thesis Fluid Membrane.pdf

7/29/2019 thesis Fluid Membrane.pdf

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Theoretical studiesof fluid membrane mechanics

Dissertation

zur Erlangung des Grades

Doktor der Naturwissenschaften

am Fachbereich Physik, Mathematik

und Informatik

der Johannes Gutenberg-Universitatin Mainz

Martin Michael Muller

geboren in Trier

Mainz

Oktober 2007


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Tag der mundlichen Prufung: 28. November 2007


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Fur meine Familie


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Zusammenfassung

Biologische Membranen sind Fettmolekul-Doppelschichten, die sich wie zweidimensionaleFlussigkeiten verhalten. Die Energie einer solchen fluiden Oberflache kann haufig mitHilfe eines Hamiltonians beschrieben werden, der invariant unter Reparametrisierungender Oberflache ist und nur von ihrer Geometrie abhangt. Beitrage innerer Freiheitsgradeund der Umgebung konnen in den Formalismus mit einbezogen werden.

Dieser Ansatz wird in der vorliegenden Arbeit dazu verwendet, die Mechanik fluiderMembranen und ahnlicher Oberflachen zu untersuchen. Spannungen und Drehmomentein der Oberflache lassen sich durch kovariante Tensoren ausdrucken. Diese konnen dannz. B. dazu verwendet werden, die Gleichgewichtsposition der Kontaktlinie zu bestimmen,an der sich zwei aneinander haftende Oberflachen voneinander trennen. Mit Ausnah-me von Kapillarphanomenen ist die Oberflachenenergie nicht nur abhangig von Trans-

lationen der Kontaktlinie, sondern auch von Anderungen in der Steigung oder sogarKrummung. Die sich ergebenden Randbedingungen entsprechen den Gleichgewichtsbe-dingungen an Krafte und Drehmomente, falls sich die Kontaktlinie frei bewegen kann.Wenn eine der Oberflachen starr ist, muss die Variation lokal dieser Flache folgen. Span-nungen und Drehmomente tragen dann zu einer einzigen Gleichgewichtsbedingung bei;ihre Beitrage konnen nicht mehr einzeln identifiziert werden.

Um quantitative Aussagen uber das Verhalten einer fluiden Oberflache zu machen,mussen ihre elastischen Eigenschaften bekannt sein. Der Nanotrommel-Versuchsauf-bau ermoglicht es, Membraneigenschaften lokal zu untersuchen: Er besteht aus einerporenuberspannenden Membran, die wahrend des Experiments durch die Spitze einesRasterkraftmikroskops in die Pore gedruckt wird. Der lineare Verlauf der resultieren-

den Kraft-Abstands-Kurven kann mit Hilfe der in dieser Arbeit entwickelten Theoriereproduziert werden, wenn der Einfluss von Adhasion zwischen Spitze und Membranvernachlassigt wird. Bezieht man diesen Effekt in die Rechnungen mit ein, andert sichdas Resultat erheblich: Kraft-Abstands-Kurven sind nicht langer linear, Hysterese undnichtverschwindende Trennkrafte treten auf. Die Voraussagen der Rechnungen konntenin zukunftigen Experimenten dazu verwendet werden, Parameter wie die Biegesteifigkeitder Membran mit einer Auflosung im Nanometerbereich zu bestimmen.Wenn die Materialeigenschaften bekannt sind, konnen Probleme der Membranmechanikgenauer betrachtet werden. Oberflachenvermittelte Wechselwirkungen sind in diesemZusammenhang ein interessantes Beispiel. Mit Hilfe des oben erwahnten Spannungs-tensors konnen analytische Ausdrucke fur die krummungsvermittelte Kraft zwischen

zwei Teilchen, die z. B. Proteine reprasentieren, hergeleitet werden. Zusatzlich wird dasGleichgewicht der Krafte und Drehmomente genutzt, um mehrere Bedingungen an dieGeometrie der Membran abzuleiten. Fur den Fall zweier unendlich langer Zylinder aufder Membran werden diese Bedingungen zusammen mit Profilberechnungen kombiniert,um quantitative Aussagen uber die Wechselwirkung zu treffen.

Theorie und Experiment stoen an ihre Grenzen, wenn es darum geht, die Relevanz vonkrummungsvermittelten Wechselwirkungen in der biologischen Zelle korrekt zu beurtei-len. In einem solchen Fall bieten Computersimulationen einen alternativen Ansatz: Diehier prasentierten Simulationen sagen voraus, dass Proteine zusammenfinden und Mem-branblaschen (Vesikel) bilden konnen, sobald jedes der Proteine eine Mindestkrummungin der Membran induziert. Der Radius der Vesikel hangt dabei stark von der lokal aufge-

pragten Krummung ab. Das Resultat der Simulationen wird in dieser Arbeit durch einapproximatives theoretisches Modell qualitativ bestatigt.


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Summary

Biological membranes consist of a bilayer of lipid molecules which behaves like a two-dimensional fluid. The energy associated with such a fluid surface is often completelydescribed by a reparametrization-invariant Hamiltonian which only depends on the sur-face geometry. Internal degrees of freedom and sometimes even bulk contributions arereadily included.Within this framework, the mechanics of fluid membranes and similar surfaces is studied.One can write stresses and torques on the surface in terms of covariant tensors. These canbe used to determine the equilibrium position of the contact line between two adheringsurfaces or different domains on one surface in a completely systematic way. With theexception of capillary phenomena, the surface energy is sensitive not only to boundarytranslations but may also correspond to changes in slope or even curvature. The resulting

boundary conditions express the balance of stresses and torques if the contact line is freeto move. At a rigid substrate, however, the variation has to follow the local substrateshape; stresses and torques enter a single balance equation and cannot be disentangled.To make quantitative predictions about the behavior of a fluid surface, its elastic prop-erties have to be known. The nanodrum setup offers a direct way to probe membraneproperties locally: it consists of a pore-spanning lipid bilayer membrane which is pokedwith the tip of an atomic force microscope (AFM). The linear behavior of the resultingforce-distance curves is reproduced in the theory if one neglects the influence of adhesionbetween AFM tip and membrane. Including it in the model via an adhesion balancechanges the situation significantly: force-distance curves cease to be linear, hysteresisand nonzero detachment forces can show up. These rich characteristics may offer a pos-

sibility to uniquely deduce membrane parameters such as the bending rigidity on thenano-scale in future experiments.Once the material parameters are known, problems of membrane mechanics can beaddressed in full detail. One particularly interesting problem in this context involvesinterface-mediated interactions. Analytical expressions for the curvature-mediated forcebetween two membrane-bound particles such as proteins can be obtained with the helpof the stress tensor. Additionally, torque and force balance yield several analyticalconditions on the membrane geometry. To see how these rather abstract analyticalexpressions can be applied, the shape of the membrane is determined exactly for thecase of two infinitely long cylinders adhering to the membrane.

When asking for the relevance of curvature-mediated interactions in the biological cell,

experiment and theory both encounter difficulties. Computer simulations then offer analternative approach with their unique ability to identify and separate individual con-

tributions to the phenomenon or process of interest. The presented simulations predict

that once a minimal local bending is realized, the effect robustly drives protein cluster

formation and subsequent transformation into vesicles with radii that correlate with the

local curvature imprint. This is confirmed qualitatively by an approximate theoretical

model.


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List of Publications

Chapter 3:

M. Deserno, M. M. Muller, and J. Guven. Contact lines for fluid surface adhesionPhysical Review E 76(1), 011605 (2007).

Chapter 4:

S. Steltenkamp, M. M. Muller, M. Deserno, C. Hennesthal, C. Steinem, and A. Jan-

shoff. Mechanical Properties of Pore-Spanning Lipid Bilayers Probed by AtomicForce Microscopy Biophysical Journal 91(1), 217226 (2006).

D. Norouzi, M. M. Muller, and M. Deserno. How to determine local elasticproperties of lipid bilayer membranes from atomic-force-microscope measurements:A theoretical analysis Physical Review E 74(6), 061914 (2006).

Chapter 5:

M. M. Muller, M. Deserno, and J. Guven. Interface mediated interactions be-

tween particles: A geometrical approach Physical Review E 72(6), 061407 (2005).

M. M. Muller, M. Deserno, and J. Guven. Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations Physical ReviewE 76(1), 011921 (2007).

M. M. Muller, M. Deserno, and J. Guven. Surfaces stresses of bulk origin inpreparation (2007).

B. J. Reynwar, G. Illya, V. A. Harmandaris, M. M. Muller, K. Kremer, and M. De-

serno. Aggregation and vesiculation of membrane proteins by curvature-mediatedinteractions Nature 447(7143), 461464 (2007).

Further Publication:

J. Guven and M. M. Muller. How paper folds: bending with local constraintssubmitted to Journal of Physics A (2007).


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Contents

Introduction 1

Dramatis Personae 3

1 The biological membrane a fluid surface 7

1.1 Biological membranes . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.1 The lipid bilayer . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Membrane proteins . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 The energy associated with fluid surfaces . . . . . . . . . . . . . . . 141.2.1 Surface tension and curvature energy . . . . . . . . . . . . . 151.2.2 Internal degrees of freedom . . . . . . . . . . . . . . . . . . . 171.2.3 Bulk energies . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Surface Mechanics 252.1 Stresses and torques in fluid surfaces . . . . . . . . . . . . . . . . . 25

2.1.1 First variation and Euler-Lagrange equations . . . . . . . . . 252.1.2 Surface stress and surface torque tensor . . . . . . . . . . . . 29

2.2 The fluid membrane . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.1 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . 342.2.2 The Helfrich membrane . . . . . . . . . . . . . . . . . . . . . 342.2.3 Lipid tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Stresses from the surrounding space . . . . . . . . . . . . . . . . . . 422.3.1 Global pressure difference . . . . . . . . . . . . . . . . . . . 422.3.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Boundary conditions at contact lines 49

3.1 The Young-Dupre equation . . . . . . . . . . . . . . . . . . . . . . . 493.2 Geometry and energy . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Determining the boundary conditions . . . . . . . . . . . . . . . . . 53

3.3.1 Continuity considerations . . . . . . . . . . . . . . . . . . . 533.3.2 Contact line variation . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Specific examples I Fluid surface adhesion . . . . . . . . . . . . . 573.4.1 Adhesion to a rigid substrate . . . . . . . . . . . . . . . . . 573.4.2 Adhesion to deformable surfaces . . . . . . . . . . . . . . . . 65

3.5 Specific examples II Domains on a vesicle . . . . . . . . . . . . . . 71

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4 How to determine elastic properties of a membrane on the nano-scale 774.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.1 Atomic force microscopy . . . . . . . . . . . . . . . . . . . . 784.1.2 Measurements on the nanodrum . . . . . . . . . . . . . . . . 79

4.2 Theoretical model of the nanodrum . . . . . . . . . . . . . . . . . . 814.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2.2 Energy considerations . . . . . . . . . . . . . . . . . . . . . 82

4.3 Solving the shape equation . . . . . . . . . . . . . . . . . . . . . . . 844.3.1 Linear approximation . . . . . . . . . . . . . . . . . . . . . . 844.3.2 Complete nonlinear formulation . . . . . . . . . . . . . . . . 87

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.4.1 No adhesion between tip and membrane . . . . . . . . . . . 91

4.4.2 Including adhesion between tip and membrane . . . . . . . . 974.5 What can be learnt in future experiments? . . . . . . . . . . . . . . 105

5 Membrane-mediated interactions between particles 1075.1 Forces and torques on particles . . . . . . . . . . . . . . . . . . . . 108

5.1.1 Balance of forces and torques . . . . . . . . . . . . . . . . . 1085.1.2 Forces and torques on the outer boundary . . . . . . . . . . 1095.1.3 Two-particle configurations with symmetry . . . . . . . . . 1125.1.4 Analytical force expressions . . . . . . . . . . . . . . . . . . 1145.1.5 Equilibrium conditions from torque balance . . . . . . . . . 123

5.2 Two cylinders on a fluid membrane exact solution . . . . . . . . 1295.2.1 Determining the profile . . . . . . . . . . . . . . . . . . . . . 1305.2.2 Conditions from torque balance . . . . . . . . . . . . . . . . 1385.2.3 Forces between the cylinders . . . . . . . . . . . . . . . . . . 1415.2.4 Two sewing needles on water a brief interlude . . . . . . . 144

5.3 The multi-body problem . . . . . . . . . . . . . . . . . . . . . . . . 1475.3.1 Cell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.3.2 Coarse-grained computer simulations . . . . . . . . . . . . . 155

Conclusions 163

Appendix 165

A Classical differential geometry of two-dimensional surfaces 165A.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A.2 Geometry at a curve on the surface . . . . . . . . . . . . . . . . . . 174A.3 Gauss-Bonnet theorem . . . . . . . . . . . . . . . . . . . . . . . . . 176A.4 Two important surface parametrizations . . . . . . . . . . . . . . . 177

A.4.1 Monge parametrization . . . . . . . . . . . . . . . . . . . . 177A.4.2 Angle-arc length parametrization . . . . . . . . . . . . . . . 180

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B Surface variations 185B.1 Derivatives of surface scalars . . . . . . . . . . . . . . . . . . . . . . 185

B.2 The stress tensor for a few standard cases . . . . . . . . . . . . . . 187B.2.1 General curvature Hamiltonians . . . . . . . . . . . . . . . . 187B.2.2 Gradients of curvature . . . . . . . . . . . . . . . . . . . . . 188B.2.3 Vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

C Solutions of the membrane shape equation 195C.1 Two cylinders on a membrane . . . . . . . . . . . . . . . . . . . . . 195

C.1.1 Outer section . . . . . . . . . . . . . . . . . . . . . . . . . . 196C.1.2 Inner section . . . . . . . . . . . . . . . . . . . . . . . . . . 198

C.2 The nanodrum in small gradient approximation . . . . . . . . . . . 208

List of Tables 209

List of Figures 211

List of Technical Points 215

Bibliography 217

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Introduction

Lipid bilayer membranes constitute one of the most fundamental componentsof all living cells. Apart from their obvious structural task in organizing dis-tinct biochemical compartments, their contributions to essential functions suchas protein organization, sorting, or signaling play an important role in nature[AJL+02, LBM+04].

Many of these tasks such as exo- or endocytosis [Mar01], the formation of vesi-cles [Rob97, MG05, Ant06], or the interaction with the cytoskeleton [LH92] sig-nificantly exceed mere passive separation or solubilization of proteins. Instead,they rely heavily on mechanical membrane properties and can be studied usingthe mathematical toolbox of theoretical physics. Various key questions are of anessentially geometrical nature: what is the shape adopted by a membrane sub-jected to some specified boundary conditions [Sei97]? How are forces and torquestransmitted through the membrane [CG02b, Guv04, LM06, Koz06]? Which in-teractions does this imply [GBP93, KPDN95, WKH98, KNO98, DFG98, KRS99,MM02, BBR02, BF03, Wei03, DOD06]?To answer questions such as these, a differential geometric description of the mem-brane has proven advantageous. On length scales much larger than its thicknessthe membrane can be described as a two-dimensional fluid surface with an energythat depends entirely on surface invariants (see Chap. 1). Consequently, stressesand torques in the membrane are expressible in terms of the local surface geome-try as well. It is possible to introduce the concept of a stress tensor well knownfrom three-dimensional elasticity theory to problems involving the mechanics ofmembranes in particular and surfaces in general (see Chap. 2).By resisting the temptation to use some specialized parametrization and ratherchoose a covariant description, many problems lead to remarkably simple geometricrelations, which are not at all obvious or even recognizable in a parametrized lan-

guage. Examples include boundary conditions at triple-lines of contacting surfaces(see Chap. 3) or interactions between membrane-bound particles (see Chap. 5).Even if the surrounding space is included in the theoretical picture, the mechanicalsurface response continues to be driven by its geometry. Hence, a geometric stresstensor is no less useful. The task is rather to couple it to potential off-surfacesstresses such as a pressure difference across the surface. There are a variety ofmethods how this may be done and we will discuss some of these in Chaps. 2and 5.

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Introduction

Knowing the connection between stresses and geometric surface properties, how-ever, is not a substitute for solving the relevant field equation if one wants to

determine the actual shape of the membrane. Unfortunately, this equation is anonlinear partial differential equation of fourth order which typically has to besolved numerically. In Chaps. 4 and 5 this will be done for problems obeying arotational or translational symmetry. For more complicated situations tailoredcomputer simulations offer an alternative approach.Combining analytical theory, numerical calculations and computer simulations in-deed seems to be the ideal way to study the mechanics of membranes theoretically.Where one approach meets its limitations the other one can step into the breachoffering a more complete view of the relevant problems. Before discussing these,however, let us start with an overview of the main characters. A detailed descrip-

tion will occur later in the text, especially in the first chapter.

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Dramatis Personae

Dramatis Personae

The cellleft: Electron micrograph of a plasma cell, a type of white blood cell that se-cretes antibodies. Only a single membrane (the plasma membrane) surroundsthe cell, but the interior contains many membrane-limited compartments, ororganelles (from [CM93] with courtesy of P. Cross); right: Schematic sketch of ananimal (eukaryotic) cell (from http://micro.magnet.fsu.edu/cells/animalcell.html).

The biological fluid membraneleft: Electron micrograph of a thin section through the membrane of a redblood cell (from [LBM+04]); right: Sketch of the membrane. It consists ofa lipid bilayer to which different kinds of macromolecules (e.g. proteins) arebound (from [GK96, Fig. 4.15 on p.99]; copyright, W. W. Norton and Co.) .

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Introduction

PO

O O

O

O O

O-

O

N+

CH3

H3CCH3

CH3 H3C

e1

e2

n

X

xy

z

Model membranes (DPPC bilayer)To model the membrane theoretically, different levels of resolution can bechosen. In all-atom computer simulations (above) every atom is representedby one bead. In coarse-grained simulations (middle) groups of atoms arecombined; their interactions can even be tuned such that the surrounding waterdoes not have to be simulated explicitly. Treating the membrane analyticallyas a two-dimensional surface is possible on length scales much larger than themembranes thickness (below).

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Dramatis Personae

z

y

x

Proteins in membranesProteins can be included into the theoretical model on all levels of resolution.For example, the BAR domain (above) can deform the lipid bilayer locally(with courtesy of P. Blood). On larger length scales such an effect can be modeledusing hard spheres that adhere to the membrane (middle and below).

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Introduction

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1 The biological membrane a fluidsurface

1.1 Biological membranes

Biological membranes consist of a bilayer of lipid molecules in which proteins

are embedded. At physiological conditions this bilayer behaves like a two-dimensional fluid, with individual molecules able to diffuse rapidly in it.

Each cell is surrounded by a closed membrane, the plasma membrane, which phys-ically separates the extracellular environment from the cells interior. Bacteria andarchae possess just this single membrane, whereas animal and plant cells also con-tain internal membrane-limited subcompartments, the organelles. The nucleus,for example, stores the genetic information, the mitochondria are the power plantsof the cell, and the endoplasmic reticulum is the factory where protein moleculesare synthesized [LBM+04, AJL+02].1

To cope with the different tasks in which membranes are involved as a crucial part,a membranes molecular composition is highly inhomogeneous. Despite that theyall have a common basic structure: the lipid bilayer.

1.1.1 The lipid bilayer

Lipid molecules are the basic building blocks of the membrane. They are am-phiphiles, that is they consist of two different parts: a hydrophilic (i.e., water-loving) head and a hydrophobic (i.e., water-fearing) tail. The hydrophobic tailis typically formed by two hydrocarbon chains originating from various fatty acids.One of them is often unsaturated. This means that at least one double bond existsbetween two of the carbon atoms (see Table 1.1). The fatty acids are attachedvia ester or amide covalent bonds to a linker group like glycerol which also bindsto the hydrophilic head group. In phospho- or glycolipids, the most abundantlipids in the cell, this head group consists of a simple phosphate or sugar group,respectively, and another group such as choline (see Table 1.2).In an aqueous environment the hydrophobic carbon tails of the lipid disturb thehydrogen bonds that exist between the water molecules close to it. These can only

1 Cells that have a nucleus and other organelles are also called eukaryotic cells.

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1 The biological membrane a fluid surface

substance #C #db chemical structure

laurate 12 0 CH3(CH2)10COOmyristate 14 0 CH3(CH2)12COOpalmitate 16 0 CH3(CH2)14COOstearate 18 0 CH3(CH2)16COOarachidate 20 0 CH3(CH2)18COO

oleate 18 1 CH3(CH2)7CH=CH(CH2)7COOlinoleate 18 2 CH3(CH2)4(CH=CHCH2)2(CH2)6COO-linolenate 18 3 CH3(CH2)4(CH=CHCH2)3(CH2)3COOarachidonate 20 4 CH3

(CH2)4

(CH=CH

CH2)4

(CH2)2

COO

Table 1.1: Common biological fatty acid anions [GKD+04, A3.8]. The number ofcarbon atoms (#C) is even because biological fatty acids are synthesized by concate-nation of C2 units. Note also that the fatty acid anions can be further divided intosaturated (no double bond (db)) and unsaturated (one or more double bonds) ions.

be maintained if the water sacrifices a part of its entropy. This phenomenon iscalled the hydrophobic effect [Tan91] and is the main reason why the hydrophobicpart of an amphiphile tries to avoid water. The hydrophilic part, however, canform hydrogen bonds with the water and therefore likes to stay in it.The schizophrenic nature of the amphiphiles causes them to aggregate spon-taneously above a certain critical concentration. At that concentration which iscalled the critical micelle concentration(cmc) it becomes energetically more favor-able for them to sacrifice some of their entropy and self-assemble into a condensedstructure, thereby shielding their hydrophobic parts from the water. The morphol-ogy of the aggregate depends on the nature of the amphiphile, ( i.e., the size of thehead group and the hydrophobic tail), but also on the solution conditions (i.e.,salt concentration, pH or temperature) [Isr92]. Typical structures that result arespherical or cylindrical micelles, bilayers and vesicles (see Fig. 1.1).

Most of the lipid molecules in cells form bilayers, a fact already discovered in 1925[GG25]. The bilayer of a biological membrane is approximately 5 nm thick andcontains typically on the order of one hundred different lipids. The lipid fractionsvary between organisms, cells, different organelles within the cell, and even betweenthe two sides of the bilayer. Still, some common features can be identified thatare essential for biological function [LBM+04, AJL+02, GKD+04]:

The hydrophobic core of the bilayer acts as a barrier to the passage of mostwater-soluble molecules. In consequence, a small number of molecular chan-

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substance chemical structure type

lysophospha- O

OR 1 C H2

C H

H2

C

H O

O P O

O

O

C H2

C H2

N C H3

C H3

C H3

lyso-tidylcholine phospholipid(lysolecithin)

phosphatidyl- O

OR 1 C H2

C H

H2

C

O

O P O

O

O

C H2

C H2

N H3

R 2

O

glycero-ethanolamine phospholipid(PE)

phosphatidyl- O

OR 1 C H2

C H

H2

C

O

O P O

O

O

C H2

C H2

N C H3

C H3

C H3

R 2

O

choline (PC)

digalactosyl-

O

CH2

O H

H H

O H

O H

H

H

H O

H

O

O

CH2

H H

O H

O H

H

HO

O

OR1 CH2

CH

H2

C

OR2

OH O

H

glycero-diglyceride glycolipid(DGDG)

sphingomyelin C H C HO H

C H

H2

C

N H

O P

R 2

O O

O

O

C H2

C H2

N ( C H2

)3

C HR 1 sphingo-phospholipid

Table 1.2: Examples of membrane lipids. R1 and R2 correspond to hydrocarbonchains of fatty acids (see Table 1.1) [GKD+04, A3.8f] (from [Mul04]).

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(a) (b) (c) (d)

Figure 1.1: Typical structures that result from self-assembly: (a) spherical micelle,(b) cylindrical micelle, (c) flexible bilayer/vesicle, and (d) planar bilayer. The blackcircles are the hydrophilic heads, the chains the hydrophobic tails of the amphiphiles.

nels and pumps can strictly regulate the exchange of chemicals between thetwo sides of the membrane.2

The cmc is very low (106-1010 M for bilayer-forming lipids as opposed to102-105 M for micelle-forming amphiphiles). The bilayer is therefore verystable: even though the exterior aqueous environment can vary widely inionic strength and pH, the bilayer has the strength to retain its characteristicarchitecture.

The biological bilayer behaves as a two-dimensional fluid at physiologicaltemperatures. The lipids can diffuse freely within each leaflet of the bilayer.3

Below a critical transition temperature Tm the bilayer loses its fluidity and becomesa gel-like solid. Mismatches between the cross-sectional areas of lipid head andlipid tail group lead to tilted or interdigitated chains and rippled surfaces [LS95,Chap. 5].The value ofTm depends, for instance, on the degree of saturation of the fatty acidchains in the bilayer. In unsaturated fatty acids the chains are kinked and cannotpack closely. Lipids with unsaturated fatty acids thus form bilayers with a lowertransition temperature than lipids with saturated fatty acids. For example, a pureDPPC (1,2-dipalmitoyl-phosphatidylcholine) bilayer has a transition temperature

of (41.3 1.8) C, whereas DOPC with one cis double bond in every chain (seeFig. 1.2(a)) is fluid above (18.3 3.6) C [KC98].4 The same rule holds forsynthetic membrane lipids. Under ambient conditions a DOTAP bilayer is already

2 An alternative way of material transport through the membrane will be discussed in the nextsection and in Chap. 5.3 A membrane lipid can diffuse the length of a typical bacterial cell (1 m) in only one second,

the length of a typical animal cell in about 20 seconds.4 The double bond cannot rotate. If the groups that are connected to it are oriented in the same

direction the molecule is referred to as cis, while when they point in opposing directions it isreferred to as trans.

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PO

OO

O

O-

O

O

O

N+CH3

CH3

H3C

H3C

H3C

(a) cis-DOPC (1,2-dioleoyl-phosphatidyl-choline)

O

OO

O

N+CH3

CH3

H3C

H3C

H3C

Cl-

(b) DOTAP (1,2-dioleoyl-3-trimethyl-ammonium-propane chloride)

Br-

N+CH3

CH3

H3C

H3C

(c) DODAB (N,N-dimethyl-N,N-dioctadecylammonium bromide)

OH

H3C H3C

CH3

CH3

H3C

(d) cholesterol

Figure 1.2: Examples of lipids with different effect on the bilayer fluidity: DOPC(a) and DOTAP (b) bilayers are fluid under ambient conditions, DODAB (c) forms agel-like bilayer. Cholesterol (d) tends to make fluid lipid bilayers less fluid.

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fluid whereas DODAB is still in the gel phase (see Fig. 1.2(b) and 1.2(c)). We willcome back to that point in Chap. 4 where these lipids will be used in an experiment

in which the elastic properties of bilayers are investigated.The fluidity of the bilayer also depends on the length of the hydrocarbon tails itcontains. A shorter chain length reduces the tendency of the tails to interact witheach other, so that the bilayer remains fluid at lower temperatures.Both effects, degree of saturation and chain length of the fatty acid tails, areexploited in nature. Under thermal stress (arising from, say, an abnormal decreasein temperature) many organisms such as bacteria or yeasts actively adjust thefatty acid composition of their membrane to maintain its fluidity [TJI98].Eukaryotic cell membranes additionally contain cholesterol for these purposes.Cholesterol consists of a rigid and relatively planar steroid skeleton with a sin-

gle hydroxyl group as the hydrophilic part (see Fig. 1.2(d)). It cannot form abilayer unless it is mixed with other lipids. By decreasing the mobility of thehydrocarbon chains it tends to make fluid lipid bilayers less fluid. At the highconcentration found in most eukaryotic plasma membranes, it also prevents thehydrocarbon chains from coming together and crystallizing thereby inhibiting thefluid-gel phase transition [LBM+04, AJL+02]. Ten years ago, it has been proposedthat both cholesterol and sphingolipids accumulate in the plasma membrane toform tiny domains of a diameter of a few tens of a nanometer [SI97, BL97, BL98].These lipid rafts are supposed to help organizing certain proteins which areimportant for biological functions such as cell signaling.Experiments with synthetic vesicles made from cholesterol, sphingomyelin and aphospholipid like POPC have shown a variety of phases, phase transitions, and co-existence regimes [DBV+01, VK05]. Typically, such a mixture can phase-separateinto micrometer-sized liquid ordered (Lo) domains which are rich in sphingomyelinand cholesterol and liquid disordered (Ld) domains of similar size. Whetherrafts exist in living cells or not, however, remains the subject of lively debate[Mun03, Nic05, Han06] as it is much harder to search for domains of nanometersize, especially in vivo. In Chap. 4 an experimental setup will be presented thatreaches the necessary spatial resolution. It may thus offer a possibility to searchfor lipid rafts in the near future.

1.1.2 Membrane proteins

Although the lipid bilayer provides the basic structure of the membrane, proteinsperform most of its biological functions. They serve as ion channels and specificreceptors, for example, or catalyze membrane-associated reactions, such as thesynthesis of adenosine 5-triphosphate (ATP), the principal carrier of chemicalenergy in cells.Proteins are macromolecules that consist of amino acids joined together by pep-tide bonds (see Fig. 1.3). One can distinguish four levels of organization in their

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HOO

O

HO

O

N

NH2 H

H2O

HO O

H2NOHHO

O

O NH2

peptidebond

(a) (b) (c)

Figure 1.3: (a) Amino acids can connect to each other via a peptide bond. Twoimportant secondary structures are (b) the helix and (c) the sheet.

structure. The linear sequence of amino acids is called the primary structure. Itis composed of 20 different types of amino acids and is encoded in the genomeof the cell. Individual amino acid residues on a protein may attract or repel oneanother. Localized parts of the peptide chain can thus fold into three-dimensionalstructures like helices or sheets (see again Fig. 1.3). These arrangements are

referred to as secondary structures. Their degree of hydrophobicity depends onthe character of the side chains of the amino acids involved. The tertiary structuredescribes the full three-dimensional organization of the chain. The protein can,however, be formed as a complex of more than one polypeptide chain. For instance,hemoglobin, the oxygen carrier of the blood, consists of four subunits. The com-plete structure is then designated as the quaternary structure [LBM+04, AJL+02].

Proteins can be bound to membranes in different ways: (i) integral membraneproteins, also called transmembrane proteins, span the lipid bilayer. They containone or more hydrophobic domains that are inserted into the core of the bilayer

as well as hydrophilic domains that extend into the aqueous medium on bothsides. (ii) Lipid-anchored membrane proteins are bound covalently to one or morelipid molecules. The hydrophobic tail of the attached lipid is embedded in oneleaflet of the membrane. The polypeptide chain itself does not enter the bilayer.(iii) Peripheral membrane proteins do not interact with the fatty acid chains ofthe bilayer. Instead they are only bound indirectly by interactions with integralmembrane proteins or directly by interactions with the lipid head groups. Howa protein associates with the membrane depends on its biological function. Onlytransmembrane proteins can function on both sides of the bilayer or transport

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molecules across it. By contrast, many proteins that function on only one side ofthe bilayer are often lipid-anchored or peripheral proteins.

Because the bilayer is fluid, all types of proteins can move laterally in the membraneand interact with one another. In Chap. 5 we will discuss membrane remodeling asone example where proteins aggregate to act cooperatively. Such processes are, forinstance, important in endocytosis in which material is ingested by invaginationof the plasma membrane and subsequent internalization in a membrane-boundvesicle.However, living cells have found ways to immobilize specific membrane proteins. Inthe previous section we have already discussed lipid rafts as one example. Besidethat, the interaction with the cytoskeleton restricts the mobility of proteins aswell. The cytoskeleton provides structural support for the eukaryotic cell. It

consists of a network of filaments build up from proteins such as actin or tubulinand permits directed movements of organelles and the cell itself.5 Some proteinsin the plasma membrane can link permanently to the cytoskeleton leaving themcompletely immobile in the membrane. Other proteins, though still mobile, areslowed down in the presence of the cytoskeleton.We will not go further into the biological details here. The interested reader isreferred once more to Refs. [LBM+04, AJL+02]. The rest of this introductorychapter will instead present a physical description of the biological membraneand similar fluid surfaces which will ultimately enable us to consider interestingproblems of fluid membrane or more generally fluid surface mechanics.

1.2 The energy associated with fluid surfaces

The energy associated with a fluid surface is often completely described bya reparametrization-invariant Hamiltonian that only depends on the surfacegeometry. Internal degrees of freedom and sometimes even bulk contributionsare readily included.

In order to choose the correct physical description of the membrane it first has tobe clear how much detail is needed to answer the questions of interest. To modelthe exact conformation of a protein embedded in the membrane, for example,

computer simulations are the right tool to use [Lea01].In this thesis, however, the membrane will be considered from a mesoscopic pointof view.6 This is applicable when the lateral extension and the size of all defor-mations of interest are much larger than its width. The lipid bilayer can then bemodeled as a two-dimensional surface embedded in three-dimensional Euclidean

5 For active transport you of course additionally need some kind of engine and fuel. In the cellthese are provided by special motor proteins and ATP, respectively.6 Only in Sec. 5.3.2 molecular dynamics simulations will be used to study the interactions of a

number of proteins bound to the membrane.

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Figure 1.4: A soap film spanning a tetrahedral frame (from [Mul04]).

space R3. Proteins or other entities that can bind to the membrane (e.g. viruses)are represented by solid particles that impose deformations in the surface.To predict the behavior of the membrane under deformations such as these, onehas to know how its energy changes. Let us therefore study possible contributionsto the energy of a fluid surface and discuss their relevance for the case of themembrane.

1.2.1 Surface tension and curvature energy

Surface tension

When surfaces occur in physical systems, the energy associated with them is oftencompletely described by a Hamiltonian that only depends on the surface geometry.The easiest and the best known examples involve capillary phenomena [RW02,dGBWQ03], in which the energy of a liquid-fluid interface is simply given by thesurface integral over a constant surface tension .7 The resulting Hamiltonian istherefore:

H =

dA , (1.1)

where is the surface domain and dA the infinitesimal area element.Hamiltonian (1.1) is not only relevant for interfaces between pure phases (suchas, for instance, water and water vapor) but also describes the behavior of soap

7 We will only consider interfaces in their ground states. Thermal fluctuations around thesestates or dynamical phenomena will not be taken into account. This implies that we do notneed to worry about entropic contributions to the free energy (for a general discussion in thecase of capillary phenomena, see again Ref. [RW02]).

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films (see Fig. 1.4) [Ise92]. Soap films consist of a 5 nm - 20 m thin layer of waterbetween two monolayers composed of amphiphilic soap ions (such as the fatty acid

anions, see Table 1.1). These ions can move laterally in the film exactly as thelipids in the membrane.The fluidity of the surface implies that it cannot resist shear stresses. Hence,static shear is not a deformation which costs the surface any energy; in fact, itcannot even be defined. As we will restrict our considerations to fluid surfaces thisproperty holds generally throughout this work.

Helfrich Hamiltonian

A bending deformation in the direction normal to the interface, however, can con-tribute to the total energy of a fluid surface. In a harmonic expansion the energy

due to any deformation of the interface is proportional to the square of the de-formation (remember Hookes law). In the particular case of bending, one has toconsider quadratic expressions of the curvature. At every point a surface has twoprincipal curvatures k1 and k2, which are the eigenvalues of the local curvature ten-sor Kba. The corresponding eigenvectors define two orthogonal principal directions(see App. A.1). One therefore has to include two independent terms in the expres-sion for the energy that depend on a quadratic combination of the two curvaturesand are furthermore invariant scalars. One convenient choice is a combination ofthe Gaussian curvature KG = k1k2 and the squared trace K

2 = (k1 + k2)2 of the

curvature tensor Kab. Including surface tension the complete Hamiltonian is then

according to Helfrich [Hel73]:

H =

dA

+

2(K K0)2 + KG

. (1.2)

where and are proportionality constants called bending rigidity and saddle-splay modulus, respectively. The constant K0 is the spontaneous curvature. Itsvalue determines how much the surface prefers to be bent in its minimal energystate. The term involving the Gaussian curvature can be written as the sum ofa topological constant and a line integral over the boundary of the surface (seeApp. A.3). The implications of this turn out to be subtle and will be discussed in

more detail in Technical Point 2.3 on page 38.A series of experimental and theoretical studies has shown that the energy as-sociated with the membrane is mainly dominated by bending (see, for instance,Refs. [BL75, FMM+89, DS95, LS95, Sei97, CDBN05]). The reason for this is thatit is the softest mode: characteristically, weak bending is a deformation whichcosts significantly less energy than, for instance, stretching.On length scales much larger than the bilayer thickness the Hamiltonian (1.2) issuitable to describe a fluid membrane.8 For typical phospholipid membranes,

8 It should be mentioned that the Hamiltonian (1.2) is also valid for other interfaces such as

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is of the order of a few tens of kBT, where kBT is the thermal energy [SL95].Furthermore, membranes exhibit tensions in a broad range from 0 up to about

10 mN/m [MH01]. At even higher values they rupture. The Gaussian bendingrigidity is rather difficult to measure. Its value is usually negative and alsosmaller than that of in the same system [GKD+04, A3.24].9

General curvature Hamiltonians

The fact that curvature (a generalized strain) enters quadratically in the Hamil-tonian (1.2) classifies this form of the bending energy as linear curvature elas-ticity (even though the resulting field equations that describe the shape of thesurface are highly nonlinear as we will see in Sec. 2.2.2). However, for sufficiently

strong bending higher than quadratic terms will generally contribute to the energydensity, giving rise to genuinely nonlinear curvature elasticity [GH96].

A general expression including all these terms is given by [Mul04]:

H[X] =

dA H(gab, Kab, aKbc, . . .) , a, b, {1, 2} , (1.3)

where the Hamiltonian density H depends exclusively on surface scalars con-structed using the metric gab, the extrinsic curvature tensor Kab, or its covariantderivatives aKbc, etc.Note in particular that Hamiltonian (1.3) is invariant under surface reparametriza-tions. In order to describe the surface in a parametrization-free way, a covariantdifferential geometric language will be used. The notation is essentially standardand is summarized briefly in Technical Point 1.1.

1.2.2 Internal degrees of freedom

So far the discussion was restricted to Hamiltonians which are exclusively con-structed from the geometry of the underlying surface. However, the surface itselfmay possess internal degrees of freedom which can couple to each other and, more

interestingly, also to the geometry. The simplest example would be a scalar field on the membrane, which could describe a local variation in surface tension orlipid composition. In that case the Hamiltonian

H[X, ] =

dA

2

(a)(a) + V() + K

(1.4)

those containing block copolymers.9 In Sec. 3.5 we will discuss an experimental setup with which the difference of the saddle-splay

moduli of two different lipid domains on a vesicle can be measured.

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Technical Point 1.1: Differential geometry

This Technical Point presents the basic notions of the differential geometry of

two-dimensional surfaces. A comprehensive summary can be found in App. A.A surface is described locally by its po-sition X(1, 2) R3, where the a (a {1, 2}) are a suitable set of local coordi-nates on the surface. The tangent vec-tors of , ea = X/

a = aX (a, b {1, 2}), form a local coordinate frame; to-gether with the extrinsic unit normal vec-tor, n = e1 e2/|e1 e2|, they definetwo geometrical tensors on the surface:

the metric gab = ea eb and the extrinsiccurvature Kab = ea bn.

e1

e2

n

X

xy

z

The symbol a is the metric-compatible covariant derivative. The trace of theextrinsic curvature tensor will be denoted by K = Kaa = Kabg

ab, which for asphere of radius r with outward pointing normal vector is positive and has the valueK = 2/r. The determinant KG = det(K

ba) is the Gaussian curvature. It can be

written as half the Ricci scalar curvature R = gacgbdRabcd, the double-contractionof the Riemann tensor. This link between intrinsic and extrinsic curvatures isa consequence of the (doubly contracted) Gauss-Codazzi equation R = K2 KabK

ab (= 2KG). As usual, indices are lowered or raised with the metric or

its inverse, respectively, and a repeated index a (one up, one down) implies asummation over a = 1, 2. More background on differential geometry can, forinstance, be found in Refs. [Car76, Spi76, Kre91].

has to be added to the geometric Hamiltonian (1.3) [CG04].10 The symbols and denote coupling constants. The last term represents an interaction betweenthe membrane curvature K and the field , whereas V() is a scalar potentialdepending on the value of . This case will not be discussed further here.Instead, let us look a little more closely at the case of an additional tangential

surface vector field ma

(see Fig. 1.5). Such a field has been introduced to describethe tilt degrees of freedom of the molecules within a lipid bilayer to accommodatethe fact that the average orientation m of the lipids themselves need not coincidewith the local bilayer normal n (see, for instance, Refs. [HP88, ML91, ML93,NP92, NP93, SSN96, HK00, MBS99, Fou99, May00, BKIM03, KZK04]).How can such a tilt be excited? In Sec. 1.1.2 it was mentioned that proteins

10 As mentioned in Technical Point 1.1 the sum convention is used throughout this work: if anindex occurs twice in a product, once as super- and once as subscript, one has to sum overthis index from 1 to 2 even though no explicit summation sign is present.

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m = maea

mn

Figure 1.5: Lipid tilt. The unit vectors n and m denote the local bilayer normal andaverage orientation of the lipids themselves. The values for ma (a {1, 2}) can beobtained by projecting m onto the tangent plane of the bilayer.

can span the entire membrane. These transmembrane proteins are not necessarilyshaped like a straight cylinder but can also vary in thickness over the width of themembrane looking, for instance, like a truncated cone. Such a membrane inclusion

distorts the lipid order close to it leading to a non-vanishing field ma.11In the presence of a field ma many additional terms for the energy emerge (for asystematic classification see Refs. [NP92, NP93]). However, the aim here is notto treat the most general case. Instead, let us focus on a simple representativeexample to illustrate how easily the present formalism generalizes to treat suchsituations.If one defines the properly symmetrized covariant tilt-strain tensors Mab and Fab

according to

Mab =1

2amb + bma

, (1.5a)

Fab = amb bma , (1.5b)one can construct (in the spirit of a harmonic theory) a Hamiltonian Hm from fourquadratic invariants:

Hm[X, ma] =

dA1

2mM

2 + mMabMab +

1

4mFabF

ab + V(m2)

, (1.6)

11 Note that membrane inclusions can also excite a change in membrane thickness if their hy-drophobic domain is much larger or smaller than the hydrophobic core of the bilayer [Kil98].This hydrophobic mismatch, however, will not be treated here.

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where M = gabMab = ama is the tilt divergence. The first two terms coincide

with the lowest order intrinsic terms identified by Nelson and Powers [NP92, NP93],

provided one restricts to unit vectors ma

.12

These terms are multiplied by newelastic constants m and m, playing the analogous role to Lame-coefficients.

13 Ifm2 = 1, a third term (also absent in usual elasticity theory [LL86]) occurs, thequadratic scalar constructed from the antisymmetrized tilt gradient; its structureis completely analogous to the Lagrangian in electromagnetism [LL00]. Finally, ifthe magnitude of ma is not fixed, one may also add a potential V depending onthe square m2 = mam

a of the vector field ma. Without loss of generality one canassume that V(0) = 0 because any nonvanishing constant is more appropriatelyabsorbed into the surface tension . IfV(x) is minimal for x = 0, then ma 0 willminimize the energy, but depending on physical conditions V may favor nonzero

values of |ma

|. In this case the lipids can acquire a spontaneous tilt below themain phase transition temperature of lipid bilayers (cf. Sec. 1.1.1).

1.2.3 Bulk energies

Surfaces do not live in vacuum. They are embedded in the surrounding space whichoften provides for additional contributions to the total energy. In this section wewill discuss two examples, a global pressure difference and a gravitational field.

Global pressure difference

In an aqueous environment any open patch of membrane costs a large amount ofenergy because at the edge the hydrophobic core of the bilayer is in direct contactwith the water. Therefore, free membrane patches usually do not exist in the cell.They close to form vesicles (see Fig. 1.1(c)).If the volume V is fixed to a constant V0, a term P(V V0) has to be added tothe energy functional H of the free surface:

HP = H P(V V0) , (1.7)

where the pressure difference P is the Lagrange multiplier for the enclosed volume.

A similar term may also be used to fix the pressure difference between the two12 Using the commutation relations for covariant derivatives (A.19) and (A.20) it is easy to see

that (amb)(bma) = (ama)2 12Rm2. If m2 = 1, the Gauss-Bonnet-theorem renders thelast term a boundary contribution (see App. A.3), and it is then easy to see that the m- andm-terms in Eqn. (1.6) are sufficient.13 In order for the Hamiltonian density (1.6) to be positive definite in the strain gradient amb,

it is necessary that m > 0 and m + m > 0. The latter differs from the usual conditionm +23m > 0 found in Ref. [LL86] because in the surface case one only has two-dimensional

tensors. The positivity of the antisymmetric term requires m < 0 since the square of anantisymmetric matrix has negative eigenvalues.

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sides of the surface:HP = H P V . (1.8)

Both ensembles (constant volume vs. constant pressure) yield the same groundstates for the shape of the surface. Questions of stability depend on which of thetwo variables is fixed: a surface shape found to be stable under constant V is notnecessarily stable under constant P. In Technical Point 2.5 on page 44 we willdiscuss this point for one example where the surface is a closed soap film ( i.e., asoap bubble).With the help of Gauss Theorem the volume V can be written as an integral overthe enclosing surface V:

V = V dV X

3=

1

3 V dA n X , (1.9)where n is the unit vector normal to the surface and X the position of the surfacein R3 as before. This allows us to express Hamiltonian (1.8) as a surface integral:

HP = H 13

P

dA n X . (1.10)

Gravity

Another energy contribution from the bulk may stem from gravity. On the sizeof a typical cell membrane ( 1 m) the gravitational field can be neglected. For

fluid surfaces on larger length scales, however, it becomes relevant as we will seein the following chapters.As an example consider a liquid-fluid interface in a homogeneous (downwardspointing) gravitational field of acceleration g (see Fig. 1.6). Gravity will make surethat in equilibrium the heavier of the two fluid phases will be below the lighter one.Furthermore, any deformation of the interface will increase the potential energyof the system, e.g. by lifting or lowering the heavier phase. This total potentialenergy can be written as

Epot = g

all fluiddV (X)h(X) , (1.11)

where (X) is the local density and h(X) is the height above some arbitraryhorizontal reference plane which fixes the zero-point of the potential energy.In the following let us assume that both phases are incompressible. Hence, thedensity in each phase will not depend on the position. Up to a constant onemay thus equivalently integrate the excess density of the heavier phase over thevolume Vhp occupied by the heavier phase:

Epot = g

Vhp

dV h(X) . (1.12)

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X

X

n zh

Vhp

g

Figure 1.6: Liquid-fluid interface in a homogeneous gravitational field.

The volume integral can then be split into an area integral over the reference planetimes a height integral perpendicular to it; the latter can be done explicitly:

Epot = g

dA

h(X)0

dz z =1

2g

dA h(X)

2 , (1.13)

where h(X) is now written as a function of the coordinates X of the reference

plane.

14

This energy depends of course on the embedding of the interface intoexterior space which is indeed a physical requirement. However, the way it iswritten it also depends on the parametrization of that surface, and this is somethingone would like to avoid if possible.In order to achieve that, any reference to some arbitrary base plane has to beeliminated. This could be done if one was able to rewrite the integrand in thevolume integral (1.12) as a pure divergence. Using Gauss Theorem again it wouldthen be possible to express it as a surface integral over the actual surface of theheavier phase rather than some arbitrary reference plane.Indeed, ifz is the upward direction, then

h = X z = 12 X2z , (1.14)

and thus the potential energy can also be written as

Epot =1

2g

Vhp

dV (X2z) = 12

g

Vhp

dA (n z) X2 , (1.15)

14 The surface parametrization that is chosen here is also called Monge parametrization. It isapplicable as long as the surface has no overhangs (see App. A.4.1).

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where the surface integral now runs over the complete surface of the lower phase.This expression is essentially parametrization-free; in particular, it does no longer

make reference to some arbitrary base plane. Notice, however, that the verticalsides of the fluid volume do not contribute to this integral (since there z n)while the lower (horizontal) surface gives a contribution which does not change ifthe upper free surface is modified. Hence, the only nontrivial contribution to theenergy stems from the upper surface integral over the fluid-liquid interface inEqn. (1.15).To obtain the full Hamiltonian HG, the potential energy has to be added to allother terms that contribute to the surface energy. In the case of a liquid-fluidinterface a tension will enter.15 Additionally, one has to take into account that thevolume of the liquid is constant. One thus obtains the Hamiltonian:

HG =

dA P(V V0) + Epot . (1.16)

Gravity will serve as an interesting and instructive but still manageable examplefor how non-constant bulk stresses act on surfaces. We therefore include it in ourdiscussion, even though it is seldomly important for the length scales prevailing atbiological membranes.However, the main focus in this work will be on the fluid membrane. As we nowknow the energies that determine its physics, questions of membrane mechanicscan be addressed, for instance, by examining the effects of forces and torques on

its shape.

15 In general of course, Hamiltonian (1.3) and possibly terms that reflect internal degrees offreedom have to be inserted.

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2 Surface Mechanics

In Sec. 1.1 we have seen that membranes in cells are not isolated free objects. Theyexperience a variety of external forces and torques. For example, proteins adhere tothem and impose deformations, actin filaments of the cytoskeleton push and pull atthem, pressure differences between the inner and outer sides may even cause theirrupture. Furthermore, lipid rafts have a higher rigidity than their surroundings, or

certain mechanosensitive membrane channels open and close gated by the tensionof the membrane.All these examples show that the mechanical properties of membranes are vitalfor the functioning of biological cells. A first step to understand these propertiesis to determine what shape the membrane adopts if it is subjected to some specificboundary conditions. Given the shape, forces and torques can be determinedwhich are imprinted in the geometry of the membrane.

2.1 Stresses and torques in fluid surfaces

The equilibrium shape of a surface is governed by a nonlinear partial dif-ferential equation, the shape equation. Stresses and torques in the surfaceare fully encoded in its geometry and described by divergence-free tensors.

In the following, the basic equations which govern the behavior of the membraneare derived. To keep the approach sufficiently general, we first consider the genericcase of a surface whose energetics can be described by the reparametrizationinvariant Hamiltonian (1.3)

H[X] =

dA H(gab, Kab, aKbc, . . .) . (2.1)

2.1.1 First variation and Euler-Lagrange equations

To determine the equilibrium (i.e., energy minimizing) shape of the surface, thefunctional (2.1) has to be minimized with respect to deformations of . Thesedeformations are described by a change in the embedding functionsX X+X.The response of the Hamiltonian can be written as [CG02b, Mul04]:

H =

dA E(H)n X+

dA aQa . (2.2)

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2 Surface Mechanics

The bulk part of this variation is a surface integral over the Euler-Lagrange deriva-tive E(H) times the normal projection of the surface variation X. Its vanishingdetermines the equilibrium shape of the interface. Hence, E = 0 is also calledthe shape equation. The second term of the variation is a surface integral overa divergence and can thus be recast as a boundary integral using the divergencetheorem [Fra03]. It originates from tangential variations1 as well as the derivativesof normal variations.The straightforward way to determine Eand Qa explicitly is to track the course ofthe deformation on X through gab,

g, Kab, and any appearing covariant deriva-

tives. This turns out to be rather tedious as gab, Kab, . . . indirectly depend on Xvia the structural relationships (see App. A.1)

gab = eaeb and (2.3a)

Kab = ea bn = ea bn , (2.3b)with

ea = X/a = aX = aX , (2.4a)

ea n = 0 , (2.4b)n2 = 1 . (2.4c)

Alternatively, one can treat gab, Kab, ea and n as independent variables, enforcingthe structural relations (2.3) and (2.4) using Lagrange multiplier functions as was

first pointed out by Guven [Guv04]. One thus introduces the new functionalHc[gab, Kab, . . . ,X, ea,n,

ab, ab,fa, a, n] given by

Hc = H[gab, Kab, aKbc, . . .] +

dA [ab(gab ea eb) + ab(Kab ea bn)]

+

dA [fa (ea aX) + a(ea n) + n(n2 1)] . (2.5)

The original Hamiltonian H is now treated as a function of the independent vari-ables gab, Kab and their covariant derivatives;

ab, ab, fa, a and n are Lagrange

multiplier functions fixing the constraints (2.3) and (2.4). The introduction ofauxiliary variables greatly simplifies the variational problem, because now we donot have to track explicitly how the deformation X propagates through to gaband Kab. As we will see in the following, this approach also provides a very simpleand direct derivation of the shape equation in which the multiplier fa, which pinsthe tangent vectors to the surface, can be identified as the surface stress tensor.

1 Away from the boundary, a tangential variation can be identified with a reparametrization ofthe surface. It always results in a pure divergence which is why it does not enter the shapeequation [CG02b].

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The Euler-Lagrange equations for X, ea, n, gab, and Kab, respectively, are givenby

afa = 0 , (2.6a)fa = (acKbc + 2

ab)eb an , (2.6b)0 = (bab + a)ea + (2n abKab)n , (2.6c)

ab = 12

Tab , (2.6d)

ab = Hab . (2.6e)Note that the Weingarten equations (A.39) an = Kbaeb have been used inEqn. (2.6b), the Gauss equations (A.41) aeb = Kabn in Eqn. (2.6c). We havealso defined

Hab := HKab

and (2.7a)

Tab := 2g

(

gH)gab

. (2.7b)

The manifestly symmetric tensor Tab is the intrinsic stress tensor associated withthe metric gab. IfH does not depend on derivatives of Kab, the functional deriva-tives in (2.7) reduce to ordinary ones.Equation (2.6a) reveals the existence of a conservation law for the multiplier fa.Using the other equations (2.6c), (2.6d), and (2.6e), it is straightforward to elim-

inate the Lagrange multipliers on the right hand side of Eqn. (2.6b) to obtainan explicit expression for fa in terms of the original geometrical variables. FromEqn. (2.6c) we find a = bab because ea and n are linearly independent; theEqns. (2.6d) and (2.6e) determine ab and ab. Thus Eqn. (2.6b) can be recast as

fa = (Tab HacKbc)eb (bHab)n . (2.8)

Once the Hamiltonian density has been specified, Eqn. (2.8) determines the con-served multiplier fa completely in terms of the geometry (see App. B.2 where

several examples are treated).The Hamiltonian (2.5) involves covariant derivatives ofX, n, and possibly Kab.To isolate the Euler-Lagrange equations (2.6), these derivatives have to be removedvia integration by parts. The resulting total derivatives must not be neglected,however, as they contribute to the boundary integral of the variation (2.2). Anadditional term due to the variation with respect to gab may appear as well becauseaKbc depends on the metric indirectly via the Christoffel symbols.2,32 Note that this is not true for aX and an as both can be written as partial derivatives.3 For instance, the covariant derivative of a scalar is given by a = a while for the

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2 Surface Mechanics

Collecting all terms together one obtains an expression for the boundary part ofthe variation (2.2):

Qa = fa X + Habeb n + Gabcgbc + KabcKbc . (2.9)This equation defines the tensors Gabc and Kabc as the boundary contributionsbelonging to the variation with respect to the metric and the extrinsic curvaturetensor, respectively. IfH is just a function of gab and Kab, Gabc and Kabc are equalto zero. If only first derivatives ofKab appear in H, these tensors will be functionsand can be written as (see App. B.2.2)

Kabc = H(aKbc) and (2.10a)

Gabc =

KacdKbd +

KcadKbd

KcbdKad . (2.10b)

In general, if derivatives of order higher than first occur in the Hamiltonian, Gabcand Kabc will be differential operators acting on gab and Kab respectively.Now let us suppose that X = a R3 is just an arbitrary constant transla-tion that of course leaves the Hamiltonian invariant, even if the surface is not inequilibrium. In this case, the first variation (2.2) can be written as4

H =

dA E(H)n a

dA a(fa a)

= a

dA

E(H)n afa

!= 0 . (2.11)

The integral must be equal to zero because a can be arbitrarily chosen. Moreover,the integrand vanishes pointwise because may be arbitrarily chosen as well.Thus,

afa = E(H)n . (2.12)If the surface is a true equilibrium surface, which implies that it is stationary withrespect to arbitrary variations, the shape equation E= 0 also holds and we recoverthe conservation law (2.6a) for fa.5

Finally, Eqn. (2.8) permits us to write E(H) completely in terms of the surfacegeometry: projecting Eqn. (2.12) onto the surface normal n and using the Gaussequations (A.41) once more, we obtain the remarkably succinct result

n afa = E(H) = KabTab + (KacKcb ab)Hab . (2.13)

contravariant tensor field Kbc one has aKbc = aKbc dab Kdc dac Kbd, where cab isthe Christoffel symbol of the second kind, defined by cab =

12

gcd(agbd + bgda dgab) (seepage 168 in App. A.1).4 The variations n, gab, and Kab vanish in this case.5 For unconstrained surfaces E(H) = 0 is indeed the shape equation. However, further con-

straints such as a global pressure difference imply additional terms as we will see in Sec. 2.3.

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Technical Point 2.1: The stress tensor in 2D and 3D

x

y

z

xxxy

xz

yyyx

yz

zx

zy

zzV

In classical elasticity theory the divergence of

the stress tensor at any point in a strainedmaterial equals the external force density ifthe whole body is in equilibrium [LL86]. Orequivalently, the stress tensor contracted withthe normal vector of a local fictitious area ele-ment yields the force per unit area transmit-ted through this area element. For a smallcubic part V of an arbitrarily shaped bodyin Euclidean space R3 the components of thetotal external force can thus be written as

(i {x,y,z})FVi =

zj=x

V

dV jij =z

j=x

V

dAj ij ,

where, for example xy is the force in x-direction acting on the unit area perpen-dicular to the y axis (see sketch). The divergence theorem was used in the secondstep and j is the partial derivative with respect to j.Comparing this with Eqn. (2.14) we see that fa is indeed the surface analog ofthe stress tensor: its divergence integrated over the surface yields a force. In fact,analogous to the 3D case, this integral can be rewritten as a boundary integral

over the contraction of f

a

with the normal vector of the local line element (seeEqns. (2.15) and (2.16)).

2.1.2 Surface stress and surface torque tensor

Identification of the stress tensor

A closer inspection of the translational variation (2.11) reveals that the conserva-tion law (2.6a) for the vector fa is simply a consequence of Noethers theorem: acontinuous symmetry (= translational invariance) implies an associated conserved

current (= f

a

) on shell (i.e., in equilibrium). To identify this current, considerthe variation (2.11) for E(H) = 0:

H = a

dA afa . (2.14)

The change in energy is given by the negative product of an infinitesimal trans-lation times an integral. Thus, the integral is a force. From this follows that thedivergence offa must be a force density. The vector fa encodes the stresses in thesurface. If we compare it with the stress tensor of a body in three-dimensional

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2 Surface Mechanics

l

l

l

l

out

13

2

1

3

2

Figure 2.1: Surface patch with 3 disjoint boundary components i and an outerlimiting boundary out.

Euclidean space, it becomes obvious that fa must be the surface stress tensor (seeTechnical Point 2.1).This identification is correct for a surface which is free to equilibrate in every point.But does fa still encode all of the surface stresses if external forces are acting on thesurface? To answer that question we have to discuss a more complicated situation:

consider a region of surface bounded by an outer limiting curve out and supposethat the bulk of this surface is in equilibrium with N particles that are attached toit (see Fig. 2.1). We now choose a curve around each of the particles, labeling thesecurves i, i = 1, . . . , N . The outer curve out together with all i encloses thesurface patch . Considering this patch first allows us to use variation (2.2) toidentify the stress tensor as no particles are attached to and every point on it isfree to equilibrate.If one of the boundaries i is translated infinitesimally by a, the boundaryvariation of the Hamiltonian (2.1) is equal to

H(i) (2.2)= a i

ds la fa = a F(i),ext , (2.15)

where the divergence theorem was used to convert the surface integral into aboundary integral. The vector l = la e

a is the unit vector which is normal to iand points out of the surface ; by construction it is tangential to (see Fig. 2.1and App. A.2). The variable s measures the arc length along i.

In Eqn. (2.15), the boundary integral is identified as the external force F(i),ext acting

on via its boundary i. The external force F(i)ext on the surface patch i is then

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simply given by F(i),ext due to force balance. Notice that this argument does notrequire us to assume that the Euler-Lagrange equation is satisfied on i. Thus,

even if external forces are acting, the conserved current fa

correctly encodes thestresses in the surface at all points not externally acted upon.

The conservation law (2.6a) for fa in fact allows us to deform the contour of

integration i to our advantage when asking for the external force F(i)ext. As

long as we do not cross any further sources of stress, the value of the integral inEqn. (2.15) will remain the same. In Chap. 5 this fact will be exploited in thecontext of interface-mediated interactions.To simplify notation, we will restrict our considerations to the patch 1 withoutloss of generality and write Fext := F

(1)ext, with

Fext = 1

ds lafa , (2.16)

where l = laea = l.

It proves instructive to look at the tangential and normal projection of the stresstensor by defining the tensors fab and fa such that [CG02b, Guv04]

fa = fabeb + fan . (2.17)

Using the equations of Gauss (A.41) and Weingarten (A.39) again, Eqn. (2.12)can then be cast in the form

afa = Kabfab + E, (2.18a)afab = Kbafa . (2.18b)

Tangential stress acts as a source of normal stress and vice versa. Both conditionshold irrespective of whether the Euler-Lagrange derivative Eactually vanishes. Infact, Eqn. (2.18a) shows that the shape equation E= 0 is equivalent to afa =Kabf

ab, while Eqn. (2.18b) merely provides consistency conditions on the stresscomponents that reflect the reparametrization invariance of the Hamiltonian. InTable 2.1 the stress tensor and the Euler-Lagrange derivative for several simple

Hamiltonians are summarized (for the calculations see App. B.2).Analogous to translational invariance, the rotational invariance of H implies theexistence of another conserved tensor ma which, as will be demonstrated in thenext section, can be identified as the torque tensor.

Identification of the torque tensor

Once again consider the surface in Fig. 2.1: under a constant infinitesimal rotation of one of the boundaries i, the position and normal vectors change according

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2 Surface Mechanics

H E fab fa

1 K gab 0

R 0 0 0

Kn

1n

K2 KabKab

nKn1 (nKab Kgab)Kn1 n aKn1

K2 K(K2 2KabKab) 2K (2Kab Kgab)K 2aK

KabKab

12 (K)2 12 (cK)(cK)

( + K2 R)K Kab(aK)(bK) 12 gab(K)2

(aK)(bK) 12 gab(K)2 KabK aK

Table 2.1: Euler-Lagrange derivative E(H) = Kabfab + afa and the componentsof the stress tensor fa = fabeb + f

an for several simple scalar surface Hamiltoniandensities H. Notice that K2 and KabKab yield identical Eand fa (see App. B.2.1).

to X= X and n = n. In this case, the boundary variation of theHamiltonian (2.1) is given by

H(i)

(2.2)=

i

ds laX fa + Habeb n

=

i

ds lama = M(i),ext , (2.19)

where the surface integral was converted into a boundary integral on the first line.The vector l = la e

a is defined as above and s again measures the arc length alongi.In Eqn. (2.19), the boundary integral is nothing but the external torque M

(i),ext

acting on via its boundary i: the corresponding change in energy is given by(minus) the scalar product ofM(i),ext with the infinitesimal rotation angle . The

external torque M(i)ext on the surface patch i is then simply given by M(i),ext due

to torque balance. Again, this argument does not require us to assume that theEuler-Lagrange equation is satisfied on i.In total analogy to the force, we will consider patch 1 without loss of generalityand write Mext := M

(1)ext, with

Mext =

1

ds lama , (2.20)

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2.2 The fluid membrane

where l = l again (see Fig. 2.1). The surface tensor

ma = X fa + Habeb n (2.21)

is the covariantly conserved torque tensor. It consists of two parts: a contributiondue to the couple of the stress tensor fa about the origin as well as an intrinsiccontribution proportional to Hab which originates only from curvature terms. Thevalue of the former does depend on the choice of origin while the latter is indepen-dent of this choice. The divergence ofma, is indeed zero in mechanical equilibriumas required by Noethers theorem:

am

a (2.4a,A.39,A.41)= ea

fa + X

(

af

a) + (

a

Hab)eb

n +

Habeb

Kcaec

(2.6a,2.8)= (Tab HacKbc) ea eb (bHab)ea n + (aHab) eb n

+ HabKca eb ec= 0 , (2.22)

where we have exploited the symmetry of Tab and Hab.The occurrence of curvature terms in the Hamiltonian is ultimately due to thefact that the physical surface is not infinitely thin but has an inhomogeneous forcedistribution along its transverse direction [LM06]. If one captures this internalstructure in the strictly two-dimensional Hamiltonian (2.1), the curvature termsgive rise to an intrinsic torque density and a non-vanishing normal component ofthe stress tensor.Note that the torque tensor, like the stress tensor, depends only on geometricproperties of the surface (modulo a contribution that is due to a shift of theorigin). It is thus always possible in principle to determine the forces and torquesoperating on a patch of surface from a knowledge of the shape of the surface alone.The snag is to determine the correct shape.


The stress tensor of a Helfrich membrane depends on local curvatures andgradients of curvature. It has a nonvanishing normal component and doesnot contain the saddle-splay modulus . A tilt degree of freedom adds furthertangential terms to the stress tensor.

Our treatment so far has been entirely general: the only properties of the Hamilto-nian we have used are reparametrization, translational, and rotational invariance.In the following, we will specialize the expressions for stress tensor, torque tensorand shape equation to Hamiltonians which are relevant for describing the mechan-ics of fluid membranes.

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2 Surface Mechanics

2.2.1 Surface tension

It is instructive to have a look at the simplest case of surface tension first beforeconsidering the entire Helfrich Hamiltonian (1.2). For the constant Hamiltoniandensity

Hcapillary = (2.23)we get: Hab (2.7a)= H/Kab = 0 and Tab (2.7b)= 2(g)1g/gab (B.4)= gab.Thus, the stress tensor is given by

fa(2.8)= ea . (2.24)

The force transmitted across a fictitious line element is normal to this line, inde-

pendent of its direction, tangential to the surface, and equal in magnitude to thevalue of the surface tension. This simple case has guided our intuition of how tothink about tensions and stresses in surfaces, but it is far too special: in general,surface stresses are not shear-free nor isotropic nor tangential nor constant as wewill see in the following example.Inserting expression (2.24) for fa into Eqn. (2.21), one obtains for the torquetensor

ma(2.21)

= X ea . (2.25)The intrinsic torque vanishes; the torque on a surface patch 1 is given by

Mext = 1

ds X l . (2.26)It is simple to interpret this expression: the tangential stress l provides a torqueper unit length at every point of the contour 1. The line integral along 1yields the total external torque on the surface patch.From Eqns. (2.18a) and (2.24), one obtains for the shape equation:

K = 0 . (2.27)

Thus, surfaces that extremize their area have vanishing mean curvature K. Suchsurfaces are also called minimal surfaces. One simple example is the catenoid(see Technical Point 2.2). In fact, it is the only surface of revolution which solvesEqn. (2.27) besides the plane [Kre91].

2.2.2 The Helfrich membrane

The energetics of a fluid membrane without internal degrees of freedom can bedescribed by the Hamiltonian density

H = Hcapillary + Hbend , (2.28)

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Technical Point 2.2: The catenoid - a surface of zero mean curvature

A soap film which is suspended between two parallel coax-

ial circular rings of equal radius a can be described by thefunction [dGBWQ03]

(h) = b coshh c

b,

where is the radial distance from the symmetry axis and his its height (see sketch down right). The parameter b is theradius at the waist of the film whereas c determines its shiftalong the h axis. A surface from this two-parameter family ofsurfaces is called a catenoid.

Soap film spanning two

rings.

(Mathematikum Gieen/photo: Rolf K. Wegst)

If the two rings are a distance d apart, the boundary valueproblem to solve is

a = b coshd

2b.

For d dcrit 1.325a two solutions exist, one with a mini-mum, the other with a maximum surface area. Ifd/a > 1.055,the surface becomes metastable as its area is now larger thanthe combined area of two soap films which separately coverthe two rings [Ise92][Arf85, Example 17.2.2]. As soon as dexceeds dcrit a connecting solution no longer existsthe soapfilm collapses.

h

(h)b

c

This is only true in equilibrium. In the photo (top right) dynamical effects play a role as well.

where

Hbend = 12

(K K0)2 + 12

R . (2.29)Compared to Sec. 1.2.1 we have rewritten the Gaussian curvature in terms of the

Ricci scalar R (A.51c)= 2KG. This allows us to apply expression (B.9) for the stresstensor of the general curvature Hamiltonian density H(K, R) to the special caseof the Helfrich membrane (see App. B.2.1). Inserting (2.28) yields6

fa =

(K K0)

Kab 12

(K K0)gab

gabeb (aK)n . (2.30)

In contrast to the capillary case, the surface stress tensor is not isotropic nortangential nor constant. The local curvatures contribute additional tangential

6 One can also read off the relevant expressions from Table 2.1.

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2 Surface Mechanics

stresses, while the gradient of the curvature creates a new normal stress; the saddle-splay modulus , however, does not enter fa.

To determine the external force on the membrane patch 1, we first simplify thecontraction offa and l:

lafa (A.56a)=

(K K0)

(Klb + Kt

b) 12

(K K0)lb

lbeb (K)n

=

(K K0)

K 12

(K K0)

l + K(K K0) t

(K)n . (2.31)

In expression (2.31) the unit tangent vector t = taea = nl is introduced: it pointsalong the integration contour 1 and is perpendicular to l and n. The symbol

= laa denotes the directional derivative along the vector l. The projectionsof the extrinsic curvature onto the orthonormal basis of tangent vectors {l, t}are given by K = lalbKab, K = tatbKab, and K = latbKab (see also App. A.2).Since the trace K of the extrinsic curvature tensor can be written as K = K+K,expression (2.31) can be further simplified and inserted into Eqn. (2.16) afterwardsto calculate the external force on 1

Fext =

1

ds

2

K2 (K K0)2

l + K(K K0) t (K)n .(2.32)

Knowing the stress tensor, the well-known shape equation of the Helfrich Hamil-tonian can be easily obtained from Eqn. (2.18a) (see, for instance, Refs. [ZcH89,Sei97, CG02b]):

K+ 12

(K K0)

(K K0)K 2KabKab

+ 2K = 0 , (2.33)

where we introduced the characteristic length

:=

, (2.34)

separating short length scales over which bending energy dominates from the largeones over which tension does.

Remarks:

Eqn. (2.33) has to be solved to determine the shape of a membrane undercertain boundary conditions. Unfortunately, this equation is a partial non-linear differential equation of fourth order in the embedding functions X; assuch, it can only be solved analytically in exceptional cases.

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In Chaps. 4 and 5 we will focus on an up-down symmetric fluid membrane forwhich the spontaneous curvature K0 vanishes

7 and solve the shape equation

numerically for special cases (see also App. C) .

Instead of trying to solve the shape equation one may also exploit that forcesand torques can be expressed in terms of the local surface geometry. InChap. 5 we will see that this fact enables one to gain a host of exact nonlinearresults, e.g. the sign of the force between two membrane-bound particles.

The Helfrich Hamiltonian density (2.28) involves the Gaussian curvatureKG. However, this term neither contributes to the stress tensor nor to theshape equation. To see whether it is relevant for torques, let us determinethe torque tensor ma of the fluid membrane and subsequently calculate the

external torque Mext on the membrane patch 1.The derivative of the density (2.28) with respect to Kab is (see also Table B.1)

Hab = (K K0) gab + (Kgab Kab) . (2.35)Thus, the intrinsic torque does not vanish and, using the expression for fa fromEqn. (2.30) again, the torque tensor can be written as

ma(2.21)

= X fa +

(K K0) gab + (Kgab Kab)eb n

. (2.36)

Inserting this result into Eqn. (2.20) yields the external torque

Mext = 1

ds 2 K2 (K K0)2 X l+ K(K K0)

X t (K)Xn

(K K0)t

, (2.37)

on the membrane patch 1. The intrinsic part of the torque tensor adds a con-tribution to the integrand in Eqn. (2.37), which is proportional to the curvature(K K0) and tangential to the contour of integration. If K is constant on 1,the intrinsic part of the torque tensor does not contribute to Mext at all. This isbecause the integral oft = X vanishes along any closed contour.Remarkably, the term M

ext = 1 ds lama originating from the Gaussian curva-ture vanishes (see Technical Point 2.3 for the details). This implies that torqueson membrane patches do not depend on the last term of the Hamiltonian den-sity (2.28) involving the Gaussian curvature KG. As shape and stresses are alsoindependent of the term, we can neglect it in the following except for Chap. 3:there, local conditions at the boundary will be discussed in which that term maybecome important (see Sec. 3.5).

7 Stress tensor, torque tensor, and shape equation for the up-down symmetric membrane can beobtained by simply inserting K0 = 0 into the corresponding expressions from this section.

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2 Surface Mechanics

Technical Point 2.3: Contribution of the Gaussian curvature

By virtue of the Gauss-Bonnet theorem the surface integral of the last term of

Hamiltonian density (2.29) can be written as a sum of a topological constant anda line integral over the geodesic curvature at the boundary of the membrane (seeApp. A.3). It does not contribute to the membrane stress tensor (2.30) and doesnot enter the shape equation (2.33). In spite of that, it gives a contribution mato the torque tensor (2.36).In Sec. 3.5 we will discuss an example where this term becomes important. Itscontribution Mext to the external torque, however, vanishes. The reason is thatits integrand can be written as a derivative with respect to the arc length s:

lama

(2.36)= la(Kg

ab Kab)eb n(A.57a)

= (Kt + Kl) (A.

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