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Shape behavior of lipid vesicles as the basis of some cellular processes.

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THE ANATOMICAL RECORD 268:215–225 (2002)
Shape Behavior of Lipid Vesicles as
the Basis of Some Cellular Processes
Institute of Biophysics, Faculty of Medicine, University of Ljubljana, and J. Stefan
Institute, Ljubljana, Slovenia
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©
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.
Received 13 February 2002; Accepted 4 April 2002
DOI 10.1002/ar.10156
Published online 00 Month 2002 in Wiley InterScience
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.
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-
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 ⫽
(A ⫺ A0)2
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
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):
Wb ⫹ G ⫽ Wb ⫹ WG ⫽ kc (C1 ⫹ C2 ⫺ C0)2dA
⫹ kG C1C2dA
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 ⫽ 4␲kG 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/4␲Rs3, 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/8␲kc) 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/8␲kc, 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
⌬A ⫽ A2 ⫺ A1 ⫽ h (C1 ⫹ C2)dA.
Integration is over the membrane area (A ⬃ A1 ⬃ A2) of
the vesicle.
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 ⫽
(⌬A ⫺ ⌬A0)2
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/8␲kc) and the reduced curvatures,
w ⫽ w b ⫹ wr ⫽
(c1 ⫹ c2 ⫺ c0)2da ⫹ (⌬a ⫺ ⌬a0)2
where da ⫽ dA/4␲Rs2, c1 ⫽ C1Rs, c2 ⫽ C2Rs, and the area
differences are reduced relative to their values for a
sphere (8␲hRs), i.e., ⌬a ⫽ ⌬A/8␲hRs and ⌬a0 ⫽ ⌬A0/
8␲hRs. Eq. [3] may also be written in terms of the reduced
⌬a ⫽
(c1 ⫹ c2)da.
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
⫹ 2 (⌬a ⫺ ⌬a0) ⫺ c0 ⫽ 0.
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
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-
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/8␲kc), 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
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
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
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
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
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 兩4␲kG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
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
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. Stable asymmetric vesicle shapes may
represent the mechanical origin of cellular polarity, and
thus of biological order.
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