Curvature instability in membranes
S. Leibler
To cite this version:
S. Leibler. Curvature instability in membranes. Journal de Physique, 1986, 47 (3), pp.507-516.
<10.1051/jphys:01986004703050700>. <jpa-00210231>
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Submitted on 1 Jan 1986
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J.
Physique 47 (1986)
507-516
MARS
1986,
507
Classification
Physics Abstracts
87.20201368.15201305.90
Curvature
instability
in membranes
S. Leibler (*)
Groupe de Physique des Solides (+) ; Ecole Normale Supérieure, 24, rue Lhomond, 75231 Paris Cedex 05, France
(Reçu le 21
aout 1983,
accepté le 31 octobre 1985 )
Un modèle thermodynamique simple a été proposé il y a quelques temps [1, 2] pour décrire les prode divers systèmes dans lesquels l’énergie de courbure élastique joue un rôle important. Ce
modèle repose sur les notions de rigidité effective et de courbure spontanée. Nous considérons ici le cas de memune source
branes bi-couches et généralisons le modèle pour des situations où de petites molécules adsorbées
possible de courbure spontanée non nulle peuvent diffuser dans la membrane. Nous montrons que dans certaines conditions celles-ci peuvent entièrement destabiliser la membrane. Cette « instabilité de courbure » peut
aider à comprendre certains changements de forme observés dans les membranes réelles, tels que l’echinocytose
de globules rouges.
Résumé.
2014
priétés physiques
2014
2014
A simple, thermodynamical model was proposed some time ago [1, 2] to describe the physical proAbstract.
perties of various systems for which the curvature elastic energy plays an important role. The basis of this model
is provided by the notions of effective rigidity and spontaneous curvature. Here we consider the case of bilayer
a possible source of nonmembranes and generalize the model for situations where small adsorbed molecules
can diffuse within the membrane. We show that under certain conditions they
zero spontaneous curvature
destabilize the membrane completely. This « curvature instability » can help to explain certain observed shape
transformations of real membranes, such as the echinocytosis of red blood cells.
2014
2014
2014
interested in their
1. Introduction
One can imagine an idealized membrane as consisting
of two homogeneous layers of phospholipids oriented
with their polar heads towards the exterior of the
membrane. Such a lipid bilayer can be in fluid state [3],
which means that the molecules diffuse freely within
the layers.
This representation is of course a crude approximation of real biological membranes, which in fact
possess a quite complicated structure, including heterogeneities in their composition, small intercalated
molecules ’such as cholesterol, and many intramembrane proteins. To describe them, one should rather
use fluid mosaic models [4], much richer and also
more
complicated
Yet, the idealized lipidic bilayers, which
can
be
produced and studied in the form of artificial vesicles,
play an important role as a starting point for modelling real biological membranes, especially if one is
simple physical properties (such as
transformations,
elasticity, mutual interactions,
shape
transport, etc.). They should be thought of as some
kind of homogenization of the plasma membranes.
Several years ago a simple model was proposed
[1, 2, 5] to explain the mechanical and thermodynamical properties of such closed, bilayer membranes.
On the basis of this model the notion of the elastic
curvature energy of the membrane is proposed. The
system is characterized by two phenomenological
parameters : the effective rigidity K and the spontaneous curvature Ho. These two parameters are easily
connected with the microscopic description of the
membrane [1] : the rigidity x is connected with the
binding of the membrane, and thus with changes in the
orientation of phospholipids, whereas the spontaneous
curvature Ho takes into account the eventual asymmetry in the distribution of the constituents in two
layers.
These important notions, introduced here by W.
Helfrich, F. Brochard and others [1, 2, 5, 6], have also
been useful for the description of other physical
systems such as microemulsions [7] or micelles [8].
Generally speaking the elastic curvature energy plays
a crucial role in all systems with small surface tension.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703050700
508
The success of the simple phenomenological model
in the study of artificial bilayer vesicles or real biological membranes has consisted mainly in :
(i) explaining a large class of possible shapes of
[6] ;
(ii) quantitatively describing some types of unusual
dynamical behaviour of the membranes, such as the
so-called flicker phenomenon observed in red blood
cells [5].
these systems
In the next section we shall give a more detailed
account of this model, its achievements and also some
of its limitations. In particular one of the simplifications
of the model is the assumption of the homogeneity
of the membrane. It for instance, a pure membrane
includes some kind of impurities (e.g. small intercalated particles, a different sort of lipid molecules, etc.),
which can eventually be a source for the nonzero
value of the spontaneous curvature, then one supposes that their distribution is homogeneous.
This assumption is not necessarily correct for the
membranes which constituents are free to diffuse
laterally. The concentration c of the « impurities »
for instance, can differ from one point to another, as
well as vary in time. The aim of this paper is to consider
such situations, and to try therefore to extend the
domain of application of the simple phenomenological
model.
In section 3 we consider a membrane which contains
a certain amount of diffusing, intercalated particles.
We suppose that their concentration c is coupled to
the local (mean) curvature H of the membrane. As a
consequence the membrane acquires a nonzero spontaneous curvature. This is not however the only effect
of the presence of the impurities. In some cases the
membrane can become unstable : the density fluctuations can couple to the curvature modes and
destabilize an initially flat membrane. We consider
both static and dynamical aspects of this phenomenon,
which we call the curvature instability. In both
approaches we derive a simple instability criterion
which gives us an insight into the physical basis of the
predicted phenomenon.
It is not impossible that
The limitations of our approach are given in
section 5. In fact, our calculation is some kind of linear
stability analysis. To describe the real systems correctly one must include the nonlinear energy terms,
as well as more details about the microscopic structure
of the membrane. We therefore sum up in conclusion
some open questions which are beyond the scope of
this paper and suggest some directions for further
experimental and theoretical studies.
2. Idealized membranes and the
mical model.
simple thermodyna-
In the
simple phenomenological model a membrane
is treated as a continuous, fluid, bidimensional
system, characterized by the following elastic energy
per unit area [1] :
where H is the mean curvature at a given point of the
membrane (H 1/Rl + I/R2, Rl,2 being the two
principal curvature radii of the surface); whereas K
and Ho are two phenomenological parameters called
rigidity and spontaneous curvature respectively. As
we have already mentioned in the introduction the
rigidity K is connected to the resistance to bending of
the membrane, while Ho takes its possible asymmetry
into account.
We shall not discuss here the nature of the approximation on which this model is based. All relevant
details can be found in several papers [1, 2, 6, 11].
Let us only recall that when writing the elastic energy
in the form (1), one is actually neglecting :
=
(i) any changes in membrane topology;
(ii) any area dilatation and is supposing the effective
surface tension to be zero ;
(iii) any shear resistance of the membrane - an
assumption reasonable as concerns our problem,
but not valid in general [11, 12] ;
(iv) other possible terms such as tilting energy or
any kind of nonlinear curvature-elastic stresses, etc.
the curvature instability [1.13];
(v) any dissipation effects within the membrane [2].
constitutes one of the mechanisms of the shape transHaving established the form of the effective surface
formations in real membranes. Shape changes occur
in connexion with many important phenomena such energy of the membrane one can apply a standard
thermodynamic treatment of the model. What is
as cell locomotion, fusion, secretion, endocytosis,
etc.
to
Thanks
its
[9].
phagocytosis
relatively simple surprising about this simple model is that it seems to
morphology and its availability, the red blood cell describe adequately many physical properties of real
constitutes a model system to study such shape/4hembranes such as erythrocytes for example :
(i) in fact Helfrich and Deuling [6] obtained a large
changes. As we have already mentioned, even
model
can
simple phenomenological
variety of the red blood shapes by simply minimizing
explain
the elastic free energy for a fixed surface and volume
4
In
se
ion
family of possible shapes of erythrocytes [6].
of the cell. In particular they were able to obtain the
we suggest that the well known phenomenon of
echinocytosis or crenation of red blood cells can be typical discoidal shape of erythrocytes (for certain
connected with the curvature instability. In particular negative values of the spontaneous curvature Ho)
as well as other shapes observed experimentally.
we recall that echinocytosis can be produced in vitro
(ii) Brochard and Lennon [2] quantitatively studied
by adding some small particles which intercalate
the so called flicker phenomenon in erythrocytes,
themselves within the membrane [10].
the
a large
509
which consists in large fluctuations (flickering) of
active red blood cells. They measured the space and
time correlations of these fluctuations by means of
phase contrast microscopy. The correlation functions
and the scaling laws which can be deduced from them
are in agreement with the predictions of the simple
model ! We shall return to the dynamical analysis of
the thermal fluctuations of the membrane in the next
section; let us only recall here the principal theoretical
results of Brochard and Lennon concerning the
fluctuation modes of a single membrane (i.e. the limit
qd &#x3E;&#x3E; 1, where q is the wavevector of the considered
dynamical mode and d is the thickness of the disco-
cyte).
If one supposes that the elastic energy of the membrane is given by equation (1), and that the liquid
beneath the membrane (inside the red blood cell)
is a simple Newtonian fluid with viscosity and
density p, then there are two fluctuation modes
present in the system :
(i)
(ii)
a
a
slow mode for which
fast mode for which
Remark : the factor i present in these formulae
that the modes are not propagating ones.
They decay exponentially with time as exp(- Wl,2 t).
The power spectrum of fluctuations is entirely
dominated by the slow mode :
possible
presence of diffusing molecules, different
from the phospholipidic constituents of the membrane,
in the simple thermodynamic model (1).
Let us suppose that the interaction of the intercalated
molecules with their neighbouring phospholipids is
« multiform », e.g. they interact much more strongly
with polar heads than with the lipidic chains and/or
they are adsorbed in the outer layer (Fig. 1). This
means that the adsorbed molecules prefer tilted
configurations of their neighbouring phospholipids,
configurations which are typical for locally curved
membranes. The intercalated molecules thus couple
to the local curvature of the membrane surface.
We can therefore take this into account by including
the following interaction term in the surface energy (1)
of the system :
where 0 is the local density of intercalated molecules
and A is a coupling constant.
The intercalated particles, which diffuse within the
membrane, interact with the phospholipidic constituents, and also among themselves. One must thus
include the third term into the free energy functional :
means
One of the important results of this calculation is the
fact that the membrane is stable against perturbations :
the frequencies w1 and OJ2 are both positive for all
wavevectors q, therefore all the perturbations decrease
exponentially with time.
In the next section we shall see that this is no longer
true if the membrane includes some free moving
molecules, which, preferring curved to flat locations,
can under certain conditions destabilize the membrane. In the static approach (as that of Helfrich and
Deuling) this will correspond to the vanishing of the
effective rigidity Keff.
where f[
Ø] can be written as :
We have supposed that there is no long range
interaction between the intercalated particles, such
as dipolar forces for example. The total surface energy
of the membrane now is :
3. Intercalated
since the membrane is supposed symmetric in the
absence of the intercalated particles (Ho
0).
The analysis of the general case given by (8) and (7)
is beyond the scope of this paper. We shall rather
The success
described in
as the demonstration that the molecules intercalated
in the lipidic membrane do not necessarily influence
its physical properties in an important way. They
would at most change the phenomenological parameters such as the rigidity x or the spontaneous
curvature Ho, but not alter the behaviour of the
membrane qualitatively. Yet, this is not always the
case. To see this we shall now take into account the
Schematic representation of how intercalated
Fig. 1.
molecules (absorbed drugs, intramembrane proteins, etc.)
can be coupled to the local curvature of the bilayer membrane.
particles and curvature instability.
of the simple thermodynamic model
the previous section would be viewed
=
-
510
consider some special situations, easier to analyse,
and then discuss possible consequences (and shortcomings) of our approximation.
Let us suppose that the
3.1 STATIC APPROACH.
functional f [0J is symmetric in tP, that is p
a3
0. This is equivalent to considering only a subspace of a general phase diagram (K, Ho, Jl, a, a3, ...).
If we describe the position of the membrane by the
function ’(x, y), then
-
=
...
=
=
where A is the Laplace operator. In this formula we
have neglected the anharmonic terms [14] and have
supposed that the membrane is not too crumpled
(i.e. VC is small, there are neither overlaps nor overhangings of the membrane, etc.).
Taking the Fourier transform we can write (9a) as :
where
and fl is the curvature corresponding to new effective
variables.
Thus, in the presence of diffusing particles, coupled
to the local curvature of the membrane, the effective
rigidity decreases. This seems indeed to agree with
well known experimental observations in different
physical systems. For example in microemulsions, the
addition of cosurfactants, which intercalate into the
surfactant-built interface, diminish its effective rigidity,
and thus the persistence length of the system [15].
Analogous phenomena also occur in other systems,
for instance in ferroelectric liquid crystals, where
polarity replaces the « impurities » concentration [16].
We also find that for A’ -+ aK some instability
phenomenon takes place. To understand its nature
better, we shall now perform a dynamical analysis
analogous to that of Brochard and Lennon [2] for a
« pure » membrane.
3.1 DYNAMICAL APPROACH.
We shall write the
local density of the intercalated molecules in the
following form :
-
We can now easily diagonalize this quadratic form
and then integrate over eigenvectors with an energy
gap E(q) - a + bq2 + ... (« hard modes »). In this
way we obtain the effective energy functional :
where Ho is the spontaneous curvature which takes
into account the presence of the intercalated molecules.
The inside of the vesicle will be supposed to be a
simple Newtonian fluid We can write down the
hydrodynamical equations for the liquid under the
membrane [ 17] :
where v is the velocity of the fluid andp is local pressure.
At the surface of the membrane the velocity of the
fluid and the displacement C are connected by the
equation :
The system is
equal
to :
subject
to the force
P.,
per unit
where Po is the mean value of the density of the particles.
The free energy functional now takes the following
form :
Thus the boundary conditions on the surface for
hydrodynamical equations (13), (14) take the following form [ 15J :
Let us now consider the intercalated molecules and
suppose that they can diffuse in the lipid membrane.
The equation for the time evolution of their local
concentration can be written in this simple form :
area
where j is the
«
current » for intercalated
particles.
511
It is
directly connected with
free energy
Fmal
+
F;nt :
Again we suppose the simple form of the free energy
Fmol
The boundary conditions (17) and the intramembrane continuity equation (21 a) applied to the hydrodynamical equations (13), (14) induce the dispersion
relation for the dynamical modes (22).
If we introduce the reduced variables [17] :
then this
The constants a and b are supposed to be positive
to exclude the situations of spontaneous demixing
or aggregation of the intercalated particles.
From equation (20) we get the simple equation
for the time evolution of c(x, y) :
equation
can
be
expressed
as :
which is a rather complicated relation between
the wavevector q of the considered mode and the
frequency m (remark that S S(q, w), y y(q),
=
p =
The first term is just a simple diffusion term;
the diffusion constant D has been measured for different molecules inside phospholipid bilayers, for proteins it is of the order of 10-9 cm’ s-1. The second
term, of entropic origin, expresses the fact that, in
the absence of all coupling with curvature H, the
intercalated molecules would prefer homogeneous
distribution over the membrane. Finally the last
term in equation (18) originates from the interaction
Fint with the mean curvature H(x, y) present in
equation (17) : the intercalated molecules induce the
curvature, but are also « attracted » by curved regions.
We can now study the stability of the membrane
against fluctuations. Let us consider a simple sinusoidal perturbation :
Equation (21) then takes
Which
by taking
the
on a
Laplace
lt(q)).
=
-
From equation (24) we can naturally -obtain the
results of Brochard and Lennon (2), (3) of slow and
fast (stable) modes OJ! 2’ if we suppose that the intercalated molecules are weakly coupled to the curvature (A = 0) or prevented from diffusing in the
membrane (D = 0).
Let us consider the slow mode (1 »S solution
of equation (24) which dominates the spectrum.
We have :
that is :
We must realize that the constant 1/F is very
small. In fact the diffusion constant of the liquid
inside the vesicle D is of order of 0.1 cm2 s-1, thus
1/F = D/D ~ 10-8 1. We shall also suppose that :
simple form :
transform
which is a very realistic assumption for real systems
such as erythrocytes. We can therefore rewrite equation (25) as :
from which
we
get :
Let us first observe that S is always real which means that the mode remains
There are two branches for the mode (01 corresponding to two signs in (26) :
non-propagating : e - St, S real.
512
The w+ branch is positive for all values of q. On the contrary the OJ - branch can become negative and thus
the slow mode becomes unstable.
In fact from equation (27) we get the following assymptotic behaviour of OJ- :
We therefore conclude that for :
the
dynamical
mode becomes unstable for wavevectors q :
Equation (29) is the simple condition of the instability of the membrane in the presence of intercalated,
diffusing molecules. It connects the static quantities
for the membrane A, a, and K. We call this instability
the curvature instability. Figure 2 schematically
summarizes the dispersion relation (27) for the slow
mode cvi. Let us recall that in the absence of the
intercalated particles w1 is of the order 10 s-1 [2]
for q - Tc/1
gm.
We have thus generalized the simple thermodynamical model of bilayer membranes for the case
when the concentration of the intercalated particles
can change locally. We have seen that the presence
within the membrane of adsorbed molecules, coupled
to the local curvature and diffusing inside the layers,
have two main consequences on the behaviour of
the system :
(i) it changes the effective spontaneous curvature
of the membrane, since, as we can see from equation (9b) : Ho -+ Ho + AipolK and it can therefore
modify the shape of the cell continuously, as predicted
by the calculations of Helfrich and Deuling [6]
(e.g. provoke the stomatocyte into discocyte transformation) ;
A similar conclusion was also drawn by Evans
within the framework of a different model [18]; in
that model, one assumes that the two membrane
layers are unconnected and then by alterning their
relative surface chemical equilibrium one induces
spontaneous curvature into the system;
(ii) it can trigger off the thermodynamical instability of the membrane : the concentration of intercalated molecules increases in the regions of higher
curvature and can provoke the changes in the shape
of the membrane. In particular the regions with higher
concentration of « impurities » will curve strongly.
The final shape of the membrane will of course depend
on many different factors, such as nonlinear interactions within the membrane, the possible presence
of long range forces, the morphological details of
the membrane (e.g. the presence of intramembrane
proteins) etc. We shall return to this point in the
next section.
Fig. 2.
Instability of the slow dynamic mode mi in the
presence of intercalated particles, coupled to the local
-
curvature
(for A2&#x3E; aK).
The idea that curvature and molecular segregation
be coupled seems to be of more general interest,
and has already appeared in other studies [19]. For
instance 2 D mixtures of smectic phases of different
symmetry can undergo lateral phase separation,
which leads to the appearance of domains. The
domain structure is accompanied by a variation
of local curvature. A simple analysis of this phenomenon done by Gebhardt, Gruler and Sackmann
was in fact based on the notion of curvature
can
energy [20].
The condition for the instability to take place
has a very simple form in our model : A2 &#x3E; xa. In
fact all this is quite intuitive : the instability is more
likely to occur if the particles are strongly coupled to
the membrane curvature (A big) and for membranes
with a small rigidity constant K (which get curved
more easily). The constant a depends on several
physical parameters : the temperature T of the system,
the mean concentration OPO of the intercalated particles, etc. Its presence in the instability condition
suggests the importance of curvature instability near
the consolute point for the mixture of molecules
in the membrane.
513
4. Possible connection with the observed
formations in real membranes.
shape
trans-
The ideal physical systems to experimentally verify
the existence of the curvature instability in membranes would be giant artificial vesicles. One could
adsorb in such pure membranes some small particles
interacting strongly with phospholipidic constituents
and inducing the curvature locally. In these systems
one could in principle control the rigidity K, the diffusion constant D and the thermodynamical coefficients a, b, a2,... (through the variations of the temperature, the concentration 410, the chemical potential,
the nature of intravesicle fluid etc.). In a certain
range of control parameters one would eventually
observe then the changes in the shape of the vesicles
characteristic for the curvature instability. To our
knowledge such experiments have not yet been performed There are however some interesting observations in real biological membranes which can be
connected with the curvature instability. As an
example, we have chosen the so called crenation of
the red blood cells, the analysis of which will naturally
enlarge the modelization of Deuling and Helfrich.
However there exist other analogous phenomena,
such as some kind of endocytosis, for which we believe,
our considerations could apply.
In fact, the « model system » on which to study
membrane shape changes is the human erythrocyte.
It has been largely investigated for many years and
is thus relatively well known [21]. This cell is one
of the simplest systems : the external bilayer membrane is connected with a thin, filamentous, protein
structure, referred to as the membrane skeleton
(cytoskeleton), but does not interact otherwise with
its interior. For various mechanical considerations
it can be viewed as a membrane « balloon » filled
with a simple, homogeneous liquid [22]. Despite
this simplicity, the human red blood cell has striking
physical properties : while it is usually found under
its familiar biconcave, discoid form, it can squeeze
through capillaries much smaller than its own diameter and rapidly revert back to its normal discoid
shape when in a wider blood vessel. However, under
the action of different chemical, mechanical or biological factors it can also alter its shape completely,
and lose its fascinating properties : these changes
take place in numerous blood diseases [23], but can
also be used by the organism to get rid of the old,
« worn-out » erythrocytes from the blood circulation [24].
The simplest kind of shape changes in the erythrocyte occurs when we place it in a hypotonic
medium : its volume increases while its area remains
constant, the cell thus becomes spherical and then
lyses [25]. In the second class of shape modification,
the volume of the cell remains practically unmodified.
These kind of changes can, roughly speaking, take
place in two different ways. Firstly the cell can
crenate [26], i.e. its membrane can sprout spicules,
and make the cell look like a small sea urchin; it is
then given the name of echinocyte. Or the cell can
become cup-shaped, and is referred to as a stoma-
tocyte [27].
of these two processes are reverthe transformation of discocytes
into echinocytes or stomatocytes becomes irreversible ; in fact the effective area of the membrane
decreases and the cell becomes approximately spherical, until eventually lyses. In the later stages of the
processes one often observes [28] the formation of
small microvesicules which separate off from the
membrane’s spicules or invaginations.
Many different causes of the crenation transformation are actually known. For example it can
be provoked in vivo by metabolic deplation of the
red cell [29]. Thus when cells are incubated in the
absence of glucose, the intracellular ATP is consumed
and crenation takes place [30]. Because the abundance
of calcium ions triggers off the deplation of the ATP
crenation can be induced by adding Ca++ [31].
The high pH of the extracellular solution [32] or
simply the presence of the glass wall near the cell
(so called « glass effect ») [33], which increases the
local pH level are other echinocytogenic factors.
While such experiments are currently performed
in the laboratories, the molecular explanation of this
interesting phenomenon is far from being definitely
established [21]. What is the most difficult to understand seems to be the metabolic shape control mechanism. The role of membrane skeleton, built from
proteins such as spectrins, actins etc. [34], in the stabilization of the cell membrane shapes has been proven
beyond doubts [22]. Until recently it was believed
that the shape of erythrocytes was controlled by the
phosphorylation of spectrins under the action of
ATP and an endogenous kinase [35]. However,
recent experiments [36] have shown that things are
far less clear : echinocytosis occurs before the phosphorylation of spectrins becomes apparent; moreover
none of the known properties of (pure) spectrins are
sensibly perturbed by this process. Research has
therefore turned to other possible explanations [21],
none of which is for the moment satisfactory. The
rpost interesting one seems the hypothesis that
ATP is required to maintain the dynamic state of the
cytoskeleton complex. The ATP depletion might
allow a cooperative clustering of spectrins, and thus,
by locally uncoupling them from the membrane
(and the intramembrane proteins in particular), could
destabilize the membrane [37].
There exists however a much simpler way to cause
crenation in vitro, and that is to put the cell in an
anionic solution : the adsorption of anionic or noncharged amphiphilic agents rapidly induce the discocyte-echinocyte transformation [10]. Several years
ago, Sheetz and Singer [38] suggested that anionic
molecules (which, in practice, are often small anaesthetic drugs) intercalate mainly into the lipid in the
The
early stages
sible, but
soon
514
exterior half of the membrane bilayer, expand that
layer relative to the cytoplasmic half, and thus cause
the cell to crenate. This was the famous bilayer
couples hypothesis, since the authors compared the
reaction of the membrane with the response of a bimetallic couple to changes in temperature. Of course,
this is a very simplified image, but the main idea
that crenation can be triggered off by the asymmetric
adsorption of amphipatic molecules nevertheless
remains true. Several predictions of this hypothesis
have been confirmed experimentally by its authors,
and other groups [38, 10]. From our point of view
the adsorbed drugs are in fact the «impurities
coupled to the curvature and diffusing inside the cell
membrane.
Another observation also supports the hypothesis
that echinocytosis could be provoked by a certain
kind of curvature instability. During crenation one
observes important changes in the lateral distribution
of the intramembrane particles, in particular the strong
accumulation of some constituents in the spicules
and the microvesicles produced during crenation [10].
Of course the echinocytosis which takes place in vivo
is much more complicated We believe however that
the dynamical « condensing » of the intercalated,
diffusing molecules can play a certain role in other
mechanisms of crenation. Let us suggest here two
possible mechanisms :
it was put forward in an interesting paper by
Nabarro et al. [39] the molecules of 1, 2-diacylglycerol,
which accumulate inside the cell -!during the ATP
depletion [31], could substitute for phosphatidylcholine in the inner layer of the membrane. Being a
fusogenic lipid, and having a similar fatty acid composition to the phosphatidylcholine but a smaller
polar head, each molecule of 1, 2-diacylglycerol can
act as an intercalated molecule in our model, (negatively) coupled to the local curvature. The strong
increase in the concentration of these molecules
has indeed been observed in the microvesicles produced during crenation [28];
(i)
as
(ii) there now exists strong experimental evidence [21, 34] that the membrane skeleton limits the
translational mobility of the erythrocyte membrane
particles, in particular the intramembrane proteins.
On the other hand during crenation one observes
important changes in the lateral distribution of these
particles within the membrane [40]. Therefore, if the
mechanism of dynamical cooperative clustering of
spectrins [21, 37] was proved true, one can imagine
that in regions which are free of cytoskeleton the
analogue of the curvature instability could take
place. In fact in these regions certain intramembrane
molecules could now freely diffuse, and destabilize
the membrane if they are coupled to the local curvature. Of course this hypothesis needs to be checked
on the molecular level;
(iii) as we have already mentioned, changes in
Fig. 3. Schematic view of the echinocytosis (or crenation
transformation) in human red blood cells. Only the first,
-
reversible state of the process is shown.
pH level of the solution can also induce echinocytosis. One of the possible explanations for this
phenomenon could be that changes in pH (as well as
other factors such as ion binding, adsorption onto
the membrane, etc.) modify the elastic properties
of the membrane. In particular the decrease of the
rigidity could in principle cause the curvature instability, as condition (29) seems to imply. Again this is
only a hypothesis needing to be verified experimentally.
5. Conclusions.
In this paper we have tried to apply the simple thermo-
dynamical model of bilayers to the case when a certain
number of small intercalated molecules can diffuse
inside the membrane. We have shown that these
molecules can not only induce the nonzero (spontaneous) curvature but also, under some conditions,
destabilize the membrane. As a consequence, the
shape of an artificial or a biological vesicle would
change completely.
It would of course be tempting to perform a detailed
calculation of the final form of such « destabilized »
vesicles. Within the framework of their model, Helfrich
and Deuling [6] obtained shapes which indeed looked
like real discocytes, stomatocytes or other red blood
cells. Could one also get, by generalizing their model,
the shapes of echinocytes, or even predict the observed
separation of the microvesicles ? This should be
possible in principle, however we encounter several
difficulties which make the case we have studied
much more complicated than that considered by
Helfrich and Deuling.
First, allowing the density of adsorbed molecules
to vary, one enlarges the space of trial shape functions
enormously. One can of course consider much simplified situations, such as the one envisaged by Nabarro
et al. [39], where the vesicles are spheres with small
semispherical spicules on them, whereas the density 0
takes only two discrete values. This seems rather
to be a crude approximation, but it can give some
insight into physical basis of the phenomenon. One
can also hope to overcome this problem posed by
515
the large variety of possible shapes and density variations by using appropriate numerical methods.
More important however are the difficulties connected with the nonlinear nature of our system. In
fact nonlinearities come here from different sources :
when perturbations start to grow the gradients
become
VC
appreciable and thus the anharmonic
terms must be included into the model of Helfrich
et ale [14];
(i)
(ii) when the concentration P of the intercalated
molecules increases in some regions their mutual
interactions become important, and thus one must
modify the energy functional (9). It should be noted,
that these interactions depend strongly on the microscopic nature of the considered system. They can
origin from the direct electrostatic or hydrophobic
forces, but also be mediated by the phospholipidic
layers or even the proteins of the cytoskeleton. These
interactions are interesting since they can, in some
cases, become repulsive [41] and thus stabilize the
nonhomogeneous distribution of the intercalated
particles;
(iii) for important deformations the relationship
between stresses and strains is no longer linear, as
it was pointed out and discussed by many authors [42];
(iv) for highly curved membranes, such as spicules,
it was shown that the shear cannot be neglected [43];
(v) for modal wave lengths of the order of the cell
dimension viscous dissipation in the membrane may
be important.
These and other nonlinearities present in the problem, as well as the fact that the real membranes
are finite, closed objects, will determined which
unstable wavevectors will contribute the most to the
final state of the membrane [44]. Thus, obtaining the
final shape of destabilized vesicles seems to be quite
a hard task. Especially that one should in principle
rather
generic situation, defined by many
potentially pertinent parameters of the problem :
p, b, a, a2, a3, ... in equation (7).
Our hypothesis that the shape changes of real
consider
a
membranes could in
some
cases, such
as
echino-
cytosis considered in the last section, have something
to do with the curvature instability is thus based
only on qualitative observations. Namely, echinocytosis can be provoked by the adsorption of small
anionic molecules (e.g. anaesthetic drugs) into the
membrane, and the concentration of these molecules
is increased in highly-curved spicules. To our knowledge, there are no quantitative and systematical
measurements of crenation nor of any other analogous phenomena. Real systems such as red blood
cells are very complex and the phenomenological
models e.g. the one built by Helfrich, Brochard and
others, are not able to describe all the physical phenomena which take place there. Yet, it would be interesting to see where the limits of their applicability are
and whether such limits, when found, can teach us
more about the role of different constituents of
biomembranes (e.g. some intramembrane proteins
the cytoskeleton etc.). We hope that further development of simple, phenomenological models, parallel
to the experiments carried out on simple artificial
systems can bring about some progress in this
direction.
Acknowledgments.
I am most gratefully indebted to Y. Bouligand,
F. Brochard, T. Dombre, P.-G. de Gennes and L. Peliti
for helpful discussions and encouragements. Part
of this work was performed during the Les Houches
conference on « Disordered Systems and Biological
Organization » ; I would like to acknowledge the
hospitality of the organizers, as well as the financial
support of DRET.
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