GROWTH INSTABILITIES OF VESICLES

Abstract

We present a model for the growth of short-wavelength instabilities of membranes with small curvature energy based on the van der Waals interaction energy.

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COLLOQUE DE PHYSIQUE
Colloque C7, suppl6ment a u n023, Tome 51, ler dgcembre 1990
GROWTH INSTABILITIES OF VESICLES
R. BRUINSMA
P h y s i c s Department, U n i v e r s i t y of C a l i f o r n i a , Los Angeles CA 90024,
U.S.A.
ABSTRACT
W e present a model for the growth of short-wavelength instabilities of
membranes with small curvature energy based on the van der Waals interaction
energy.
I.
INTRODUCTION
>S is well known,% the macroscopic shape of lipid membranes is largely
controlled by a combination of the curvature energy K, spontaneous curvature,
temperature, and the interaction between membranes. Heasured curvature
energies are usually considerably in excess of 8-1 = kBT. The surface of the
membrane is approximately . l a t on length scales less than the persistence
length Ep - exp BK.
If we do reduce BK, then Ep becomes smaller and smaller
while the membrane becomes rougher and rougher.
For @K $ i, membranes are
believed to be unstable due to thermal fluctuations. The phase-transition
between the L , and L3 phases may be an example of such an instability.'
Recently, E. Evans devised an Ingenious experiment which suggests an
alternative scenario for the evolution of instabilities of low X membranes. H e
dissolved the sub-surface protein scaffolding which had maintained the rigidity
of a (spherical) vesicle3 and then watched the evolution. The initial
curvature energy was quite low as testified by visible thermal fluctuations i n
the vesicle shape immediately following the dissolution. After a while, the
vesicle grew "buds" which led t o new smaller vesicles.
The new vesicles were
stable.
Host interestingly, he observed on occasion during this process tubular
buds and these tubes showed a bead-like instability. In other words, the
cross- section of the tube was modulated periodically.
Instabilities of liquid
cylinders are quite familiar since Rayleighq but there the driving force is the
surface tension. For vesicles, the surface area is a fixed quantity because of
the surfactant action of the lipid molecules.
Spontaneous curvature effects
can also be ruled out.
The instability would be understandable as the consequence of a long-
range attractive interaction between the walls of the tube.¶ A long-range
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990705

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COLLOQUE DE PHYSIQUE
attractive force would favor a large number of small vesicles over a single
large vesicle (with the same total area) since more sections of the membrane
would be in close proximity in the former case. In that sense, an attractive
force has an effect somewhat similar to surface tension. The observed Rayleigh-
type instability would then also be understandable. This long-range attraction
must compete with the curvature energy.
vesicle -- with no spontaneous curvature -- into N smaller vesicles then the
curvature energy is increased by roughly (N-114n (~K+E)
If w e transform a single spherical
with
the Gaussian
curvature energy.
The dominant long-range attraction for membranes is provided by the van der
Waals interactione.
The attractive van der Waals energy between two f l a t
parallel layers of thickness 6 a distance z apart is well known t o be of order
W62/z4 with W the Bamaker constant (-10-2' - 10'22
J.) If we use for 6 the
membrane thickness (- SOA) then this attraction is miniscule compared to the
curvature energy as long as r(z) >> 6. This would appear to be a fatal
objection t o the proposed explanation of the observed instability. However,
for the experiment discussed earlier there is no reason for the dielectric
constants of the solvents i n the interior and exterior of the vesicle to be the
same. Because of the dissolved protein scaffolding i n the interior they could
indeed be significantly different. The van der Waals interaction per unit area
between two parallel sheets enclosing a medium of dielectric constant E* with a
medium of dielectric constant cB on the outside is of order W/zz.
proportional t o (E, - eB)Z/(cA + eB)2. The attraction has increased by a
Here, W is
factor (z/6)z compared t o the direct interaction and can now be of the same
order of magnitude as the curvature energy.
In this article we w i l l
investigate the stability of a vesicle for which the van der Waals self-energy
is comparable t o the Helfrich curvature energy.

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11. CURVATURE AND VAN DER W A A L S ENERGIES.
The free energy of a closed vesicle with no spontaneous curvature is
= 1 K
{R~-' + %-l)t + -
j3
Jd3r. -
1
exterior interior
(1)
The f i r s t term is the standard Helfrich curvature energy. W e dropped the
Gaussian curvature term as it is independent of the vesicle shape and we also
assumed zero spontaneous curvature. The second term in Eq.1 is the non-
retarded van der Waals self-energy in the de Boer-Hamaker approximation as
W e are implicitly assuming i n Eq.1 that --
notvithstanding the different solvents -- there is no appreciable osmotic
discussed in the Appendix.
pressure difference between interior and exterior of the vesicle.
W e now should minimize F vith respect t o the vesicle shape for a given
fixed surface area. From dimensional considerations w e should expect for K >>
W to find a spherical shape since then the Helfrich term dominates.
For K << W , the van der Waals self-energy dominates.
W e saw i n the
introduction that two parallel f l a t sheets attract each other.
If K << W we
thus should expect the vesicle to be crumpled in some way.
The general
minimization of F with respect t o shape is clearly a quite difficult problem.
W e w i l l consider only a special case, motivated by Evans' experiment, namely
that of a rotationally invariant vesicle.
Assume a tubular membrane with a position dependent radius r(z) attached to
a micron-size vesicle of radius R >> r (see Fig.1) which acts as a reservoir.
The length of the tube is L and the 2 axis coincides with the tube. The free
energy Eq.1 then simplifies t o

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with A g.6W.
The derivation of the van der Waals term in Eq.2 is given in
the Appendix but it's form is obvious from dimensional considerations: for
r f z ) constant, F must be proportional t o both L and W so F a WL/r.
The second
term in Eq.2 is an obvious generalization of this.
Equation 2 is only valid for the,non-retarded van der Waals interaction.
If r 2 X with X the dominant adsorption wavelength of the solvent (- SOnm),
then we must replace ~ r - l by ~r-' with B = XA (the retarded van der Waals
interaction).
The qualitative features of F are easily understood. For a cilindrical tube
with r(z) = rO, Eq.2 gives F = L(Kn - A)/ro. For K > A/n, we can minimize F by
reduclng L and increasing ro. For K < A/n, we minimize F by reducing ro and
increasing L. W e thus expect that for K > A/n the vesicle is stable while for
K < A/n it w i l l spontaneously develop tubular protrusions.
L(Xn-B/ro)/ro so there is then a critical radius R* = 8B/3nK such that for ro L
R* 2 X the tube w i l l swell and vanishes while for X 5 ro 5 K* it w i l l shrink
For ro ?.,X,
F =
and elongate (the reason for the numerical factor
w i l l become clear later).
The necessary condition X 2 R* for this instability is just K 5 A/n as before.
W e conclude that K* = A/n marks the threshold of a growth Instability which we
w i l l now proceed t o investigate in more detail.
I11 . DWAHICS
The growth-rate of the tube w i l l be controlled by the flow of solvent
material.
To see why, assume w e have a tube of length L and uniform radius r.
A s it shrinks, its total surface area S a r(t)L(t) must remain (roughly) .fixed.
This means that the tube v o l ~
V ( t ) a rZ(t)L(t) must decrease as V ( t ) a S r ( t ) .
This in turn implies that there must be flow from the tube into the vesicle.

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As a result, the region where the tube is connected to the vesicle moves
towards the vesicle (see Fig. 1).
W e w i l l focus on this contact region and in
particular look for "steady-state" solutions with the tube shape fixed but
moving left.
There w i l l be contributions to the viscous energy-dissipation by the
solvent flow both from the vesicle interior and exterior. Viscous losses from
the interior w i l l dominate because of the velocity gradients imposed by the
boundary conditions a t the vesicle surface.
+
flow velocity v, w e w i l l set :
As our boundary condition on the
i 0 a t the membrane. This is again because the
membrane has a fixed surface area. If w e wish t o maintain, or even increase,
the length of the tube during the flow, then the membrane molecules cannot be
carried along by the flow towards the vesicle so G must be zero at the surface
of the tube. W e also w i l l assume that there is no solvent transport across the
membrane.
-f
+
Let v(p,z) be the solvent flow velocity. To compute v, we use the
Poisseuille approximation:
with P(z,p) the pressure i n the tube and q the viscosity. W e assumed i n Eq.3
-3
r(z) t o be a slowly varying function of z.
A s a consequence v is roughly
a23
azz . The
parallel t o the tube axis
("lubrication approximation")e and ap2 >> -
solution of Eq.3 is then
v(p,z) P -
(r2 - p2)
2q az
Finally, w e demand mass conservation: