The later remark explains why we did not try to find an exact model describing our observations of strongly fluctuating vesicles of high topological genus.
These vesicles can be schematically described as two concentric spheres connected by little tubes or necks. The experimental observation is that the position of these necks varies a lot (up to 50 % of the mean distance between tubes) and rapidly (thus excluding the conformal diffusion, which is a slow phenomenon) (see Figure 7) .
Figure 7: Observations of giant fluctuations in two genus 2 vesicles (a) and (b). Arrows indicate necks which vary from one snapshot to the other (a few seconds apart). Bar indicate 10 µm.
Such giant fluctuations are striking in vesicles, where thermal fluctuations ordinarily only slightly perturbate the overall shape. Noting that the shape of the necks was similar to that of a catenoid, we tried to understand the energetics of parallel membranes connected by catenoidal passages. In order to take into account the finiteness of the vesicles, we moreover restricted our theoretical study to a periodical network of pieces of parallel membranes. Thus, a unit cell could be identified with a finite volume and finite area vesicle.
The matching of a catenoidal shape with an almost flat membrane is a very complicated problem, which can only be solved either numerically or with approximations. We explored both ways, in collaboration with Bertrand Fourcade.
The numerical approach consisted in calculating the equilibrium shape of a neck connecting two periodical membranes, and led to a relationship between the relative inner diameter of the neck and the curvature energy. It was used to validate the results of the analytical approach.
The analytical approach consisted in distinguishing between two regions: an inner region in which the shape varied a lot (in order to connect the two separated membranes) and was assumed to be locally a catenoid (thus with a zero curvature energy); and the other outer region, where the shape varied very slowly, thus permitting to neglect small derivatives. It turned out that doing so, the problem was reduced to an electrostatic one: determining the electrical field in a two dimensional network of electrical charges screened by a continuous distributions of opposite charges.
The solution is thus elementary, and one can get the mean curvature as a function of the inner diameter of the neck, requiring the matching of the catenoidal inner solution and the outer solution. The result for the mean curvature in the outer region is:
where r0 is the inner radius of the catenoid. The curvature energy of the whole elementary cell is given approximately by the constant curvature energy of the outer region (the inner one being by hypothesis one of zero curvature, and the matching region being of reduced extension):
This results fits remarkably well with the numerical one, thus validating our analytical approach. Figure 8 shows our numerical solution for a single neck.
Figure 8: Four elementary cells of a network of necks, with r0/L = 0.1.
The analytical approach can easily be extended to more than one neck per elementary cell. In this case, distinction between inner and outer regions around each neck is valid as long as the necks are sufficiently far away one from the other.
The outer solution, which gives the main contribution to the curvature energy is again obtained as the solution to an electrostatical problem. Minimizing the total curvature energy under the constraint of a fixed reduced volume leads to the result that all necks should have the same inner radius. The outer region curvature is then fixed by the sum of the inner radii, not depending on the precise position of the necks. So, as long as the necks are sufficiently far away, they do not interact, at least in our first order approximation.
If one tries to find out what would be the solution, if two necks were to be brought close to each other, one finds that no minimal surface could play the role of a catenoid for two necks. As the property of minimal surface is to have a zero curvature energy, any solution for the two necks would thus increase the curvature energy, leading to the idea of a short range repulsion of the necks.
In conclusion, however approximated, our theoretical description correctly explains the behavior of the observed strong thermal fluctuations of necks connecting two nearly spherical concentric vesicles.
We also observed even more stranger vesicles. Schematically, they look like many concentric vesicles interconnected by many necks. This structure is reminiscent of that of the endoplasmic reticulum or of some membrane structures observed in plant chloroplasts. We explored the preceding approach to understand their stability.
Our model of a periodic stack of connected membranes can be extended to more than two membranes: the electrostatics analogy remains valid, but one has to distinguish between positive and negative charges, depending on the direction of the neck with respect to the reference membrane. However, in this complicated topology, other catenoid-like minimal surfaces exist: that means that two or more necks connecting three or more parallel membranes could be frozen into a bound state of necks, connected smoothly as a catenoid to the outer solution. The Costa surface is an example of such a bound state of four necks locally connecting three membranes (see Figure 9) .
Figure 9: the Costa surface, a minimal surface connecting 3 (asymptotic) membranes.
In the limit when the number of necks as well as the number of membranes grow to infinity, one recovers a phase known as the sponge phase. As such, our distinction between highly fluctuating necks when their concentration is low, and bound states when it reaches a critical density could account for the observed transition between the disordered sponge phase and more ordered ones like the cubic phases.
|page realised by Xavier Michalet||last revised: september 16th,1997|