CRITICAL BEHAVIOR OF VESICLES AND MEMBRANES

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CRITICAL BEHAVIOR OF VESICLES AND MEMBRANES

R. Lipowsky

To cite this version:

R. Lipowsky. CRITICAL BEHAVIOR OF VESICLES AND MEMBRANES. Journal de Physique

Colloques, 1990, 51 (C7), pp.C7-243-C7-248. �10.1051/jphyscol:1990724�. �jpa-00231123�

COLLOQUE DE PHYSIQUE

Colloque C7, suppl6ment au n023, Tome 51, ler d6cembre 1990

CRITICAL BEHAVIOR OF VESICLES AND MEMBRANES

R. LIPOWSKY

I n s t i t u t fur Festktirperforschung. Forchungszentrum Jiilich, 0-5170 Jiilich, F.R.G.

Abstract

-

Recent theoretical work on the critical behavior of vesicles and membranes is reviewed.

Vesicles can exhibit a wide variety of different shapes and shape transformations which can be continuous or discontinuous. In the presence of an attractive surface, a vesicle can undergo shape transformations between two different free states, between a free and a bound state, and between two different bound states. In solution, membranes undergo shape jZuctuations on many length scales which are characterized by the roughness exponent

C.

For polymerized membranes,

C

= 112 has been obtained from scaling arguments and Monte Carlo simulations. In contrast to previous simulations, our Monte Carlo data are consistent with a finite value of the shear modulus on large scales. The shape fluctuations affect the adhesion of membranes and lead to unbinding transitions. Detailed renormalization group studies show that both fluid and polymerized membranes unbind in a continuous manner for sufficiently short-ranged interactions.

I. Introduction

Membranes such as lipid bilayers are very thin and highly flexible sheets of molecules which exhibit many fascinating and unique properties. On length scales which are large compared to the size of the molecules, these membranes can be regarded as 2-dimensional surfaces embedded in 3-dimensional space.

The behavior of these surfaces leads t o a variety of critical phenomena.

First, consider a free segment of a lipid bilayer in aqueous solution. In order to prevent any contact between the hydrocarbon chains of the lipid and the water, a sufficiently large segment.wil1 form a closed surface or vesicle. As discussed in Sect. 11, these vesicles can undergo continuous or discontinuous shape transformations. Some of these transformations have been recently observed by light microscopy.

In solution, a membrane undergoes thermally excited undulations or shape fluctuations. These fluctuations can be characterized by the roughness exponent

C

which describes their scale invariance. For polymerized membranes, the value of this exponent is somewhat controversial. As explained in Sect. 111, we have recently found

C

⁼ 112 from scaling arguments and Monte Carlo (MC) simulations while previous simulations of tethered networks gave

C

^{= 0.64}

*

0.04. The latter value for

C

would imply that the shear modulus is scale-dependent and goes to zero on large scales. In contrast, our MC data are consistent with a finite shear modulus.

Next, consider a vesicle which interacts with an attractive surface. It can then undergo shape transformations (i) between two different free states, (ii) between a free and a bound state, and (iii) between two different bound states, see Sect. IV. Adhesion can also lead to topological changes of the vesicle such as vesicle rupture and vesicle fusion.

The adhesion of membranes is strongly affected by their shape fluctuations. Indeed, these fluctuations renormalize the interaction of the membranes. As a result, the system may undergo an unbinding transition.

The critical behavior at such a transition depends, to some extent, on the details of the interaction potential. However, for sufficiently short-ranged potentials, the transition should be continuous both for fluid and for polymerized membranes, see Sect. V.

11. Shape transformations of vesicles

Vesicles formed by lipid bilayers exhibit many different shapes which can be transformed into one another by changing, e.g., the osmotic conditions or the temperature. Most of the experimental work on these shapes involves relatively complex systems containing, e.g., charged lipids or additional solutes such as sugar. It has been recently shown, however, that even a relatively simple binary system, consisting of a single lipid (DMPC) in water, can exhibit a large variety of different shapes and shape transformations. / l /

If one ignores possible topological changes of the vesicle, its shape is primarily determined by thc mean curvature, Cl

+

C2

,

of its surface where Cl and C2 denote the two principal curvatures. More precisely, the free energy functional for a vesicle with enclosed volume V and total surface area A is given by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990724

COLLOQUE DE PHYSIQUE

where X, C O , P, and C are the bending rigidity, the spontaneous curvature, the difference between the outside and the inside pressure, and the lateral tension, respectively.

For a given set of parameters, the most relevant vesicle state is the ground state, i.e., the state of lowest energy. It is important to realize that one may choose different sets of independent parameters which correspond to different statistical ensembles. Indeed, the model (2.1) involves three pairs of conjugate variables, namely (i) P and V, (ii) B and A, and (iii) CO and M ^I

4 b~

(Cl+C2). One variable out of each pair can be taken as an independent parameter.

We have systematically studied two different ensembles /2/: (i) The (V,A,CO)

-

ensemble which is equivalent to the spontaneous curvature model introduced by Helfrich /3,4/, and (ii) The (V,A,M) - ensemble which is identical with the so-called bilayer coupling model /5,6/. For both ensembles, we have made a detailed calculation of the phase diagram, i.e., we have determined the variation of the ground state with a change of the independent parameters. /2/ Several features of the phase diagrams are qualitatively different for the two ensembles; in particular, shape transformations are usually continuous in the bilayer coupling model while they are typically discontinuous in the spontaneous curvature model.

In the experiments on giant DMPC vesicles, the shape transformations were induced by changing the temperature which effectively changes the volume to area ratio. Three different routes have been found and analysed in detail / l / : (i) Transformations from a symmetric dumbbell to an asymmetric pear-shaped state and back to a symmetric dumbbell, see Fig. l(a); (ii) Budding, i.e., the expulsion of a smaller vesicle from a larger one, see Fig. l(b); and (iii) Transformations from a discocyte to a stomatocyte, see Fig. l(c).

We have found that all of these shape transformations can be explained theoretically within the bilayer coupling model provided one assumes that the two monolayers of the lipid bilayer exhibit a small difference in their thermal expansivities. / l / Thus, we were able to calculate the experimentally observed shapes using only one single fit parameter, y, which measures this asymmetry. The shapes shown in Fig. 1 represent, in fact, theoretical shapes which are, however, indistinguishable from the experimentally observed ones.

Fig.1 - Shape transformations of free vesicles / l / : (upper sequence) From a symmetric dumbbell to an asymmetric pear and back to a symmetric dumbbell; (middle sequence) Budding, i.e., the expulsion of a small vesicle from a larger one; and (lower sequence) From a discocyte to a stomatocyte shape. The shapes axe axisymmetric with respect t o the broken line.

111. Shape fluctuations of membranes

Vesicles undergo thermally excited shape fluctuations which can be directly observed in the light microscope. In this way, one can probe shape fluctuations with wavelengths of the order of the vesicle radius

which is typically 1-10 pm. It is important t o realize, however, that there are many more length scales involved in the shape fluctuations. Indeed, the smallest wavelength of the membrane fluctuations is set by the size of the lipid molecules which is of the order of a few nm. Thus, the fluctuations are composed of many modes with wavelengths from the molecular size up t o the membrane dimension.

The shape fluctuations lead t o a rough membrane state characterized by the roughness exponent : a membrane segment of linear size L will form a hump of longitudinal and transverse extension, L,, and LL, with

LII L and L* d~~~~ d~~ with 0 5 ( 5 l (3.1)

where the amplitude A has the dimension of

The roughness exponent depends on the internal structure of the membrane. For fluid membranes with bending rigidity ^{K ,} one has

C

⁼ 1 and J& = ( T / K ) ~ / ~ . /7,S/ For polymerized membranes, on the other hand, which are characterized by the (bare) bending rigidity K, the (bare) shear modulus p, and the (bare) area compressibility modulus KA, the roughness exponent should satisfy < 1. /9,10/

V K

If < 1, the bending rigidity becomes scaleedependent and grows as L for large L with qK = 2 - 2C.

Likewise, the shear modulus should behave as L -B for large L with

v,,

= 4 5 ^- 2

.

/11/ Thus, the shear modulus goes t o zero on large scales provided the roughness exponent satiifies C > 112. Such a behavior has been found in perturbative renormalization group calculations: (i) For a dl14imensional membrane embedded in d = d

+

¹ dimensions, one finds j x 1 - G ( 4 4 ) / 2 5 for small ( 4 5 ); /11/ and (ii) For a

I I I I I I

2-dimensional membrane, one finds :: 1 - i / d for large cl. 1121 It is not possible, however, t o obtain in this way a reliable estimate for real membranes in d = 2+1 since these expansions for

C

are only asymptotic and presumably have zero radius of convergence.

On the other hand, scaling a.rguments for polymerized membranes in d = 2+1 show that the amplitude d in (3.1) is given by d =

/

T ( ~ - ( ) / ~ where l a 4 p K A

/

(p

+

KA) denotes the 24imensional Young modulus. 1131 Obviously, the roughness amplitude d must increase with decreasing bending rigidity ^K which implies

5 >

1/2.

Several computer simulations have been performed in order to determine the roughness exponent for polymerized membranes. For tethered networks, Monte Carlo (MC) and molecular dynamics simulations

Fig.2 ^- Scaling plot of Monte Carlo data for a polymerized membrane subject t o an external pressure, p.

1131 The roughness exponent follows from the scaling relation

C

⁼ 2 $

/

^{(1 -} $).

C7-246 COLLOQUE DE PHYSIQUE

gave the value ( = 0.64

*

0.04

.

114-161 More recently, we have performed MC simulations of the continuum model for a solid-like elastic sheet. 1131 A scaling analysis of these data reveals that the roughness exponent has the value 1 = 112. Some of our scaled data are shown in Fig. 2. One should note that the rescaled pressure, p, which is used to confine the shape fluctuations in a controlled manner, has been varied over more than 10 decades !

Obviously, the value ( = 112 as found in our work is quite different from the values

<

^{= 0.64}

*

0.04 as obtained for tethered networks

.

We have shown that this discrepancy arises from a pronounced crossover 1131: for relatively large bending rigidity or relatively small shear modulus, the small scale excitations of the membrane are characterized by

C

⁼ 1 as for fluid membranes, and one has to probe undulations beyond a certain crossover scale in order to see the true asymptotic behavior with

C

⁼ 112. Thus, the values for

1

obtained from previous simulations of tethered networks represent effective exponents which reflect this crossover.

Our result = 112 has important consequences 1131: (i) The partial resummation of perturbation theory performed by Nelson and Peliti /g/ ives, in fact, the correct value of (

.

This is quite unexpected ; (ii) It implies that the critical exponent 7 for the shear modulus is zero since 7 = 4C - 2 as mentioned. In such a

/I P

case, the shear modulus could still vanish with a weak logarithmic scale dependence. However, our data do not give any indication of such a logarithmic behavior and thus are consistent with a jnite value of the shear modulus on large scales; and (iii) Shape fluctuations of red blood cells have been experimentally studied for a long time. 117,181 We have argued that the crossover between fluid-like and solid-like behavior of the undulations should be accessible in such flicker experiments.

IV. Adhesion of vesicles

The shape of giant vesicles which adhere to another surface can be experimentally studied by micropipet aspiration techniques. /19/ From a theoretical point of view, this shape is determined by the interplay of adhesion and bending energies. This interplay can be theoretically studied starting from a simple model in which the membrane experiences a contact potential of strength W arising from the attractive surface. /20/

The free energy functional for this model consists (i) of the terms for a free vesicle as in (2.1) and 11) of the

area A*.

I'

additional term, - W A*. The latter term represents the adhesion free energy of the membrane wit contact

Fig.3 - Shape transformations of vesicles in the presence of an attractive surface. Starting from a spherical shape with area A,, the transformations are induced by changing area, A, while the volume is kept constant. Such a trajectory corresponds to a change in temperature. For the upper and the lower sequence, the attractive surface has strength W = 22.6 K / A ~ and W = 6.28 %/Ao

,

respectively.

We have determined the phase diagram of this adhesion model both for the (P,A,CO)

-

ensemble /20/

and for the (V,A,C,) _" - ensemble 1211. We find that a vesicle attracted to another surface can undergo shape transformations (i) between two different free states, (ii) between a free and a bound state, and (iii) between two different bound states. Shape transformations for two different values, W, of the attractive potential are shown in Fig. 3. In both cases, we initially start with the same sphere of area A,. The shape transformations are then induced by a change in area, A, while the volume V is kept constant.

In the limit of strong adhesion, a bound vesicle with constant volume attains the shape of a spherical cap.

In this limit, W is related to the lateral tension C via the Young-Dupre equation, W = C (1

+

cos t+beff), where

t+beff

denotes an effective contact angle. /20,21/ If the tension C exceeds a certain threshold, Cmax, it will disrupt the membrane. Thus, very strong adhesion with W

2

Cmax will always lead to vesicle rupture.

As a result, the closed vesicle will be transformed into a disk-like membrane segment with a free edge.

SO far, isolated vesicles (or disks) have been considered. Now, let us imagine to increase the surface coverage, e.g., by increasing the bulk concentration of the lipid. It will then happen that bound vesicles (or disks) come into contact and fuse. For two fiee vesicles (with P = O), the curvature model predicts that the energy gain from fusion is given by AEfv -. = 47r (26

+

lig) where represents the Gaussian curvature modulus. Thus, fusion of two free vesicles is energetically favorable for > - 2 ~ . /22/ Quite generally, adhesion acts to increase this energy gain. For example, the energy gain obtained from the fusion of two bound vesicles of identical size, R, behaves as

for large R with g E 2.8. 1231 Thus, in this limit, AEbv is positive irrespective of the sign of K~ Therefore, two vesicles can fuse in their bound state even if they cannot fuse in their free state because ^lig

<

- 2 ~ . V. Unbinding or adhesion transition

The overall shape of a bound vesicle as discussed in the previous section can be observed through a light microscope. Now, let us imagine to use a microscope with a much larger resolution and let us focus on the region of contact between the vesicle and the second surface. Within this contact region, the two surfaces are separated by a small water gap and experience a variety of direct interactions arising from the intermolecular forces. Quite generally, these direct interactions are renormalized by thermally excited shape fluctuations of the membranes.

The renormalized interaction may be attractive or repulsive at large membrane separation corresponding to a bound or an unbound state of the membranes. These two different states are separated by a phase boundary at which the system undergoes an unbinding or adhesion transition. Such transitions were first predicted on the basis of renormalization group calculations. /8,10/; their existence has been confirmed by Monte Carlo simulations 1241 and by experiments with sugarlipid membranes 1251.

The most interesting behavior is found in the so-called strong fluctuation regime, i.e, for interaction potentials V(9 which satisfy IV(9

I

^l/lT for large l with T = 2 / ( . For fluid membranes, this regime contains, for example, realistic van der Waals interactions. First, consider potentials V(l) within this regime which have on151 one minimum at finite (or infinite) l . In this case, the membrane is predicted to undergo a continuous unbinding transition characterized by universal critical exponents. For example, the mean separation, <b, of the membrane from the other surface grows as

as the unbinding temperature, T,,, is approached from below. The critical exponent

-

Il, is independent of the parameters of the direct interaction, and has the presumably exact value $ = 1 for fluid membranes, and the value

t+b

N 0.69 < 1 for polymerized membranes. /10/

Now, assume that the interaction potential V(l) has a more complicated shape and exhibits two local minima at l = l. and l = m with V($) < V(m). In the absence of shape fluctuations, the membrane is then bound at l = l.

:

On the other hand, sufficiently strong shape fluctuations should again lead to an unbinding transition from l ⁼ l. to l ^{= m}

.

It is not obvious, however, if this transition is first - or second

-

order, i.e., if <R jumps to infinity or diverges in a continuous manner as in (5.1).

In order to determine the order of the unbinding transition for general interaction potentials, we have

C7-248 COLLOQUE DE PHYSIQUE

recently performed a detailed renormalization group (RG) study in which we have determined all RG fixed points for unbinding transitions. /26/ The fixed point structure exhibits a complex dependence on the decay exponent r = 2/<. This dependence implies that interacting membranes can undergo first - order transitions even in the strong fluctuation regime provided r > rs5! ./26,27/ We have used two different RG

--

transformations in order t o estimate the value of rS3. and found 4 < rS2 < 5 from both schemes. 1261 Therefore, within the strong fluctuation regime, unbinding transitions should alivays be of second ^- order both for fluid membranes with r = 2 and for polymerized membranes with r = 4.

One should note, however, that these results apply to tensionless membranes. In the presence of a lateral tension, membranes are only marginally rough with r = m . /17,28,26/ In this case, the shape fluctuations have a relatively small effect on realistic van der Waals interactions. In fact, these interactions then belong to the mean-field or t o the weak-fluctuation regime for which the unbinding transition can be jrst-order for all r ) 2. 1291

I am indebted to Karin Berndl, Marc Girardet, Stefan Grotehans, Joseph Icas, Erich Sackmann, and Udo Seifert for enjoyable collaboration. I thank the organizers of this conference for their invitation. Partial support by the Deutsche Forschungsgemeinschaft through the SFB 266 and by the H8chstleistungsrechen- zentrum Jiilich is gratefully acknowledged.

/ l / I<. Berndl, J. Kas, R. Lipowsky, E. Sacltmann, and U. Seifert, Europhys. Lett. 13 (in press) / 2 / U. Seifert, I<. Berndl, and R. Lipowsliy, submitted to Pllys. Rev. A

/3/ W. Helfrich, Z. Naturforsch. 28c, 693 (1973)

/4/ H.J. Deuling and W. Helfrich, J. Physique 37, 1335 (1976) /5/ E. Evans, Biophys. J . 14,923 (1974)

/6/ S. Svetina and B. Zelis, Eur. Biophys. J. 17, 101 (1989) /7/ W. Helfrich, Z. Naturforsch. 33a, 305 (1978)

/S/ R. Lipowsky und S. Leibler, Phys. Rev. Lett 56, 2541 (1986), and 59, 1983 (E) (1987) /g/ D.R. Nelson and L. Peliti, J. Physique 48, 1085 (1987)

/10/ R. Lipowsky, Europhys. Lett. 7, 255 (1988), and Phys. Rev. Lett. 62, 704 (1989)

/11/ J . Aronowitz and T.C. Lubensky, Phys. Rev. Lett. 60, 2634 (1988); J . Aronowitz, L. Golubovic, and T.C. Lubensky, J. Phys. France 50,609 (1989)

/12/ F. David and E. Guitter, Europhys. Lett. 5, 709 (1988); E. Guitter, F. David, S. Leibler, and L.

Peliti, J. Phys. France 50, 1787 (1989)

1131 R. Lipowsky and M. Girardet, Phys. Rev. Lett. (in press) /14/ S. Leibler and A. Maggs, Phys. Rev. Lett. 63, 406 (1989) /15/ J.-S. Ho and A. BaumgLtner, Europhys. Lett. 12, 295 (1990) /16/ F.F. Abraham and D.R. Nelson, Science 249, 394 (1990) 1171 F. Brochard and J.F. Lennon, J. Phys. (Paris) 36, 1035 (1975)

/IS/ A. Zilker, H. Engelhardt, and E. Sackmann, J . Pllys. (Paris) 48, 2139 (1987) /19/ For a review, see E. Evans, Coil. and Surf. 43, 327 (1990)

/20/ U. Seifert and R. Lipowslcy, Phys. Rev. A 42, 4768 (1990)

1211 R. Lipowsky and U. Seifert, in Flw.ctuations in lamellae and membranes, ed. by W.J. Benton and L.A.

Turkevich, (ACS boolis, in press)

/22/ W. Helfrich and W. Harbich, in Physics of Amphiphilic Lavers, ed. by. J . Meunier, D. Langevin, and N. Boccara, Springer Proc. in Physics, Vol. 21 (Springer-Verlag, 1987)

/23/ R. Lipowsky and U. Seifert, Mol. Cryst. Liq. Cryst. (in press) 1241 R. Lipowsky and B. Zielinsl<a, Phys. Rev. Lett. 62, 1572 (1989) /25/ M. Mutz and W. Helfrich, Phys. Rev. Lett. 62, 2881 (1989) /26/ S. Grotehans and R. Lipowsky, Phys. Rev. A 41,4574 (1990) 1271 F. David and S. Leibler, Phys. Rev. B 41, 12926 (1990)

/28/ W. Helfrich and R.M. Servuss, Nuovo Cimento D 3, 137 (1984); see also R. Lipowsky, Phys. Rev.

Lett. 52, 1429 (1984) for related results on wetting transitions.

1291 For reviews, see R, Lipowsky, Physica Scripta T 29, 259 (1989) and in Random Fluctuations and Growth, ed. by H.E. Stanley and N. Ostrowsliy (Icluwer Academic Publishers, Dordrecht 1988)