Team:Paris/Transduction modeling The Fec Operon as used in our system : chemical equations and kinetics

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Chemical Equations Description

To try and understand our system, we decided to divide the global in simple reactions traducing the chemical steps of the transduction cascade ; each one of these reactions is described with a cinetic law, thus allowing to run both deterministic and stochastic simulation (see here for further informations on these simulations). In our work, we assumed that the crossing of the periplasm is the limiting step ; consequently, we decided to distinguish two ways to describe this step:


Chemical Equations : no complexation in the periplasm

This trancriptional cascade can be described thanks to simple chemical equations traducing the chronological and chemical steps of our tranduction and activation system.


Description of main reactions

These equations are described on the scheme below :

Fec nocomplex.png


  • The FecA molecules coming from the vesicules are clowly diffusing on the lipid bilayer of the outer membrane while vesicules are fusioning ; they finally reach the bacteria outer membrane where they are in proper conditions to be able to activate the FeR molecules. This "crossing" from OMV to the outer membrane is explicited as a first reaction :
(1) FecA_OMV ---> FecA_OM


  • Then, once in the outer membrane, these FecA molecules are able to activate the FecR proteins constitutively present in the receiver ;the FecA molecule directly activates FecR, and the phenomenon is described in a single reaction :
(2) FecA_OM + FecR ---> FecA_OM + FecR*


  • Once FecR is activated, it can activate the FecI molecule ; to avoid an to much variability in the approaches, we decided to consider that this reaction occured as a single step (with no intermediate complex formation) :
(3) FecR* + FecI ---> FecI* + FecR*


  • Then, FecI* can activate the pfec promoter in a reversible reaction :
(4) FecI* + pfec ---> pfec*


  • After the pfec activation, transcription and translation can start ; the creation of proteins is materialised by the single reaction :
(5) pfec* ---> FecI + FecR_Nterm + GFP

where FecR_Nterm correspond to the residues ??-??? of FecR which make this molecule constitutively active (the FecR_Nterm is able to activate FecI without needing to be in presence of FecA)


  • Finally, the constitutively active FecR_Nterm can activate the FecI proteins :
(6) FecR_Nterm + FecI ---> FecI* + FecR_Nterm


  • It is also compulsory to consider some dilution and degradation reactions for the molecules FecA_OM, FecI, FecI*, FecR_Nterm, FecR* and GFP. The natural FecR is considered as a molecule having reached a steady state level ; as a consequence the only noticeable changes are the results of a reaction of activation.
FecA_OM ---> ∅
FecI ---> ∅
FecI* ---> ∅
FecR_Nterm ---> ∅
FecR* ---> ∅
GFP ---> ∅


Kinetics Equations

As we do not have much information on the kinetic laws, we have decided to chose a mass action law for each reaction. Each reaction has a kinetic constant k determining the reaction rate given the concentratin of reactants. This will lead us to write a differential system used for a deterministic resolution.


The first thing to do is to write the reaction rate of each chemical step for the two systems of reactions :

  • A system without an intermediary step in the FecR activation
Reaction Number Kinetics Constants Reaction Reaction Rate
1 kf_1 FecA_OMV ---> FecA_OM kf_1*[FecA_OMV]
2 kf_2 FecA_OM + FecR ---> FecA_OM + FecR* kf_2*[FecA_OM]*[FecR]
3 kf_3 FecR* + FecI ---> FecI* + FecR* kf_3*[FecI*]*[FecR*]
4 kf_4 FecI* + pfec <---> pfec* kf_4*[FecI*]*[pfec]-kr_4[pfec*]
5 kf_5 pfec* ---> FecI + FecR_Nterm + GFP kf_5*[pfec*]
6 kf_6 FecR_Nterm + FecI ---> FecI* + FecR_Nterm kf_6*[FecR_Nterm]*[FecI]
7 kf_7 FecA_OM ---> ∅ kf_7*[FecA_OM]
8 kf_8 FecI ---> ∅ kf_8*[FecI]
9 kf_9 FecI* ---> ∅ kf_9*[FecI*]
10 kf_10 FecR_Nterm ---> &#x2205 kf_10*[FecR_Nterm]
11 kf_11 FecR* ---> ∅ kf_11*[FecR*]
12 kf_12 GFP ---> ∅ kf_12*[GFP]

Chemical Equations : Complexation in the periplasm

This trancriptional cascade can be described thanks to simple chemical equations traducing the chronological and chemical steps of our transduction and activation system.


Description of main reactions

These equations are described on the scheme below :


Fec Complex.png


  • The FecA molecules coming from the vesicules are clowly diffusing on the lipid bilayer of the outer membrane while vesicules are fusioning ; they finally reach the bacteria outer membrane where they are in proper conditions to be able to activate the FeR molecules. This "crossing" from OMV to the outer membrane is explicited as a first reaction :
(1) FecA_OMV ---> FecA_OM


  • Then, once in the outer membrane, these FecA molecules are able to activate the FecR proteins constitutively present in the receiver ; FecA and FecR form a complex which is then destructed to release an activated FecR protein ; this mechanism is described through the 2 following reactions
(2) FecA_OM + FecR ---> FecA_OM-FecR
(3) FecA_OM-FecR ---> FecA_OM + FecR*


  • Once FecR is activated, it can activate the FecI molecule ; to avoid an to much variability in the approaches, we decided to consider that this reaction occured as a single step (with no intermediate complex formation) :
(4) FecR* + FecI ---> FecI* + FecR*


  • Then, FecI* can activate the pfec promoter in a reversible reaction :
(5) FecI* + pfec ---> pfec*


  • After the pfec activation, transcription and translation can start ; the creation of proteins is materialised by the single reaction :
(6) pfec* ---> FecI + FecR_Nterm + GFP

where FecR_Nterm correspond to the residues ??-??? of FecR which make this molecule constitutively active (the FecR_Nterm is able to activate FecI without needing to be in presence of FecA)


  • Finally, the constitutively active FecR_Nterm can activate the FecI proteins :
(7) FecR_Nterm + FecI ---> FecI* + FecR_Nterm


  • It is also compulsory to consider some dilution and degradation reactions for the molecules FecA_OM, FecI, FecI*, FecR_Nterm, FecR* and GFP. The natural FecR is considered as a molecule having reached a steady state level ; as a consequence the only noticeable changes are the results of a reaction of activation.
FecA_OM ---> ∅
FecI ---> ∅
FecI* ---> ∅
FecR_Nterm ---> ∅
FecR* ---> ∅
GFP ---> ∅


Kinetics Equations

As explained above, we do not have much information on the kinetic laws, we have decided to chose a mass action law for each reaction. Each reaction has a kinetic constant k determining the reaction rate given the concentratin of reactants. This will lead us to write a differential system used for a deterministic resolution.


  • A system of equations with an intermediary reaction in the activation of FecR
Reaction Number Kinetics Constants Reaction Reaction Rate
1 kf_1 FecA_OMV ---> FecA_OM kf_1*[FecA_OMV]
2 kf_2 & kr_2 FecA_OM + FecR <---> FecA_OM-FecR kf_2*[FecA_OM]*[Fec_R] - kr_2*[FecA_OM-FecR]
3 kf_3 FecA_OM-FecR ---> FecA_OM + FecR* kf_3*[FecA_OM-FecR]
4 kf_4 FecR* + FecI ---> FecI* + FecR* kf_4*[FecR*]*[FecI]
5 kf_5 & kr_5 FecI* + pfec <---> pfec* kf_5*[FecI*]*[pfec] - kr_5*[pfec*]
6 kf_6 pfec* ---> FecI + FecR_Nterm + GFP kf_6*[pfec*]
7 kf_7 FecR_Nterm + FecI ---> FecI* + FecR_Nterm kf_7*[FecR_Nterm]*[FecI]
8 kf_8 FecA_OM ---> ∅ kf_8*[FecA_OM]
9 kf_9 FecI ---> ∅ kf_9*[FecI]
10 kf_10 FecI* ---> ∅ kf_10*[FecI*]
11 kf_11 FecR_Nterm ---> ∅ kf_11*[FecR_Nterm]
12 kf_12 FecR* ---> ∅ kf_12*[FecR*]
13 kf_13 GFP ---> ∅ kf_13*[GFP]





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