I. The Physics of Fluid Membranes and the equilibrium shapes of vesicles





Vesicles are small tenuous objects easily prepared by mixing small amounts of surfactant molecules like phospholipids with pure water. Observed with a phase contrast microscope, they most of the time look like slightly fluctuacting deformed spherical bags, with their size ranging from 1 to 100 micrometers.


Their membrane is made of a bilayer of the surfactant molecules. These are asymmetric molecules, with one polar (hydrophilic) head, and one or more hydrophobic tails, the tails gathering together inside the bilayer, protected from the water by a « sandwich » made by the heads. This very thin membrane (around 5 nanometers) is fluid, meaning that all the molecules are free to wander along the membrane, thus allowing shear of the membrane at no cost. Its similarity to cellular membrane freed of its embedded proteins, makes vesicles simple models for the study of some of the physico-chemical properties of cell as well as of their shapes.


The physics of fluid membranes was first developed in order to study emulsions and micro-emulsions, binary or more complicated mixtures which are used in almost all fields of human activity, from cooking, painting, etc, to oil extraction. Vesicles, which form a subclass of the spontaneously organized phases of these mixtures, are also used in cosmetics, in drug delivery, etc, in addition to them being a simple model of the cellular membrane.


It turns out that the first non negligible energetic term which enters the physical description of vesicles, is a curvature term. As stated above, shear stresses lead to no energetical cost, due to the fluidity of the membrane, and vesicles being closed and freely suspended in solution, the residual internal tension has negligible effects, when the deformations of the membrane remain small, which is almost always the case.


The resulting elastic energy, first studied by Canham, Evans and Helfrich, reads as a surface integral on the membrane [1-3]:


Equation 1 (1)



where k is the bending elastic modulus, usually of the order of some tens of kBT for phospholipid bilayers (k ~ 10-12 erg), and H = 1/2 (C1 + C2) is the local mean curvature of the surface S. A term taking into account the gaussian curvature G = C1C2 is usually omitted, being constant for all shapes within the same topological class.


An obvious property of (1) is its scale invariance, which states that two vesicles which differ only by their size, not by their shape, have the same elastic energy. As a consequence, it is not the real value of the volume V and the area A of a vesicle which are important, but there adimensional ratio, called the reduced volume:


Equation 2 (2)



Vesicles are characterized by a remarkable stability: their volume and area remain stable during days. As a result, to study their equilibrium shapes means to minimize (1) given the reduced volume constraint (2).


The first calculation done using this scheme aimed to find a purely physical basis to the red blood cell discoid shape. In fact, the red blood cell shape can be obtained, but only by taking into account the asymmetry of the bilayer membrane: due to the curvature, there are less molecules on the inner side of the bilayer than on its outer side. This number difference being conserved on relatively long time scale, it can be taken into account introducing a new geometrical parameter, the area difference, which again because of the scale invariance of (1), can be replaced by a dimensionless parameter, the reduced area difference:


Equation 3 (3)



Using these two geometrical parameters as physical constraints set up at the time of vesicular equilibration, one obtains a wide variety of vesicle shapes, among which those of the red blood cells [4].


Another interesting shape observed upon a temperature variation is that of a budding daughter vesicle linked to a bigger one by an infinitesimal neck, as seen during exocytosis [5]. This morphological change can be described in the framework described previously by supplementing a simple law for the area dilation with temperature [6,7].


Finally, let us state that this model allows a direct measurement of the bending modulus k from the thermal fluctuations of the vesicular shape [8].

page realised by Xavier Michalet last revised: september 16th,1997