Aggregates of Two-Dimensional Vesicles: Rouleaux, Sheets, and Convergent Extension

Using both numerical and variational minimization of the bending and adhesion energy of two-dimensional lipid vesicles, we study their aggregation, and we find that the stable aggregates include an infinite number of vesicles and that they arrange either in a columnar or in a sheetlike structure. We calculate the stability diagram and we show that the sheetlike aggregate can be transformed into the columnar aggregate via vesicle intercalation, which makes the transformation reminiscent of the process of convergent extension observed in some biological tissues.

Figure 1

Representative numerically obtained vesicle and aggregate shapes. Top: Free vesicles of small ($a=0.2$), intermediate ($a=0.5$), and large reduced areas ($a=0.8$). Center: Doublet, columnar triplet, and columnar quadruplet for $a=0.5$ and reduced adhesion strength $\gamma =0.5$. Bottom: The energy of a trefoil exceeds that of a columnar triplet unless $a$ and $\gamma$ are large. The $a=0.5$, $\gamma =0.5$ trefoil is metastable, its $a=0.95$, $\gamma =20$ counterpart is stable. Also shown is the metastable columnar triplet for $a=0.95$ and $\gamma =20$.

Figure 2

Rouleaux of ten $a=0.2$ vesicles for $\gamma =25$ and 40. In both cases, the outer contact lines of the central pair are nearly parallel, showing that the pairs can serve as sigmoud-contact–flat-contact repeat units of an infinite rouleau.

Figure 3

Model $a=0.4$ vesicles in sigmoid-contact rouleau [(a) pear-shaped, (b) $S$-shaped version; $\gamma =1$] and in flat-contact rouleau [(c) $\gamma =19$]. Also shown are examples of model vesicles in sheetlike aggregate introduced in the text [(d) hexagons, (e) rectangles; $\gamma =45$]. Regular tilings are special cases of the sheet geometry, which is generally disordered as argued in the text.

Figure 4

Stability map of adhering 2D vesicles is divided into regions of free vesicles, infinite rouleaux, and infinite sheetlike aggregates. The sigmoid-contact–flat-contact rouleau transition is shown by the dashed line; the large-area limit of rouleau stability is infinitesimally smaller than 1 as at $a=1$ the vesicles can only be free. The largest reduced areas supporting periodic sheet tilings with three, four, and sixfold coordination correspond to the two dotted lines and the right-hand boundary of the sheet region, each labeled by the respective schematic repeat unit. Also indicated are the modes of the sheet-rouleau transition described in the text. For $a<0.27$, free vesicles reduce to two tether-connected buds, and in the parameter space qualitatively outlined by the crossed region they form aggregates more complex than those discussed here.