue
5ecoli.FecR_a + ecoli.FecI <-> ecoli.FecI_a + ecoli.FecR_akFecIActivation*ecoli.FecR_a*ecoli.FecI - kFecIDeactivation*ecoli.FecI_a*ecoli.FecR_atrue
6ecoli.FecR + ecoli.[ligand:FecA-PrhA] <-> ecoli.FecR_a + ecoli.[ligand:FecA-PrhA]kFecRActivation*ecoli.FecR*ecoli.[ligand:FecA-PrhA] - kFecRDeactivation*ecoli.FecR_a*ecoli.[ligand:FecA-PrhA]true
7plant_cell_wall.ligand + ecoli.[FecA-PrhA] <-> ecoli.[ligand:FecA-PrhA]kCellBinding*plant_cell_wall.ligand*ecoli.[FecA-PrhA] - kCellUnbinding*ecoli.[ligand:FecA-PrhA]true

Simulations

The following figure shows the typical evolution of the output of the system (the generated chemoattractant) againts the inputs (the wall cells ligand and the FecA-PrhA components on the outer membrane)

Simbiology model

Analysis

Sensibility

Simbiology allows to compute the sensibility of the system against the different parameters.

The following figure shows the sensibility of all state variables (molecules of the different species considered) with respect to all the parameters.

Simbiology model

What the analysis reveal is that the system is quite insensitive to changes in the parameters. This is due mainly to the nature of the transduction signals. The promoters act as a kind of "switch". This means that, provided these promoters reach certain levels, the other parts of the circuits are activated, even if the levels are not equal.

This analysis is referred to the steady-state of the system. Some parameters do not affect the final steady-state number of molecules, but on the other hand affects the velocity of the system in the transition. This can be seen in the following paragraphs.

Scanning of Parameters

If we perform a scan over several values of some parameters it can be seen the influence of these parameters on the output. For instance, in the following figure it can be seen how the parameter kCellBinding (the "force" of the binding with the cell wall) affects the final output of the system (medium.L_aspartate, the amount of chemoattractant), for several scales of magnitude.

Simbiology model

This parameter affects the velocity of the system, but not in great deal. In the following, the same analysis is done for the four main steps in the generation of the chemoattractant (for quite different orders of magnitude):

  1. Transcription of the aspA, promoted by FecI_a (active)
  2. Translation of aspA
  3. Activation of FecI, induced by the activation of FecR
  4. Activation of FecR induced by FecA-PrhA
Simbiology model

If we analyze the reaction:

ecoli.FecR + ecoli.[ligand:FecA-PrhA] <-> ecoli.FecR_a + ecoli.[ligand:FecA-PrhA]

We change the names of some quantities for the sake of clarity:
NameVariable Name
ligand:FecA-PrhAbind
kFecRActivationk9
kFecRDeactivationk10

The differential equation considering the quantities involved is given by:

Simbiology model

Knowing that $FecR+FecRa=FecR_{0}$, the initial amount of $FecR$, and some simple calculations, then:

Simbiology model
If we assume that $bind$ is constant, then the equation is just a first order differential equation, and the evolution of the system is:
Simbiology model

The steady-state solution is $\frac{k_{9}\cdot FecR_{0} }{k_{9}+k_{10}}$, and it does not depend on $bind$. The time constant of the system (its speed) is governed by $k_{9}+k_{10}$ and $bind$.

In the actual system, $bind$ evolves with time. If its evolution is much quicker than this system, reaching its steady-state, it can be considered constant and the same analysis can be applied.

If the evolution is , there is an actual solution. A simplified analysis can be done assuming that $bind$ is constant for small amounts of time, and applying the same idea. The main effect of bind is to accelerate or decelerate the evolution

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