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Growth instabilities of vesicles
R. Bruinsma
To cite this version:
R. Bruinsma. Growth instabilities of vesicles. Journal de Physique II, EDP Sciences, 1991, 1 (8), pp.995-1012. �10.1051/jp2:1991122�. �jpa-00247570�
J Phys II France 1 (1991) 995-10I2 AOOT 1991, PAGE 995
Classification Physics Abstracts
82 70 64.60F 05.40
Growth instabilities of vesicles
R. Bruinsma
Physics Department, University of Cahfornla, Los Angeles, Los Angeles, CA 90024, US A
(Received 27 July 1990, revised I February 1991, accepted 18 April 1991)
Abstract. We present a model for the growth of short-wavelength instabilities of membranes
~Amth small curvature energy based on the van der Waals interaction energy
1. Introduction.
The Shape Of lipid vesicles is currently believed tO be controlled by a combination Of
macroscopic factors [I] Such as the bending energy K, the spontaneous curvature
co Of the lipid membrane, and the osmotic pressure lZ As these parameters are varied, a broad range Of morphologies is encountered If thermal fluctuations are not too strong, then the vesicle-shape can be found by mm1mlzing the bending energy [2]
F~ = K ld~«iRi~(«) + Ri~(«) coi~ (ii)
Here, Rj
~(ar are the pnncipal radii of curvature at the point ar of the vesicle surface, while co iS the Spintaneous radius of curvature. We will only consider from hereon the
case of zero osmotic pressure In minimizing F~, the vesicle surface area S is to be kept fixed assuming
that the number of lipid molecules constituting the vesicle does not change and that the area per lipid molecule is fixed We assume implicitly m equation (I.I) that the membrane
molecules are m the liquid State.
If one vanes the Spontaneous curvature m F~, then an initially Sphencal vesicle becomes
unstable as we increase co For example, a sphencal vesicle of radius R has a bending energy F~ = 2 wK (2 coR)~ We can compare this with the bending energy of two vesicles each of
radius RI /
i e with the same total surface area where F~
=
2 w K (2 /
coR)~. For
Rco > I + /), the
energy is lowered by fissioning the ongmal vesicle (neglecting Gaussian curvature contributions) If we do not allow breakage then instead the vesicle will produce
« buds » attached to the original vesicle.
The full vesicle-shape « phase-diagram » is quJte nch. Recently Miao et al. [5] predicted a variety of budding instabilities Typically, they find that with increasing spontaneous
curvature a vesicle first resembles a double-sphere configuration connected by a short, narrow tube. The tube radius then goes to zero (« kissing spheres ») followed by fissioning The shape
instabilities have the character of first-order phase transitions. Shape instabilities indeed are
996 JOURNAL DE PHYSIQUE II N 8
observed expenmentally [3, 4] for erythrocytes and for DMPC (lecithin) They may play a
role m biological processes such as endocytosis
In the following we will restnct ourselves to bilayer membranes For bilayer membranes,
we would expect co = 0 by symmetry yet budding mstabihtles are expenmentally observed for this case [3, 4]. One possible scenano for bilayer budding instabilities was proposed by Seifert
et al. [6] based on the model of Svetina and Zeks [7] if the two lipid layers of the bilayer
contain different numbers of molecules then, for large enough area difference, similar
budding instabilities are encountered. Interestingly, the shape transformation now bears the character of a continuous phase transition.
We will investigate m this paper a different mechanism for budding instabilities which relies instead on the presence of long-range attractive forces. More fundamentally, long range
attractive forces will provide a lower bound for K. Intuitively it is clear that both budding and
fissioning are favored by long-range attraction, because more parts of the membrane are put into closer proximity, while (modest) repulsive forces would tend to stab1llze sphencal
vesicles The dominant long-range attraction for membranes m solution is the van der Waals force [8]. The van der Waals force is well known to be significant for vesicle morphologles. It
is for instance responsible for membrane-membrane adhesion as observed m lecithin [9] To
see whether the van der Waals interaction is strong enough to lead to budding, we first recall that two sheets of thJckness b and separation z » b feel an attractive van der Waals potential
U(z) m
~~'
b~/z~ (1.2)
per unit area. Here, W is the Hamaker constant which is of the order
((e~ ej)/(e~ + sj))~k~ T with e~ and ej the dielectric constants of respectively the material outside and inside the sheets In the present case, with water and lipid, W 0 75 k~ T Now, take an initially sphencal vesicle of radius R » b and deform it into a flat disc of radius
RI The separation
z between the top and bottom bilayers of the disc is assumed small
compared to R. The energy cost AF(z) of the deformation easily follows from equations (I. I)
and (1.2)
AF(z)
=
K(«~ll)(Riz) wa2R21z4 (1.3)
If, m equation (1 3), we decrease z starting from z
= R, then, initially, AF(z) increases with
decreasing z The sphencal shape is thus at least locally stable. However, as we decrease z further, AF(z) passes over a maximum and for small z it decreases. For z s Wb~R/K)~'~, the disc shape actually has a lower energy then the sphere, so the sphencal shape, although locally stable, would appear to be globally unstable against collaps induced by the van der Waals
attraction.
There are however a number of obvious requirements which would have to be met
,
first of
all, since z cannot be less then b, R has to exceed bK/W for the collaps to take place
However, for K ~10-100 k~ T, and b 30-50 I, this condition is easily satisfied for typical
vesicles with R ~10
~L Next, repulsive forces such as electrostatic repulsion [10], hydration
forces [11] or Helfnch entropic repulsion [12, 13] could overcome the van der Waals
attraction for small z. However, since membrane-membrane adhesion is actually observed [9]
thJs m general need not be the case
Nevertheless, this form of collaps has not been reported and there are (at least) two reasons
why it is not physical First of all, we mentioned that AF(z) goes through a maximum with
decreasing z, so the collapse is apparently an acttvated process The energy barner AE preventing collapse is found by maximizing AF(z). The result is
AE/K = 8 w + 2 (w~/3 /)~'~ (K/@j~'~ (Rib
)~'~ (l.4)
M 8 GROWTH INSTABILITIES OF VESICLES 997
To allow thermal fluctuations to overcome thJs energy barrier. we must require AE to be
comparable to k~ T but it is clear from equation (1.3) that even for the floppiest vesicle (K W~ k~ T), AE greatly exceeds k~ T because of the factor (Rib )~'~.
The second reason why global » collapse is not possible is of a dynamical origin When
one minimizes F~, the vesicle area S but not the vesicle volume V are kept fixed (as
membranes are reasonably permeable against water but have a fixed surface area). ThJs is
however the proper procedure only under conditions of thermodynamic equihbnum. The
cntical shape fluctuation which must carry the vesicle over the barrier require a large change
AV m vesicle volume (i e., AV V) Even for a highly permeable membrane, we would have to wait on the order of 101~ s to see a shape fluctuation with AV of the order of V [13]. For
dynamical problems such as bamer activation we evidently cannot allow much volume change
at all For dynamic purposes we have to treat V as a conserved quantity as well.
The fact that a global collapse of the above nature is impossible does not however
completely shield our vesicle from the underlying instability. In the next sections we are going to search for local shape instabilities with activation barners comparable to k~ T and with no
changes m vesicle volume, i-e-, AV
= 0 Before we do this, we first note here that equations (12-14) only apply if the dielectric media inside and outside the vesicle are the same. For
laboratory-prepared vesicles this should be the case, but for a typical plant or animal cell the
cell intenor consists of about 30ili macromolecules and 70fb water. Because of this
difference m chemical composition there can be a considerable dielectric contrast Let eA and e~ be the dielectnc constants of respectively the vesicle interior and extenor The van der Waals potential per unit area (Eq (1.2)) now reads [8]
U(z) m W/12 wz~ (1 5)
with W~ [(e~ e~)/(s~ + e~)]~k~ T Thls potential is both considerably stronger and longer-ranged. The condition that R has to exceed bK /W to see collapse remains unchanged
but the activation barner AE is reduced to
AEj~ = 8 « + (3j4) « 2(~j~~/~ (1.6)
In the following we will consider the two cases separately Our first aim will be a reexamination of the energetics for localized deformations of the shape of a vesicle if we include the van der Waals attraction
2. Energefics.
Start with a closed vesicle of volume V and surface area S We will treat both S and V as fixed quant1tles for the reasons given m the introduction. The vesicle will be assumed to initially
have an approximately sphencal shape so V~'~/S~'~ is somewhat less then [(4/3) w ]~'~/[4 w ]~'~
the corresponding value for a perfect sphere. Let b~ WAS « S be the excess area which
would have to be removed to turn the vesicle into a perfect sphere If we minimize the
bending energy F~ (Eq (I.1)) with co = 0, then the resulting vesicle typically resembles a
somewhat deformed sphere with a mean radius R close to (3 V/4 w)~'~ and a bending energy
close to 8 arK.
What is the effect of including the van der Waals attraction ? The minimization of the full free energy becomes m general a daunting mathematical problem because of the non-local
nature of the van der Waals attraction. Assume, for simplicity, that we have collected all the
excess area AS into a cylindrical tube protruding from an otherwise spherical vesicle (see
Fig. I) Let r(z) be the cylinder radius with z running from z
= 0, where the tube connects to the vesicle, to z = L, with L(» ro) the tube length The tube is assumed to be capped by a
998 JOURNAL DE PHYSIQUE II M 8
~V(p,i)'
~fi
l
~
, z
/# '
~'
~/~~~
',
/ L(I)
R
'
al )
Fig I -
Tubular appeanng on the surface of sphencal vesicle of radius R. The
tube
rotationally invariant around
the z axes a) First-Stage Growth The tubeincreases its length L for a
fixed
mean
radius ro The flow velocityvanes little cross the tube ross.section Both lipid and
solvent
matenal is
transferred from the vesicle into the tube. b) Second-Stage Growth The tube has reached its
maximum surface area AS The radius
hemisphere of radius r(L) The vesicle free energy, including the van der Waals «self- energy », is
F
= 12 «
K + II dz K «,(z) S A ~
A ~ ~ r-
(z)I
~A = ~B (2 la)
F =12 wK + j~ z~Kwr(z) ~~( ~ Ar~ (z)~ e~ # s~ (2.lb)
o dz ~(Z)
for a slowly varying r(z) « R. The first term m equation (2,I) is the sum of the bending
energies of the sphencal part of the vesicle (~8wK) and of the hemisphencal cap
(~ 4 wK ). The second term is the bending energy of the tube. Note that the pnnclpal radii of curvature of the tube have opposite sign. The last term m equation (2 1) is the van der Waals
attraction without (Eq. (2.la)) and with (Eq (216)) dielectnc contrast. The denvation is
given in the appendix (both A' and A are of the order of Rj. The vesicle radius R was assumed large enough so we could neglect the van der Waals self-energy of the spherical part of the vesicle. Equation (2. I) is only valid for non-retarded van der Waals interaction, I-e-, for r(z) less then the dominant wavelength A for light adsorption (A 50 nm). For r(z) > A, we
must replace Air by B/r~ with B
~
AA
The equilibnum shape of the tube is found by minimizing F under the constraint that
~
AS
= dz 2 wr(z) (2 2)
0
M 8 GROWTH INSTABILITIES OF VESICLES 999
is kept fixed. We do not need to fix the tube volume as solvent can be freely exchanged
between tube and spherical vesicle. If necessary, we can always adjust L at the end of the calculation to ensure that the vesicle volume is unchanged We thus can thJnk of the vesicle as
a solvent reservoir of zero chemical potential.
This minimization problem has no stable solution. We can ascertain thJs claim by
considenng a tube with constant radius r(z)
= ro From equation (2.2), AS
= 2 wro L The free energy is for ro
< A :
~ ~~ ~~ ~ ~~j~ 1TK A'(b/ro)~ (8A
= 88)
~~ ~~
°
ITK A (8A # sB)
while for ro > A and s~ # s~,
F 12 irK + (Ljro)(irK Biro) (2.4)
If K
> AIv and e~ # e~, then F is lowered by continuously increasing ro and decreasing L
Eventually, ro reaches R after which the tube construction becomes meaningless. For large
ro, we can neglect the van der Waals contribution so F F~. As a result we would recover the old « deformed sphere solution. We conclude that for K > A/w, the van der Waals self- energy does not play a significant role and we do not expect any tubular growth instabilities. If
K < AIv and e~ # e~, then F is lowered by continuously decreasing ro and increasing L (with
AS
= 2 wroL). This will continue until ro ~
b when higher Order non.linear terms in the
bending energy will prevent further shnnkage of the tube radius Finally, if there is no
dielectnc contrast, then the tube construction lowers the free energy provided
wK s A'( biro)~ for ro > b (see Eq. (2.4)) ThJs leads us to the same critenum : K must be less then W for a tubular protrusion to lower the free energy.
We conclude that under conditions of fixed volume, a vesicle becomes unstable against local tubular growth if the bending energy K drops below a critical value of order the Hamaker
constant W, i-e if K s kb T. This cnticality must be contrasted with the situation for the
global collapse where for any K we found an instab1llty One of the reasons why global collapse was not possible was because of the large volume changes involved We have seen that this objection is obviated by the local instability scenario
How about the problem of the high energy barriers preventing the instability ? Just as for global collaps, a tubular instability must overcome an energy banner AE. To estimate AE, fix ro m equation (2.3) and increase L until L
= AS/2 wro. Let AE(ro) be the energy maximum
encountered dunng this process. Afterwards, we can minimize AE(ro) to find the lowest
possible energy barrier
Start with e~ # e~. If ro > r*, with r * B/K, then F always increases with increasing L and
no energy is gained (see Eq. (2.4)) We conclude that r* forms a maximum radius for tubular protrusions For K of order k~ T, r* is of order 1000i For
r less then r*, the energy decreases linearly with L (see Eq (2.3)) provided K <A/w. The maximum energy value
occurs for the minimum L value (L ro) and, for K W, it is of the order of the bending
energy of the hemispherical cap (~ 4 w K For e~ = e~, we find a similar result We conclude
that thermal fluctuations indeed can overcome the activation bamer for tube growth
provided, Once again, that
K is of Order k~ T
It is clear from the preceding discussion that the actual shape of the tube cannot be established from purely static arguments as long as r(z) is large compared to b. We will see that the shrinkage of r(z) to b is a very slow process, so actual tube shapes must be determined from dynamic considerations We will do this m the following sections Before doing this, it could be pointed out that a tube is not necessarily the most advantageous local growth shape
1000 JOURNAL DE PHYSIQUE II N 8
to exploit the van der Waals attraction. For instance an equatonal annulus would gain more
van der Waals energy then a tube. The reason for not considering thJs possibility was that the associated energy barrier is of the order KR/z with z the annulus layer spacing which is
large compared to k~ T We expect to encounter dunng a first-order growth mstabihty that
growth shape which minimizes the activation barrier (by analogy with the nucleation and
growth of crystals during supercooling)
3. Dynamics.
31 GENERAL. We have seen that vesicles are unstable against tubular growth once the
bending energy is less then a cntical value which is of the order of the Hamaker constant.
From now on, we will always assume to be in thJs unstable region of the vesicle phase diagram. The basic parameters we used for describing the tube shape were the tube radius and the tube length For reasons which will become clear below, we can treat the tube length (L as a fast dynamical vanable and the mean tube radius (r0) as a « slow » variable. Note
that at all times 2 wroL s AS
The tube growth can be divided into two stages The first stage starts with a thermal activation event over the energy barrier AE followed by rapid growth of L for some value of ro less then r*. The first stage ends when L reaches its maximum value AS/2 wr0 During this
stage, both area and volume of the tube increase with time so both lipid and solvent material must be transferred from the sphencal part of the vesicle to the tube part
Dunng the second stage, the tube surface area is fixed at AS and there is no further
exchange of lipid material between vesicle and tube The tube energy is lowered by reducing
ro while L is slaved to ro by L
= AS/2 wro. The tube length thus continues to increase. Because the tube volume wrjL decreases proportional to ro, solvent material must now be pumped
from the tube back into the vesicle
To see why L is a fast variable compared to ro, we must consider the dissipative losses suffered dunng the tube growth In general, energy is dissipated both by lipid and solvent material. This dissipated energy must be compensated by the work done by the van der Waals force as we change the vesicle shape. The hydrodynamic viscous losses m the solvent can be
computed from the Navier-Stokes equation. The viscous losses inside the bilayer are due to the motion of the top lipid layer over the bottom lipid layer of the bilayer when lipid material
is transported from the vesicle to the tube. To compute these losses, we will model the bilayer
as a sheet of thickness b with a (high) viscosity ~~. If ~s is the solvent viscosity then the total power dissipation £ is given by
«
= ~ s
d3r v Av + ~
~
d3rv Av (3.i)
vesicle biiayer
with v (r) the flow velocity We only included solvent viscous losses inside the vesicle because the flow outside the vesicle is unconstrained by boundary conditions on v,(see below) so flow
gradients are expected to be very small
As is suggested by figure la, if we increase L for fixed ro then viscous losses are mainly
restricted to the region near the tube onfice because there are no flow gradient inside the tube (« plug flow »). However, when during the second stage we reduce the tube volume, then there are viscous losses all along the tube (see Fig 16) and not just at the onfice. For
comparable flow velocities, the dissipative losses are increased by a factor L/r0 according to
equation (3.I). The growth velocity is thus expected to be greatly reduced compared with the first stage We will venfy this heunstlc argument later on
