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:

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:

#	Reaction	ReactionRate	Active
1	ecoli.ammonia + ecoli.fumarate + ecoli.AAL <-> ecoli.L_aspartate + ecoli.AAL	k1*ecoli.ammonia*ecoli.fumarate*ecoli.AAL - k2*ecoli.L_aspartate*ecoli.AAL	true
2	ecoli.L_aspartate <-> medium.L_aspartate	kWallDiffusion*ecoli.L_aspartate - kWallDiffusionBack*medium.L_aspartate	true
3	ecoli.DNAaspA + ecoli.FecI_a -> ecoli.ARNm_aspA + ecoli.DNAaspA + ecoli.FecI_a	kTranscript*ecoli.DNAaspA*ecoli.FecI_a	true
4	ecoli.ARNm_aspA -> ecoli.AAL + ecoli.ARNm_aspA	kTranslation*ecoli.ARNm_aspA	true
5	ecoli.FecR_a + ecoli.FecI <-> ecoli.FecI_a + ecoli.FecR_a	kFecIActivation*ecoli.FecR_a*ecoli.FecI - kFecIDeactivation*ecoli.FecI_a*ecoli.FecR_a	true
6	ecoli.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
7	plant_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)

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.

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.

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

The system can be seen as a set of smaller subsystems that are coupled by the enzymatic promoters. This subsystems are qualitatively quite similar.

If we analyze the FecR activation reaction:

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

The differential equation govering the reaction is:

were we have changed the names of some quantities for the sake of clarity:

Name	Variable Name
ligand:FecA-PrhA	bind
kFecRActivation	k9
kFecRDeactivation	k10

Knowing that FecR+FecRa=FecR₀, the initial amount of FecR, and some simple calculations, then:

If we assume that the inducer bind is constant, then the equation is just a first order differential equation, and the evolution of the system is given by:

The steady-state solution is k₉FecR₀/(k₉+k₁₀), and it does not depend on bind.

The time constant of the system (its speed) is governed by (k₉+k₁₀) 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. It acts as an activator of the system; moreover, the higher the quantity, the quicker the system.

If bind evolves with time the solution of the equation is bit more involved. A simplified analysis can be done assuming that bind is timewise constant for small amounts of time, and applying the same idea. The main effect of bind is to accelerate or decelerate the evolution.

The simulation of the FecR system, with no approximation, is very similar to the first order system described: