Elastic theory of shapes of phospholipid vesicles (1) is a very

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1 Flat and sigmoidally curved contact zones in vesicle vesicle adhesion P. Ziherl* and S. Svetina* *Jožef Stefan Institute, SI-1000 Ljubljana, Slovenia; and Department of Physics and Institute of Biophysics, Faculty of Medicine, University of Ljubljana, SI-1000 Ljubljana, Slovenia Edited by Joseph F. Hoffman, Yale University School of Medicine, New Haven, CT, and approved November 15, 2006 (received for review September 1, 2006) Using the membrane-bending elasticity theory and a simple effective model of adhesion, we study the morphology of lipid vesicle s. In the weak adhesion regime, we find flat-contact axisymmetric s, whereas at large adhesion strengths, the vesicle aggregates are nonaxisymmetric and characterized by a sigmoidally curved, S-shaped contact zone with a single invagination and a complementary evagination on each vesicle. The sigmoid-contact s agree very well with the experimentally observed shapes of erythrocyte aggregates. Our results show that in identical vesicles with large to moderate surface-to-volume ratio, the sigmoid-contact shape is the only bound morphology. We also discuss the role of sigmoid contacts in the formation of multicellular aggregates such as erythrocyte rouleaux. lipid vesicle vesicle sigmoid-contact rouleau Elastic theory of shapes of phospholipid vesicles (1) is a very successful model. Its phase diagram, now explored in considerable detail, comprises a broad spectrum of shapes such as stomatocytes, discocytes, dumbbells, pears, torocytes, starfish, rackets, etc. (2 5). A large majority of the predicted shapes has been observed experimentally (6), and some of them correspond very closely to the different normal and abnormal forms of a mammalian erythrocyte, a simple anucleate eukaryotic cell. If the theory is extended to include the shear elasticity of the membrane skeleton, the agreement between the calculated and the actual shapes is truly striking, even in very deformed erythrocytes such as echinocytes (7). These results give hope that the approach can be extended to describe not only single vesicles but also their aggregates. With some exceptions, theoretical studies of aggregates rely on the simplest model of the intermembrane attraction where the adhesion energy is proportional to the contact area (8 10). In the first analyses of erythrocyte s and rouleaux, the contact zone was assumed to be flat (10 12), but at large adhesion strengths, this hypothesis was found to disagree with experiments (10); so far no explanation of the observed shapes has been available. Here we fill this gap by studying vesicle vesicle adhesion within a fully numerical model free of all symmetry constraints, and we focus on vesicle s as the most elementary aggregates. Our central result is a morphology with a sigmoid shape of the contact zone, which closely reproduces the large-scale features of erythrocyte s (10 16). In vesicles of volume and area of a human erythrocyte, this is the stable shape at large enough adhesion strengths, whereas immediately beyond the aggregation threshold, a flat-contact is found. We show that with increasing surface-to-volume ratio, the range of adhesion strengths corresponding to the flat-contact should diminish and eventually vanish, leaving the sigmoid-contact as the only stable bound morphology. These findings also provide an insight into the mechanics of aggregates of more than two vesicles, which may be relevant for many biological systems and processes involving attraction of flexible membranes, such as cell adhesion (17), cell fusion (18), and tissue formation (19). Theory Doublet Energy. The theoretical description of the phospholipid vesicle adopted here is based on the bending energy of the membrane (20, 21): W b k c (C 2 1 C 2 ) 2 da, [1] where k c is the local bending constant [typically between 0.5 and J in phospholipid vesicles (22) and Jin erythrocyte membrane (23)], C 1 and C 2 are the principal curvatures, and the local bending energy density k c (C 1 C 2 ) 2 /2 is integrated over the surface of the vesicle. Any vesicle morphology is subject to the fixed-area and fixed-volume constraints as the membrane is virtually unstretchable, closed, and impermeable. Another parameter that codefines vesicle shapes is the difference between the areas of the outer and inner monolayers, which is given by A h (C 1 C 2 )da; here, h is the separation of the monolayers neutral surfaces. Minimizing the bending energy at a fixed vesicle area, volume, and A gives the complete set of possible stationary shapes of a free vesicle (2, 3); the stability of each shape also depends on its nonlocal bending energy, corresponding to the relative stretching of lipid monolayers and conventionally written as k r ( A A 0 ) 2 /2h 2 A 0, where k r is the nonlocal bending constant [2k c to 3k c in vesicles (24) as well as in erythrocytes (23)], A 0 is the vesicle area, and A 0 is the relaxed monolayer area difference of a free vesicle (25). The bending energy and the nonlocal bending energy together constitute the area-difference-elasticity (ADE) model (25, 26), and the set of stable shapes predicted by this model is controlled by the ratio of the two bending constants, k r /k c. In this introductory study of nonaxisymmetric vesicle aggregates, we would like to identify the main qualitative features of the possible morphologies. To this end, we focus on two limiting cases of the ADE model and analyze (i) shapes characterized by a given area, volume, and A, which correspond to the bilayercouple model (2), and (ii) shapes with unconstrained A. Within this framework, the energy of a is a sum of the vesicles bending energies, W b,1 and W b,2, and the adhesion energy assumed to be proportional to the contact area A c, W W b,1 W b,2 A c, [2] Author contributions: P.Z. and S.S. performed research and wrote the paper. The authors declare no conflict of interest. This article is a PNAS direct submission. Freely available online through the PNAS open access option. Abbreviation: ADE, area-difference-elasticity. To whom correspondence should be addressed at: Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia. primoz.ziherl@ijs.si. The ADE model can be extended further to allow for the spontaneous curvature of the membrane (21), which does affect the phase diagram, but the set of stationary shapes remains unchanged (6). In the limit of k r/k c 3 0, the ADE model reduces to the so-called spontaneous-curvature theory (1, 3) by The National Academy of Sciences of the USA cgi doi pnas PNAS January 16, 2007 vol. 104 no

2 where is the adhesion strength. This model of the adhesion potential, which typically includes a combination of electrostatic, van der Waals, depletion, and specific intermolecular bridging interactions (27), is clearly rough. Nevertheless, the microscopic details of these interactions can be represented consistently by a measurable (28) effective adhesion strength provided that they do not give rise to membrane inhomogeneities on length scales larger than the local radius of curvature or the size of the contact zone, whichever is smaller. With the same proviso, the above adhesion energy also can be considered to include the effects of membrane fluctuations (29, 30). The elastic energy is conventionally represented in a dimensionless form based on a characteristic length scale R s corresponding to the radius of a sphere such that its area is equal to the area of a single vesicle A 0 : R s A 0 /4. The vesicle and the contact areas then are normalized to da da/a 0 1 and a c A c /A 0, and the vesicle volume is given relative to the volume of the reference sphere, i.e., v V/(4 R s 3 /3). The bending energy of a vesicle normalized by the bending energy of the sphere, 8 k c, reads w b (1/4) (c 1 c 2 ) 2 da, where c i C i R s. The reduced monolayer area difference is a 1 2 (c 1 c 2 )da, [3] and the reduced adhesion strength is defined by R s 2. [4] 2k c To find the equilibrium shapes, we have minimized the total energy of a numerically by using Surface Evolver (31) (the Surface Evolver package is available at facstaff/b/brakke/evolver/evolver.html) adapted to describe adhering vesicles; the estimated numerical accuracy of our results is 1%. Fig. 1. Cutaway view of the representative bound morphologies, which both consist of vesicles of identical shapes. The flat-contact (Upper; v 0.6, a 1.04, and 3) is axisymmetric, the symmetry axis being perpendicular to the contact zone, which is shown in a lighter shade of gray. In the sigmoidcontact (Lower; same v and a but 6), the contact zone of either vesicle is characterized by an invagination and a complementary evagination both diametrically off-center such that the main feature of the mirror-plane cross-section is the S-shaped partition. The cutting plane is the mirror plane of the, and the twofold axis is perpendicular to it. Sigmoid-Contact Doublet. We first have concentrated on vesicles of a reduced volume of v 0.6 and a These values of the parameters correspond to the discocyte shape of a normal human erythrocyte (2, 32). If is small but larger than the threshold for adhesion at a flat substrate (8, 9), the contact zone is planar and circular as if each of the vesicles would stick to a wall (10), the shapes of the two vesicles are the same, and the thus belongs to the C h symmetry class. But if the adhesion strength is increased further, the stable consists of identical vesicles joined at a sigmoidal, S-shaped contact zone with an invagination and a complementary evagination on each vesicle (Fig. 1). This is nonaxisymmetric, its only symmetry elements being a twofold axis and a mirror plane perpendicular to the axis (C 2h ). Qualitatively, the vesicles can be visualized as stomatocyte derivatives with an off-center invagination and an arm on the rim. The external appearance of a sigmoid-contact is quite different from the flat-contact : the vesicles do not sit directly on top of each other but are displaced alongside the contact zone, and the outer surface of the is much less concave than in flat-contact s. As is increased, the lateral displacement of vesicles decreases, and the outer surface becomes convex and more and more curved. Simultaneously, the rim of the contact zone is increasingly more tilted with respect to the top and bottom caps. We also have analyzed some shapes for other values of a, keeping v 0.6 and 10. For a 1.04, where the free vesicles are stomatocytes, s are qualitatively the same as the sigmoid-contact shape described above. However, for a 1.04, where the free vesicles are racket-like, prolate, or budded, depending on a, s consist of irregular starfish with their bodies back-to-back and arms intertwined. The simplest representative of this class is the tennis-ball of two U-shaped dumbbells joined at a saddle-like contact zone. Specifically, we have searched for axisymmetric s formed by two stacked stomatocytes reported in a numerical analysis of vesicles interacting with a square-well potential (33), but we have not found them; it is possible that this shape is stabilized by a finite range of the adhesion potential. It also is conceivable that, in certain cases, the topology of the contact zone could be more complicated, annular, disconnected, etc. So far, we have not succeeded in delineating the parameter ranges where such s would be stable. Phase Diagram. For v 0.6 and a 1.04, the vesicles are free until reaches 0.3, and the discontinuous flat-contact/sigmoidcontact transition takes place at 4.1. A few typical s represented by their mirror-plane cross-sections are shown in Fig. 2 along with the shape sequence of vesicles with a relaxed membrane, i.e., unconstrained a. In the relaxed-membrane model, the free vesicle/flat-contact transition is at 0.4, and the sigmoid-contact s are stable for 3.5. The relaxed s are evidently much more bent, their outer surfaces are more convex, and the sigmoid shape of their contact zones are more pronounced than in unrelaxed s with the same. We also have calculated the reduced monolayer area difference of vesicles in relaxed sigmoid-contact s, a r. Within the range of adhesion strengths discussed here, it depends only weakly on and is typically 0.94, indicating that the preferred shape for the formation of sigmoid contact is a stomatocyte. As implied by Eq. 2, the reduced energy w should be a roughly linearly decreasing function of adhesion strength if the bending energy and the contact area do not vary dramatically with. Our results show that the linear dependence of w( ) is indeed a good approximation, which is substantiated by Fig. 3 where we have plotted the reduced contact area of the bound morphologies, a c A c /A 0, as a function of. Because the sigmoid-contact has a larger contact area than the flat-contact, it must be stable at large adhesion strengths, although its bending energy is larger than in flatcontact s at all. Fig. 3 also shows that the contact area of relaxed s is larger than in those with a The same holds for unrelaxed s with other a values not too cgi doi pnas Ziherl and Svetina

3 a c spherical biconvex v = 0.77 Fig. 2. Cross-sections of free vesicles and s with a 1.04 (Left) and relaxed a (Right)at 0, 3, 6, and 9. In sigmoid-contact s where the cross-section lies in the mirror plane, the dotted line indicates the rim of the contact zone. Besides the relative position of the vesicles, the orientation and shape of the rim are the most obvious external signs of the shape of the contact zone. In sigmoid-contact s, the rim generally is wavy and nonplanar (as in the above relaxed s), but it also may be planar and tilted with respect to the largely flat top and bottom caps (as in the s with a 1.04 shown here). The axes of rotational symmetry of free vesicles and flat-contact s run along the vertical of the cross-section plane. The double, straight, and wavy vertical lines next to the two shape sequences indicate the ranges of stability of free vesicles, flat-contact, and sigmoidcontact s, respectively. close to the relaxed value a r, and the larger the a a r, the larger the difference of the contact areas. Given that ( w/ ) v, a a c (v, a, ), this illustrates an important feature of the energy landscape: the energy of relaxed s decreases more rapidly with than does the energy of unrelaxed s. In other words, the minimum of w versus a, which corresponds to the relaxed shapes, becomes more and more pronounced as the adhesion strength is increased. Limiting Shapes. The difference between the flat-contact and sigmoid-contact s can be elucidated further in terms of limiting shapes they would assume at very large adhesion a c flat-contact sigmoid-contact γ Fig. 3. Reduced contact area versus adhesion strength for s with a 1.04 (dashed line; bound for 0.3) and for relaxed s (solid line; bound for 0.4); v 0.6. The contact area of a sigmoid-contact is larger than its flat-contact counterpart. At the transition, a c jumps by 10% in vesicles with a 1.04 and by 12% in relaxed vesicles. Thus, at large, the sigmoid-contact must be the stable morphology. Note that a c increases rather slowly with, which shows that the adhesion energy of both types of s is an approximately linearly decreasing function of the adhesion strength; in relaxed vesicles, its slope is larger than in unrelaxed vesicles v Fig. 4. Reduced contact area of the limiting s as a function of reduced volume. The cross-sections at v 0.5, 0.77, and 0.9, drawn to scale, illustrate the evolution of the two types of limiting morphologies. The S-shaped partition of the spherical, calculated by minimizing the bending energy of the contact zone as described in the text, is consistent with the nonlimiting sigmoid-contact s. The dashed line indicates the rim of the contact zone. strengths ( 3 ). To maximize the contact area, the outer surface of the limiting s should consist of spherical caps. The two obvious choices are a of plan-convex halves and a perfectly spherical with a nonplanar contact; the latter first was suggested 25 years ago (10), and it only exists for reduced volumes no larger than 4/3 3/ In Fig. 4, the reduced contact areas of the two limiting shapes are shown as a function of v. As expected, the biconvex is inferior to the spherical at all v 0.77, and so in this volume range, the latter is the stable shape at very large adhesion strengths. The difference of the contact areas of the limiting shapes increases steeply with decreasing reduced volume. For v 0.6, it amounts to as little as 0.026, but at v 3 0, it approaches 0.5. Thus, the adhesion strength of the flat-contact/sigmoid-contact transition must decrease very rapidly with v. On the other hand, the phase diagram of a vesicle with zero spontaneous curvature adhering to a flat substrate shows that the free vesicle/flatcontact transition is shifted to ever larger adhesion strengths as volume is decreased beyond v 0.6 (8, 9); a similar behavior is to be expected in the bilayer-couple model. Together, these two facts suggest that, at small enough volumes, the flat-contact s should be replaced completely by the sigmoid-contact s, rendering the latter the only bound morphology of two vesicles at any. The exact value of the minimal reduced volume needed to stabilize the flat-contact s depends on the theoretical framework that best describes a given system [the bilayer-couple, ADE, or spontaneouscurvature model (3)] and the vesicle parameters, but our numerical results indicate that it should typically be 0.5. Discussion Experimental Reports of Sigmoid-Contact Zone. Despite continuing interest (34 37), the experimental insight into the morphology of lipid vesicle aggregates is still far from systematic. In contrast, much more relevant work has been done on erythrocyte erythrocyte adhesion (10, 12 14, 16). Although our theory does not include the shear elasticity provided by cytoskeleton, it does represent a basic model of aggregates of erythrocytes that do not depart much from the discocytic shape because the cytoskeleton does not play a significant role for deformations with length scales larger than k c / 0.28 m (where J/m 2 is the shear modulus of the cytoskeleton) (38). Thus, it can be expected that the s of erythrocytes that are not too deformed can be described by vesicle morphologies discussed here. Ziherl and Svetina PNAS January 16, 2007 vol. 104 no

4 Fig. 5. Sigmoid contact in erythrocytes in dextran. Transmission electron micrograph of a with localized adhesive contacts between membranes shows remarkable agreement with the cross-sections of relaxed vesicle s. [Reproduced with permission from ref. 13 (Copyright 1987, European Biophysical Societies Association).] To establish correspondence with observations, we first look at the adhesion strengths typically encountered in experiments. For erythrocytes in plasma and in dextran, peaks over an order of magnitude between 1 and 10 J/m 2 (12, 39). Given that the erythrocyte surface area is 140 m 2 and that the bending constant of its membrane is J (23), the reduced adhesion strength corresponding to a rather low value of of 1 J/m 2 is 30, which means that these experiments were performed far beyond the flat-contact/sigmoid-contact transition described above. Indeed, the vastly predominant shape reported is of the sigmoid-contact type shown in Fig. 5 (10, 13 16), especially when we take into account that the electron micrographs need not show the most distinctive mirror-plane cross-section. Our numerically predicted sigmoid-contact shape is remarkably similar to the observed large-scale structure even in cases where the membranes are not contiguous along the contact zone because of the instability attributable to the intermembrane water layer (40). A comparison of shapes of unrelaxed and relaxed s with those seen in experiments suggests that, upon adhesion, some relative stretching of monolayers within a membrane does take place. To estimate the importance of this effect, we have calculated the energy of shapes with a close to a r, the monolayer area difference of a vesicle in the relaxed, and we have fitted it with a parabola of the form w r p( a a r ) 2, where w r is the energy of the relaxed ; w r, a r, and p all depend on v and. We find that the minimum corresponding to the relaxed is typically rather sharp, i.e., p 1, and that p increases with adhesion strength. In s with v 0.6, p 60 at 6, whereas at 10, p 90. The large value of p shows that any departure from the relaxed shape is energetically far more costly than the nonlocal bending energy of the two vesicles 2q( a a 0 ) 2 with q k r /k c 3 as in erythrocytes (23). Given that at v 0.6, the monolayer area difference of free vesicles a and a r (almost independent of ) is 0.94, we conclude that at adhesion strengths beyond an estimated 10 the predictions of the ADE model essentially reduce to relaxed s. Note that is well below the typical experimental values of 30, which explains the good agreement of the observed sigmoid-contact erythrocyte s with our relaxed shapes. The sigmoid-contact zone is seen not only in s but also in multicellular aggregates (14), and certain signatures of this contact morphology are reflected on the outer surface of the aggregate. For example, the zig-zag arrangement of cells in weakly bound erythrocyte rouleaux, shown in Fig. 6 (10), implies that the building blocks may be sigmoid-contact s. To see how this could work, note that none of the shapes shown in Fig. 2 appears suited to readily accept a third discocyte such Fig. 6. Shape of cells in a rouleau. The zig-zag arrangement of erythrocytes in a rouleau with concave caps (a) suggests that the building blocks are nonaxisymmetric biconcave cells very similar to those in a sigmoid-contact at small adhesion strengths. A mirror-plane cross-section of such a (b)(v 0.6, a 1.04, 3) shows that it could readily accept a third discocyte as indicated. Upon adhesion, another sigmoid contact is formed at a low extra bending energy cost. The two possible types of triplets (c) obtained by repeating the central part of the : the zig-zag and the staircase stackings should be roughly equivalent. Note that the rims of the contact zones (dashed lines) are tilted with respect to the lengthwise axis of the rouleau just as in a. [The micrograph in a is reproduced with permission from ref. 10 (Copyright 1981, Biophysical Society).] that the two contact zones of the triplet would be as equivalent as possible: the outer surfaces of the flat-contact and the sigmoid-contact are too concave and convex, respectively. However, at small adhesion strengths where they are metastable, sigmoid-contact s consist of biconcave vesicles. For a certain range of, the invaginations in the contact zone and the cap side are very similar to those on a free discocyte, and a third discocyte therefore could easily dock with the (Fig. 6). The little reshaping needed indicates that the energy barrier to reach the triplet should be low. It is possible that the derived triplet could be the stable configuration at the relatively small adhesion strengths 1, which correspond to the typical physiological conditions that can induce erythrocyte aggregation in the blood (10). The process could be repeated, each step lowering the energy by the same amount, which would lead to a more or less linear but nonaxisymmetric rouleau. The concave caps of the observed erythrocyte rouleaux and the zig-zag arrangement of cells (10) support this hypothesis. At large adhesion strengths, however, the caps become convex, and upon adhesion, each vesicle should assume a stomatocytic shape (41). Universality of Sigmoid Contact. Given the experimental evidence of the sigmoid contact and its agreement with the theoretical picture, it is tempting to ask how general this structure is. Our preliminary insight into this question is twofold. Firstly, we have analyzed the shape of the contact zone in limiting s of identical vesicles. Because the rim of the contact zone carries a very large bending energy, it should be as short as possible and thus lie on a great circle. With this constraint, we find that the sigmoid-contact zone with one invagination and one evagination minimizes the bending energy at all reduced volumes down to v 0.3, indicating that, in s of identical vesicles, this type of partition should be stable within a broad range of volumes. Secondly, we also have studied shapes of s of dissimilar vesicles. Presently, we chose to keep their areas and monolayer area differences identical, and we varied the ratio of their volumes at constant total volume, which makes the shape of the sigmoid-contact asymmetric but qualita- At volumes below v 0.3, the contact zone is still sigmoidal but becomes folded and approaches the self-intersecting regime associated with topological changes cgi doi pnas Ziherl and Svetina

5 tively unchanged. In particular, the partition of the is still sigmoidal but the contact-zone evagination of the more inflated vesicle is larger than its invagination; the opposite is true for the less inflated vesicle. The range of volumes where the sigmoid contact is found depends both on volume ratio as well as on average volume. At 9, a 1.04, and v (v 1 v 2 )/2 0.6 (where v 1 and v 2 are the reduced volumes of the vesicles), the asymmetric sigmoid-contact is unstable only when v 1 /v 2 is increased beyond 1.8, whereas for v 0.7, it is unstable when v 1 /v For v 1 /v 2 1, axisymmetric cup-pear s are found at small adhesion strengths instead of the flat-contact s with v 1 /v 2 1. The cup-pear s are characterized by a nonplanar contact zone formed by a single invagination on the less inflated vesicle and a matching evagination on the more inflated vesicle, and the transition between them and the asymmetric sigmoid-contact s takes place at a larger value of the adhesion strength than at v 1 /v 2 1. For example, at a 1.04, v 0.6, and v 1 /v 2 1.7, the asymmetric sigmoid-contact s are stable for larger than 12, a value 3 times as large as in identical vesicles with the same a and v. These results suggest that the sigmoid-contact is reasonably robust and that a moderate variation of the relative vesicle area would not affect its qualitative features. However, at very large area and volume differences, the s should 1. 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Summary In this study, we have demonstrated that, in vesicle aggregates, the contact zone of the adhering flexible membranes may be curved. We have concentrated on the simplest aggregate, the, which can be readily compared with the experimental data, showing that the approach adopted here could be used to address several related problems. In particular, it would be interesting to work out the phase diagram of bound vesicles in detail, extending it over a range of volumes and to multivesicular aggregates. Within this context, the qualitative features of s reported above could serve to construct an analytic parametrization of aggregates and look for equilibrium shapes variationally, which would provide an efficient way of scanning the phase diagram. Another important finding of this analysis is that, in vesicle aggregates, the bending energy and the adhesion energy typically depend on the monolayer area difference much more strongly than the nonlocal bending energy does. 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