Metabolic model

Overview

Enzymes regulation:
This 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

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

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. It's evolution can also be considered linerar, using a mean production rate Fur0.


In this model, we only track the free Fe-FUR and not those which are 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 FeFUR that represses it, thhe LacI can be expressed as a logistic fuction of the FeFUR:

Ki1 is a non-dimensional parameter that repesents the inhibition power, and Kf is the fixation rate of the FeFUR 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:
d[Fer]/dt = p*Ferext – Kfur[Fur][Fer] + dff [FeFur] d[Fur]/dt = Pfur*Fur0 – Kfur[Fur][Fer] + dff[FeFur] d[FeFur]/dt = Kfur*[Fur][Fer] – dff*[FeFur] – 1/(Na*V)*dLacI/dt dLacI/dt = Ki1/Kf*d[FeFur]/dt*LacI(1-LacI/Nbrpla1) dLacO/dt = Ki2*dLacO/dt*LacO(1-LacO/Nbrpla2) d[mRNA]/dt = Kr*LacO – Dmrna*[mRNA] d[Enz]/dt = Kp*[mRNA] – Denz*[Enz]

Parameters

Results

Conclusion