Curvature instability in membranes

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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�

507

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 )

Résumé.

Un modèle thermodynamique simple ^{a été} proposé ^{il y a} quelques temps [1, 2] pour décrire les pro-

priétés physiques ^{de 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 mem-

branes bi-couches et généralisons ^le modèle pour des situations où de petites ^{molécules adsorbées}

^{une source} possible de courbure spontanée ^{non nulle}

peuvent diffuser dans la membrane. Nous montrons que dans ^cer- taines 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.

Abstract.

A simple, thermodynamical ^{model was} proposed ^some time ago [1, 2] ^to describe the physical ^pro- 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

membranes and generalize the model for situations where small adsorbed molecules

a possible ^{source of non-}

zero spontaneous ^curvature

^can diffuse within the membrane. We show that under certain conditions they

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.

J. Physique ⁴⁷ (1986) ^{507-516 MARS} 1986,

Classification Physics ^Abstracts 87.20201368.15201305.90

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} approxi-

mation of real biological membranes, which in fact possess ^a quite complicated structure, including ^hetero- geneities ^{in their} composition, small intercalated molecules ’such as cholesterol, ^and many intramem- brane 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 model-} ling ^real biological membranes, especially ^{if one is}

interested in their simple physical properties (such ^as shape transformations, elasticity, ^mutual interactions, 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 thermodyna-

mical 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} sponta-

neous 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 _asym- metry ^{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

The success of the simple phenomenological ^model

in the study ^of artificial bilayer ^{vesicles or real} biologi-

cal membranes has consisted mainly ^{in :}

(i) explaining ^a large ^{class of} possible shapes ^of

these systems [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].

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 interca- lated particles, ^{a different sort} of lipid molecules, etc.),

which can eventually ^{be a source for the nonzero}

value of the spontaneous curvature, then ^one sup- poses 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} spon- taneous curvature. This is not however the only ^effect

of the _presence of the impurities. ^{In some cases the}

membrane can become unstable : the density ^fluc-

tuations 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 curvature} instability

constitutes one of the mechanisms of the shape ^trans-

formations in real membranes. Shape changes ^occur

in connexion with _many important phenomena ^such

as cell locomotion, fusion, secretion, endocytosis, phagocytosis ^etc. [9]. ^{Thanks to its} relatively simple morphology ^{and its} availability, the red blood cell constitutes a model system ^to study ^such shape changes. ^{As we have} already mentioned, ^even the

simple phenomenological ^{model can} explain a large

family of possible shapes of erythrocytes [6]. ^In se ion ⁴

we suggest that the well known phenomenon ^of echinocytosis ^or crenation of red blood cells can be connected with the curvature instability. ^In particular

we recall that echinocytosis ^{can be} produced in ^vitro by adding ^{some small} particles which intercalate themselves within the membrane [10].

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 ^cor-

rectly ^{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 simple thermodyna-

mical model.

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 approxi-

mation 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.

[1.13];

(v) any dissipation effects within the membrane [2].

Having established the form of the effective surface energy of the membrane ^{one can} apply ^{a standard} thermodynamic ^{treatment of the model. What is} surprising about this simple ^{model is that it seems to}

describe adequately many physical properties ^{of real} /4hembranes ^{such as} erythrocytes ^for example :

(i) in fact Helfrich and Deuling [6] ^{obtained a} large variety of the red blood shapes by simply minimizing

the elastic free _energy for a fixed surface and volume of the cell. In particular they ^{were able to obtain the} typical ^discoidal shape ^of erythrocytes (for ^certain negative values of the spontaneous ^curvature Ho)

as well as other shapes ^observed experimentally.

(ii) ^Brochard and Lennon [2] quantitatively ^studied

the so called flicker phenomenon ⁱⁿ erythrocytes,

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 >> 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 mem-

brane 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) ^a slow mode for which

(ii) ^a fast mode for which

Remark : the factor i present ⁱⁿ these formulae

means 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 :

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 mem-

brane. In the static approach (as that of Helfrich and

Deuling) ^{this will} correspond ^{to the} vanishing ^{of the}

effective rigidity Keff.

3. Intercalated particles ^{and curvature} instability.

The success of the simple thermodynamic ^model

described in the previous section would be viewed

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 para- meters 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

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 ^consti-

tuents, and also _among themselves. One must thus include the third term into the free _energy functional :

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 :

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

Fig. ^1.

^Schematic representation ^{of how} intercalated molecules (absorbed drugs, intramembrane proteins, etc.)

can be coupled ^{to the local curvature of the} bilayer ^mem-

brane.

consider some special situations, ^{easier to} analyse,

and then discuss possible consequences (and ^short- comings) ^{of our} approximation.

3.1 STATIC APPROACH.

Let us suppose that the functional f [0 J is symmetric ^{in tP, that} is p ⁼ a3 ⁼

...

= 0. This is equivalent ^to considering only ^{a sub-}

space 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 over-} hangings ^{of the} membrane, etc.).

Taking the Fourier transform we can write (9a) ^{as :}

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

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 :}

where Po ^{is the mean} value of the density ^{of the} par- ticles.

The free _energy functional now takes the following

form :

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 subject ^{to the force} P., per unit area

equal ^{to :}

Thus the boundary conditions on the surface for

hydrodynamical equations (13), (14) take the fol-

lowing ^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 :}

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 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) :

The first term is just ^a simple diffusion term;

the diffusion constant D has been measured for diffe- rent molecules inside phospholipid bilayers, ^for pro- teins 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 ^sinu-

soidal perturbation :

Equation (21) then takes on a simple ^{form :}

Which by taking ^the Laplace ^transform

The boundary conditions (17) and the intramem- brane continuity equation (21 a) applied ^{to the} hydro- dynamical equations (13), (14) induce the dispersion

relation for the dynamical ^modes (22).

If we introduce the reduced variables [17] :

then this 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 = 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 inter- calated molecules are weakly coupled to ^{the cur-}

vature (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 :

which is a very realistic assumption ^{for real} systems such as erythrocytes. ^{We can} therefore rewrite _equa- tion (25) ^{as :}

from which we get :

Let us first observe that S is always real which means that the mode remains non-propagating : e - St, ^{S real.}

There are two branches for the mode ₍₀₁ corresponding ^{to two} signs ⁱⁿ (26) :

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 insta-

bility 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 ^thermo- dynamical ^{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} equa- tion (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 ^trans- formation) ;

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 ^insta- bility of the membrane : the concentration of inter- calated 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 inter-} actions 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> aK).

The idea that curvature and molecular segregation

can 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} pheno-

menon done by Gebhardt, Gruler and Sackmann

was in fact based on the notion of curvature energy [20].

The condition for the instability ^{to take} place

has a very simple ^{form in our model : A2} > 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 _par-

ticles, ^{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 shape ^trans-

formations in real membranes.

The ideal physical systems ^to experimentally verify

the existence of the curvature instability ^{in mem-}

branes 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 diffu-}

sion constant D and the thermodynamical ^coeffi-

cients a, b, a2,... (through the variations of the tempera- ture, 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 per- formed There are however some interesting ^observa-

tions 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 ^mem-

brane 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 dia- meter and rapidly ^{revert back to} its normal discoid

shape ^{when in a} wider blood vessel. However, ^under the action of different chemical, mechanical or bio-

logical 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 circu- lation [24].

The simplest ^{kind of} shape changes ^{in the} ery-

throcyte ^{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].

The early stages ^{of these two} processes ^{are rever-}

sible, ^{but soon} the transformation of discocytes

into echinocytes ^or stomatocytes becomes irrever- sible ; in fact the effective area of the membrane decreases and the cell becomes approximately spheri- cal, ^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 trans- formation 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 under- stand seems to be the metabolic shape control mecha- nism. The role of membrane skeleton, built from

proteins ^{such as} spectrins, ^{actins etc.} [34], ^{in the sta-}

bilization 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 phospho- rylation ^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 non-

charged amphiphilic agents rapidly induce the dis-

cocyte-echinocyte transformation [10]. ^Several years ago, Sheetz and Singer [38] suggested that anionic molecules (which, ⁱⁿ practice, ^are often small anaes-

thetic drugs) intercalate mainly ^{into the} lipid ^{in the}

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 bime- tallic 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 :

(i) ^{as 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 phosphatidyl-

choline in the inner layer of the membrane. Being ^a fusogenic lipid, ^and having ^{a similar} fatty ^acid compo- sition to the phosphatidylcholine ^{but a smaller} polar head, ^each molecule ^of 1, 2-diacylglycerol ^can

act as an intercalated molecule in our model, (nega- tively) coupled ^to the local curvature. The strong increase in the concentration of these molecules has indeed been observed in the microvesicles _pro- duced during ^crenation [28];

(ii) ^{there now exists} strong experimental ^evi-

dence [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 cur-}

vature. ^{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 echino-

cytosis. 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 insta-} bility, ^{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 (spon- taneous) ^{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 sim-

plified 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 ^varia-

tions by using appropriate numerical methods.

More important ^{however are} the difficulties con-

nected with the nonlinear nature of our system. ^In fact nonlinearities come here from different sources :

(i) ^when perturbations ^{start to} grow the gradients VC become appreciable and thus the anharmonic

terms must be included into the model of Helfrich

et ale [14];

(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 micro-} scopic ^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} pro-

blem, ^{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

consider a 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}

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, ^echino- cytosis ^{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 know-} ledge, ^{there are no} quantitative ^and systematical

measurements of crenation nor of _any other analo- gous 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 pheno-

mena which take place ^there. Yet, it would be interest-

ing ^{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 develop-

ment 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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