THE ANATOMICAL RECORD 268:215–225 (2002) Shape Behavior of Lipid Vesicles as the Basis of Some Cellular Processes SAŠA SVETINA* AND BOŠTJAN ŽEKŠ Institute of Biophysics, Faculty of Medicine, University of Ljubljana, and J. Stefan Institute, Ljubljana, Slovenia ABSTRACT The basic principles that govern the shape behavior of phospholipid vesicle shapes are discussed. The important membrane parameters of the system are defined by presenting the expressions for the relevant contributions to the system’s mechanical energy. In the description of the rather unique shape behavior of lipid vesicles, the emphasis is on providing a qualitative understanding of the dependence of vesicle shape on the parameters of the system. The vesicle shape behavior is then related to biologically important phenomena. Some examples are given of how the results of the shape behavior of lipid vesicles can be applied to the analysis of cellular systems. Red blood cell shape and shape transformations, vesicle fission and fusion processes, and the phenomenon of cellular polarity are considered. It is reasoned that the current biological processes that involve changes of membrane conformation may have their origin in the general shape behavior of closed lamellar membranes. Anat Rec 268:215–225, 2002. © 2002 Wiley-Liss, Inc. Key words: lipid vesicles; shape behavior; membrane elasticity; bilayer couple; cellular processes Lipid vesicles are macroscopic objects defined by a lipid membrane enclosing an aqueous medium and separating it from the external aqueous medium. The division of space into internal and external compartments is also the basic characteristic of living cells. Lipid vesicles are thus a simple model system with which to study at least some of the cellular properties related to the phenomenon of compartmentalization. In some water-lipid mixtures, lipid vesicles form spontaneously (Lasic, 1993), and thus they have been suggested to be one of the important basic structures involved in biogenesis (Deamer, 1997; Luisi et al., 1999). Lipid vesicles possess a variety of properties that may have been employed in the evolution of present biological systems. The topic of this work is the shape behavior of lipid vesicles. Studies of lipid vesicle shapes have constituted a considerable part of experimental and theoretical vesicle research (Lasic, 1993; Lipowsky and Sackmann, 1995; Petrov, 1999; Luisi and Walde, 2000) since the discovery of these structures in the mid 1960s (Bangham and Horne, 1964). It was recognized early on that lipid vesicle shape behavior reflects to a large extent the bilayer nature of lipid membranes (Evans, 1974; Helfrich, 1974). Further analyses of vesicle shapes revealed some general features of lipid vesicle shape behavior that depend on the layered membrane structure, and not on the structural and com© 2002 WILEY-LISS, INC. positional details of the membrane monolayers (reviewed in Svetina and Žekš, 1996; Seifert, 1997). This notion also has important implications for biogenesis, because one can ascribe the general features of vesicular systems to prebiotic vesicles even without knowing from what type of lipids they were made. Our main aim here is to describe the shape behavior of vesicular structures involving layered membranes, and in so doing to elucidate in particular its general features. In addition, we indicate how the vesicle shape behavior could have served as an elementary basis for some cellular processes involving changes of membrane conformation. The present work consists of two parts. In the first part we discuss the basic principles that govern phospholipid vesicle shape behavior. To define the important parame- *Correspondence to: Saša Svetina, Institute of Biophysics, Faculty of Medicine, University of Ljubljana, Lipičeva 2, 1000 Ljubljana, Slovenia. Fax: ⫹386-1-4315127. E-mail: sasa.svetina@biofiz.mf.uni-lj.si Received 13 February 2002; Accepted 4 April 2002 DOI 10.1002/ar.10156 Published online 00 Month 2002 in Wiley InterScience (www.interscience.wiley.com). 216 SVETINA AND ŽEKŠ ters of the system, we first present expressions for the relevant contributions to the system’s mechanical energy. We then describe the rather unique shape behavior of lipid vesicles. The emphasis will be on providing a qualitative understanding of the dependence of shape on the parameters of the system. Therefore, we avoid any description of the formalisms that are used in theoretical determinations of vesicle shape (Svetina and Žekš, 1996; Seifert, 1997). In the second part we discuss some biologically important vesicle phenomena to which lipid vesicle shape behavior can be related. Special attention is given to the relationship between vesicle shape transformations and vesicle fission and fusion processes, and to the phenomenon of cellular polarity. We also discuss the functional significance of the shape of the red blood cell. We conclude by suggesting that some biological processes may have their origin in the general shape behavior of closed lamellar membranes. MECHANICAL BASIS OF LIPID VESICLE SHAPE FORMATION Lipid vesicles form when lipid molecules, because of their amphiphilic nature and geometry, associate in an aqueous environment to form membranes. Typical of these lipid membranes are phospholipid membranes. In these membranes an adequate contact of phospholipid molecules with water is established by arrangement of their polar heads at the membrane surface, and by their hydrophobic tails oriented in the direction of the membrane interior. The thermodynamically stable bilayer membrane is obtained by the hydrophobic side of one such monolayer being covered by the hydrophobic side of another, oppositely-oriented monolayer. A piece of a bilayer membrane would have the hydrophobic parts of the molecules at its edges still in contact with the water. However, because the membrane of a vesicle forms a closed surface, there are no edges; consequently, vesicles are more stable than membrane pieces. In an unilamellar phospholipid vesicle, a single bilayer membrane separates the external and internal water solutions (Fig. 1). There are different prescribed conditions for the spontaneous formation of phospholipid vesicles from a mixing of water and phospholipids (Lasic, 1993). The resulting vesicles may thus have different sizes: ⬃10 nm for small phospholipid vesicles (SPVs), ⬃0.1 m for large phospholipid vesicles (LPVs), and ⬃10 m for giant phospholipid vesicles (GPVs). The size influences vesicle behavior by phenomena that depend, for example, on both vesicle volume and membrane area, as is the case with the characteristic time for transmembrane diffusion transport, or on the ratio between the membrane thickness (⬃5 nm) and vesicle diameter. Among vesicles of different sizes, GPVs deserve special attention because their dimensions are comparable to the dimensions of cells. As such, they can also be visualized by an optical microscope. For a given area of the vesicle membrane (A), the vesicle volume (V), being practically equal to the volume of the internal solution, can have any value between nothing and the volume of a sphere with radius Rs ⫽ (A/4)1/2. Vesicle volume may be the result of the process of vesicle formation and the processes occurring during its subsequent history. It can also be monitored by the osmotic state of the inside and outside solutions. For any vesicle volume smaller than the volume of the sphere, the vesicle is flaccid and can assume an infinite number of shapes. How- Fig. 1. A schematic representation of a phospholipid vesicle. a: The cross-section of a spherical vesicle. b: The axial cross-section of a vesicle with an axisymmetric shape exhibiting a protuberance and resembling a pear. Rs is the radius of the sphere and R m is the meridianal principal radius. The two examples of Rm indicate that the membrane principal radii are defined to be positive at the convex parts of the membrane and negative at its concave parts. In both vesicles the structural features of phospholipid membranes are shown schematically for the indicated membrane section. Phospholipid molecules are shown as composed of heads (circles) and two tails. Dashed lines represent neutral surfaces of the membrane monolayers, with their positions defined through the requirement of independent lateral expansion and bending deformational modes. The distance between the neutral surfaces is denoted by h. The arrows in the section of vesicle b indicate the relative shifts of the positions of phospholipid molecules in the two monolayers when the protuberance forms. ever, experimental determination of shapes indicates that they are limited to certain distinct shape types. In Figure 2 are the cross-sections of some characteristic GPV shapes that have been obtained from optical microscopy. Two characteristic oblate shapes are the disc shapes (shape 4) and cup shapes (shapes 1–3), and two characteristic prolate shapes are the cigar shapes (shape 5) and pear shapes (shapes 6 – 8). Shapes 9 –12 are characteristic of shapes with lower volumes, and shapes 13–16 are those with narrow necks. It can be seen that phospholipid vesicle shapes exhibit some symmetry characteristics, which indicates that their formation obeys certain rules. It can also be deduced from Figure 2 that different shapes can exist at the same vesicle volume. This implies that there are systemic properties other than the vesicle volume that influence vesicle shape. Elastic Properties of a Membrane Described as an Elastic Sheet The outside and inside vesicle solutions are liquids; therefore, the formation of vesicle shapes can be, in the absence of external forces, related only to the mechanical properties of their membranes (Evans and Skalak, 1980). Because of their relatively small thickness, phospholipid membranes as a mechanical system resemble a thin elas- 217 SHAPE BEHAVIOR OF LIPID VESICLES Fig. 2. A series of vesicle shapes as observed by phase contrast microscopy. This microscopy senses the parts of vesicles in which the path of the optical beam through the membrane is the longest; therefore, the equatorial contours of vesicles are seen representing the equatorial cross-sections of vesicles. In the first row are three shapes belonging to the cup-shape class (1–3) and a shape belonging to the disc-shape class (4). In the second row are shapes belonging to the cigar-shape (5) and pear-shape (6 – 8) classes. In the third row are some examples of shapes with a relatively small vesicle volume/membrane area ratio. Shape 9 is termed a codocyte, shape 10 is a torocyte, shape 11 is a starfish, and tic sheet. Thin sheets can be treated elastically in terms of independent elastic deformational modes, i.e., their inplane elasticity and bending. The in-plane elasticity of phospholipid membranes is specific, in that phospholipid molecules can exchange their lateral positions and can therefore move freely within the plane of the membrane. Consequently, phospholipid membranes behave as twodimensional liquids. As such they do not exhibit in-plane shear and are laterally isotropic. However, membranes exhibit compressibility properties. When a membrane is laterally compressed or expanded, its elastic energy increases in a way that can be approximated by the area expansion energy term shape 12 is a worm shape. The fourth row shows shapes characterized by narrow necks connecting nearly spherical vesicle parts. Shape 13 has two invaginated spheres within a large sphere. Shape 14 is composed of a large sphere and two small evaginated spheres. Shape 15 has a small sphere in between two large spheres, whereas shape 16 has (in addition to a large mother sphere) five small spheres arranged in a row and a single small sphere connected to it at another position. Data are from: shapes 1– 4, 13, and 14 (Käs and Sackmann, 1992), 5– 8 (Käs et al., 1993), 9 and 16 (Svetina et al., 2001), 10 –12 (J. Majhenc, unpublished data), and 14 (Farge and Devaux, 1992). WA ⫽ K (A ⫺ A0)2 2A0 [1] where K is the area expansivity modulus (reciprocal of the compressibility modulus), and A0 the equilibrium area of the membrane. The area expansion energy term (Eq. [1]) is particularly important when a vesicle is in a swollen state, i.e., its volume is larger than the volume of the sphere with an area of the unextended membrane A0. In the opposite case, when vesicle volume is smaller than this volume, the membrane does not respond correspondingly by being 218 SVETINA AND ŽEKŠ compressed, but rather assumes a nonspherical shape. This happens because the energy cost due to membrane bending is in general much smaller than the energy cost needed for membrane lateral compression. When a vesicle is not spherical, its membrane is curved differently at different locations on the vesicle surface. The two principal curvatures (C1 ⫽ 1/R1 and C2 ⫽ 1/R2, with R1 and R2 being the principal radii) thus differ and vary over the vesicle surface (Fig. 1b). The vesicle bending energy, which can be expressed in terms of principal curvatures, is therefore obtained by integrating the local bending contributions over the whole membrane area. For a thin sheet with isotropic elastic properties, this integration is the sum of the local bending term (Wb) and the Gaussian bending term (WG) (Helfrich, 1973): 冕 1 Wb ⫹ G ⫽ Wb ⫹ WG ⫽ kc (C1 ⫹ C2 ⫺ C0)2dA 2 冕 ⫹ kG C1C2dA [2] where kc is the local bending modulus, kG the Gaussian bending modulus, and C0 the spontaneous curvature. The nonzero spontaneous curvature C0 reflects the possible intrinsic property of the membrane that would cause its unsupported piece to assume mechanical equilibrium at a curved membrane conformation. Spontaneous Curvature Model of Vesicle Shapes It has been proposed that the shapes of vesicular structures, such as phospholipid vesicles, correspond to the smallest possible value of the membrane bending energy (Canham, 1970). Such shapes can be predicted by a mathematical procedure (Deuling and Helfrich, 1976) in which the shape with the minimum energy is found essentially by scanning theoretically over all possible shapes. In the minimization procedure, the only role is played by the first term in Eq. [2], Wb, because for a vesicle of given topology the Gaussian contribution to the bending energy has a constant value. However, it has to be kept in mind that this value can be finite (WG ⫽ 4kG for the spherical topology) and therefore it must be taken into consideration in processes in which the number of vesicles is changing, as, for example, in vesicle fission and fusion processes. In the flaccid vesicle the membrane area is scarcely expanded. It is therefore possible for flaccid vesicles to assume that A ⬃ A0, and to obtain their shapes by minimizing the membrane local bending energy Wb under the constraint of constant membrane area. It can also be recognized that the minimum energy shape does not depend on the value of the bending constant kc which is just a constant factor in the varied local bending energy term. The shapes are also usually determined under the constraint of a fixed vesicle volume. Therefore, in their determination of a catalogue of vesicle shapes by minimizing Eq. [2], Deuling and Helfrich (1976) could express their results in terms of the reduced volume v ⫽ 3V/4Rs3, with Rs the radius of the sphere now corresponding to the area of the unextended membrane (A0/4)1/2, and the reduced spontaneous curvature c0 ⫽ C0Rs. The shapes of vesicles Fig. 3. Membrane local bending energy in units of the bending energy of the sphere (wb ⫽ Wb/8kc) as a function of the reduced vesicle volume for minimum energy shapes in the spontaneous curvature model (adapted from Svetina and Žekš, 1989). The value of the spontaneous curvature c0 is taken to be zero. The three curves represent cigar-, disc-, and cup-shape classes. Typical examples of the corresponding shapes are presented. in this, the so-called “spontaneous curvature” model, thus depend only on the values of v and c0. In Figure 3 the bending energies are shown, expressed in terms of the bending energy of the sphere, wb ⫽ Wb/8kc, together with some calculated shape cross-sections of vesicles, with the smallest possible bending energy as a function of the reduced volume v for the value of the reduced spontaneous curvature c0 ⫽ 0. Nonlocal Bending Energy In a more complete description of shapes of phospholipid vesicles, the fact that phospholipid membranes are composed of two monolayers has to be taken into consideration (Fig. 1). The two monolayers of a phospholipid bilayer can, in the first approximation, be considered as compositionally independent, because the transbilayer movement of phospholipids is slow, with typical half-times for phospholipid equilibrium exchange being on the order of hours or days (Wimley and Thompson, 1991). As already stated, because of the hydrophobic effect, the two monolayers are in a contact. Thus their positions are geometrically related in that they are aligned in a parallel manner. By assuming that the distance between the neutral surfaces of the two monolayers is the same all over the membrane surface, the area of the neutral surface of the outer layer (A2) is larger than the area of the neutral surface of the inner layer (A1) by the integral of the sum of the membrane principal curvatures over the whole surface, multiplied by the distance between the two neutral surfaces (h). The difference between the areas of the two monolayers is thus 冕 ⌬A ⫽ A2 ⫺ A1 ⫽ h (C1 ⫹ C2)dA. [3] Integration is over the membrane area (A ⬃ A1 ⬃ A2) of the vesicle. 219 SHAPE BEHAVIOR OF LIPID VESICLES For a proper description of the mechanical behavior of phospholipid membranes with closed surfaces, it is important to note that the two monolayers are free to relax their strains in their lateral direction. This is because the tails of phospholipid molecules of one monolayer do not restrain the tails of the molecules of the other monolayer when they need to be shifted into more favorable positions. Thus the monolayers can both move laterally past each other, and relax their elastic strains independently (Fig. 1b). The elastic expansion and bending energies can therefore be described separately for each leaflet by the energy terms given by Eqs. [1] and [2], respectively. The parameters K, A0, kc, and C0 must be renamed as Ki, Ai,0, kc,i, and C0,i, with i ⫽ 1 and 2 for the inner and the outer monolayers, respectively. The elastic energy of a bilayer is thus essentially the sum of the elastic energies of the two monolayers. Because the two monolayers can only be bent concomitantly, their bending modes are coupled, and therefore the number of independent deformational modes of the bilayer is three and not four. In defining independent deformational modes of a bilayer, it is convenient to retain the two deformational modes (as defined above) for a single layered membrane, i.e., lateral expansion and bending deformational modes. It can be shown (Svetina et al., 1985) that in the corresponding energy terms we have K ⫽ K1 ⫹ K2, kc ⫽ kc,1 ⫹ kc,2 and C0 ⫽ C0,1 ⫹ C0,2. The third independent deformational mode can then be derived by taking into consideration Eq. [3]. It is conveniently (Svetina and Žekš, 1996; Seifert, 1997): written as Wr ⫽ kr (⌬A ⫺ ⌬A0)2 2A0h2 [4] where kr is the nonlocal bending constant, and ⌬A0 is the difference between the two monolayer equilibrium areas A2,0 and A1,0. Equation [4] represents the nonlocal contribution to the membrane bending term because its variable, the area difference ⌬A, is proportional to the integral of the sum of the membrane principal curvatures over the whole membrane area (Eq. [3]). In its equilibrium state, each phospholipid molecule occupies a given equilibrium area, ␣i,0. The equilibrium area of a monolayer with a single phospholipid species is therefore Ai,0 ⫽ Ni ␣i,0, where Ni is the number of phospholipid molecules. In a more general sense, the equilibrium area difference ⌬A0 is given by the numbers of different molecules occupying the two leaflets, and by their equilibrium areas. In comprehending the significance of the nonlocal bending energy term (Eq. [4]) in establishing phospholipid vesicle shapes, it is important to realize that the area difference ⌬A depends on the shape, whereas the equilibrium area difference ⌬A0 depends on the composition of the two membrane monolayers and their interaction with the surroundings. In general, the values of these two quantities are not equal, and the two monolayers are stretched relative to each other. The importance of the nonlocal bending term for the shape behavior of phospholipid vesicles has been proven in tether-pulling experiments (Bo and Waugh, 1989; Waugh et al., 1992). In those experiments, which were designed to measure membrane elastic parameters, a thin cylindrical tether was pulled out of an aspirated vesicle. Depending on the pulling force and the aspiration pressure used, it could be stabilized at a certain length. It was shown (Božič et al., 1992) that such a system cannot be stable without including the nonlocal bending term involving a value of the nonlocal bending constant kr larger than a certain critical value. Bilayer Couple Models of Vesicle Shapes A realistic model for determining phospholipid vesicle shapes, based on the sum of the local and nonlocal bending energies (Eqs. [2] and [4]) is termed the “generalized bilayer couple” model or the “area difference elasticity” model. For a concise analysis of vesicle shapes predicted by this model, the sum W ⫽ Wb ⫹ Wr has to be rewritten in terms of the reduced bending energy (relative to the bending energy of the sphere for zero spontaneous curvature, w ⫽ W/8kc) and the reduced curvatures, w ⫽ w b ⫹ wr ⫽ 冕 1 kr (c1 ⫹ c2 ⫺ c0)2da ⫹ (⌬a ⫺ ⌬a0)2 4 kc [5] where da ⫽ dA/4Rs2, c1 ⫽ C1Rs, c2 ⫽ C2Rs, and the area differences are reduced relative to their values for a sphere (8hRs), i.e., ⌬a ⫽ ⌬A/8hRs and ⌬a0 ⫽ ⌬A0/ 8hRs. Eq. [3] may also be written in terms of the reduced quantities, ⌬a ⫽ 冕 1 (c1 ⫹ c2)da. 2 [6] Equation [5] can be minimized conveniently by first obtaining for all possible values of ⌬a the bending energy for zero spontaneous curvature (c0 ⫽ 0). Then, by inserting the resulting dependence wb(⌬a) into Eq. [5], minimization with respect to ⌬a is performed, giving (by taking into consideration Eq. [6]) the equation dwb(⌬a) kr ⫹ 2 (⌬a ⫺ ⌬a0) ⫺ c0 ⫽ 0. d⌬a kc [7] The parameters on which the shapes depend are thus v, c0, ⌬a0, and kr/kc. There are actually only three independent parameters, because the shapes obtained are the same for any combination of the parameters c0, ⌬a0, and kr/kc that gives the same value of the constant term in Eq. [7], the expression 2(kr/kc)⌬a0 ⫺ c0. To describe the shape behavior of phospholipid vesicles in terms of the generalized bilayer couple model, it is convenient to treat first the limiting case of the infinitely large ratio kr/kc. This limit is called the “strict bilayer couple” model and was originally introduced (Svetina et al., 1982; Svetina and Žekš, 1989) by the requirement of incompressible membrane monolayers, expressed by the consequent relationship ⌬a ⫽ ⌬a0. In the strict bilayer couple model, the shapes depend only on the reduced volume, v, and on the reduced difference between the areas of the two membrane monolayers, ⌬a. For each value of these two parameters the shape with the smallest value of the local bending energy (Eq. [2]) can be predicted. The results showed that in different regions of the v and ⌬a values, the symmetries of the shapes obtained 220 SVETINA AND ŽEKŠ Fig. 4. The v-⌬a phase diagram of vesicle shapes. The regions are shown where in the strict bilayer couple model, the shapes with the lowest local membrane bending energy belong to cup-, pear-, nonaxisymmetrical-, cigar-, and pear-shape classes (Svetina and Žekš, 1989, 1990; Heinrich et al., 1993). One set of class boundaries are the lines that give for the limiting shapes the dependence of their reduced volume (v) on their reduced area difference (⌬a). They are drawn by full lines and are given for (A–F) some indicated limiting shapes. The limiting shapes shown are compositions of spheres connected by infinitesimally narrow necks. Limiting shapes at ⌬a ⬍ 1 (A and B) have invaginated spheres. Other sets of class boundaries are the symmetry-breaking lines defined by the v and ⌬a values where the shapes with an equatorial mirror symmetry (disc and cigar shapes) become unstable. They are drawn by dashed lines. Points 1–16 represent the positions in the v-⌬a phase diagram of the shapes presented in Fig. 2. The positions of shapes 1–9 are obtained by comparison of the observed shapes with the corresponding calculated shapes. Positions of other vesicles are estimated. are different. The shape classes can be defined as the domains within the v-⌬a phase diagram where shapes of the same symmetry are obtained by continuous shape transformations caused by continuously varying parameters v and ⌬a (Svetina and Žekš, 1989; Seifert et al., 1991). In Figure 4 some regions in the v-⌬a phase diagram are presented in which, in the strict bilayer couple model, the shapes of some shape classes have the lowest values of the local bending energy. To date, the shape classes have been well characterized primarily for the v and ⌬a values in regions that are not too far from the point representing the sphere (⌬a ⫽ 1, v ⫽ 1). For smaller reduced volumes v, only some types of shapes have been characterized theoretically (Wintz et al., 1996). Some, but not all, shape classes are comprised of axisymmetric shapes, including those shapes that have also equatorial mirror symmetry (e.g., disc and cigar shapes), and those without such symmetry (e.g., pear and cup shapes). In the intermediate region between the oblate (lower ⌬a) and prolate (higher ⌬a) shapes, there is the region of nonaxisymmetric shapes (Heinrich et al., 1993). In order to provide more detailed insight into the characteristics of vesicle shape behavior, we present the strict bilayer couple predictions for the shape behavior of the cigar- and pear-shape classes. In Figure 5 the bending energies of these two classes are given as a function of the area difference ⌬a for two values of the reduced volume v (0.85 and 0.95). The axial cross-sections of the shapes of the corresponding shape series are also presented in this figure. For both considered reduced volumes the shape with the absolute minimum energy belongs to the more symmetric cigar class. However, at continuously increasing ⌬a, a point is reached where there is a continuous transition to the pear shape, i.e., the shape with the lower symmetry, because it has no equatorial mirror symmetry. Another significant feature of the system is that limiting shapes at higher ⌬a boundaries of the pear-shape class are composed of a large and a small (evaginated) sphere connected by an infinitesimally small neck (Fig. 5, shapes 6 and 12). We now show that for the finite values of the ratio kr/kc, i.e., within the generalized bilayer couple model (Heinrich et al., 1993; Miao et al., 1994), some of the stable shapes of the strict bilayer couple model become unstable. For this purpose we have to solve Eq. [7]. It is convenient and instructive to do this graphically. In the same graph (Fig. 6) we plot the derivative of the local bending energy obtained numerically from the results for the wb(⌬a) dependence as presented in Figure 4 (in Fig. 6 the results are shown only for v ⫽ 0.85), and the line with the slope – 2kr/kc, which intersects the abscissa at a point defined by the chosen values of ⌬a0 and c0. The solutions of Eq. [7] are the points at which the derivative dwb(⌬a)/d⌬a and the line intersect. In Figure 6 this procedure is performed separately for three values of the ratio kr/kc. For kr/kc ⫽ 20 (Fig. 5c) there is a single intersection of the two curves and thus a single solution of Eq. [7] at all relevant values of ⌬a0 and c0. At kr/kc ⫽ 6 (Fig. 5b) there are, within a certain interval of the values of ⌬a0 and c0, three solutions of Eq. [7]. The intersections show that one of these solu- SHAPE BEHAVIOR OF LIPID VESICLES 221 Fig. 6. A graphical method for solving Eq. [7]. For the relative volume v ⫽ 0.85, the ⌬a dependence of the derivative of the local bending energy by the area difference, dwb(⌬a)/d⌬a (full lines) is plotted together with the line representing the negative value of the sum of the second and third terms of Eq. [7] (dashed line). This is done separately for three values of the ratio between the nonlocal and local bending constants: (a) kr/kc ⫽ 0, (b) 6, and (c) 20. The steep, straight part of the derivative dwb(⌬a)/d⌬a curve belongs to the cigar-shape class, and the other part of the curve is of the pear-shape class. The solutions of Eq. [7] are at the ⌬a values corresponding to the intersections between the full and dashed lines. Fig. 5. Membrane local bending energy wb, in units of the bending energy of the sphere (wb ⫽ Wb/8kc), is presented as a function of the relative difference between the areas of the membrane monolayers (⌬a) for two indicated values of the relative vesicle volume (v) (adapted from Svetina and Žekš, 1989, 1996). The results comprise cigar- and pearshape classes, as indicated. The cigar shapes are stable only at ⌬a values in the vicinity of the ⌬a value of the absolute minimum energy shape. The axial cross-sections of some characteristic shapes are given for the indicated ⌬a values. tions belongs to the class of cigar shapes, whereas the other two are within the class of pear shapes. The intermediate intersection characterizes an unstable shape, whereas the other two intersections represent two stable shapes. The occurrence of multiple solutions indicates that the shape transformation from the cigar shapes to the pear shapes is discontinuous. This is, in fact, the case. By keeping, for instance, c0 constant, there is a value of ⌬a0 at which the energies of the cigar and pear shape are equal. Below this value of ⌬a0 the cigar shape has the lowest energy, whereas above this value of ⌬a0 the pear shape has the lowest energy. Accordingly, on increasing ⌬a0, vesicle shape changes discontinuously from the cigar to the pear shape. The same type of behavior is also predicted (by varying c0; Fig. 5a) by the spontaneous curvature model (Seifert et al., 1991) described above, which is actually the limiting case of the generalized bilayer model in which the ratio kr/kc has the value zero. General Discussion of Vesicle Shape Behavior In this subsection we give a physical interpretation of the general properties associated with the shape behavior of vesicular structures, and then discuss in this context the observations of phospholipid vesicle shapes presented in Figure 2. A notable general characteristic of vesicle shape behavior is that vesicles may have different symmetries. This is not an obvious result because it is more plausible to expect that a vesicle with the lowest energy has the highest possible symmetry. In this sense the flaccid vesicles should be axisymmetric and exhibit equatorial mirror symmetry. In fact, for reduced volumes larger than v ⫽ 0.58, the phospholipid vesicles with the absolutely lowest local bending energy do have such symmetries (Fig. 3). Then why, at some values of the reduced area difference ⌬a or at smaller reduced volumes v, is the lower symmetry preferable? The answer to this question lies in the constraints arising due to the geometrical contact of the two monolayers (Eqs. [3] and [6]). Simply put, for some prescribed values of v and ⌬a, the shapes with a lower symmetry are less restrictive geometrically than the corresponding shapes exhibiting higher symmetry. This is seen, for instance, in the middle of the v-⌬a phase diagram in Fig. 4, where the shapes with the lowest energy are nonaxisymmetric, and also in the prolate and oblate regions with higher and lower ⌬a values, respectively, where the shapes with the lowest energy are axisymmetric but have no equatorial mirror symmetry. We shall present a more thorough qualitative analysis for the prolate case and base it on the limiting shape behavior. The so-called limiting shapes are obtained as the extreme values of the cell volume at fixed ⌬a, or of the area difference ⌬a at fixed v, as exemplified in Figure 5 (Svetina and Žekš, 1989). They were shown to consist essentially of spherical and cylindrical parts, and as such they are often compositions of spheres connected by infinitesimally narrow necks. Independently of the number of spheres composing such a limiting shape, the spheres (or parts of spheres) can have only two possible values of their radii (Fig. 4). For ⌬a ⫽ 1 the limiting shape is a single sphere with a reduced volume v ⫽ 1. The next simple limiting shape is composed of a large and a small sphere, as it occurs in the class of the pear shapes at ⌬a ⫽ 1.30 for v ⫽ 0.85, and ⌬a ⫽ 1.117 for 222 SVETINA AND ŽEKŠ v ⫽ 0.95, and has the value of the reduced local bending energy wb ⫽ 2 (Fig. 5). The simplest limiting shape of the shape classes exhibiting the equatorial mirror symmetry (disc and cigar shapes) is composed of one large and two small spheres, and thus has the value of the reduced bending energy wb ⫽ 3. At v ⫽ 0.85, for instance, the ⌬a values of the limiting shapes with one large and either one or two small spheres (limiting shapes C and D in Fig. 4) are 1.30 and 1.44, respectively. The cigar shape has to raise its energy from the lowest value wb ⫽ 1.35 at ⌬a ⫽ 1.09 (Fig. 5) to wb ⫽ 3 in the ⌬a interval 0.35, and this rise is considerably larger than that needed for pear shapes to reach, in the ⌬a interval 0.21, the value wb ⫽ 2. It is therefore natural to expect that the pear shapes are the shapes with the lowest bending energy within the whole range of their ⌬a values. The geometrical restrictions due to the ⌬a constraint appear to be even more pronounced in the regions of smaller volumes. As shown in Figure 3, at vesicle volumes smaller than v ⫽ 0.58 the bending energies of oblate and prolate shapes exhibiting equatorial mirror symmetry (disc and cigar shapes) increase above the value 2, whereas the minimum energy shapes of the cup-shape class, which are very similar to the limiting shapes of the cup-shape class, now have the absolutely lowest local bending energy because their local bending energy is smaller than the energy of the limiting shape, which is equal to wb ⫽ 2. Consequently, at small reduced volumes the oblate cup shapes exhibiting no equatorial mirror symmetry are energetically more favorable than the corresponding shapes exhibiting equatorial mirror symmetry. The next interesting feature of vesicle behavior is that, for some values of the parameters of the system (such as the ratio kr/kc), the shape transformations are continuous, whereas for some other values they are discontinuous. This general property of the system can be understood on the basis of the detailed dependence of the local bending energy wb on ⌬a. The change of the bending energy upon change of the area difference ⌬a is different for different shapes, i.e., it depends on ⌬a. For the pear-shape class (Fig. 6), this dependence is strongest at the ⌬a value at the cigar- to pear-shape transition point. Then, at increasing ⌬a, the derivative dwb(⌬a)/d⌬a first decreases, then attains a minimum value, and increases again when it approaches the limiting shape (Fig. 6). In Figure 6 it can also be seen that the system depends qualitatively on the ratio between the nonlocal and local bending constants kr/kc. When this ratio is large enough, all shapes predicted by the strict bilayer couple model can be realized, whereas when its value is below a certain critical value, some of these shapes are unstable. The critical value of the ratio kr/kc thus determines whether the system exhibits strict bilayer couple model-like behavior or spontaneous curvature model-like behavior. The critical value of the ratio kr/kc is determined by the largest value of the derivative dwb(⌬a)/d⌬a for the pear-shape class, which is at the cigar- to pear-shape transition. It depends on the reduced vesicle volume and is larger at higher volumes; for instance, it is 18 for v ⫽ 0.85. The behavior of the system in the regions of disc and cup shapes is analogous to the described behavior in the region of the cigar- and pearshape classes (Svetina et al., 2001). The occurrence of discontinuous shape transitions depending on the ratio kr/kc and the reduced volume v has been also analyzed in the regions of disc, nonaxisymmetric, and cigar shapes (Heinrich et al., 1993). For unilamellar phospholipid vesicles, the ratio between the nonlocal and local bending constants kr/kc has been estimated to be between 2 and 3 (Waugh et al., 1992; Svetina et al., 1998). Therefore, the phospholipid vesicles exhibit the spontaneous curvature model-type behavior. In order to demonstrate the validity of this, in the v-⌬a phase diagram in Figure 4, the points corresponding to the values of the parameters v and ⌬a are given for the shapes in Figure 2. These values can be obtained by comparing the measured contours with the ones determined theoretically. This can be done for shapes 1– 4 (Svetina and Žekš, 1989), shapes 5– 8 (Käs et al., 1993), shape 9 (Svetina et al., 2001), and the starfish shape (Wintz et al., 1996). Other values are estimated on the basis of the nearby limiting shapes. An inspection of the distribution of these points in general supports the predictions of the generalized bilayer couple model for the experimental value of the ratio kr/kc. For instance, many shapes appear in the v-⌬a phase diagram (Fig. 4) close to the lines of the limiting shapes, as predicted by the generalized bilayer couple model. The observed exemptions (shapes 6 and 7) can be interpreted as long-lived shape fluctuations (Döbereiner et al., 1997). However, such shapes could still occur if the value of the ratio kr/kc were larger. One possibility for having a vesicular system with larger values of the ratio kr/kc are vesicles with multilamellar membranes. The mechanical properties of such membranes are describable by the same elastic deformations as the properties of the unilamellar membrane; however, the ratio kr/kc increases strongly as the number of lamellae increases (Svetina and Žekš, 1992). Another possible interpretation of larger effective kr/kc ratios lies in a more complex membrane composition. It was indicated recently (Svetina and Žekš, 2001) that when the membrane inclusions are thermodynamically distributed between the membrane and the surrounding solutions, and when their binding constant depends on the lateral density of lipid molecules, the effective value of the nonlocal bending constant may attain a value that considerably exceeds the value for the simple single-component phospholipid membrane. Biological Implications of Vesicle Shape Behavior Many cellular processes, notably endocytosis, exocytosis, and cytokinesis, comprise changes of membrane conformations and thus depend on membrane mechanical properties. The role of membrane mechanics is also indicated by characteristic cellular shapes, such as the disc shape of the resting red blood cell. It seems, therefore, natural to extend the concepts developed in studies of lipid vesicles to analysis of the shape behavior of cells and subcellular vesicles, as well as multicellular layered systems involving closed surfaces (Svetina and Žekš, 1991). However, cellular processes are in general highly controlled, and their successful performance depends on the intactness of many proteins and their complexes. The connection between the shape behavior of simple vesicular structures, such as phospholipid vesicles, and of cellular systems might therefore be fortuitous. Here we consider the opposite possibility by asserting that some of the general characteristics of the shape behavior of simple vesicles may have served as the basis of present-day cellular processes. This notion is reasonable because of the high SHAPE BEHAVIOR OF LIPID VESICLES probability that vesicular structures were part of the prebiotic systems from which cellular life emerged (Deamer, 1997; Luisi et al., 1999). In this section we give some examples of how the results of the shape behavior of lipid vesicles can be applied in the analysis of cellular systems at the phenomenological level, and, in parallel, develop the idea that the shape behavior of lipid vesicles has played a role in the development of cellular life. The phenomenological similarities between the shape behavior of cells, or cellular organelles, and of lipid vesicles (Menger and Angelova, 1998) indicate that the same (or at least very similar) rules govern the behavior of these systems. This notion can be supported by the characteristics of the shape behavior of vesicular objects described in the previous section that do not depend on the compositional or structural details of the membranes involved. Accordingly, the shape behavior of phospholipid vesicles can serve as a framework for the interpretation of complex cellular processes. The macroscopic parameters that play a role in these processes can be identified, and a deeper understanding of the functional aspects of cellular shapes can be gained. In the following an attempt is made to relate the shape behavior of cells and their membranous organelles to that of lipid vesicles. We concentrate on the cellular phenomena that could be based on principles common to those governing the shape behavior of lipid vesicles. We provide some examples in which the cause of shape transformation can be identified as identical or at least analogous to the cause affecting the shapes of lipid vesicles. In particular, we consider the role of cell shape in the function of the red blood cell, the mechanisms responsible for its shape transformations, vesicle fusion and fission, and a possible role of shapes in the establishment of cellular polarity. We begin with the discussion of the shape of the red blood cell. Mammalian red blood cells and lipid vesicles are similar because they both comprise a single membrane that divides two solutions. The resting shape of the normal red blood cell is a disc, which means that it corresponds (considering that the typical reduced volume of normal red blood cells is around 0.6) to the minimum of the membrane local bending energy (Fig. 3). Thus it looks as if the shape of the red blood cell is determined solely by the mechanical properties of its membrane. However, the red blood cell membrane is much more complex than a simple lipid membrane. In its bilayer part there are many embedded integral proteins, and on its cytoplasmic side it has a protein-based membrane skeleton. The shape of the red blood cell depends subtly on the intactness of its membrane composition and structure (Mohandas and Evans, 1994), and also on cell metabolic activity. Therefore, the correspondence between the shape behavior of red blood cells and lipid vesicles may not be so simple. Nevertheless, it is possible to speculate about the functional significance of the red blood cell shape. In performing its function, the red blood cell has to squeeze through capillaries that are much narrower than the cell diameter. In order to attain the required elongated morphology, a red blood cell must be easily deformable. For simple geometrical reasons it is easier to deform vesicular objects with smaller reduced volumes. In Figure 3 we can see that on progressively decreasing the reduced cell volume, the shape with the lowest energy is first a prolate cigar shape, then a disc shape, and finally a cup shape. The cup is not a very convenient shape for squeezing a cell into the capillary 223 because of its low area difference value. When the elongated “capillary” shape of the red blood cell, which has a relatively high value of the area difference, is formed, the system has to pay for the change of the nonlocal bending energy (Eq. [4]), and this change is larger for the initial cup shapes than for the initial disc shapes. By having the reduced volume around 0.6, the red blood cell meets both these criteria: it has the reduced volume as low as possible, but its shape is not a cup. It can be concluded that the red blood cell shape evolved into a disc shape because it is functionally favorable for this cell to have such a shape. The shape of a red blood cell can change under a variety of altered physical and chemical conditions. Shape changes into cup and crenated shapes have been conveniently interpreted by the bilayer couple model (Sheetz and Singer, 1974). This model explains the isovolume shape transformations of red blood cells by the asymmetric changes of the areas of the layers of the bilayer part of the red blood cell membrane. The changes of these areas can, for instance, be caused by the asymmetrical binding of different compounds to the bilayer part of the red blood cell membrane (Sheetz and Singer, 1974), or by the chemical modification of membrane lipids (Ferrell and Huestis, 1984). The bilayer couple model can be associated with ⌬A0 changes, and it was actually this model that stimulated the corresponding theoretical studies (Svetina et al., 1982; Svetina and Žekš, 1989) presented in the previous section. The shape transformations of the red blood cell can be paralleled by the shape behavior of phospholipid vesicles in that some of their shape transformations have an identical course. The red blood cell shape transformation from the disc shape to the cup shape is identical to the vesicle shape transformation demonstrated in Figure 2 (shapes 1– 4). They can both be interpreted by the decrease of the reduced difference between the areas of the outer and the inner membrane layer (⌬a). It must be noted that, at larger values than the disc ⌬a values, red blood cells transform into echinocytes; thus, in this region of the v-⌬a phase diagram they do not follow the same course as phospholipid vesicles. The difference in the two behaviors can be understood by the influence of the red blood cell membrane skeleton (Waugh, 1996). The shape behavior of lipid vesicles can be related directly to the processes of membrane budding and vesiculation. Membrane budding is a frequent intermediate step in vesicular membrane trafficking. The process of endocytosis (Riezman et al., 1997), for instance, can be dissected into the formation of an internalized membrane bud and its fission. Membrane budding can be understood on the basis of the characteristic feature of vesicular objects in that they exhibit the limiting shapes composed of spheres connected by narrow necks (Fig. 2, shapes 8 and 13–16). The budding process can be interpreted by assuming that in its first step the area difference ⌬A is changed such that the budded limiting shape, composed of connected spheres, is reached. For instance, the shape transformation of pear shapes, due to an increase of either v or ⌬a, leads to a limiting shape that is composed of a large mother sphere and a smaller daughter sphere connected by an infinitesimally narrow neck (Fig. 5). It might not be a coincidence that the cellular constituents, e.g., phosphoinositides, which by their metabolism may affect the parameters ⌬A0 and c0, play a role in membrane transport (Simonsen et al., 2001). Specifically, it has been demonstrated that compounds that cause changes of the area 224 SVETINA AND ŽEKŠ difference affect the rate of endocytosis (Rauch and Farge, 2000). The budding process can end up by the vesiculation process, which consists of the release of a vesicle. Membrane fission requires the structural rearrangement of the membrane in the neck. It has to be borne in mind that when the limiting shape is reached, some other processes may occur. A vesicle can either transform its shape into another one (such as shapes 13 and 14 in Fig. 2) or it can lyse. In this respect it can be speculated that one of the roles of protein machineries involved in endocytosis and other vesicle budding processes is to make the processes deterministic, i.e., to decide always on the same pathway. The opposite of vesiculation is fusion of two vesicles into a single vesicle. Vesicle fusion is a frequent biological process; however, it also occurs in simple systems such as lipid vesicles. This means that the fusion process is at least partly governed by the properties of the lipid bilayer (Lentz et al., 2000). Fusion of simple lipid vesicles can occur upon aggregation of two vesicles. Assuming that these vesicles are spherical, the aggregate looks like shapes 6 and 12 in Figure 5, except that the two membranes are separated. Vesicle fusion occurs by the fusion of the two separated membranes into a single membrane. The membrane conformation thus obtained is essentially the conformation of the proper limiting shape, where the two spheres are connected by a narrow neck, with the energy higher for the value 兩4kG兩, because the Gaussian bending constant is in general negative. After fusion of the two membranes, the obtained vesicle attains the shape that corresponds to the ⌬A0 value of the fused membrane. For the termination of the vesicle fusion process, it is convenient that the energy of the final shape of the fused vesicle is lower than the energy of the initial state of the system. This requirement is well fulfilled if the ⌬A0 value of the final vesicle is in the region of the lowest local bending energies (shapes 1 and 7 in Fig. 5). This reasoning (Svetina et al., 1994) shows that for fusion to occur, it is important that the equilibrium area difference of the fused vesicle is such that its shape falls into a domain in the v-⌬a phase diagram where vesicles are flaccid. It is of interest to note that an addition of calcium ions in the case of negatively charged fusing membranes enhances the fusion process by both properly adjusting the equilibrium area difference, and straining the membrane in its initial state (Svetina et al., 1994). When the internal vesicle fuses with the membrane, as is the case in exocytosis, fusion will occur if the final vesicle has a higher ⌬a value corresponding to the cup-shape class. Exocytosis is an extremely well regulated process and involves a complex protein machinery. Again, one can reason that while with simple vesicles the process of vesicle fusion is stochastic and occurs with a certain probability, in a cell this machinery ensures that membrane fusion happens only when it must happen, due to the proper signals. In the context of this work, it is also interesting to note that the signals usually affect the system parameters that can produce ⌬A0 changes, e.g., calcium ions and phosphoinositides (Simonsen et al., 2001). Cell shape transformations can also be associated with the establishment of cellular polarity, i.e., with the process in which the cell constituents distribute in a polar manner (Drubin and Nelson, 1996). Cellular polarity is a property of many unicellular organisms and isolated cells of multicellular organisms, and demonstrates their ordered state. A hypothesis can be put forward that the polar distribution of cell constituents arises as the consequence of polar cell morphology. It was demonstrated in the previous section that vesicles may have polar equilibrium shapes (Figs. 4 and 5). A mechanical mechanism for the establishment of cellular polarity (Svetina and Žekš, 1990) would consist of adjusting the values of the shapedetermining parameters to stabilize a polar cell shape. This can be realized by the variation of cell parameters such as areas and mechanical constants of different membrane layers. There are many physical and chemical means by which a cell can control these parameters. The polar cell shape having been attained, the membrane curvature varies over its surface in a polar manner. Consequently, any curvature dependent interaction between membrane constituents will cause their lateral density to vary correspondingly (Kralj-Iglič et al., 1996, 1999). Some of the general characteristics of the shape behavior of lipid vesicles may have served as the origin of the corresponding cellular processes. The assumption is that many cellular processes that occur today are in essence an upgraded and controlled version of the corresponding physical process that occurs at the level of pure vesicles. An illustrative example of this is cytokinesis. It was recently shown in a model system (Berclaz et al., 2001) that, under given conditions in the population of vesicles incorporating the membrane material, the number of vesicles increases. It was concluded that this is due to vesicle fission. This behavior can be understood in terms of the vesiculation processes discussed above. Initially, a spherical vesicle, as a result of the incorporation of new molecules into its membrane, decreases its reduced volume. If the trajectory within the v-⌬a phase diagram (Fig. 4) is within the class of pear shapes, a limiting shape can be reached, composed in general of two spheres of different radii. The necessary conditions for the appropriate changes of the reduced area difference ⌬a and the reduced volume v can in principle be satisfied by the appropriate ratios between the rate of membrane area increase, the rate of increase of the difference between the areas of membrane monolayers, and the rate of water uptake into the vesicle-defining vesicle volume. Contemporary cytokinesis may have developed from such processes by evolution of the corresponding controls that would give a well defined ratio between the sizes of daughter cells. In conclusion, lipid vesicles and corresponding cellular systems differ in that the processes with vesicles are in general stochastic, whereas the processes in a cell are deterministic. One implication is that, in the course of evolution, a certain basically stochastic process evolved into a deterministic process based on a complex protein machinery. SUMMARY Studies of the shape behavior of phospholipid vesicles have revealed the basic principles of how the shapes of vesicles characterized by closed, layered membranes are determined. A significant influence of the local bending energy on the detailed behavior of vesicle properties can be recognized. By studying a relatively simple lipid system, the mechanical parameters that also determine the shapes of more complex vesicular objects can be identified. The shape behavior of vesicular objects enclosed by lamellar membranes involves some properties that do not depend on the structural and compositional details of the composing layers. Two properties that may be particularly SHAPE BEHAVIOR OF LIPID VESICLES important bases for possible physical mechanisms underlying different biological phenomena are the occurrence of stable asymmetric shapes and the propensity of the system for budding. 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