Team:Paris Bettencourt/Modeling/Assisted diffusion/Membrane tension
From 2011.igem.org
![](https://static.igem.org/mediawiki/2011/1/19/Tubeornottube.png)
Membrane tension calculation
Motivation
Let's evaluate the constraints that impact the membrane. As it is a Gram positive bacteria, the external sugar envelope imposes the shape of the bacteria. On the other hand, the osmotic pressure is pushing the membrane against the sugar wall. At every moment, the number of phospholipids inside the membrane is fixed, but this number can fluctuate depending on the phospholipid production. So we can pretend that our system is evolving through a series of quasi-equilibrium states.
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The membrane tension is caused by interaction between phospholipids on the surface of bacteria. Closer the phospholipids are to each other, more the membrane is tense.
Can we describe the interaction between two phospholipids in the membrane? The question is complicated, but we will demonstrate that by describing properly the phospholipid, as it is reasonable to fit a Lennard-Jones potential energy to this interaction. The energy of binding is smaller and the speed of sliding of one phospholipid against another is slower than the internal vibration of the chemical bounds and the internal conformers rotation of the CH2 tails. Though, in Intermolecular Forces by Israelachvili, the author shows that a phospholipid trapped in a bilayer can be aproximated by a section of cone. The section of cones is an individual, and the others are sections of cone joining the previous one. The shape of the sections are giving the shape of the global structure.
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Speaking in terms of energy, once we have these cones, it is reasonable to fit a Lennard-Jones potential to this interaction. If the cones are interpenetrating, a steric repulsion keeps the molecules apart, and the hydrophobic interaction that joins one lipid with the other acts as an attractive force keeping the coherence of the membrane. We will discuss later about the value of the two coefficients.
The membrane tension calculation
Some hypotheses used :
- We deal with a system under fixed temperature, so we can reason in canonical ensemble terminology
- Every bacterium in this model has a perfectly spherical form, so each phospholipid on it has 4 degrees of freedom
- In this model we will consider phosphlipids equally distributed on the membrane and only the initial density of this distribution can change
![](https://static.igem.org/mediawiki/2011/f/fa/Surface_per_phospholipid.png)
![](https://static.igem.org/mediawiki/2011/c/c6/Characteristical_length_between_2_phl.png)
![](https://static.igem.org/mediawiki/2011/2/27/Partition_function.png)
Using statistical physics approach, we can say that the phase space has 4N dimensions: two dimensions of impulsion and two dimensions of coordinate for each particle (N is the number of particles in the system).
where the Hamiltonian is :
![](https://static.igem.org/mediawiki/2011/b/be/Hamiltonian.png)
![](https://static.igem.org/mediawiki/2011/c/cf/Free_energy.png)
![](https://static.igem.org/mediawiki/2011/4/49/Free_energy_total.png)
![](https://static.igem.org/mediawiki/2011/f/f4/Tension.png)
![](https://static.igem.org/mediawiki/2011/f/f2/Tension_1.png)
![](https://static.igem.org/mediawiki/2011/8/87/Tension_2.png)
Assisted diffusion section.