Vesicles and Biomembranes - Max Planck Society · 2007. 8. 24. · each biomembrane consists of a specific mix-ture of many different amphiphilic molecules that reflect its diverse

VESICLES AND BIOMEMBRANESREINHARD LIPOWSKY, Max-Planck-Institut für Kolloid- und Grenzflächenforschung,Teltow-Seehof, Germany

1.1.1

1.21.3

1.42.

2.12.22.3

2.43.

3.13.24.

4.14.24.3

5.

5.1

5.1.1

5.1.25.1.3

5.1.4

5.1.55.2

5.2.15.2.2

5.2.35.2.4

5.35.3.15.3.26.

6.1

6.1.1

6.1.2

6.2

6.3

7.

7.1

7.2

7.2.17.2.2

7.2.3

7.37.4

8.

8.18.2

8.3

8.3.1

8.3.29.

9.19.2

9.3

Encyclopedia of Applied Physics, Vol. 23 © 1998 WILEY-VCH Verlag GmbH3-527-29476-7/98/$5.00 + .50

199

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 200From Cells to Vesicles . . . . . . . . . 201Plasma Membrane andInternal Membranes .............. 201Evolution of Biomembranes ... 201Universal ConstructionPrinciple .............................. 202Fluidity of Biomembranes ..... 202Molecular Structure of LipidBilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202Lipid Molecules in Water ...... 202Self-Assembly of Lipids ......... 202Lateral Diffusion and "Flip-flops" ................................... 203Transport across Bilayers ...... 203Elastic Properties of FluidMembranes . . . . . . . . . . . . . . . . . . . . . . . . . 203Stretching Deformations ........ 203Bending Deformations ........... 204Related but DistinctSystems .............................. 204Soap Films ........................... 204Surfactant Layers ................. 205Solidlike or PolymerizedMembranes .......................... 205The Morphology ofVesicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Shape of HomogeneousMembranes .......................... 205Curvature and BendingEnergy ................................. 205Asymmetric Membranes ........ 207Constraints on Area andVolume ................................ 207Shape Transformations andLimit Shapes ........................ 207Vesicles with Handles ........... 207Shape of InhomogeneousMembranes .......................... 208Composition and Shape ......... 208Domain-Induced Budding andFission ................................. 208Bud Size .............................. 209Domain-Induced Budding ofBiomembranes ..................... 209

Adhesion of Vesicles ............. 209Contact Potential .................. 209Adhesion Threshold .............. 209Shape Fluctuations ofMembranes . . . . . . . . . . . . . . . . . . . . . . . . . 210Bending Modes orUndulations ......................... 211Roughness Arising fromBending Modes .................... 211Measurement of the BendingRigidity ............................... 211Thermal Fluctuations onMolecular Scales ................... 211Persistence Length on LargeScales .................................. 211Generic Interactions of TwoMembranes . . . . . . . . . . . . . . . . . . . . . . . . . 211Direct Interactions betweenRigid Membranes ................. 211Renormalization by BendingUndulations ......................... 212Repulsive Interactions ........... 212Attractive Interactions andContinuous Unbinding .......... 212Potential Barriers andDiscontinuous Unbinding ....... 213Tension-Induced Adhesion ..... 213Repulsive Interactions atSmall Separations ................. 213Bunches and Stacks ofMembranes . . . . . . . . . . . . . . . . . . . . . . . . . 213Free Membranes in Solution .. 214Membranes at a RigidSurface ................................ 215Limit of Lyotropic LiquidCrystal ................................. 215Power-Law Peaks of theScattering Intensity ............... 215Melting of Lamellar States ..... 216Polymer-DecoratedMembranes . . . . . . . . . . . . . . . . . . . . . . . . . 216Anchored Polymers ............... 216Dilute and SemidiluteRegimes ............................... 216Polymer-Induced Curvature .... 217

200 Vesicles and Biomembranes

9.410.

10.110.210.3

10.4

11.

INTRODUCTION

The membranes and vesicles consideredhere are ultrathin and highly flexible sheetscomposed of lipids and other amphiphilicmolecules (which have both a water-soluble,hydrophilic part and a water-insoluble, hy-drophobic part). In biological systems, thesemembranes represent complex interfacesthat partition space into different compart-ments and, thus, are responsible for theamazing architecture of these systems. Oneexample of this architecture is shown inFig. 1.

Biomembranes have been studied for a

long time in biology, pharmacology, andmedicine. On the micron scale, these mem-branes exhibit a unique combination ofproperties:

1. They form closed surfaces without edges;2. they are highly flexible and, thus, can eas-

ily adapt their shape to external pertur-bations; and

3. in spite of this flexibility, they are ratherrobust and keep their structural integrityeven for strong deformations.

This combination of stability and flexibilityis a consequence of their molecular struc-

FIG. 1. Some internal membranes of a liver cell. The labyrinth of membrane sheets and tubes defines the en-doplasmic reticulum; within the lamellar region, one sees the membranes of three mitochondria (M) and of onelysosome (L). (Krstic, 1976.)

Polymer-Induced Adhesion ..... 217Applied ElectromagneticFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218Alternating Electric Fields ..... 218Electroporation ..................... 218Electrofusion ........................ 219

Optical Tweezers .................. 219

Outlook on Applications . . . . . . 219

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Works Cited . . . . . . . . . . . . . . . . . . . . . . . . 221

Further Reading . . . . . . . . . . . . . . . . . . 222

Vesicles and Biomembranes 201

ture. When viewed on the nanometer scale,each biomembrane consists of a specific mix-ture of many different amphiphilic moleculesthat reflect its diverse biological functions.However, in spite of this chemical complex-ity, all biomembranes are organized accord-ing to the same universal construction prin-ciple: their basic building block is providedby a bilayer of lipid molecules. The lattermolecules are essentially insoluble in theaqueous solution, which ensures membranestability. In addition, these lipid bilayers aremaintained in a fluid state, which is themain mechanism underlying their enormousflexibility.

This article focuses on fluid bilayers thatcontain only one or a few lipid components.Even the simplest model membranes of thiskind already exhibit several levels of self-organization that are reminiscent of bio-membranes: The lipid molecules arrangethemselves into bilayers; the bilayers sponta-neously form closed bags or vesicles withoutedges; the vesicles adhere to one another andform various multilayer structures.

In addition, lipid bilayers in their fluidstate can easily adapt to external forces byreorganization of their supramolecular struc-ture. Indeed, one intriguing aspect of thesemembranes is their ability to undergo mor-phological changes: Depending on tempera-ture and osmotic conditions, free mem-branes and vesicles exhibit a large variety ofdifferent shapes and shape transformations;similar shape changes can be induced by theformation of intramembrane domains; inter-acting membranes undergo transitions be-tween bound and unbound states. A newlevel of self-organization is obtained whensuch bilayers are "decorated" with anchoredpolymers. These polymers form "mush-rooms," "pancakes," or "brushes," which tendto curve the bilayers and to change their in-teractions.

Just like biomembranes, lipid bilayers areflexible but stable structures, which makes itpossible to isolate them and to manipulatethem in various ways: One can suck theminto micropipettes, attach them to other sur-faces, and grasp them with optical tweezersgenerated by focused laser beams. These bi-layers can even tolerate local perturbationsthat lead to the formation of small holes:The bilayers restore their structural integritysince the holes close again spontaneously as

long as the membranes do not experience alarge lateral tension.

1. FROM CELLS TO VESICLES

1.1 Plasma Membrane and InternalMembranes

All living matter is built up from cells.Each cell is enclosed by its outer plasmamembrane that controls the interaction be-tween the cell and its environment. This ap-plies both to the relatively small cells of bac-teria or prokaryotes, which have no cellnucleus, and to the much larger cells of eu-karyotes, which have such a nucleus. Thelatter class of organisms includes all animalsand plants as well as single-celled microor-ganisms such as amoeba or yeast. In addi-tion to the outer plasma membrane, all eu-karyotic cells contain internal membranesthat represent the boundaries of the internalorganelles such as the nucleus, mitochon-dria, chloroplasts, etc. The membranesshown in Fig. 1, for example, represent theinternal membranes that bound the endo-plasmic reticulum and some mitochondria ofa liver cell (Darnell et at., 1990; Alberts et al.,1994).

The total membrane area of an eukaryoticcell is relatively large. The membranes of asingle liver cell, for example, have a totalsurface area of about 1.1 × 105 µm2 whileits volume is about 5 × 103 µm3. About 98%of this large area belongs to the internalmembranes and only 2% to the outer plasmamembrane of the cell.

1.2 Evolution of Biomembranes

Membranes are quite old and thus haveevolved over a long period of time. Indeed,membranes defined the boundaries of thefirst cells on earth and thus played a crucialrole in the origin of life. The oldest microfos-sils that are viewed today as remnants ofcells have an age of about 3.5 × 109 years.As a result of this long evolution, we nowhave an enormous variety of different cellsand organelles, which live in very differentenvironments. The interactions between thecells or organelles and their environmentsare mediated by their membranes. The basicfunction of these membranes is to act as


highly selective barriers for the exchange ofmolecules between the different spatialregions and compartments. In this way, theysustain concentration gradients betweentheir two sides, which are used, for example,in order to produce energetic molecules orto propagate localized excitations along themembranes. Likewise, these membranes actas transducers for chemical, optical, or me-chanical signals and as supporting surfacesfor anchored polymers and polymer net-works.

1.3 Universal Construction Principle

In order to fulfill its specific biologicalfunctions, each biomembrane is composed ofa specific mixture of hundreds of differentmolecules. However, in spite of this complexchemical composition, all biomembranes ex-hibit the same universal construction princi-ple: a bilayer of lipid molecules, which pro-vides a two-dimensional solvent for thehydrophobic anchors of membrane proteins.Therefore, the simplest model systems forbiomembranes are lipid bilayers without anyproteins. When dissolved in water, these bi-layers form closed bags or vesicles that re-semble the closed compartments as formedby biomembranes.

1.4 Fluidity of Biomembranes

Under physiological conditions, biomem-branes are in a fluid state; i.e., the mem-brane molecules can diffuse rapidly alongthese membranes. On the molecular scale,this fluidity is necessary for the proper func-tioning of membrane proteins. On the supra-molecular scale, it ensures that the biomem-brane is highly flexible and can undergoshape changes such as the formation ofsmall spherelike buds. Therefore, both pro-karyotic and eukaryotic cells adjust the lipidcomposition of their membranes in such away that they remain in a fluid state irre-spective of the ambient temperature and ofother external conditions. Prokaryotic cellsachieve this by increasing the number of un-saturated double bonds within the hydrocar-bon chains of the lipid molecules. These un-saturated bonds act as defects within thebilayer that prevent the freezing of thesemembranes. Eukaryotic cells, on the other

hand, increase the concentration of choles-terol within their membranes in order tomaintain the fluidity.

2. MOLECULAR STRUCTURE OF LIPIDBILAYERS

2.1 Lipid Molecules in Water

Lipids are amphiphilic molecules with ahydrophilic head group and usually two lipo-philic (= hydrophobic) hydrocarbon chains.The head group of phospholipids and glyco-lipids contains a phosphate group and somesugar groups, respectively.

Those lipid molecules that are containedin biomembranes are essentially insoluble inwater. More precisely, single lipid moleculescan be dissolved in water only up to a criti-cal monomer concentration . This concen-tration is very small and decreases with in-creasing length of the hydrocarbon chains.For phospholipids with two identical chainscontaining carbon atoms, one has

at room temperature, as can bemeasured for relatively small values of10 (Cevc and Marsh, 1987). For the lecithinDPPC (dipalmitoyl phosphatidyl choline)with , this leads to the estimate

, i.e., less than one monomer per 10µm3 or per 10- 8 mL of water.

2.2 Self-Assembly of Lipids

As soon as the lipid concentration in theaqueous solution exceeds the critical mono-mer concentration , lipid molecules as-semble into supramolecular structures. Thesestructures are built up from lipid bilayers inwhich the polar head groups of the lipidmolecules form the two lipid/water interfaceswhereas the hydrophobic chains are buriedwithin the bilayer and have essentially nocontact with the aqueous solution. This pro-cess is an example of the so-called hydropho-bic effect, which has a large entropic contri-bution arising from the configurationalentropy of the hydrogen bond networkswithin the water (Tanford, 1991). Because ofthis effect, the lipid bilayers also arrangethemselves in such a way that they have noedges and thus form closed vesicles or lipo-somes.

In practice, there are several preparation


methods in order to produce lipid vesiclesstarting with a lipid/water solution (see, e.g.,Lasic, 1993). Small vesicles with a relativelynarrow size distribution can be preparedfrom larger bilayer structures by sonificationwith ultrasound or by millipore extrusion.These vesicles are usually too small to bevisible in the optical microscope. Relativelylarge vesicles with a linear size of the orderof 10 µm are obtained if ordered stacks ofmembranes are swollen in a controlled way.In general, one then obtains a mixture ofboth unilamellar vesicles, consisting of a sin-gle bilayer, and multilamellar liposomes,consisting of several, closely packed bilayers.

In principle, the bilayers exchange lipidmolecules with the aqueous solution sur-rounding them. However, since the mono-mer concentration is so small, the ex-change of molecules between the bilayersand the solution is extremely slow. As longas one considers phenomena that are fastcompared to this rather slow exchange pro-cess, the chemical equilibrium that wouldarise from this exchange of molecules isblocked, and the number of lipid moleculeswithin each bilayer is practically constant.

2.3 Lateral Diffusion and "Flip-flops"

Lipid bilayers are essentially two-dimen-sional systems: Their thickness is 4-5 nm,whereas their lateral extension often exceeds10 µm. A two-dimensional system can ex-hibit distinct thermodynamic phases. Lipidmonolayers at the water-air interface, for ex-ample, exhibit a large number of differentphases, which have been studied by opticalmicroscopy and x-ray and neutron scatter-ing. Likewise, lipid bilayers always exhibit afluid phase at high temperatures and one orseveral gel or solidlike phases at low tem-peratures. This article focuses on the fluidstates of bilayers since these are the relevantstates of biomembranes.

Within a fluid phase, the molecules canfreely diffuse along the bilayer. The corre-sponding diffusion coefficients are typicallyof the order of 10- 7-10- 8 cm2/s. This impliesthat each lipid molecule covers of the orderof 1 µm in 1 s.

While lateral diffusion within the fluid bi-layer is rather rapid, "flip-flops," i.e., thetransverse diffusion between the two mono-layers, are much slower. In bilayers consist-

ing of a single phospholipid, it takes usuallyseveral hours and more to exchange half ofthe phospholipid molecules between the twomonolayers. In multicomponent bilayers, onthe other hand, smaller molecules such ascholesterol can flip-flop more easily, and thecorresponding time scale may be of the or-der of minutes.

2.4 Transport across Bilayers

Even though lipid bilayers are very thin,they provide selective barriers for the diffu-sive transport of molecules. Water and smalluncharged molecules such as CO2 or N2 canpermeate the bilayers: They first dissolve intotheir hydrophobic interior, cross it by simplediffusion, and finally dissolve into the aque-ous solution on the other side of the mem-branes. The water permeability can be di-rectly measured if one studies mixtures ofH

2O with isotopically labeled water such as

DHO or THO. On the other hand, lipid bilay-ers are essentially impermeable to ions andto larger uncharged molecules such as glu-cose. The transport of these latter moleculesthrough biomembranes is facilitated by spe-cial membrane proteins such as ion channelsand carriers. Some carriers represent ionpumps that provide an active, i.e., energy-consuming, transport mechanism against theconcentration gradients (Darnell et al., 1990;Alberts et al., 1994).

3. ELASTIC PROPERTIES OF FLUIDMEMBRANES

Fluid membranes have rather special elas-tic properties: Any shear deformation relaxesby flow within the membrane; i.e., the (zero-frequency) shear modulus of such a mem-brane vanishes. Thus, a fluid membrane ex-hibits only two types of elastic deformations:stretching and bending (Canham, 1970; Hel-frich, 1973; Evans, 1974).

3.1 Stretching Deformations

The stretching of lipid bilayers is limitedto rather small deformations, since they startto rupture as soon as their area is changedby about 1%. If a membrane segment witharea experiences the lateral tension ,


one has the area increase with. The area compressibility modulus

is of the order of , as hasbeen measured in micropipette aspiration ex-periments; see Fig. 2. By this method, onecan also determine the tension of rupture,

, which is of the order of a few(Evans and Needham, 1987).

3.2 Bending Deformations

Consider a planar membrane segment andlet us focus on a cut perpendicular to such amembrane. If this membrane is bent, thesurface of the cut is tilted; i.e., it is subjectto a torque. This torque is proportional tothe curvature of the membrane and to thelength of the cut; the proportionality con-stant represents the bending rigidity , whichhas the dimension of an energy. For phos-pholipid bilayers, the bending rigidity is ofthe order of as deducedfrom the thermally excited flickering of vesi-cles, see Sec. 6.1 (here and hereafter, thetemperature has energy units; i.e., is ashort-hand notation for Boltzmann constant

temperature in kelvins).As mentioned, the admixture of choles-

terol to lipid bilayers increases their fluidity.At the same time, it also increases the bend-ing rigidity of these membranes. For phos-

pholipid bilayers, for example, the bendingrigidity was measured to increase by a factor3-4 if the bilayer contained 30% cholesterol.These combined effects of cholesterol arerather remarkable and must be the result ofa long optimization process during evolution.

4. RELATED BUT DISTINCT SYSTEMS

4.1 Soap Films

The behavior of fluid membranes as dis-cussed below is often counterintuitive. Thisis understandable since there is no macro-scopic analog for such systems. The onlyfluid surfaces in our macroscopic world aresoap films and soap bubbles. A soap filmconsists of a thin layer of water that isbounded by two monolayers of surfactantmolecules. The latter molecules are also am-phiphilic but they are soluble in water, atleast to a certain extent: Their critical mono-mer concentration typically varies be-tween 105 and 107 molecules per 1 µm3 orper 10- 9 mL of water.

Each surfactant monolayer represents anair-water interface that is characterized by arelatively large interfacial tension. The lattertension represents the free energy per inter-facial area and ensures that soap bubbles at-

FIG. 2. Lipid vesicles may be sucked into small glass pipettes and can then be manipulated mechanically. Thevesicle radii are about 10 µm. The top vesicle is almost spherical since it is exposed to a relatively large suc-tion pressure. (Courtesy of E. Evans.)


tain a spherical shape. In addition, this inter-facial tension is also responsible for theintrinsic instability of soap films: If onepunches small holes into them, the tensionacts to enlarge these holes and, thus, to rup-ture the soap films.

4.2 Surfactant Layers

Fluid monolayers of surfactant also formin multicomponent systems containing waterand oil. Such mixtures exhibit many differ-ent phases in which these monolayers sepa-rate oil from water domains. Likewise, bi-nary mixtures of water and surfactant maylead to liquid phases in which the surfactantmolecules form bilayers separating two wa-ter domains. Since the critical monomerconcentration of the surfactant moleculesis relatively large, these mixtures relax rela-tively fast toward states of thermal andchemical equilibrium, which are character-ized by constant temperature and constantchemical potentials of the different compo-nents. These thermodynamic phases havebeen studied for a long time in physicalchemistry and chemical engineering (for re-cent reviews, see Kahlweit and Lipowsky,1996).

Surfactant layers are even more flexiblethan lipid bilayers, but they are also less sta-ble and more sensitive to external perturba-tions. For example, it is usually not possibleto transfer a single surfactant layer into asurfactant-free solution since this layer willdissolve rather rapidly. Likewise, it is hardlypossible to manipulate surfactant layers bymicropipettes or optical tweezers without de-stroying their structure. Therefore, in con-trast to lipid bilayers, the behavior of singlesurfactant layers is not accessible to experi-mental studies.

for such membranes in our macroscopicworld (such as an ordinary piece of paper, athin film of rubber, or a fishnet). Solidlikemembranes have a nonzero shear modulus,which leads to a coupling between bendingand stretching deformations. Indeed, allbending deformations that change the Gaus-sian curvature of a solidlike membrane arenecessarily coupled to strong stretching de-formations of this membrane. Therefore, ifthe membrane (such as a piece of paper) isessentially unstretchable, it is impossible todeform it smoothly, i.e., without creating lotsof folds or defects, from a planar into aspherical state. In contrast, such a shapetransformation can be obtained easily for afluid membrane; see Fig. 3.

5. THE MORPHOLOGY OF VESICLES

As mentioned, one can prepare relativelylarge vesicles with a linear size of the orderof 10 µm that are bounded by a single bi-layer. These vesicles can be directly observedin the optical microscope. One then findsthat these vesicles exhibit a large variety ofdifferent shapes and various shape transfor-mations as illustrated in Fig. 3.

A vesicle that is osmotically swollen expe-riences a lateral tension and, thus, attains aspherical shape just like a fluid droplet or asoap bubble. However, since water can per-meate the membrane, the vesicle can adaptits volume and relax the tension. In this way,the membranes can attain an essentially ten-sionless state, the shape of which is governedby bending energies; compare Sec. 3 above.These shapes can be determined from sys-tematic theories based on the relation be-tween bending and curvature as described inthe following section.

4.3 Solidlike or Polymerized Membranes

Another class of membranes that has beenrecently studied in the theoretical physicscommunity but will not be discussed in whatfollows consists of solidlike or polymerizedmembranes (for reviews, see Nelson et al.,1989; Lipowsky, 1991). These latter mem-branes, which are characterized by a fixedconnectivity of their building blocks, aremore familiar since there are many examples

5.1 Shape of Homogeneous Membranes

First, consider a homogeneous fluid mem-brane with uniform elastic properties. As ex-plained, the only elastic deformations thatare relevant for fluid membranes are bendingdeformations governed by curvature.

5.1.1 Curvature and Bending EnergyIn general, each point of the membrane sur-face is characterized by its mean curvature


FIG. 3. Shape transformation of a single vesicle as observed via phase-contrast microscopy and calculatedfrom curvature models. The transformations were induced by an increase in temperature that leads to an in-crease of the surface area of the vesicle. (a),(c) Budding, i.e., expulsion of a small vesicle from a larger one viathe transformation from a prolate to a pear; (b),(d) inverse budding (or "endocystosis") via the transformationfrom a discocyte to a stomatocyte. The shapes are axisymmetric with respect to the broken line. In both cases,the bud is connected to the large vesicle by a small neck.


(1)

and by its Gaussian curvature , whereand are the two principal curvatures,which are equal to the inverse curvature ra-dii.

If the two sides of the membrane areidentical, the total bending energy has theform

(2)

up to second order in the principal curva-tures and . The surface integrals extendover the whole membrane surface, and isthe intrinsic area element. The two parame-ters and have the dimensions of energyand represent the bending rigidity and themodulus of the Gaussian curvature, respec-tively. For closed membranes without edges,the integral over the Gaussian curvaturedoes not depend on the shape of the surfacebut only on its topology, as follows from theGauss-Bonnet theorem.

The mean curvature termis a dimensionless and thus scale-invariantquantity. In fact, it is even invariant underarbitrary conformal transformations of three-dimensional space (Willmore, 1982).

5.1.2 Asymmetric Membranes In gen-eral, the two monolayers of the bilayer mem-brane may differ in their chemical composi-tion, and the two sides of the bilayer mayface different surroundings. This asymmetrycan be described by a spontaneous meancurvature (Helfrich, 1973). Furthermore,the difference in density between the twomonolayers adapts locally to the curvature:If the bilayer is bent, one of the monolayersis compressed and the other one is stretched(Evans, 1974).

The coupling between the density differ-ence and the mean curvature leads to a con-straint on the total mean curvature

(3)

which is proportional, for the closed bilayer,to the area difference of the two monolayers.This area difference can be changed bymembrane proteins, which actively produceflip-flops of lipids between the two monolay-

ers (Farge and Devaux, 1992). On the otherhand, if flip-flops play no role, the totalmean curvature would like to attain the pre-ferred value, which corresponds to an un-constrained packing of the lipid moleculesfor which they all occupy the same (optimal)area (Miao et al., 1994).

5.1.3 Constraints on Area and VolumeIn practice, vesicles exhibit a great variety ofnonspherical shapes. This polymorphismarises, to a large extent, from two globalconstraints that are present for real vesicles:

1. The area of the bilayer membrane is con-stant (at constant temperature) since theexchange of molecules between the mem-brane and the solution can be neglected;and

2. the volume of the vesicle does not ad-just freely but is determined by the os-motic pressure arising from those sol-utes that cannot permeate the bilayer;compare Sec. 2.4.

If is the solute concentration outside ofthe vesicle and is the number of solutemolecules within the vesicle, one has

for small solute concentra-tions.

5.1.4 Shape Transformations and LimitShapes As one changes a control parame-ter such as the osmotic pressure or the tem-perature, the shape of minimal bending en-ergy usually evolves in a smooth way.However, for certain values of the controlparameter, the shape undergoes a transfor-mation that can be continuous or discontin-uous. The shape transformations shown inFig. 3 represent continuous transformationsbetween a discocyte and a stomatocyte shape[Figs. 3(b) and 3(d)] and between a prolateand a pear shape at which the up-downsymmetry of the shape is broken [Figs. 3(a)and 3(c)].

In addition, the shape may evolve intolimit shapes in which different segments ofthe membrane surface start to come intocontact. One type of limit shape consists oftwo segments connected by an infinitesimalneck that costs no bending energy; see thelast shapes in Fig. 3.

5.1.5 Vesicles with Handles Recently,toroidal vesicles with one or several handleshave also been studied both theoretically and


experimentally. For vesicles with two ormore handles, the shape of minimal bendingenergy is degenerate even if one includes thevarious constraints on the vesicle shape (Jü-licher et al., 1993). In this degenerate region,theory predicts a diffusion process in shapespace along a conformal mode with constantarea, volume, and total mean curvature asshown in Fig. 4. Such a process has been re-cently observed by phase-contrast micros-copy (Michalet and Bensimon, 1995).

5.2 Shape of Inhomogeneous Membranes

In general, a bilayer is composed of dif-ferent types of molecules. In such a multi-component system, the composition can be-come inhomogeneous, which affects theelastic properties and thus the shape of themembrane. Several cooperative phenomenathat are driven by this coupling betweencomposition and shape have to be distin-guished. One rather general effect is pro-vided by the budding of intramembrane do-mains.

5.2.1 Composition and Shape Smallvesicles that are composed of two differentlipids often exhibit a strong asymmetry inthe composition of the two monolayers. Thiscompositional asymmetry reduces the mis-match or "frustration" between the sponta-neous curvatures of the two monolayers.Likewise, conelike molecules within themembrane tend to diffuse toward membraneregions with an appropriate curvature. Thus,curvature may induce phase segregation.

The phase diagram of a multicomponentbilayer usually exhibits one-phase regimes

and two-phase coexistence regimes depend-ing on the temperature and the membranecomposition. When such a bilayer is pre-pared in the two-phase regime, it will un-dergo phase separation, which leads to theformation of lateral domains or patcheswithin the membrane. These intramembranedomains often form ordered patterns as orig-inally observed by freeze fracture and elec-tron microscopy (Sackmann, 1990). Thetheoretical work on these domain patterns isreviewed in Lipowsky (1995b).

5.2.2 Domain-Induced Budding andFission Now, let us focus on a single do-main within the membrane. In general, thisdomain may have a spontaneous curvaturethat differs from the spontaneous curvatureof the surrounding matrix. In addition, theedge of the intramembrane domain has anenergy that is proportional to its length.Therefore, the domain has a tendency to at-tain a circular shape in order to minimize itsedge energy.

However, a flat circular domain does notrepresent the state of lowest edge energysince the length of the edge can be furtherreduced if the domain forms a bud: The do-main edge now forms the neck of the bud,and this neck narrows down during the bud-ding process; see Fig. 5. Because of the edgeenergy, the domain must bud as soon as itslinear size exceeds a certain critical size evenif it has no spontaneous curvature. Simpletheoretical estimates also show that the budis likely to break off from the membrane ma-trix. There is some evidence for such bud-ding and fission processes from experimentson vesicles that contain mixtures of phos-pholipids and cholesterol.

FIG. 4. Conformal diffusion of a vesicle with two handles: all three shapes have the same bending energy andthe same area, volume, and total mean curvature.


FIG. 5. Budding of the membrane domain embed-ded in the membrane matrix . The domain edge isindicated by the full-broken line. The length of thisedge decreases during the budding process from (1)to (3).

5.2.3 Bud Size A vesicle that consistsof two domains and could be obtained,for example, as a result of complete phaseseparation within the membrane. For such avesicle, the shape of minimal bending energyexhibits a bud for a large range of parame-ters. This budded state attains a limit shapein which the bud consists of a closedsphere that is connected to the mother vesi-cle by an infinitesimal neck. The domainboundary with line tension is contained inthis neck. The mean curvature of thisbud and the mean curvature of themother vesicle adjacent to this neck satisfythe general condition

(4)

where the bending rigidities and andthe spontaneous curvatures and ofthe and domains may be different (Jü-licher and Lipowsky, 1996).

The size of the bud is given by . Ifthe spontaneous curvatures are negligible,the bud size becomes in thelimit of a relatively small domain. In thislimit, the size of the bud is determined bythe elastic properties of the domain alone,and domain-induced budding then repre-sents a local mechanism.

5.2.4 Domain-Induced Budding ofBiomembranes In biological cells, bud-ding of intramembrane domains representsthe first step in the production of vesicles forthe intracellular transport between differentcell compartments. These domains can growby diffusion-limited aggregation of molecules

within the membrane or by the adsorptionof molecules from the surrounding medium.

The budding of these domains could begoverned by their spontaneous curvature.The aggregated molecules may have mem-brane-spanning anchors that induce such acurvature in the bilayer. Likewise, the ad-sorption of coat proteins onto one side of thebilayer leads to an asymmetric membrane.In any case, the reduction of the edge energyof these domains during the budding processwill always act to facilitate this process.

5.3 Adhesion of Vesicles

A vesicle that is attracted toward a sur-face gains adhesion energy but increases itsbending energy. For large vesicles, the adhe-sion energy must dominate since it is pro-portional to the contact area whereas thebending energy is scale-invariant and thusindependent of the vesicle size.

5.3.1 Contact Potential A vesicle neara wall or substrate experiences various mo-lecular forces; see Sec. 7. The typical rangeof these forces is usually small compared tothe size of the vesicle. In order to study theoverall shape of the bound vesicle, the mi-croscopic interaction can be replaced by acontact potential. This potential is describedby a single parameter , which is equal tothe adhesion energy per unit area. The valueof the potential strength is determined bythe competition between direct molecularforces and fluctuation-induced forces; seeSec. 7.

5.3.2 Adhesion Threshold Since thebound vesicle is curved more strongly thanthe free vesicle, the contact potential hasto exceed a certain threshold

(5)

before the vesicle starts to adhere to thewall. The dimensionless coefficient de-pends on the reduced volume and isof order 1 (Seifert and Lipowsky, 1995).

For an ensemble of vesicles with differentsizes, the relation (5) implies that large vesi-cles with surface area stick tothe wall, whereas vesicles with smaller sur-face area do not. This difference in the sizedistribution of bound and free vesicles is ac-cessible to reflectivity measurements.


6. SHAPE FLUCTUATIONS OFMEMBRANES

The membranes of vesicles undergo ther-mally excited shape fluctuations, which canbe directly observed in the light microscope.For adhering vesicles, one can use reflection

interference microscopy as shown in Fig. 6(Rädler et al., 1995). In this way, one canprobe fluctuations with wavelengths betweenabout 0.4 µm and the vesicle size. There are,however, many more length scales involvedin these fluctuations, which have wave-lengths down to molecular dimensions.

FIG. 6. Membrane of a bound vesicle as observed by reflection interference contrast microscopy. (a) Relativelylarge lateral tension, which suppresses all shape fluctuations, and (b) pronounced shape fluctuations for rela-tively small tension. (Courtesy of J. Rädler and E. Sackmann.)


6.1 Bending Modes or Undulations

On scales that are large compared to themembrane thickness, the typical shape fluc-tuations of fluid membranes are bendingmodes or undulations in which the surfacearea of the membrane remains unchanged.

6.1.1 Roughness Arising from BendingModes An almost planar membrane seg-ment that has bending rigidity and experi-ences the lateral tension undergoes bend-ing undulations that can be expanded inFourier modes. The amplitude of these exci-tations depends on the temperature For asegment of linear size , these undulationslead to the membrane roughness with

(6)

provided the crossover lengthis large compared to the molecular scales(Brochard and Lennon, 1975). Thus, one hasa rigidity-dominated regime for and atension-dominated regime for

6.1.2 Measurement of the Bending Ri-gidity Since the roughnesswithin the rigidity-dominated regime, mea-surements of this roughness can be used todetermine the bending rigidity . In practice,this is done by optical microscopy of nearlyspherical vesicles for which the undulationsare expanded in spherical harmonics. In thisway, one finds the typical values

for phospholipid bilayers (for a recentlist, see Seifert and Lipowsky, 1995). It is,however, difficult to obtain high-precisionvalues for . This is understandable since thebending rigidity of lipid bilayers is affectedby small changes in their molecular struc-ture arising, e.g., from conelike defects orfrom the molecular roughness of the lipid-water interfaces.

6.2 Thermal Fluctuations on MolecularScales

The concept of a bending mode is nolonger meaningful as soon as its wavelengthbecomes of the order of the membranethickness. On these small scales, the molec-ular structure of the lipid-water interfaceshould be taken into account. This interface

is roughened by thermal fluctuations, as hasbeen observed in computer simulations (Pas-tor, 1994) and has been deduced from scat-tering experiments (König et al., 1992; Wie-ner and White, 1992; McIntosh and Simon,1993). These thermal fluctuations correspondto relative displacements or deformations ofthe lipid head groups: Since they will, ingeneral, change the surface area of the lipid-water interface, they are governed by an ef-fective tension . The roughness arisingfrom these fluctuations is set by the lengthscale , which is expected to be ofthe order of a few angstroms.

6.3 Persistence Length on Large Scales

On sufficiently large scales, a fluid surfacethat does not experience any lateral tensionstarts to crumple, i.e., to behave as a randomsurface without any average orientation. Thishappens as soon as its size exceeds theso-called persistence length (de Gennesand Taupin, 1982; Gompper and Kroll,1995). For phospholipid bilayers with bend-ing rigidities , the persistence lengthis, however, very large compared to the larg-est accessible sizes of these bilayers. There-fore, under normal circumstances, lipid bi-layers (and biomembranes) do not behave asrandom surfaces with no average orienta-tion.

7. GENERIC INTERACTIONS OF TWOMEMBRANES

The behavior of interacting membranes isgoverned by the interplay of direct interac-tions arising from the forces between themolecules and shape fluctuations such asbending undulations that act to renormalizethese interactions.

7.1 Direct Interactions between RigidMembranes

The direct interaction between tworigid membranes at separation can be mea-sured by the surface force apparatus consist-ing of two mica surfaces onto which themembranes are immobilized (Marra and Is-raelachvili, 1985). The simplest example isprovided by lipid bilayers that


1. are electrically neutral and2. interact across a water layer that contains

no macromolecules or colloids.

In this case, the interaction potential iscomposed of a repulsive hydration and anattractive van der Waals interaction and hasthe schematic form shown in Fig. 7(a).

Lipid bilayers may become charged by ad-sorption of ions from the solution or by dis-sociation of their head groups. They then ex-hibit electric double layers, which usuallylead to repulsive interactions between thesurfaces as predicted by the classical Pois-son-Boltzmann theory. The combination ofvan der Waals and electrostatic interactionsoften leads to a potential barrier as shown inFig. 7(b).

7.2 Renormalization by BendingUndulations

Two interacting membranes that are inthermal equilibrium with the surroundingliquid undergo bending undulations. Sincethe configurational entropy of the undula-tions increases with the membrane separa-tion, these fluctuations lead to a repulsiveforce that drives the membranes apart and,thus, reduces the attractive part of their in-teraction. It will be tacitly assumed here thatthe interaction potential is effectivelyshort-ranged and decays faster than forlarge Van der Waals forces eventually de-cay as and thus belong to this strong-fluctuation regime.

7.2.1 Repulsive Interactions If the in-teraction between the membranes is ef-fectively repulsive, the membranes can bekept together by an osmotic pressure aris-ing, e.g., from macromolecules in the sur-rounding solution that cannot permeate themembranes. In this case, the membranes arebound for finite but unbind in thelimit of zero . The mean separation be-haves as for small with the un-binding exponent . As the membranesunbind, the probability for pair contactsor two-membrane collisions decays to zeroas with the contact exponent

. This is completely analogous to the be-havior of one-dimensional strings, which aregoverned by a finite line tension.

The behavior of the mean separationcan be obtained from the entropic or fluctu-ation-induced interaction , as de-duced by Helfrich (1978) using a heuristicscaling picture even though this picture im-plies the incorrect behavior forthe contact probability.

7.2.2 Attractive Interactions and Con-tinuous Unbinding For a direct interac-tion as shown in Fig. 7(a) that is dominatedby attractive van der Waals forces, the mem-branes undergo a continuous unbinding tran-sition from a bound state at low temperaturesto an unbound state at high temperatures, aswas first predicted theoretically (Lipowskyand Leibler, 1986). If the interaction poten-tial is parametrized by the effective po-tential depth and the effective potentialrange , the unbinding temperature is givenby

FIG. 7. Direct interaction between two planar membranes as a function of the membrane separation (a) hy-dration and van der Waals interaction between electrically neutral membranes; (b) hydration, van der Waalsand electrostatic interaction between electrically charged membranes leading to a potential barrier.


(7)

As one approaches from below, themean separation of the membranes di-verges as , and the strength

of the renormalized potential vanishes as. The same critical behavior

is found for two interacting strings in two di-mensions where the unbinding process rep-resents a wetting transition.

The adhesion of large vesicles is also di-minished by thermally excited fluctuations;compare Fig. 6. From the relation (5) andthe behavior of the adhe-sive strength, one obtains the unbinding tem-perature for large vesi-cles with surface area

7.2.3 Potential Barriers and Discontin-uous Unbinding The direct interaction

may exhibit a potential barrier as shownin Fig. 7(b). In this case, the unbinding tran-sition can be continuous or discontinuousdepending on the relative strength of the po-tential barrier. If the potential barrier is rela-tively weak, the undulations "tunnel"through the barrier and the unbinding tran-sition is continuous. This is again analogousto one-dimensional strings in two dimen-sions, which tunnel, however, through anypotential barrier that decays faster thanfor large In contrast, two-dimensionalmembranes cannot tunnel through suffi-ciently strong barriers. In the latter situation,they are trapped by the barrier, and the un-binding transition is discontinuous.

7.3 Tension-Induced Adhesion

A lateral tension acts to suppress thebending undulations and thus to decreasethe fluctuation-induced repulsion. In fact,this interaction now becomes short-rangedand decays exponentially as

for separations . Thus,for sufficiently large separations, this fluctu-ation-induced repulsion cannot compete withthe attraction arising from van der Waalsforces, and the membranes form a boundstate in the presence of lateral tension. Inthe limit of vanishing tension, the mem-branes unbind provided the temperature ex-ceeds the unbinding temperature .

It has been argued by Helfrich (1989) thatsome lipid bilayers show a more complex be-

havior in the low-tension regime and thatthese layers develop a superstructure thatacts as an additional reservoir for membranearea. This issue remains to be clarified.

7.4 Repulsive Interactions at SmallSeparations

It has been well established by many ex-periments that lipid bilayers in aqueous so-lution experience a strong repulsion at smallseparations of the order of 1 nm (Rand andParsegian, 1989). The corresponding forceper unit area or disjoining pressure, , is ob-served to decay exponentially as

, where the effective decay length is ofthe order of a few angstroms; see Fig. 8. Itwas originally thought that this short-rangedrepulsion represents a hydration effect andreflects the perturbed water structure infront of the polar head group (Marcelja andRadic, 1976). On the other hand, a similarrepulsive force can also arise from the mo-lecular roughness of the lipid-water inter-faces (Israelachvili and Wennerström, 1990).In general, both mechanisms should contrib-ute, and their relative importance will de-pend on the temperature and on the lipid-solvent composition (Lipowsky and Grote-hans, 1994).

8. BUNCHES AND STACKS OFMEMBRANES

Lipid bilayers in solution often formbunches in which several membranes are, onaverage, parallel to each other. Two differentgeometries must be distinguished:

1. free bunches in solution, as shown in Fig.9(a), and

2. bunches that adhere to a rigid surface orwall; see Fig. 9(b).

The latter geometry is obtained, e.g., byspreading a concentrated lipid solution on aglass slide.

The structure of such a bunch can becharacterized by its density profile, whichdepends on the mean separations of themembranes within the bunch. If the shapefluctuations of these membranes are strong,they drive the membranes apart and lead toloosely bound or highly swollen states (Li-powsky, 1995a).


FIG. 8. Disjoining pressure of phos-phatidyl choline (PC) bilayers as a func-tion of the mean separation of lecithinor PC bilayers for the three solventswater, formamide, and 1,3-propanediol(PDO) (McIntosh et al., 1989).

8.1 Free Membranes in Solution

For a free bunch of identical mem-branes as shown in Fig. 9(a), all membraneshave the same bending rigidity , and eachadjacent pair of membranes interacts withthe direct interaction . If the bunch iskept together by short-ranged attractive in-teractions arising, e.g., from van der Waalsforces, it undergoes a unique unbinding tran-sition as observed

1. in experiments for up to 20 sugarlipid

membranes (Mutz and Helfrich, 1989)(see Fig. 10), and

2. in Monte Carlo simulations for up to fourmembranes (Netz and Lipowsky, 1993).

The unbinding temperature of the freebunch is independent of and thus identicalto the unbinding temperature of two mem-branes.

Analytical work on bundles of strings(Hiergeist et al., 1994) leads to the predictionthat the mean separation of the adjacent

FIG. 9. Bunches of fluctuating membranes: (a) Free bunch in solution; (b) bunch adhering to a rigid surfaceor wall.


FIG. 10. Unbinding or adhesion transition as experimentally observed for a bunch of eight lipid bilayers in aque-ous solution. (Left) For , the membranes undulate very strongly and then appear as thick fuzzylines. (Right) For , the membranes form a bound state that corresponds to the sharp dark line. Thewater between the membranes has been squeezed into the large water pocket. The bars represent 10 µm.(Courtesy of W. Helfrich.)

membranes labeled by and is givenby

(8)

Thus, all length scales should diverge withthe universal unbinding exponent butshould exhibit an amplitude that dependsstrongly on and on the position withinthe bunch. This implies that the temperatureinterval in which the behavior is dominatedby fluctuations shrinks as for large .Therefore, for large the continuous un-binding transition will appear to be discon-tinuous.

8.2 Membranes at a Rigid Surface

In the case of membranes attracted to-ward a rigid surface or wall, as shown inFig. 9(b), the unbinding behavior depends onthe relative strength of the surface-mem-brane and the membrane-membrane attrac-tion. There are essentially two possibilities:

1. If the attraction toward the substrate issufficiently weak, one has a sequence of

two transitions: the whole bunch un-binds from the substrate at the unbindingtemperature and then un-dergoes the transition toward completelyunbound membranes at

2. If the attraction toward the rigid surfaceis relatively strong, one has a sequence of

unbinding transitions and thus a se-quence of unbinding temperatures with

. At each of these transi-tions, a single membrane peels off fromthe bunch. Such peeling processes shouldalways occur during the formation of ves-icles from oriented samples. Thus, in thiscase, the presence of the rigid surface isstill felt by the outermost membrane evenin the presence of many intervening mem-branes.

8.3 Limit of Lyotropic Liquid Crystal

In the limit of large , the membranestack becomes a lyotropic liquid crystal,which can be studied by x-ray and neutronscattering.

8.3.1 Power-Law Peaks of the Scatter-ing Intensity The renormalization of thedirect interaction by bending modes


with small wavelengths leads to the effectivepotential , which determines the meanseparation via . Expandingthis effective potential up to second order inthe displacements, one obtains a harmonicmodel, which leads to the prediction that thescattering intensity exhibits the power-law behavior

(9)

in reciprocal space along the directionclose to with

and at . Thesepower-law peaks are known as Landau-Peierls singularities and have been studiedby x-ray scattering on lamellar phases of oil-water-surfactant mixtures (Meunier et al.,1987).

8.3.2 Melting of Lamellar States La-mellar phases of oil-water-surfactantmixtures often melt into isotropic phases. Ifthe melting is initiated by bending undula-tions, one may assume that the lamellarphase becomes unstable as soon as theroughness of the membrane separationsatisfies (de Gennes and Taupin,1982) (note that the roughness of a single"tracer" membrane within the liquid crystalincreases logarithmically with the size of themembrane). This relation corresponds to theso-called Lindemann criterion for the melt-ing of solids. Within the harmonic model,this implies that the lamellar phase becomesunstable as soon as the bending rigidity be-comes too small and satisfies

9. POLYMER-DECORATED MEMBRANES

In addition to the lipid bilayer, biomem-branes contain a complex mixture of macro-molecules. These molecules, which are an-chored in the membranes, are usuallyconnected to a network of relatively stiff,rodlike filaments that belong to the so-calledcytoskeleton within the cell. Likewise, theouter plasma membrane is often covered bya "coat" of polysaccharides, i.e., of branchedpolymers, which are also anchored withinthe membranes. These systems can be mod-

eled, to a certain extent, by the decoration oflipid bilayers with polymers.

9.1 Anchored Polymers

Any polymer with one or several hydro-phobic segments that fit into the hydropho-bic interior of the lipid bilayer can be an-chored to this bilayer. The simplest geometryis obtained for a polymer with a single an-chor attached to one of its ends. Artificialsystems of this kind are water-soluble poly-mers covalently bound to a single lipid mol-ecule (Lasic, 1994) or block copolymers witha short hydrophobic block. Another possibil-ity is to use polymers with hydrophobic sidegroups (Simon et al., 1995). In this case, thepolymers can be attached by several anchors.Such a system is shown in Fig. 11. In con-trast to polymers grafted to solid surfaces,these anchored polymers can diffuse laterallyalong the fluid membrane.

If the nonanchored segments of the poly-mer are effectively repelled from or attractedto the membrane surface, the anchored poly-mer is in a desorbed or in an adsorbed state,respectively. In the desorbed state, a polymerthat is attached to the membrane by a singleanchor has the form of a "mushroom," thesize of which is comparable to the dissolvedpolymer. Weak adsorption leads to squashedmushrooms; strong adsorption, to polymer"pancakes" that are tightly bound onto themembrane surface.

9.2 Dilute and Semidilute Regimes

First, consider effectively repulsive inter-actions between the anchored polymer andthe membrane surface. If the concentrationof anchors within the membrane is suffi-ciently small, the anchored polymers formwell-separated mushrooms. This dilute re-gime applies up to the overlap concentrationat which the membrane surface becomescompletely covered by anchored polymers. Ifone increases the coverage beyond this over-lap value, the polymers start to squeeze andto stretch each other and one enters thebrush regime. This brush regime extends upto a maximal coverage at which the loss ofentropy arising from the stretched state ofthe polymers exceeds their anchoring energy.

A similar distinction applies to anchored

.


FIG. 11. Shape transformation of (a) a tu-bular vesicle into (b) a string of spheres asinduced by the addition of polymers with hy-drophobic side groups. This transformation ispresumably governed by the interaction ofthese side groups with the lipid bilayer.(Courtesy of H. Ringsdorf.)

polymers that are adsorbed onto the mem-brane. In the dilute regime, one haswell-separated pancakes. Beyond the overlapconcentration, these pancakes start to inter-penetrate each other and to form a semi-dilute adsorption layer.

9.3 Polymer-Induced Curvature

A bound polymer or mushroom suffers aloss of entropy compared to the free state.This entropy loss depends on the shape ofthe membrane; see Fig. 12. Therefore, themushroom exerts entropic forces on themembrane that bend the membrane awayfrom the polymer (Lipowsky, 1995b).

In the unbound state, a linear flexiblepolymer consisting of monomers of linearsize forms a coil of linear dimension

with for good solvents. In thebound mushroom state, the polymer inducesthe "spontaneous" mean curvature

(10)

of the decorated membrane. In a similarway, curvature is induced by a polymerbrush that is attached to the membrane in anonsymmetric way. If the polymer is ad-sorbed onto the membrane, on the otherhand, it induces the opposite sign of the cur-vature: The membrane now bends towardthe membrane in order to maximize thenumber of contacts.

9.4 Polymer-Induced Adhesion

The adhesion of cell membranes is medi-ated by rodlike polymers that have a linearextension of 10-30 nm. In the absence ofthese sticky rods, the cell membranes expe-rience an effectively repulsive interactionarising from the electric charges on theirsurfaces. The adhesive strength mediated by


FIG. 12. A polymer attached to the membrane by a single anchor forms a mushroom that bends the membraneby entropic forces. For the three shapes of the membrane segment shown here, the cone leads to the highestentropy for the anchored polymer.

the sticky rods depends on the concentrationof their anchors within the membrane.Since this strength must exceed a certain

threshold value in order to overcome the re-pulsion arising from the membrane undula-tions, the anchor concentration must also ex-ceed a critical value . If the rod anchorsattract one another, the bilayers may un-dergo phase separation into polymer-rich do-mains with and polymer-depleted do-mains with . In the latter case, themembranes will only adhere via the polymer-rich domains.

10. APPLIED ELECTROMAGNETICFIELDS

For the phenomena discussed in the pre-vious sections, the molecular structure of thelipid bilayer is not perturbed in an essentialway. In contrast, the application of pulsedelectric fields or of optical tweezers built upfrom a laser beam strongly perturbs the bi-layer structure, which leads to new coopera-tive effects.

10.1 Alternating Electric Fields

The lipid membrane has a dielectric con-stant (or permittivity) , which is smallcompared to the dielectric constantof water. Likewise, the electrical conductivity

of water is large compared to the conduc-tivity of the lipids. This implies that theelectric field is amplified within the mem-brane. This is the basic mechanism that is

responsible for the electroporation and elec-trofusion of lipid bilayers and biomembranes(Neumann et al., 1989; Chassy et al., 1992).

If the alternating electric fieldis applied in the water and the two lipid-wa-ter interfaces are treated as dividing surfaceswith no intrinsic width, the transmembranepotential across a planar membrane isgiven by

(11)

where represents the thickness of themembrane (this relation follows from Max-well's equations and the continuity equationfor the charge carriers). For water withoutsalt, the square root prefactor is about 5× 103.

10.2 Electroporation

If the lipid membrane is exposed to atransmembrane potential that exceeds

for an extended period of time, themembrane ruptures in an irreversible way.On the other hand, if one applies a potential

of the order of for a relatively shorttime period within the range

, one obtains a reversible electricalbreakdown of the lipid bilayer: The perme-ability of the bilayer is strongly increasedright after the pulse has been applied butthen relaxes back to its original value. Thisbehavior is observed both for artificial lipidbilayers and for biomembranes. In both


cases, the high-permeability state arises fromthe formation of pores within the lipid mem-brane. These pores have been directly ob-served by freeze-fracture electron micros-copy.

10.3 Electrofusion

Two membranes that are in close contactcan be fused by applying a short electricalpulse. It is widely believed that this processof electrofusion is induced by the simultane-ous electroporation of both membraneswithin the contact area. It has also beenfound, however, that fusion can even occurwhen the membranes are treated with elec-trical pulses before being brought into closecontact. Electrofusion is a universal fusionmechanism that has been applied to manydifferent types of biomembranes. An exam-ple in shown in Fig. 13.

The initial fusion step presumably con-sists in the juxtaposition of two pores in thetwo adjacent membranes. The next stepshould be the formation of a neck or "worm-hole" that bridges the gap between the twomembranes. Several intermediate structuresthat could be involved in this topologicaltransformation have been proposed.

10.4 Optical Tweezers

Optical tweezers, arising from a laserbeam that is focused through a microscopeobjective, provide localized light traps and

can be used to manipulate various objects inthe micron range. Recently, such tweezershave been directly applied to lipid bilayersand were found to induce novel shape trans-formations. When applied to cylindrical ves-icles consisting of a single bilayer, the tweez-ers induce a pearling instability, whichspreads from the trap and eventually pro-duces a chain of pearls that again resemblesthe Plateau-Rayleigh instability of capillarytubes. When applied to bunches of severalbilayers, the light trap acts to unbind themembranes locally and to induce a charac-teristic shape of the loosely bound structure(Bar-Ziv et al., 1995).

It is presently believed that the focused la-ser beam destroys the bilayer structure lo-cally and acts to pull lipid molecules into thelight trap. This process should generate a lat-eral tension for the rest of the membrane,which is not exposed to the beam. A theo-retical model has been developed that isbased on this interpretation and has beenused to calculate the selected wavelength ofthe pearling instability (Goldstein et al.,1996).

11. OUTLOOK ON APPLICATIONS

The tendency of lipids to form closedcompartments such as vesicles is used inpharmacology, food industry, and cosmeticsin order to enclose and transport drugs, en-zymes, vitamins, moisturizers, and other in-

FIG. 13. Electrofusion of two lipid vesicles as observed by phase contrast microscopy: (a) The two vesicles,with diameters of about 45 and 37 µm, are brought into close contact by an alternating electric field; (b) fusedvesicle about 1 s after the application of a short electrical pulse; (c) the fused vesicle, which has a diameter ofabout 51 µm, becomes roughly spherical when the alternating field is switched off. (Courtesy of U. Zimmer-mann.)


gredients. The delivery of drugs via vesiclescan be improved by the decoration withpolymers that are attached to the lipid headgroups and that prevent the uptake of thesevesicles by lymphocytes in the blood. Fur-thermore, vesicles composed of lipidmixtures might be used in order to transportdrugs directly through the skin and, thus, toreplace the use of hypodermic needles.

Other interesting areas are attempts to at-tach lipid layers to solid substrates in orderto build up new types of biosensors, or tocover artificial bones by such layers in orderto improve their biocompatibility.

The mechanisms of electroporation andelectrofusion have been applied to many ar-eas of biotechnology. For example, electro-poration is now widely used in order totransfer genes into all kinds of cells.

In summary, flexible membranes alreadyplay an important role in various soft-mattertechnologies. It seems rather likely that animproved understanding of the physics ofthese systems will lead to new technologicalapplications: The future is in the flexible!

GLOSSARY

Aggregation Colloids: Supramolecularstructures that arise from the self-assemblyof single molecules or monomers in solution.

Amphiphilic Molecules: Molecules thathave a lyophilic part, which dissolves in thesolvent, and a lyophobic part, which is insol-uble in the solvent. One important exampleis molecules dissolved in water that have apolar hydrophilic part and a nonpolar hydro-phobic part.

Anchored Polymer: Polymer chain at-tached to the bilayer membrane by one orseveral hydrophobic segments that fit intothe hydrophobic interior of the bilayer. If thenonanchored segments of the polymer are ef-fectively repelled or attracted to the mem-brane surface, the anchored polymer attainsthe form of a "mushroom" or "pancake," re-spectively.

Area Compressibility Modulus: Elasticmodulus of membranes that governs theirresistance against stretching.

Bending Rigidity: Elastic modulus ofmembranes that governs their resistanceagainst bending.

Bilayer: Membrane consisting of twomonolayers that are arranged in such a waythat the polar head groups of the lipid mol-ecules form the two lipid-water interfaces,whereas the hydrophobic chains are buriedwithin the bilayer and have essentially nocontact with the aqueous solution. The thick-ness of these bilayers is 4-5 nm.

Budding: Shape transformation in whichthe membrane produces a small bud that isessentially spherical and connected to theoriginal membrane via a small neck.

Conformal Diffusion: For vesicles withtwo or more handles, the shape of the vesiclecan undergo a diffusion process along degen-erate states that are related by conformaltransformations.

Critical Monomer Concentration: Thisconcentration represents

1. the upper limit for the concentration ofamphiphilic monomers in solution and

2. the critical concentration at which theamphiphilic molecules start to aggregateinto supramolecular structures.

It is also called critical micelle concentrationsince many surfactant molecules assembleinto micelles.

Curvature: Any point on a two-dimen-sional surface that is embedded in three-di-mensional space can be characterized by twoprincipal curvatures, and , which areequal to the inverse curvature radii. Themean curvature is given by

Decorated Membrane: Lipid bilayer withanchored polymers. For fluid bilayers, thesepolymers can diffuse along the membranesurface.

Direct Interaction: Interaction betweentwo rigid membranes arising from the forcesbetween the molecules. This interaction canbe measured by the surface-force apparatusin which the membranes are immobilizedonto two mica surfaces.

Electrofusion: Two adjacent membranesthat are separated by a small water layer canbe fused by the application of an electricalpulse, which is believed to induce poresthrough both membranes.

Electroporation: Pores that go through abilayer can be induced by the application ofa short electrical pulse with a voltage of theorder of 1 V.


Flip-flops: The slow exchange of mole-cules between the two monolayers of a bi-layer membrane.

Fluid Membranes: Membranes for whichshear deformations relax by flow within themembrane. The molecules within such mem-branes undergo rapid lateral diffusion.

Hydrophobic Effect: The tendency ofamphiphilic molecules in water to form su-pramolecular structures in which the hydro-philic groups of these molecules form theoutside interface toward the aqueous solu-tion, whereas the hydrophobic groups areburied inside and have essentially no contactwith the water; the corresponding free-en-ergy change has a large entropic contribu-tion arising from the configurational entropyof the hydrogen bond networks in the water.

Internal Membranes: Biomembraneswithin the cell that cover many intracellularorganelles such as the cell nucleus, the en-doplasmic reticulum, mitochondria, etc.

Lipids: Amphiphilic molecules for whichthe hydrophobic part consists of two hydro-carbon chains and the hydrophilic part con-sists of a polar head group.

Lyotropic Liquid Crystal: Liquid crystalsthat can be swollen by the addition of sol-vent. In the present context, these liquidcrystals consist of stacks of bilayer mem-branes.

Monolayer: Monomolecular layer of am-phiphilic molecules that can be found, e.g.,at the interface between two immiscible liq-uids such as water and oil.

Optical Tweezers: Light trap with a lin-ear size of about 0.5 µm, which is generatedby a focused laser beam and which can beused to grasp and to move a single lipidmembrane or a whole bunch of membranes.

Plasma Membrane: Biomembrane thatrepresents the boundary between a biologicalcell and its environment; it consists of a lipidbilayer with many anchored proteins.

Polymer-Induced Curvature: Polymersthat are anchored onto the membrane exertentropic forces that induce a "spontaneous"curvature of the decorated membrane.

Shape Fluctuations: Membranes flickerand thus undergo thermally excited undula-tions that are directly visible in the opticalmicroscope.

Shape Transformations: The shape ofvesicles can be transformed in a controlledway by changing the temperature or the os-

motic conditions. Such shape transforma-tions can occur in a continuous or discontin-uous fashion.

Surfactants: Amphiphilic molecules forwhich the hydrophobic part usually consistsof one hydrocarbon chain.

Tension of Rupture: Lateral tension, ofthe order of a few mJ/m2, at which the lipidmembrane ruptures.

Unbinding Transition: Phase transitionof interacting membranes between a boundadhesive state at low temperatures and anunbound nonadhesive state at high tempera-tures.

Vesicle: Closed bilayer without edges,which can attain a large variety of differentshapes. An onionlike structure consisting ofseveral closed bilayers is usually called a li-posome.

Works Cited

Alberts, B., Bray, D., Lewis, J., Raff, J. M., Rob-erts, K., Watson, J. D. (1994), Molecular Biology ofthe Cell, 3rd ed., New York: Garland.

Bar-Ziv, R., Menes, R., Moses, E., Safran, S.-A.(1995), Phys. Rev. Lett. 75, 3356-3359.

Brochard, F., Lennon, J. F. (1975), J. Phys.(Paris) 36, 1035-1047.

Canham, P. B. (1970), J. Theoret. Biol. 26, 61-81.

Cevc, C., Marsh, D. (1987), Phospholipid Bilay-ers: Physical Principles and Models, New York: Wi-ley.

Chassy, B. M., Saunders, J. A., Sowers, A. E.(Eds.) (1992), Guide to Electroporation and Electro-fusion, San Diego: Academic.

Darnell, J., Lodish, H., Baltimore, D. (1990),Molecular Cell Biology. Scientific, New York: W. H.Freeman.

de Gennes, P.-G., Taupin, C. (1982), J. Phys.Chem. 86, 2294-2304.

Evans, E. A. (1974), Biophys. J. 14, 923-931.Evans, E., Needham, D. (1987), J. Phys. Chem.

91, 4219-4228.

Farge, E., Devaux, P. F. (1992), Biophys. J. 61,347-357.

Goldstein, R. E., Nelson, P., Powers, T., Seifert,U. (1996), J. Phys. II (France) 6, 767-796.

Gompper, G., Kroll, D. M. (1995), Phys. Rev. E51, 514-525.

Helfrich, W. (1973), Z. Naturforsch. 28c, 693-703.

Helfrich, W. (1978), Z. Naturforsch. 33a, 305-315.

Helfrich, W. (1989), Liquid Cryst. 5, 1647-1658.


Hiergeist, C., Lässig, M., Lipowsky, R. (1994),Europhys. Lett. 28, 103-108.

Israelachvili, J. N., Wennerström, H. (1990),Langmuir 6, 873-876.

Jülicher, F., Lipowsky, R. (1996), Phys. Rev. E53, 2670-2683.

Jülicher, F., Seifert, U., Lipowsky, R. (1993),Phys. Rev. Lett. 71, 452-455.

Kahlweit, M., Lipowsky, R. (Eds.) (1996), Mi-croemulsions—Experiment and Theory, Berichteder Bunsengesellschaft für Physikalische ChemieNo. 100, 181-393.

König, S., Pfeiffer, W., Bayerl, T., Richter, D.,Sackmann, E. (1992), J. Phys. II (France) 2, 1589-1615.

Krstic, R. V. (1976), Ultrastruktur der Säugetier-zelle, Berlin: Springer.

Lasic, D. D. (1993), Liposomes: From Physics toApplications, Amsterdam: Elsevier.

Lasic, D. D. (1994), Angew. Chem. Int. Ed. Engl.33, 1685-1698.

Lipowsky, R. (1991), Nature 349, 475-481.Lipowsky, R. (1995a), Z. Phys. B 97, 193-203.Lipowsky, R. (1995b), Curr. Opin. Struct. Biol

5, 531-540.Lipowsky, R., Grotehans, S. (1994), Biophys.

Chem. 49, 27-37.

Lipowsky, R., Leibler, S. (1986), Phys. Rev. Lett.56, 2541-2544.

Marcelja, S., Radic, N. (1976), Chem. Phys. Lett.42, 129-130.

Marra, J., Israelachvili, J. N. (1985), Biochem-istry 24, 4608-4618.

McIntosh, T. J., Simon, S. A. (1993), Biochem-istry 32, 8374-8384.

McIntosh, T. J., Magid, A. D., Simon, S. A.(1989), Biochemistry 28, 7904-7912.

Meunier, J., Langevin, D., Boccara, N. (Eds.)(1987), Physics of Amphiphilic Layers, SpringerProceedings in Physics vol. 21, Berlin: Springer.

Miao, L., Seifert, U., Wortis, M., Döbereiner,H.-G. (1994), Phys. Rev. E 49, 5389-5407.

Michalet, X., Bensimon, D. (1995), Science 269,666-668.

Mutz, M., Helfrich, W. (1989), Phys. Rev. Lett.62, 2881-2884.

Nelson, D., Piran, T., Weinberg, S. (Eds.)(1989), Statistical Mechanics of Membranes andSurfaces, Singapore: World Scientific.

Netz, R. R., Lipowsky, R. (1993), Phys. Rev.Lett. 71, 3596-3599.

Neumann, E., Sowers, A. E., Jordan, C. A.(Eds.) (1989), Electroporation and Electrofusion inCell Biology, New York: Plenum Press.

Pastor, R. W. (1994), Curr. Opin. Struct. Biol. 4,486-492.

Rädler, J., Feder, T. J., Strey, H. H., Sackmann,E. (1995), Phys. Rev. E 51, 4526-4536.

Rand, R. P., Parsegian, V. A. (1989), Biochim.Biophys. Acta 988, 351-376.

Sackmann, E. (1990), Can. J. Phys. 68, 999-1012.

Seifert, U., Lipowsky, R. (1995), in: R. Li-powsky, E. Sackmann (Eds.), Handbook of Biologi-cal Physics, Vol. 1, The Structure and Dynamics ofMembranes, Amsterdam: Elsevier.

Simon, J., Kühner, M., Ringsdorf, H., Sack-mann, E. (1995), Chem. Phys. Lipids 76, 241-258.

Tanford, C. (1991), The Hydrophobic Effect: For-mation of Micelles and Biological Membranes, Mal-abar, FL: Krieger.

Wiener, C. M., White, S. H. (1992), Biophys. J.61, 434-447.

Willmore, T. J. (1982), Total Curvature in Rie-mannian Geometry, Chichester: Ellis Horwood.

Further Reading

Bloom, M., Evans, E., Mouritsen, O. G. (1991),Quart. Rev. Biophys. 24, 293-397.

Cevc, C., Marsh, D. (1987), Phospholipid Bilay-ers: Physical Principles and Models, New York: Wi-ley.

Chassy, B. M., Saunders, J. A., Sowers, A. E.(Eds.) (1992), Guide to Electroporation and Electro-fusion, San Diego: Academic.

Gennis, R. (1989), Biomembranes: MolecularStructure and Function, New York: Springer-Ver-lag.

Lasic, D. D. (1993), Liposomes: From Physics toApplications, Amsterdam: Elsevier.

Lipowsky, R., Sackmann, E. (Eds.) (1995),Handbook of Biological Physics, Vol. 1, Structureand Dynamics of Membranes, Amsterdam: North-Holland.

Seifert, U. (1997), "Configurations of Fluid Ves-icles and Membranes," Adv. Phys. 46, 13-137.

Shinitzky, M. (Ed.) (1993), Biomembranes:Physical Aspects, New York: VCH Publishers.

Documents

Vesicles and Biomembranes - Max Planck Society · 2007. 8. 24. · each biomembrane consists of a specific mix-ture of many different amphiphilic molecules that reflect its diverse