Metabolic model

Overview

This model aims to simulate the entire pathway of the enterobactin production in our Iron Coli. It includes the synthetic sensing system, and the chemical reactions leading to the enterobactin.
This model has two goals:

To the delay needed to produce siderophores starting from the iron-sensing
To determine the ideal plasmid configuration

The enterobactin production regulation is based on two consecutives inhibitions, which, in the end, is an activator with a certain delay. The model will follow this principle.

Assumptions

The only limiting reagant in the pathway is the chorismic acid, which we consider being in excess amount
The FUR is also considered in excess amount
All the enzymes (and the mRNA involved) needed are assimilated to a single variable, since they all play equivalent roles

Model Description

Variables:

[Fe] : Iron concentration inside the bacteria
[Fur] : FUR concentration inside the bacteria
[FeFur] : Iron-FUR complex concentration inside the bacteria
LacI : Number of inhibited LacI
LacO : Number of non-inhibited LacO
[mRNA]: mRNA (from LacO) concentration
[Enz] : Enzyme concentration : EntA,-B,-C,-D,-E,-F

All those concentrations are expressed in mmol/L

Fe, FUR and FeFUR:
The iron-FUR complex is simply formed that way:

We reduced this equation to:

Which is not annoying, since we just have to divide our [FeFur] by to to get the real complex concentration.
We can easily write down both the formation (v) and the dissociation (v') speed:

We chose to model the iron input in the bacteria using a linear function of the external iron concentration Ferext, the factor p being the cell-wall permeability for iron.
The FUR on the other hand is produced by the bacteria. Its evolution can also be considered as linear, using a mean production rate Fur0.

In this model, we only track the free Fe-FUR complex and not those attached to a FUR Binding Site. As LacI is the number of inhibited LacI, we can use this number to express how much Fe-FUR does bind to a FBS per unit of time.

LacI-LacO system:
To simulate the inhibition phenomenon, we chose to use our logistic function under its differential form. Since it is the Fe-FUR that represses it, the LacI can be expressed as a logistic fuction of the Fe-FUR:

Ki1 is a non-dimensional parameter that repesents the inhibition power, and Kf is the fixation rate of the Fe-FUR on the FBS. Finally, Nbrpla1 is the number of pasmids containing the LacI.
In the same way, LacO is modeled with a logistic funtion. LacO is repressed by LacI:

Ki2 is the inhibition power and Nbrpla2 is the number of plasmimds containing LacO.
LacI and LacO are both ruled by a normal logistic function. If we were to track the number of expressed LacI/LacO, we would be using two inverted logistic fuctions to model a double inverter. The thing is, since LacI represents the number of repressed genes and LacO the number of expressed genes, the double inverter is still there, but the calculations are easier.

mRNA and Enzymes:
The [mRNA] and [Enz] equations are alike. The prodction rates are Kr for the mRNA and Kp for the enzymmes, and both variables have a negative degadation term:

Finally:

Parameters

Name	Value	Unite	Description	Reference
p	0.1	min^-1	Permeability of cell wall	-
KfeFUR	0.01	M^-1.s^-1	Formation constant of FeFur complex	-
Dff	0.001	min^-1	-	-
Kp	0.5	min^-1	translation rate	-
KT	2.0	M.min^-1	transcription rate	-
milliNa	6.02.10^20	mol^-1	Avogadro's constant	-
V	6.5.10^-16	m^3	Volume of a call	-
Dmrna	0.001	min^-1	mRNA degradation rate	-
Denz	0.001	min^-1	Enzyme degradation rate	-
Kf	10^-4	min^-1	fixation rate of FeFUR	-
Fur0	0.01	mM.min^⁻1	Fur Production	-
vE1	0.01	mM.min^⁻1	Chorismate production	-

Enzymatic Parameters:

KcatA	5550	min^-1	-	-
KcatB	600	min^-1	-	-
KcatC	173	min^-1	-	-
KmA	300	M	-	-
KmB	14.7	M	-	-
KmC	14	M	-	-

Note that we don't have any informations about the KI1 and KI2 parameters, since they are purely artificial. We thus had to run simulations to study the effects of different sets of values on the model.

Results

High copy?
Here are 9 gaphs pointing out the relevance of different plasmid configurations. The goal of this simulation is to try answering a biological question: "should we increase the number of plLacO plasmids or the LacI plasmids?"
The horizontal axes are the time (minutes) and the iron concentration (logarithmic scale). The vertical axis is the Enzyme production (mmol/L).