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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.

Fig2: If there is an excess of phospholipids, the membrane is being compressed	Fig3: If there are not enough phospholipids, the membrane is being extended

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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Chemical structure of one phospholipid	Schematics of the cylindrical approximation made for the phospholipid

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

Considering one membrane a sphere of the radius R, on which the phospholipid double layer is uniformally distributed, we can find approximately a part of the whole sphere surface per one phospholipid :

The caracteristic distance between two neighbor phospholipids on the membrane can be written like this :

where N is the number of phospholipids on the membrane. From this moment on we will work in canonical ensemble. The temperature is fixed and so is the number of particles in the system N. We also consider that the phospholipids can move only on the sphere surface that means that we will study the evolution of our system in 2N dimentional phase space. The partition function will look like :

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 :

Knowing the partition function is a great deal. We can derive a free energy

of each sphere, total free energy and the membranian tenstion from it :

For two spheres we get :

and