Apart from having also been observed in our laboratory, vesicles with two holes raised new theoretical interest once the conformal invariance of the curvature energy was fully understood by physicists.
Mathematicians have only recently conjectured the solution of the minimization of (1). That its solution for surfaces of topological genus one (that is, of toroidal topology) is the Clifford torus (and its associated Dupin cyclides), is a result due to Willmore [17]. Lawson later discovered candidates for surfaces of higher topological genus, and numerical experiments performed using Surface Evolver led Kusner and collaborators to the conjecture that the Lawson surfaces were indeed the solution of the minimization problem [18,19].
For instance, in the case of surfaces with two handles, the Lawson surface L (shown in Figure 4) is associated with an infinite set of surfaces obtained by sphere inversions of L. But now, a single parameter like the reduced volume is not enough to describe the whole set of minimal surfaces. It turns out, as pointed out by Jülicher, Seifert and Lipowsky, that even adding the reduced area difference, there is still, in general, an infinity of minimal Lawson-Kusner surfaces: the Lawson-Kusner family is a three parameters one [20].
Figure 4: Some Lawson-Kusner surfaces minimizing the elastic curvature energy for topological genus 2. The Lawson surface L is described in the text. All these surfaces are obtained from one another by conformal transformations, and are characterized by the same elastic curvature energy.
From a physical point of view, this has a striking theoretical consequence: if a vesicle is fully defined by its reduced volume and area difference, there should be no reason for it to adopt one rather than another member of the Lawson-Kusner surfaces with the same reduced volume and area difference. One might thus expect that all the shapes with the same geometrical parameters and same curvature energy should be equally probable, and that thermal excitation alone should be sufficient to slowly drive the vesicle in each minimal state, one after the other.
As we had a good experience of how vesicles of complicated genus looked like, we rapidly observed this phenomenon, called conformal diffusion [21]. A sequence of different views of the same vesicle exhibiting conformal diffusion is shown in Figure 5. The numerical shapes were calculated using the Surface Evolver program, starting from a sphere inverted Lawson surface whose cross section was similar to the observed one. Applying successive infinitesimal conformal transformations preserving reduced volume and area difference as explained in reference [20], the other shapes were obtained, which compare well with the observed evolution.
This observation thus definitively sets the number of relevant parameters for the description of vesicles to two. Indeed, if another constraint had to be taken into account, one Lawson-Kusner surface only would correspond to the triplet characterizing a vesicle, and no conformal diffusion would have been observed.
Figure 5: Snapshots of the same vesicle exhibiting conformal diffusion, separated by around 30 seconds. Bar indicates 10 µm.
Stable shapes with curvature energy greater than the minimal one set by the Lawson-Kusner surfaces also exist: we observed all the predicted types of shapes, among which the most spectacular is probably the button surface shown on Figure 6 [21]. The calculated shape was obtained with Surface Evolver, starting from the button Lawson-Kusner surface and trying different reduced volume and area difference constraints until a good correspondance with the observed shape was obtained.
Figure 6: A button vesicle and its numerical model. The energy is 1.75 times the minimal one of the Lawson-Kusner surfaces.
The observation of vesicles, stables or conformally diffusing ones was also done for vesicles with three holes and more. Needless to say, they are difficult to describe, and no comparison with theoretical calculations was attempted (reference [22]). However, one must point out that topological genus 3 and higher have one more particularity: in these cases, the Lawson-Kusner family is not unique...
page realised by Xavier Michalet | last revised: september 16^{th},1997 |