II. Toroidal vesicles

Toroidal vesicles are not excluded by the theoretical model presented above. Of course, due to the cellular division process, which has no trivial solution in the case of a toroidal shape, the study of such shapes is of no biological relevance. However, it led to new insights into the beauty of this simple experimental model.

II.1. First observation of toroidal vesicles.

Toroidal vesicles were first observed in our laboratory, using polymerizable phospholipids [9,10]. At first sight, it was not clear why (unpolymerized) polymerizable phospholipids should be necessary to observe toroidal shapes. As it later turned out, when we finally observed toroidal vesicles (though in a much smaller amount) using standard non polymerizable phospholipids, this was not related to the polymerizable sites of the phospholipids, but simply to the fact that they first auto-assembled as tubules, that is as small isolated amount of phospholipids, which then could turn into vesicles of random topology by swelling into water.

In the case of standard phospholipids, one has to start from a relatively huge amount of deposited molecules, from which vesicles have to extract themselves during the swelling process. The existence of greater mechanical stresses in this case, could disadvantage the formation of complicated shapes like tori. The first observed toroidal vesicles where of two kinds: axisymmetric ones, and non axisymmetric ones (see Figure 1). Among the axisymmetric ones, all of them had a circular cross section and a constant reduced volume vred = 0.71, though not the same volume and area.

Clifford Torus & Dupin Cyclide

Figure 1: A Clifford torus and one of its associated Dupin cyclides. Theoretical shapes are on the left. Two different views of each vesicle are shown, as observed by phase contrast microscopy. The reduced volume is indicated for each shape. Bar indicates 10 µm.

II.2. Clifford torus, Dupin cyclides and conformal invariance.

After the publication of the theoretical works of Duplantier and Seifert, it turned out that these shapes where Clifford tori for the axisymmetric ones, and Dupin cyclides for the non axisymmetric vesicles (see Figure 1) [11,12].

These shapes are known by mathematicians as the surfaces minimizing (1) with toroidal topology. All of them are related by a geometrical transformation called a sphere inversion. A sphere inversion of center O maps every point M of a surface on M' defined by:

sphere inversion (4)

These sphere inversions together with rotations, translations, and dilations form the group of conformal transformations, which actually leaves the curvature energy (1) invariant, a property called conformal invariance of the curvature energy. That is the reason why the Clifford torus and the Dupin cyclides, which are conformal transforms one from the others, have the same, minimal, curvature energy.

II.3. Axisymmetry breaking by temperature change.

The former properties led us to perform a crucial experimental test of the curvature energy model: as in the case of budding, changing the temperature has a direct impact on the reduced volume. The thermal expansivity of the bilayer leads to a non negligible change in the area of the membrane, while the vesicular volume is hardly modified. For instance, starting from an observed Clifford torus at a given temperature, any decrease in temperature yields a reduction of the membrane area, which following (2) turns into a increase of the reduced volume.

At higher reduced volume, the Clifford torus can no more exist, but for each value 0.71 < vred < 1, there exist one shape with the same minimal energy among the Dupin cyclides family: so one should observe a move of the hole off the center, together with its shrinking [13]. Our observations, described in reference [14] fully confirm this prediction. Moreover, this shape change is perfectly reversible, when one increases again the temperature (see Figure 2).

temperature changes

Figure 2: Temperature decrease transforms a Clifford torus into its associated Dupin cyclides. This process is reversible.

II.4. Non minimal tori.

Clifford tori and Dupin cyclides do not represent all theoretically predicted toroidal shapes, as first pointed out by Seifert [12]. A detailed phase diagram of the predictable shapes can be calculated, in terms of the reduced volume and the reduced area difference, a work done by Jülicher, Seifert and Lipowsky [15].

Apart three new families characterized by non circular cross sections for their axisymmetric members, the main result of this study is that non axisymmetric shapes should only exist for sufficiently great reduced volume. These shapes are obtained as local minima of the energy (1).

Our observations led us to the discovery of all the predicted families but one, which however might be very difficult to distinguish from a mere spherical vesicle. We observed both axisymmetric and non axisymmetric vesicles, with some of the shapes not predicted in the work cited above. As some of its predictions relied on extrapolation of infinitesimal Taylor expansions, we decided to check the stability of the observed though unpredicted shapes in the theoretical framework presented above.

Starting from the observed shapes, we first built numerical triangulated approximations of them. Then using a public domain program (Surface Evolver) written by Kenneth Brakke and collaborators of the Center for Geometry, we calculated the nearest stable shape minimizing (1) under the constraints of the starting reduced volume and area difference [14,16]. The resulting shapes looked very much like the starting one (see, for example, Figure 3), thus proving again the validity of the curvature energy model, and the richness of the equilibrium shapes it predicts.

non axisymmetric discoid vesicle

Figure 3: An unpredicted non-axisymmetric toroidal shape, which however turns out to be a solution of the curvature energy model. Bars on the observed phase contrast images indicate 10 µm.

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