Team:Paris/Production modeling

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Modeling

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A. Genetic Network Regulation

The first thing that we wanted to study in modeling was the efficiency of the construction chosen to create a delay. Our first approach was a deterministic analysis of the system using differential equations. The regulation of promoters was described using the Hill functions and the methods described by U. Alon in his book An Introduction to Systems Biology.

When the system is activated, there is Arabinose in the medium and the pBad promoters are activated. And the system can be described by this system of differential equations :

Differential System

As a first approximation, we assumed that :

• When the pBad promoter is induced, the concentration of arabinose in the medium is very high and constant during he whole study ; as a consequence, we will consider that the creation rate of Protein and LacI* is constant during the experiment :

• We considered that all the binding constants are identical and of an average of 40nM which correspond to approximately 40 monomers per cell ; we can write that :

• We chose identical intrisinc promoter activities, β , all equal to 4000 proteins/cell cycle :

• All the time units were expressed in units of cell cycle (approximately half an hour) ; as a consequence we chose a dilution rate γ of 1 for protein without special tags. For the Laci protein with a LVA tag, the dilution rate is multiplied by 3:

B. Modeling Vesicles creation.

Our model is based on three different physical phenomenon:

• The lipid surface conformation, Tol-Pal proteins diffusion and the increase of the osmotic pressure in the periplasm

• The Lipid conformation of the outer membrane is a well known problem: at 35°c the lipid bilayer behaves like a liquid which conformation character is ruled by an energy called the bending energy. This energy represent the fact that the lipid bilayer will search a special conformation linked to the shape and the chemical properties of its constituents.

The conformation of the lippopolysaccahrides and the normal phospholipids can be seen has a lipid bilayer which conformed itself as a liquid crystal at the growth culture. The conformation of lipid bilayer has been well studied and a lot of theorical results have been given. Some of them are of great interest for our project. This conformation is something which can enable us to understand the way vesicle can create themselves and how they will be receive. In our case we first notice that from a basic calculus we can obtain some interesting results on the way outer membrane will create themselves.

We first use the bending energy has a rough shape for our model and its understanding

This formula give use the abilities to explain the way lipids will assembles themselves together. E is the energy of a whole lipid bilayer (or monolayer).K_b and are Bending and Gaussian modulii which can be obtain by experimentation is the intrinsic curvature of the outer membrane describe the local form of a lipid bilayer when itself is at the lowest state of energy the more stable. Our first work was to calculate the energy of two different shapes of membranes.

First the E.coli Shape before the division and then the vesicles shape.

We approximate E.coli shape has a cylinder of rayon r =0.3 μm and infinite length because we wanted to approximate this energy on the division region of E.coli before septation.With this approximation is equal to 2/r and γ=0 in this case

For the vesicles we approximate their basic shape to a sphere of rayon r’ their bending energy by lipid area units:

Thous as the area of a sphere is known and is independent of the location on the sphere we can write: .

In the same way we can write for an area of E.coli lipids that the bending energy is:

Those considerations can lead us toward a basic vision of the statistical repartition of vesicles length in case of absence of integrity control system in the outer membrane.

The first energy is the potential energy of the lipid area in E.coli outer membrane necessary to the construction of a vesicle and the second one the potential energy of the same lipid aerea but in the conformation of a vesicle shape.

So the energy which must be given to the whole system to create a vesicle is:

We can suppose that the most easily created vesicles will be the ones which will present a minimum . By derivation we find that the minimum his obtain for:

Hence as we know that the range of created vesicles radii is 25 nm to 175nm we can suppose that the r’ is somehow about 100 nm and so

• Tol and Pal are membrane proteins which are located respectively in the outer and the inner membrane. The diffusion of protein in those lipid bilayers can be represented by a probabilistic Brownian movement. This diffusion model can give us the law of probability for the location of Tol and Pal on the membranes. It has been observed that the Tol and Pal proteins interact and this interaction is linked to the membrane stability: indeed the Tol and Pal will linked inner and outer membrane and furthermore stabilized the outer membrane using the peptidoglycan rigidity.

• The osmotic pressure in the periplasm is the same that the medium pressure in normal time. But during the division period of the bacteria the peptidoglycan is degraded to be recycled in a new cell wall. During this phenomena of turn-over a part of the peptidoglycan is released in the periplasm which increased the osmotic pressure in the periplasm.

• We can thous consider that if there is not enough Tol-Pal linked proteins the outer membrane will deform itself to create a beginning of vesicle. But in this part of the membrane the tol pal proteins will not have the possibility to bind themselves and they will be free to diffuse in other parts of the membranes. The surface shape will guide the proteins to the border of the vesicle and stabilized the shape of the vesicle.