Lecture notes for IOL - 2008 version 0.4January 12, 2008

Thermo-elasticity of membranes.

JH Ipsen

MEMPHYS - Center of Biomembrane Physics

Department of Physics and Chemistry

University of Southern Denmark

Odense, Denmark

Abstract:

The lectures will focus on the thermo-mechanical aspects of simple lipid bilayer membranes

and biological membranes. The topics will cover a range of material properties of membranes

from nano- to macroscopic length scales, i.e membrane elasticity, membrane interactions,

membrane material modifiers and the effects of membrane inclusions. Both experimental

and theoretical approaches will be taken. Some experimental techniques will be introduced

with emphasis on membrane micro mechanics. The theoretical level will vary from simple

thermodynamic modeling of lipid bilayers to statistical mechanics of continuum models of

membranes.

Outline

1. Introduction.

2. Thermodynamics of squeezed and stretched membranes.

3. Bending membranes.

4. Measuring elastic constants of membranes.

5. Stiffening membranes by steroles.

6. Lipid anchors.

7. Electrostatics and mechanics of membranes.

8. Membrane domains and bending.

9. Deformation of ”solid” membranes.

10. Membrane skeletons.

For the toturials consider the exercises in sections 2 and 3 andin the section on membrane domains and bending. We can alsodiscuss practical implementations of experiments and about howto get theory and experiments to ”talk”.

1 Introduction

The present incomplete lecture notes are covering a 3-hour lecture on membrane thermo-

mechanics at the International School on Biomembrane Physics Interphase of Life, Chennai

2008. There will only be space for a few selected topics in lectures, which are mostly

chosen from my interests around 1990 and again in the recent years. The lectures will be

held at an introductory level, although some basic understanding of thermodynamics and

mechanics is required. Also, there will be some statistical mechanics of thermal fluctuations

in the lecture, but only at the level of Gaussian statistics. The characteristic length scale

of description is above the nanometer range, where molecular details have vanished and

are only appearing in large-scale material characteristics of the membrane. Continuum

modeling of membrane properties has developed into a relatively large activity in statistical

mechanics with high degree of sofistication and abstraction in the description. However,

I find that there is a major need of experimental input to the field to turn theory and

hypothesis into real knowledge. The physical modeling of membranes has apparently been

very successful in the sense that modern molecular and cell biology has borrowed many

ideas and terminologies from membrane biophysics. That is e.g. the case for the biological

concepts of ”curvature sensing” proteins, ”rafts” and fusion. This is indeed very motivating

for studies of membrane biophysics, but our criteria of success should be a little different: to

be able to describe the observed membrane phenomena in terms of basic physical principles.

This requires a continuing dialog between theory and physical experiments. Therefore, I will

restrain myself to topics where some connections between theory and experiments have been

established. A range a phenomena in membrane thermo-mechanics, which has been subject

to my interest will thus be missing in these notes, e.g. crumpling of membranes, critical

unbinding, topological transitions and dynamics of membrane conformations will only be

sparsely covered here, as well as many potential applications to biological problems.

2 Thermodynamics of stretched and squeezed mem-

branes.

The lipid monolayer

The thermo-mechanical properties of membranes has been under investigation since the

earliest days of lipid research, while the development in our understanding of the phenomena

has been surprisingly slow. Already in 1925 Gorden and Grendel

F (Ns, A, T ) = γA/WA+ Σ(Ns, A) = Nsf(a, T ) (1)

where Ns, A, T is the surface area, number of lipids in the interface and the temperature.

The surface free energy F is a contribution to the total Gibbs free energy, since temperature,

pressure and the overall composition is the control variables. γA/W is the air-water interfacial

2

tension (' 72mJ/m2 for pure water). A term like γA/WA will always be present in problems

of interfaces between isotropic phases (no preferred orientations). Σ(Ns, A) is the free energy

contribution from the lipids in the interphase. A monolayer investigation will usually circle

around the properties of Σ(Ns, A) through analysis of the surface pressure Π versus area A

isotherms, which relates to the free energy through the equation of state:

γ = γA/W − Π =∂f

∂a

)

Ns,T

, Π = −∂Σ(a)

∂a

)

Ns,T

(2)

A complete isotherm of a single lipid component will usually contain features of several mono-

layer phases and phase transitions (se figure ??). At low densities the gas-liquid transition

is well described by real gas models like the van der Waal equation of state. The liquid state

of a lipid monolayer is relatively compressible, and it was early on recognized that the big

liquid phase region found in studies of lecithin must be the relevant phase in biomembranes,

and many breakthrough studies of membrane properties was conducted in monolayers of

lecithin, e.g. J.Chr. Skou’s first discovery of a membrane pump Na+ K+-ATPase (1957)(7).

A popular, but poor model of the surface pressure in the liquid phase with easy to predict

consequences is given by (3; 4).

Πh =C1

a2(3)

Πh gives an approximative description of the contributions to the surface pressure from steric

and electrostatic interactions between the headgroups. The lipid chains gain conformational

entropy when the area is expanded, giving rise to another repulsive contribution to the sur-

face pressure. This can be described in an explicite calculation of the chain conformational

statistics and the lateral pressure profile π(z) (6), or it can be approximated from the as-

sumption that the lipid chain behaves as an ideal polymer by a simple scaling argument

(5).

Πt(a, T ) = ρ

∫

π(z)dz ' C2

a3(4)

The mechanical properties of the monolayers can be derived from the measured isotherm or

from calculated pressures Π(a, T ), e.g. the comprssibility modules

KA = −a ∂Π∂a

)

Ns,T

(5)

which is the inverse of the compressibility χA.

Numerous studies was conducted during the 1950 and 1970’s where the behavior of Π

and KA for different lipid species. E.g. already in 1958 it was shown that cholesterol has a

condensing effect on phospholipids with saturated acyl-chains in the liquid state(? ).

The most notable phase transition in the monolayer isotherm of a single phospholipid is

the so-called liquid-expanded to liquid-ordered phase transition, which involves a significant

ordering of the hydrocarbon chain and a concominant reduction in the cross-sectional area

3

per lipid in the monolayer. (figure ..) Due to problems with purity and humidity control it

was first in 1976 (? ) a clear experimental foundation for this transition was established. The

transition is manifested as a flat isotherm at a well defined surface pressure. In flourescense

microscopy or AFM of transferred monolayers to a solid support the transition is appear as

coexistence of macroscopic domains of composition corresponding to the phase boundaries

Fig.( 1). At high temperatures beyound a critical temperature Tc this transition vanishes.

Close to the critical region the macroscopic phase coexistence disappear and a characteristic

ramified domain pattern emerge with fractal characteristics Fig.(3. This transition has a

counterpart in lipid bilayers to be considered below.

The lipid bilayer

The lipid bilayer is an even better model system of a biological membrane, and it forms

spontanously, when lipids are hydrated. The understanding of the stability the lipid bilayer

in solution came first in the late 1960’s and 1970’s with the theory of aggregation (4; 3) and

liquid crystal elasticity theory (? ). An important difference between the lipid monolayer

and a free standing lipid bilayer is that it is a self-assembled structure, i.e. a one-component

bilayer composed of Nl lipid will equilibrate at a well-defined cross-sectional area ao per lipid.

Generalizing Eq.(2) to the bilayer gives

γ = 2γL/W − Π =∂f

∂a

)

Ns,T,ao

= 0 (6)

where γL/W defines the interfacial tension of the water and the oily lipid chains and Π is

the surface pressure modified from the air-lipid-water interphase. The last equation simply

follows from the fact that the free energy is minimal in equilibrium with an equilibrium area.

This statement of a vanishing surface tension for a membrane bilayer is sometimes called

Shulmann’s criterium (9). This observation has vast consequences for the desription of lipid

bilayer properties. An imediate consequence for the free energy of dilating a lipid bilayer is

the approximative form

f(a, T ) = f(ao, T ) +KAao

2

(

a− aoao

)2

(7)

KA is basically two times the corresponding compression modules for the monolayers due to

the bilayer nature. KA is today mostly measured by micro-pipette technique to be described

in section 4. Typical values for a fluid membrane KA ∼ 100− 500mJm2 and ao ' 0.5− 0.7nm2.

Let us now for simplicity start to consider a one-component lipid bilayer. Many of the popular

lipids for model membrane studies and present in biological membranes exibit a melting phase

transition, the main transition, in the interesting temperature region 0−100 ◦C, where water

is in the liquid state. The melting involves both disordering of the acyl chains as well as

lattice melting. We will here consider the energetically most dominating process, the acyl

chain disordering processes, while lattice melting will be of importance as we discuss the

4

effect of sterols in section 5. However, to make the notation consistent we will introduce the

high temperature phase the liquid-disordered phase (ld) referring to the molten state of both

sets of degrees freedoms. The low-temperature phase is similarly called the solid-ordered

(so) phase(27).

The basic structural parameters of a bilayer, like the membrane hydrophobic volume

and thickness d can be obtained experimentally by X-ray and neutron scattering techniques

(10). Another important techniques is deuterium nuclear magnetic resonance spectroscpy2H-NMR. In Fig.(4) is shown a the spectrum from 2H-NMR spectroscopy of oriented lamellar

deuterated lipid membranes (35). The first moment, M1, of the spectrum is a reflection of

the average quadrupolar splitting from the deuterons along the acyl chains and is defined as

M1 =1

2

∫ +ωL

−ωLf(ω)|ω| dω

∫ +ωL

−ωLf(ω) dω

(8)

where ω is the frequency shift from the central (Larmor) frequency, f(ω) is the spectral

intensity, and ± ωL are the frequency limits of the spectrum. M1 is related to the average

order parameter by SCD =∑n

i=2 SCD(i)

M1 ∝e2qQ

h〈|SCD|〉 (9)

where SCD(i) = 12〈3cos2(θi) − 1〉, θi is the angle between the C-D bond of the ith carbon

position and the axis of symmetry of rapid motion of the acyl chain, and e2qQ/h is the static

quadrupolar coupling constant. For fluid phases with fast rotational averaging of the acyl

chains, the average order parameter is affinely related to the hydrophobic thickness of the

membrane(11; 29)

d

dmax

= α′〈|SCD|〉 + β ′ (10)

where the coefficient in Eq.(10) obey 0.5α′ + β ′ = 1 and depends on the deuterated chain.

As mentioned in the monolayer section above the acyl chain melting may vanish as

a cooperate transition under appropriate system conditions. It was shown experimentally

experimentally in 1980’s and theoretically about 1990 that the main transition is near critical

or pseudo-critical, i.e. at the melting transition there is a nearby critical point (30; 31). This

leads to a softening of the bilayer with demishing KA close to Tm, precisely as the volume

compressibility reduces close to a liquid-gas critical point. This interesting consequence of

lipid bilayer elasticity of the main transition as well as others will be considered in section

8. In the following we will first consider some elementary effects of bilayer elasticity. In

a typical chemistry notation the equilibrium free energy would be denoted the standard

chemical potential µ0 (T ) = f(ao, T ) for the lipid component. Close to the main transtion

we can thus write

µ,α0 (T ) = µ

0 (Tm) − sα(T − Tm), α = ld, so (11)

5

where sα = − ∂µ∂T

is the specific entropy of the lipid in the bilayer phase α. It is related to the

enthalphy change at the transition, which can be measured calorimetrically by Differential

Scanning Calorimetry (DSC) (? ), by

∆h(Tm) = Tm(sld − sso) = Tm∆s(Tm) (12)

By use of the fact that the hydrophobic volume can be considered as constant, i.e. a·d = cst,

where d is the hydrophobic by layer thickness, Eq.(7) can be phrased in two equivalent forms

µ,α(a, T ) = µ,α0 (T ) +

KαAa

αo

2

(

d− dαodαo

)2

(13)

= µ,α0 (T ) +

KαAa

αo

2

(

a− aαoaαo

)2

here dαo is the equilibrium thickness in phase α, a measureable quantity from X-ray diffraction

and H2NMR. This simple relationship find many applications. We will just consider a few

of them:

Two-component mixture: The simplest possible model free energy of a mixture of two lipid

components forming a bilayer in phase α can be written:

gmix,α(T, x, d) = µ,α1 (d, T )x+ µ,α

2 (d, T )(1.− x) + kBT (x ln(x) + (1 − x) ln(1 − x)) (14)

where x is the molar fraction of component ”1” in the mixture and µα1 and µα2 for pure

components is given by the Eq.(14). The last term is the ideal entropy of mixing. Appar-

ently, there is no energy of mixing involved, e.g. a term ∝ x(1 − x). However, there is an

important assumption hidden in Eq.(14), the hydrophobic thickness d is common for the

two component. This is the principle of Matching of hydrophobic thickness. In the case of

lipid mixtures, it is justified in terms of interaction strengths (8). The actual hydropho-

bic thickness for a particular composition is then given by minimal free energy condition∂gmix,α

∂d= 0,i.e. mechanical equilibrium condition. When both phases ld and so are taken

into account, this principle is sufficient to predict the phase behaviour of non-charged bi-

nary lipid mixtures Fig.(5). So, the non-ideal mixing behaviour observed for binary lipid

systems can mostly be understood in terms of elastic strain build up due to the hydrophobic

matching. The first quantitative model of this type, The Matress Model, is 25 years old

to describe the solubility of integral proteins in membranes with a hydrophobic thickness

mismatch(13). It can predict how the main phase transition is changed due to the presence

of integral membrane proteins and has been verified in numerous studies (? ? ? ). The

Matress Model is a thermodynamic model and the thus assuming local homegeneity of the

membrane around the protein, which is not a realistic configuration. In many cases it is

useful to consider local variations in cross-sectional or thickness, e.g. in the investigations

of the local membrane profile around a protein (14). Let us turn to the one-component

6

system again. A straigthforward generalization of Eq.(7) to a flat membrane with thickness

variations takes the form

F = Nf(ao, T ) +∑

i

KαAao2

(

ai − aoao

)2

+∑

〈j,i〉

c′(

ai − ajao

)2

(15)

H[φ(x)] = Nf(ao, T ) +

∫

d2x

(

c

2(∇φ(x))2 +

KA

2φ(x)2

)

(16)

The sum is over all the molecules in the plane, and the second term in Eq.(15) is the local

free-energy cost of a local inhomogeneity in a between nearest neighbors. Eq.(16) is the

continoum version where the local relative area deformation is represented by a dimension

less field φ(x), where the vector x labels the position in the plane. The two-dimensional

gradient is ∇ = ( ∂∂x1

, ∂∂x2

). The constant KA is the same as discussed above, while c in

front of the gradient term is in general not directly measurable. We expect it to be of the

order 10kBT . The local density variations can be measured in scattering experiments, in

particular Small Angle Neutron Scattering experiments (15). The observable in such an

experiment is the thermally average correlation function 〈φ(k)φ(−k)〉, where φ(k) is the

Fourier amplitudes of φ(k):

φ(x) =∑

k

φ(k) exp(ikx), (17)

By application of Fourier transformation and the equipartition theorem on Eq.(16):

〈φ(k)φ(−k)〉 =kBT

c(k2 +KA/c)(18)

which is the wellknown Lorenzian structure factor with a characteristic length scale ξ =√

c/KA. Since the main transition is pseudo-critical, i.e. is deminished for T → T+m , ξ

becomes large in the main transition region. This is observed in experiments (15) and in

Fig.(7) is shown their manifestation in a computer simulation as so-like domains with a

characteristic dimension ξ in the ld-phase. Much experimental and theoretical research in

MEMPHYS of the 1990’s focussed on membrane active drugs and proteins close to the main

phase transition, because it provided a very simple membrane model system to study the

effects of lateral membrane domains on functional aspects of the membrane. In this notes

only the effects on the mechanical properties will be considered in section 8.

Note:

Chaiken and Lubensky’s (CB) book is a good reference for this section, e.g. CB-chapter 2

for structure functions and Fourier transformation, CB-chapter 3 for the transtion between

thermodynamics to contnuom descriptions and the properties of functionals.

7

Exercises:

Exercise 1: Verify the scaling relation leading to Eq.(4). Notice that the hydrophobic volume

is constant.

Exercise 2: Consider a one-component lipid membrane under the influence of a external

lateral tension σ. Find a relation between the melting temperature Tm and σ. You can e.g.

apply the relations

σ =∂µα

∂a, µlo(Tm, σ) = µld(Tm, σ) (19)

Exexcise 3: Consider a protein embedded in a single component bilayer with a mismatch in

the hydrophobic thickness. Calculate the membrane thickness profile away from the protein

provided there is perfect hydrophobic matching at the interface between lipid and protein.

The problem can e.g. be simplified by considering a long protein in the membrane, so

the problem become one-dimensional. Notice that the minimum free energy condition for

Eq.(16) fulfill: c∆φ(x) = KAφ(x) (Use (CB) Appendix 3A).

3 Bending Membranes

In the previous sections we have discussed the mechanics of membranes as if membranes

always can be considered as purely planar problems. The reality is that lipid membranes

take the most fantastic surface shapes to be imagined. However, for the problems in the

previous sections, the energetics of the overall membrane configuration is a minor effect. We

saw that the usual cappilary term (γA) in the free energy plays small role in describing the

free energy and thus the configuration of a membrane. So, other usually insignificant terms

in the interfacial free must emerge as the dominant ones. This problem was solved at the

same time in the begnning of 1970’s by three authors Evans, Canhamm and Helfrich, who

suggested that curvature elasticity as the main determinant of membrane shapes from quite

different standpoints (? 17; 18).

H[X] =κ

2

∫

dA(2H − Co)2 (20)

where X is the 3-D configuration of the membrane. This quantity H is the mean curvature,

which is a measure of the local curvature characteristics of a surface. There are several ways

of expressing H

2H =

(

1

r1+

1

r2

)

= −n · ∆X (21)

r1 and r2 are the two principal curvature radii, the normal line curvatures along the two

principal curvature axis in the plane of the surface. This is a generalization of the usual

curvature concept for curves. It is more evident with the second formulation of the mean

curvature in Eq.(20), where ∆X is the Laplacian on the surface position, and n is the local

outward normal vector of the oriented surface (some more details are given in Appendix

8

on surface geometry to be included in the end). Co is the spontanous curvature, which

set the preferred mean curvature of the membrane caused by asymmetric conditions of the

two bilayer halves. In most model membrane studies, we can set Co = 0 because the two

monolayers are identical. The form Eq.(20) measures the elastic energy involved in the

local deformations of 2H away from Co integrated up over the whole surface area A. The

elastic constant κ is called the bending rigidity. It has the dimension of energy taking

values in the range κ ∼ 10 − 100 kBT . Eq.(20) is a free energy functional involving the

surface geometry like the cappilary free energy. This is a manifistation of the length scale

separation, the overall size and shape of the membrane is much larger than the thickness of

the membrane. The material properties come into play through the constants κ and Co. It

was demonstrated that Eq.(20) for a closed membrane combined with a volume constraint

can explain the sequences of shape changes observed in Red Blood Cell (RCC) shapes e.g.

in osmotic swelling experiments (21). Helfrich’s formulation of the membrane surface free

energy (17) has yet another term involving the surface curvature:

HGauss[X] =κ

2

∫

dAG = κχ (22)

where G = 2r1r2

is the Gaussian curvature. The integral over the Gaussian curvature is

given by the Euler characteristics of the surface χ, which for a closed surface is a topological

invarient χ = 2(1 − h), where h is the number of handles of the surface, h = 0 for sphere

topology, h = 1 for torus topology, h = 2 for a 2-torus and so on. The coupling constant κ

has dimension of energy. However, convincing experimental determinations of its magnitude

has not yet been made, except for its sign. κ > 0 favours high topology structures like

cubic phases or ”plumbers nightmare phases”, while κ < 0 promotes low-topology vesicles

or lamellar phases. We will not dig further into this exciting field here, but just complete

the modeling of the large-scale conformation of membranes by looking at the stability of the

total area of a membrane of finite area A. This is e.g. of interest if the membrane is subject

to stretching. First, we will reformulated the dicrete sum over the area in Eq.(15) (see (20)),

N∑

i=1

KAa0

2

(

ai − a0

a0

)2

=KA

2Na0

(

a− a0

a0

)2

+KAao

2

N∑

i=1

(

ai − a

ao

)2

(23)

where we have introduced the mean area a per molecule:

a =1

N

N∑

i

ai, (24)

With this manueuvre the elastic free energy in Eq.(15) is composed into a global term

denoting the collective deviations from the equilibrium area, Ao = Nao, and a local term

denoting the fluctuations about the average area per molecule. The last term in Eq.(23) is

apparently not affected by the external conditions. However, we will include it in the total

desription of the large-scale conformation of membranes, because it can have an influence

9

when ξ (see Eq.(18) becomes large as we will see in section 8. In continoum formulation the

free energy functional of a closed vesicle become

H[X] = γA+κ

2

∮

dA(2H)2 +KaAo

2

(

A− AoAo

)2

+

∮

dA

(

c

2(∇φ)2 +

KA

2φ2

)

(25)

The membrane is here considered to be symmetric between the two monolayers, Co = 0.

The first term in Eq.(25) is very small for membranes, but will play a role when ”framed”

membranes are considered in the next section.

Exercises

Verify Eq.(23)!

4 Measuring elastic constants of membranes.

Measuring κ

There are two major experimental techniques for measurements of κ. X-ray diffraction of

lammellar membranes, which is born out of the analysis of the structure factor from smectic

(lamellar) liquid crystals from 1970 (22), and the analysis of membranes flicker, a century

old observation in the microscopy of membranes RBC These random dynamic movements

of the membrane found in 1975 (19) an explanation as Brownian motion of an interface

subject to bending stiffness. This techniques is also based on the construction of a structure

function, based on very large observable surface ondulations (Fig.(8). A third techniques is

the microppette aspiration techniques on Giant Unilammellar Vesicles (GUV) in the low-

tension regimes (26). All techniques suffer from problems with the practical implementations

of the theory, but major progress has taken place.

First, let us consider a nearly flat membrane, where the surface configuration can be

written as X = (x1, x2, u(x1, x2)), the so-called Monge representation. The deviations from

the flat configuration is just represented as a real function u(x1, x2) with two arguments. In

this representation the ingredients of the surface integrals of Eq.(25) can for small deviations

u approximatively be written

2H ' ∆u =∂2u

∂x21

+∂2u

∂x22

, (26)

dA ' (1 +1

2(∇u)2)d2x =

(

1 +1

2

(

(

∂u

∂x1

)2

+

(

∂u

∂x2

)2))

dx1dx2 (27)

The first two terms of Eq.(25) can now be written in a harmonic (Gaussian) form.

10

H[u(x1, x2)] = γAp +

∫

d2xAp

(γ

2(∇u)2 +

κ

2(∆u)2

)

(28)

where Ap is the reference area and x = (x1, x2). The procedure is now the same as in the

previous section, where we developed the structure factor for the density-density correlation

function. We introduce the Fourier decomposition of u

u(x) =∑

k

u(k) exp(ixk) (29)

and express Eq.(28) in a diagonal form in terms of u(k):

H[u(k)] = σAp +∑

k

(γ

2k2 +

κ

2k4)

u(k)u(−k) (30)

i.e. as a sum of harmonic spring energies labelled by k. Then we apply the equipartition

theoren, i.e. the contribution to 〈H〉 to each fluctuating Gaussian variable is kBT2

. This leads

to

〈u(k)u(−k)〉 =kBT

γk2 + κk4(31)

Eq.(31) allow us to measure 〈u(k)u(−k)〉 and then afterwards fit the equation to the expres-

sion on the right-hand-side to abtain γ and κ. The determination of κ in MD-simulations of

membrane patches is usually based on Eq.(. This is basically the procedure in quasi-sperical

flicker analysis (? 25).

In the quasi-spherical description of vesicles the internal parametrization is given by

the spherical angular coordinates (θ, φ), and the membrane configuration is represented by

a displacement-field u(θ, φ), which measures the local deviation from a spherical reference

surface. The surface positional vector thus become:

X(θ, φ) = R (1 + u(θ, φ)) · Nr(θ, φ) (32)

where Nr is the normal of the unit sphere and R is the radius of the chosen reference frame.

A popular choice of R is the equivalent volume sphere radius RV =(

3V4π

)1

3 . Further, we

assume that the surface undulations are small with no overhangs,( e.i. |∇u(θ, φ)| � 1 and

|u(θ, φ)| � 1,) and as a consequence the first two terms in Eq.(25) can be approximated

by a Taylor expansion up to second order in u(θ, φ) (25; 24) in this limit. The harmonic

approximation (or Gaussian) of Eq.(25) involves the bending energy functional

H[u] ' γ4πR2 + γR2

∫ π

0

∫ 2π

0

(

2u+ u2 +1

2(∇u)2

)

sin(θ) dθ dφ (33)

+ 8πκ+κ

2

∫ π

0

∫ 2π

0

(

4u∆u− 4∆u+ (∆u)2 + 2(∇u)2)

sin(θ) dθ dφ

where ∇ and ∆ are the gradient operator and the Laplacian on a sphere. With the Gaussian

approximation the model has been turned into a symmetric quadratic form and its properties

11

are most conveniently described by diagonalization. Due to the spherical geometry the

relevant basis for diagonalization is the spherical harmonics:

u(θ, φ) =lmax∑

l=0

l∑

m=−l

ulmYlm(θ, φ) (34)

〈ulmu∗l′m′〉 =kBT

κ(l − 1)(l + 2)[

l(l + 1) + Σ]δll′δmm′ (35)

where Σ = γR2

κ. In phase contrast light microscopy of an ondulating vesicles it is only

its cross-section with the focal plane we observe, usually the equatiorial plane of the vesicle

where the contrast is largest. How do we establish the connection between the 3-dimensional

statistical model and the measured 2-dimensional observed contours. This problem was

solved by Faucon et al. (25) by introducing the maximum cross section radius

ρ(φ, t) = R(

1 + u(θ0, φ, t))

. (36)

where θ0 is the azimutal angle defining the position of the maximum crosssection. In case the

vesicle is exposed to symmetric inside-outside solutions the maximum crosssection is located

at θ0 = π/2, but non-symmetric conditions (∆ρ ≥ 0) leads to θ0 ≥ π/2. On basis of Eq.(36)

the cross-sectional angular correlation function is defined

ξ(γ, t) =1

R2

[ 1

2π

∫ 2π

0

ρ(φ + γ, t)ρ∗(φ, t) dφ− ρ2(t)]

, (37)

The thermal average of ξ(γ, t), which experimentally is represented by the the time average

of a long time series of contours, can by use of the spherical harmonic sum-rule can be

rewritten in Legendre polynomial basis Pl(x)l≥0

〈ξ(γ)〉 = 〈B0〉P0

(

cos(γ))

+lmax∑

l=2

〈Bl〉Pl(

cos(γ))

. (38)

The coefficients of Eq.(38) is given by

〈Bl〉 =2l + 1

4π〈ul0u∗l0〉 for l > 1 (39)

This is the equation to be fitted in analysis of membrane flicker of quasi-spherical vesicles,

some time called Vesicle Fluctuation Analysis (VFA) (48).

In X-ray diffraction of multilamellar stacks of membranes, the starting point of the analysis

is a modification of Eq.(28) to account for the stacking:

H[u(x1, x2, z)] =

∫

d2xdz

(

κ

2(∆u)2 +B

(

∂u

∂z

)2)

(40)

12

where ∇ apply in the plane of the membrane, z is perpedicular on the stacking direction.

u(x1, x2, z) is the local displacement along the z-direction. An interesting consequence of

Eq.(40) is that the scattering intensity of the n′th order Bragg peak in the z-direction takes

a power law form

S(0, 0, qz) ∝(

qz − n2π

D

)−2+ηn

(41)

where D is the repeat distance, and the power contains information about the elastic moduli

κ and B(22; 23)

ηn =π

2D2

kBT√κB

n2 (42)

The structure factor S(q, qz) is only part of the total scattering intensity

I(q, qz) = |F (q, qz)|2S(q, qz) (43)

The form factor |F (q, qz)| depends on the details of the scattering centers in the lipid bilayers.

Much effort goes into detailed modelling of |F | to gain information about κ as well as a range

of structural parameters including the hydrophobic thickness d(? ).

Measurement of KA (and κ)

Micro-pipette techniques is another commonly used method for micro-mechanical charac-

terization of membrane material properties. One version of the basic experimental setup

is shown in Fig.(. It consist of an inverted microscope with phase contrast or Hoffmann-

modulation optics. The observation chamber is a thin slit between two coverslips, where the

solution in the chamber is kept in place by capillary effect. The micro-pipette can now be

introduced to the observation chamber in a plane perpendicular to the optical path, ensuring

that the whole pipette can be chosen to be in the focal plane. The pressure between the in-

side of the pipette and the chamber is controlled by a pressure stage. Vesicles in the chamber

can now be caught and aspirated in to the pipette and the response of the vesicle geometry

to the change in aspiration pressure can be monitored (see Fig.(9) ). The first published mi-

cropipette expriment on RBC was made in 1964 by Peter Rand (), but is was first in the mid

1970’s it became a reliable technique to characterize RBC mainly due to the development of

Evan Evans and coworkers (? ). In 1980’s the characterization of synthetic vesicles sparked

off, also mainly in Evan Evans laboratory. Today, the use of micro-pipette techniques in

the characterization of membrane material properties quite modest, while micro-pipettes as

micro-manipulation tools for vesicles are widespread. The most common approach is to use

generalised Laplace equation

4p = 2σ

(

1

R1

− 1

R2

)

(44)

13

to identify the tension σ in the membrane due to the aspiration pressure 4p. R1 and R2 is the

pipette radius and the outer radius of the aspirated vesicle (see Fig.( 10). The relative change

in area will then repond in an approximatively linear fashion, and KA can be determined

according to the equation of state Eq.(25) as shown in Fig.(11),

σ = KAA− A0

A0

(45)

The micro-pipette techniques has also been in the determination of κ (26). Here it is assumed

that the membrane can be considered as incompressible at low tension levels, so all the

applied work goes into pulling out all the thermal membrane ondulations, i.e. an entropic

spring effect. The equation of state in this regime takes the approximate form (see below)

4AA

' kBT

8πκln(σA) (46)

from which κ can be estimated. Many bending measurements have been made based on this

approximation. However, in (? ) it is pointed out that great care is nessecary in the use

of Eq.(46) in measurements of κ. Sometimes there is corrected for the compressibilty of the

membrane by the combined expression:

4AA

' kBT

8πκln(σA) +

σ

KA

(47)

but still it is nessecary to be very careful with the data intepretation. The full analysis is

fairly involved, so only some principles will be discussed here. The starting point for the

analysis of the effect of strecting a membrane is Eq.(25) and the following substitution:

H = Hc +KAA0

2

(

A− A0

A0

)2

−→

H(J) = Hc + JA− A0

(

J +J2

2KA

)

(48)

where Hc is the bending part of Eq.(25), γ = 0 and the last term in Eq.(25) will be ignored

here, since we consider density fluctuations to be less important here. The variable J is

chosen so the free energy is minimal. For simplicity we assume that the stretching takes

place in the plane, so that we can apply the form Eq.(28), i.e. we consider small deviations

u away from the flat configuation. The free energy depends on the frame area Ap

F (J,AP ) = −kBTTr [exp(−β(Hc + JA))] − A0

(

J +J2

2KA

)

(49)

where

Tr [...] =

∫

D[u(x)] [...] =∏

x

∫

du(k) [...] (50)

14

Minimizing Eq.(49) w.r.t. J gives the relation

Js =KA

A0

(〈A〉 − A0) (51)

Comparing with Eq.(45) indicate that Js is to be identified with the tension σ.

To be completed in the final version!

5 Stiffening membranes by steroles

In the destription of the mechanical properties of membranes described in Section 3, we had

mostly the mechanical properties of single component or binary phospholipid bilayers. A

third major type of components in eucaryotic plasma membranes is the steroles. For ver-

tebrate membranes cholesterol can constitute up to half of the lipid content. The sterols

are structurally quite different from the phospholipids by the presence of the multiple hy-

drocabon rigid ring structure. As mentioned in the monolayer section, it has been known

for long that cholesterol is capaple of modifying the membrane properties dramatically. The

behaviour was puzzling: The melting temperature is reduced for small cholesterol concentra-

tions indicating a usual freezing point depression, while measurements of the cross-sectional

area or lipid chain order suggested that the ordered phase was stabelized. However, mea-

surements J. Davies (NMR) (? ) and E.Evans (micropipette)(? ) clearly indicated that

at high cholesterol levels the membrane has is fluid and homogenous. These indications

and other experimental input led to the construction of a theoretical phase diagram with

a new type of membrane phase, the liquid ordered phase lo (27) (Figure), indicating it is

a 2-D liquid with relatively ordered lipid chains. The phase diagram is characterized by a

low-temperature so − lo coexitence region, an ordinary melting transition of the solid (so)

and a high-temperature ld− lo coexistence region between two liquid phases. The two coex-

istence regions are separated by a three-phase line. This topology of the phase diagram of

binary lipid-cholesterol appeared to generic to a variety of lipids with saturated and mono-

unsaturated chains (33). It is possible to understand the action of cholesterol from molecular

modeling (27; 29), however at the thermodynamic level cholesterol has the capacity to split

the two coupled transition constituting the main transition in single component membranes

into a melting of the solid (so − lo) and a disordering of the chains ld − lo. The ld − locoexistence region terminates in a critical point, where the difference between the two liquid

phases disappears. Usually, macroscopic phase separation is not observed in the binary sys-

tems due to critical fluctuations. The lo phase and ld − lo has found its way into molecular

and cell biology through the ”raft” hypothesis (? ) as a possible principle of lateral orga-

nization of the membrane. The so-called ”raft-mixtures”, usually ternary systems of lipids

and cholesterol (? ), display a large ld − lo coexistence with macroscopic domain observable

in flourescence microscopy far away from critical points.

15

The ordering of liquid lipid membranes is easily seen in the measurements of the lipid

chain order parameter by 2H-NMR and the compression modules by micropipette aspiration

techniques, shown in figure ... for the three sterols with the ordering capability of POPC

in the order cholesterol ¿ lanosterol ¿ ergosterol (35). When plotting KA versus M1 an

interesting collapse of data emerge (fig...). It tells us that KA in these lipid-cholestorol

mixtures are solely determined by the lipid chain order. The differences among the sterols

to stiffen the membrane is thus due to differences in the ability to order the lipids.

It follows that the canonical free energy is the relevant thermodynamic potential

F (Nl, Ns, al, T ) = Nf(x, al, T ) (52)

where N = Nl + Ns is the total number of sterols and lipids and x = Ns

Nthe mole fraction

of sterols. As described in Chapters 2 and 3, the total area, A = Nlal +Nsas, is conjugated

to the lateral tension

τ =∂F

∂A

)

Nl,Ns,T

=∂al∂A

)

Nl,Ns,T

∂F

∂al

)

Nl,Ns,T

=1

1 − x

∂f

∂al

)

x,T

(53)

where the cross-sectional area of cholesterol as ' 32A2. For a free standing membrane

in mechanical equilibrium, τ = 0 the POPC cross-sectional area is a0l ' 60A2. The area

expansion modulus is

KA = Ao∂τ

∂A

)

Nl,Ns,T

=aol (1 − x) + asx

1 − x

∂τ

∂al

)

x,T

(54)

where A0 is the total area in equilibrium. The experimentally determined correlation between

lipid order (and thus d and al) and area expansion modulus for all three mixtures shown in

Fig. ?? can be expressed as KA(al, x) = KA(al(x)), i.e. KA has no explicit x-dependence.

With the definitions Eq.(53-54) we get extract the grneral form of f(x, al, T ).

∂KA

∂x

)

al,T

= 0 ⇒ f(x, a) =1 − x

aol + asx

1−x

(

U(x)al + V (al))

+W (x) (55)

U(x), V (al) and W (x) are some general differentiable functions to be determined, where

some properties of U and V are obtained from the experimental results1. The first term

represents contributions to the chemical potential of the lipids. U captures the interfacial

tension and the interaction between lipid and sterol molecules. Contributions to

V can derive from, for example, chain conformational energy, the entropy confinement of

floppy lipid chains, and Flory-Huggins-like entropy of mixing. In this representation, the

equilibrium condition is V ′(al) = −U(x) and the area expansion modulus Ka = V ′′(al).

1W may contain contributions from interactions between sterols and entropy of mixing.

16

6 Membrane domains and bending: The Standard Model

In this section are going to combine some of the considerations of sections 3 and 4 and discuss

the coupling between membrane curvature and lateral density or compositional fluctuations

and its relation to experiments. Let us as a starting point take the considerations leading

to Eq.(16). It was argued that local variation in the density can be described by a single

scalar field φ. However, these variations have their origin in the cooperative behaviour of

the coupled monolayers. A more precise description is thus to represent the fluctuations by

two fluctuating fields φ+ and φ− corresponting to density variations in the upper and lower

bilayer leavelets:

Hφ[φ(x)] =

∫

d2x

(

cm

2

(

(∇φ+(x))2 + (∇φ−(x))2)

+KmA

2

(

φ+(x)2 + φ−(x)2)

)

(56)

=

∫

d2x(

cm(

(∇δφ(x))2 + (∇φ(x))2)

+KmA

(

δφ(x)2 + φ(x)2))

(57)

where δφ = φ+−φ−2

and φ = φ++φ−2

are local density mean and difference fields. cm and KmA

are monolayer parameters. A local density difference between monolyers may cause a local

spontanous curvature, thus a contribution

HδφH = λ

∫

A

dA · δφ · (2H) (58)

' λ

∫

d2x∆u(x)δφ(x) (59)

Eq.(59) was first used by S. Leibler in 1986 (36) in a study of the effects of asymmetric

membrane proteins on membrane conformation. It has since been very popular in theo-

retical model studies, but the experimental verification has been difficult to achieve, i.e.

determination of λ has been difficult. For a single-component bilayer it is expected to take

the values in the range λ ∼ 10−10Jm−1 (38; 40). The shape of a free-standing unperturbed

membrane is Eq.(28)

Hc[u(x)] =

∫

d2xκ

2(∆u)2 (60)

So, adding the contributions Eq.(57,59,60) the ∆φ and u fields gets coupled. The effect of

this copuling can e.g. be seen in measurements of the bending rigidity. The model prediction

of this measurement can be obtained by calculating the effective configurational free energy

functional

17

Heffc [u(x)] = −kBT ln(

∫

d[δu(x)]

exp(− 1

kBT

∫

d2x(κ

2(∆u)2 +Km

A δφ(x)2 + λ∆u(x)δφ(x))

=1

2

(

κ− λ2

2KmA

)∫

d2x(∆u)2 + const. (61)

where we left out the cm term for simplicity. The effective bending regidity

κeff = κ− λ2

2KmA

(62)

is thus reduced compared to κ. From the discussion in section 2 it follows that KmA is dem-

inished as the main transion is approached in PC-bilayers, thus a substatial reduction in

κ is expected. This is indeed what is observed in measurments of the bending rigidity by

VFA (37) as shown in Fig.() . This softening of the lipid bilayer elasticity has a spectacu-

lar consequence for multilammelar membranes, the swelling and unbinding of multilamellar

membranes or anormalous swelling (41; 42) Fig.(??). A simplified discussion of this phe-

nomenon can take its starting point in Helfrich’s analysis of the smectic layer elasticity

leading to Eq.(40). He concluded (43) that the interplay of steric repulsion and the con-

formational fluctuation of the membrane give rise the an effective inter-membrane repulsive

potential:

Vst(z) = const.(kBT )2

κ

1

z2(63)

Where z is the inter-membrane distance. Clearly, the repulsion is amplified as κ is lowered.

A more sofisticated discussion of membrane adhesion by funtional renormalization group

techniques by Leibler and Lipowsky () showed the possibility of a continous unbinding tran-

sition from an adhered to a non-adhered state beween membranes with z ∼ (κ − κcrit)−1.

Several experimental works support that anormalous swelling has critical unbinding charac-

teristics (? ? ). The coupling term Eq.(59) has also proven to have interesting consequences

for the dynamics of membrane conformation (39), which have led to experimental estimates

of λ and the inter-monolayer friction coefficient (40) for a single component PC-membranes.

Recently, it was possible to estimate λ in the context of lipid-peptide interactions, which is

somehow closer to Leibler’s original idea. In (44) it is shown that the membrane bending

rigidity is strongly deminished by a small presence of the anti-microbial peptide magainin on

the membrane. It is a soluble 23 amino acid peptide which takes random coil conformations

in solution and bind, while is in an amphiphatic alpha helix conformation parallel to the

membrane (Fig.(??). In Fig.(17) is shown how the apparent bending rigidity is changing

with surface coverage of magainin. Even when less than 1% of the membrane area is covered,

the membrane rigidity is strongly reduced. The difference from the above consideration is

18

that the density of inserted peptides is very low, but each of the inserted peptides is strongly

perturbing the local membrane packing in the inserted monolayer, creating a local assymetry:

HCoupling = λ

∫

d2x · δρ(x) · ∆u(x) (64)

where δρ = ρ+ − ρ−. The density field in Eq(64) is not scaled with a reference density as in

Eq.(59) since ρ(x)± is close to zero. The peptide densities is thus approximatively governed

by the ideal gas entropy

H0Gas = kBT

∫

d2x ·(

ρ+ · ln(ρ+ · a2) + ρ− · ln(ρ− · a2) − 1)

+ cst

≈∫

d2x ·(

2kBT · ρ0 · ln(ρ0 · a2) + kBT · ρ0 · (δρ

2ρ0

)2 + cst)

(65)

where ρ0 = (ρ+ + ρ−)/2. The first virial correction to Eq.(65) is

H1Gas =

t

2

∫

d2x · (ρ2+ + ρ2

−) + s

∫

d2x · ρ+ · ρ−

= t

∫

d2x ·(

ρ20 + (

δρ

2)2)

+ s

∫

d2x ·(

ρ20 − (

δρ

2)2)

(66)

An analysis similar to the above Eq.(62) of the combined model Eq.(60,64,65,66) gives an

effective bending rigidity

κeff = κ− 4λ2 · ρ0/kBT

1 + (t− s) · ρ0/kBT(67)

In this description a saturation in κeff sets in if local densities on the two bilayer halves can

partially restore the symmetry. A fit of Eq.(67) against the measured bending rigidities gives

λ = 3.210−28 J.m, which corresponds to a very high local curvature associated with a peptide

∼ nm−1. Similarly, (t − s) = 4.510−36 J.m2, which is a high value compared the expected

values from excluded volume and electrostatic interactions between peptides t ∼ 10−38 J.m2.

This suggest a considerable effective attractive interaction s < 0 between peptides across

the membrane, a possible origin of the barrier breaking ability of magainin and other similar

microbial peptides (44).

Exercise

Verify Eq.(62). Use e.g. the Gaussian integral:

∫ ∞

−∞

exp

(

1

2Cx2 + hx

)

dx =

(

2π

Cexp

(

h2

2C

)

(68)

19

7 Lipid anchors

Major classes of cell proteins have their membrane affinity regulated by lipidic membrane an-

chors. (45) This includes e.g. structural proteins like Lamin B and Coat proteins, and signal

transduction proteins like γ−G protein subunits and Ras proteins. In the protein biosyn-

thesis the lipid moeties are post-translationally attached and the evolution has picked very

few pathways for this lipidations. The most common lipid anchors involves palmetoylation,

myristate or palmetate chain attachment, or prenylation, farnesyl or geranyl-geranyl anchors

(figure). These two groups of lipidic membrane anchors have received considerable attention

during the recent years because they show very different behaviours depending on the mem-

branes under consideration. We will here consider aspects of their partitioning properties

into model membranes and their effect on the machanical properties of membranes.

In general posttranslational attachment of a hydrophobic lipid chain is essential for

protein-membrane association, but not sufficient (? ? ). The the role of lipidation is thus

to enhance the partititioning into the membrane. Further, it appaers that prenyl chains

provides a weaker binding to phospholipid vesicles than saturated acyl chains (? ? ? ). In

a DMPC membrane in pure water small neutral lipidic compounds seems to be insoluble in

water (46; 47? ? ) and fully partitioned into the membrane, and structurally organized in

the membrane as expected with the lipidic chains aligned with the phospholipid acyl chains.

The types of lipid achors affect the main transition very differently. While the small com-

ponents with a palmetoyl chain like MOG(? ), increases the main transition significantly

and stablizes the low-temperature so, the small membrane compound with a farnesyl chain,

like farnesol, has the opposite effect an is stabelizing the fluid ld phase (? ). These different

affinities to the ordered and disordered lipid environments found in thermodynamic studies

also shows up in flourescense microscopy (49) and partition studies (? ) and have therefore

become candidates for the main sorting mechanisms between disordered and ordered do-

mains (”raft”) in the membranes. Numerous membrane proteins contains both prenyl and

acyl chains. The biophysical studies thus supports that enzymatic attachment and removal

of lipidic chains have the capacity to direct and redirect proteins to specific membrane en-

vironments in the cell and within a membrane. Further, the mixed chain peptides may act

as surfactants in the plane of the membrane, stabelizing micro-domains or forming a two-

dimensional micro-phase separation (or micro-emulsion), as flourescence microscopy studies

indicate (? ).

The ld phase is perturbed surprisingly little by large uptake of of farnesol. 2H-NMR results

show that the lipid chain order is hardly affected up to 20 mole% farnesol in the membrane

(47) and measurements of the bending rigidity shows at most a modest decrease, if any, up to

25 mole% (figure?). The relatively large amounts of prenyl chains found in some membranes,

e.g. the inner nuclear membranes have probably little effect on the material properties on

the membrane.

20

The Membrane κ(×10−19 J)DMPC 1.40 ± 0.03

uncharged Ac-C-(fsl)-OMe (5.0 mol %) 1.32 ± 0.07farnesol (2.5 mol % ) 1.34 ± 0.03farnesol (5.0 mol % ) 1.34 ± 0.04farnesol (25.0 mol % ) 1.30 ± 0.05

charged Ac-NKNC-(fsl)-OMe (2.5 mol %) 1.60 ± 0.08Ac-NKNC-(fsl)-OMe (5.0 mol %) 1.59 ± 0.03Ac-NKNC-(fsl)-NH2 (5.0 mol %) 1.72 ± 0.03

Table 1: Summery of the effects of farnesylated peptides and farnesol on the bending rigidityof DMPC GUVs at 37◦ C. The cited concentrations refer the bulk peptide/farnesol concen-tration of the stock solution.

8 Electrostatics and mechanics of membranes

The measurements of the effect of electrostatics on membrane mechanical properties is noto-

riously difficult. This is mainly due to the difficulties in preparation procedures, separating

the direct electrostatic effects from derived effects. E.g. for charged lipids at low added

salt concentrations, liquid-liquid phase separation can take place, lipid headgroup repulsion

and the effect on lateral compressibility is difficult to quantify due to ion partitioning in

aspiration experiments, while studies at high salt levels up to now have been hampered by

the difficulties in making Giant Unilamellar Vesicles (44). A possible way around these dif-

ficultes is to add a charged lipidated peptide to the solution so a small partitioning peptide

concentration builds up a surface charge of the vesicle. The system thus self-regulates into a

characteristic distribution of ions at the membrane interface, in the bulk solution and in the

vicinity of the membrane. If we further those a prenylated membrane anchor, which appar-

ently in inself gives rise to an insignificant perturbation of the membrane bending rigidity,

we should have a chance to identify an electrostatic contribution to κ. Also, at low charge

concentrations mean field theory should apply, e.g. Poisson Boltzmann theory works and

charge correlations effects should be neglegiable (? ). Let us consider a small, cationic, far-

nasylated peptide (figure) and their anionic counterions, both of which are monovalent. As

aforementioned the hydrophobically driven adsorption of charge peptide to the membrane

results in the accumulation of charge at the interface and free charges in the bulk redistribute

accordingly. This results in an inhomogenous Boltzmann distribution of charges as both the

charged peptides and their counter ions in solution seek to simultanously optimize both elec-

trostatic interactions and their random motion. This delicate balance between electrostatic

interaction and translational entropy results in the so-called diffusive or electrostatic double

layer and an electrostatic potential ψ(X). Assuming that the membrane can be considered

as a flat surface, symmetry dictatates that ψ(X) only depends on the direction z perpen-

dicular to the surface. A full mean-field treatment of the electrostatic problem results in a

21

set of self-consistent equations governing the form of ψ(z), and the charge distribution. The

shape of ψ(z) is set by Possion-Boltzmann’s equation

∂2ψ

∂x2=

8πecbε

sinh

(

eψ

kBT

)

(69)

where cb is the concentration of peptide in bulk solution, and e the unit charge. Gaus’ law∂2ψ∂x2 = −4πe

εn+

0 , assuming only the cationic peptides bind to the membrane and the electro-

neutrality condition ψ(z → ∞) = 0 provide the nessecary boundary conditions to solve

Eq.(69) and an exact solution can be found(? ). The surface density of farnesylated peptide

is given by Davies’ adsorption isotherm.

ρ+0 =

ρ+b

exp(

−(α + βρ+0 − eψ(0))/kBT

)

+ ρ+b

(70)

where the concentrations are expressed as scaled concentrations ρ+0 = (a+)2n+

0 and ρ+b =

(a+)3cb. α is here the free energy of partitioning of a farnesol chain into the bilayer and β is

the 1st virial correction from the interaction between peptides in the membrane, which we will

ignore here. A representation of typical solution of the equations is illustrated in Fig.(19)

where some typical lenght scales has been introduced: Bjerrum length lB = e2

εkBT' 7A,

the distance at which the electrostatic energy between two unit charges equal the thermal

energy, Gouy Chapman length lG = 1/(2πlBen+0 ), the distance at which the interaction of

a counterion and the charhed interface reaches the thermal energy, and the Debye length

lD =√

kBTε/8πe2cb = 1/√

8πlBn+0 is the lenght scale characterizing the decay of the

electrostatic potential to zero, setting the length scale of the diffuse double layer.

This diffuse double layer is expected to contribute to the electrostatic bending rigidity and a

number of approaches have been taken to calculate this contribution (reviewed in (52) which

all boils down to a simple scaling form of the electrostatic bending rigidity:

κ = κ0 + κel

κel = c · kBT (n+0 )2lBl

3D (71)

where c is some prefactor of order unity and κ0 is the bending rigidity without the electro-

static effect. Notice, that when the concentration of peptide is increased, n+0 will increase,

while lD will decrease. To judge which of these effects will dominate in κel needs a full

solution of Eq.(69) and Eq.(70). In Fig (??) is shown such a calculation for realistic pa-

rameters. Except for a steep increase in κel at very low surface concentrations, its stays at

an elevated nearly constant level of several kBT over a wide range concentrations, where

both α and a+ and the constant c are significant. This is in good agreement with what is

observed experimentally. Three small farnesylated peptides were prepared (Table 1), where

two of them are cationic, and one neutral. The neutral Ac-C-(fsl)-OMe partitioned all into

the membrane like farnesol, and likewise farnesol, only a modest decrease in bending rigidity

22

was observed. Other biophysical studies confirm that this peptid behaves like farnesol in the

membrane(54). For the two charged peptides, only a fraction of the peptid is partitioning

into the membrane corresponding to molar fractions in the range 0.1-1%, but a significant

increase in bending stiffness is observed (55) ∼ 5 − 8kBT is observed. Since we can exclude

a stiffening effect from the prenyl chain, we can attribute the found increase in κ to κel.

9 Deformation of ”solid” membranes

To be completet in the final version

10 Membrane skeletons

To be completet in the final version

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23

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26

Figure 1: Lipid monolayer isotherm

Figure 2: Lipid monolayer isotherms including the critical isotherm.

Figure 19: (a The partitioning of charged farnesylated peptides into the membrane resultsin an accumulation of surface charge and the formation of an electrostatic double layer. (b)Within the Gouy-Chapman length, lG, the potential ψ and counter ion density is elevated. ψdecays exponentially as x → ∞. In the linear Debye-Huckel approximation, lG is not takeninto account and the full decay goes as exp(−x/lD), where lD is the Debye screening length.

Figure 21: Comparison between the experimental decrease of the bending elasticity κ as afunction of the peptide area coverage.

27

Figure 3: LE-LC domains observed by AFM of DMPC-monolayer transferred to solid sup-port.

(a)

(b)

(c)

0 20 40-20-40kHz

Figure 4: Spectra obtained by 2H-NMR for POPC-d31 membranes containing (a) 10 mol%;(b) 20 mol%; and (c) 30 mol% lanosterol at 25◦C. Acquisition parameters are documentedin the text.

28

(a) (b)

29

(c) (d)

Figure 5: Phase diagrams for binary mixtures of pholipid bilayer. Experiments and predic-tions from hydrophobic matching principle.

Figure 6: Specific areas and compressibilities from computer simulations of DMPC, DPPCand DSPC in a temperature range going through the main transition.

30

Figure 7: Snapshots from computer simulation data of DMPC, DPPC and DSPC at relativeteperatures just above and below the main transition.

Figure 8: Shapshot of a contour taken from Gian Unilamellar Vesicle of SOPC illustratingthe experimental proceedure in Vesicle Fluctuation Analysis.

31

Figure 9: Vesicle in pipette, aspirated with increasing pressure.

Figure 10: Cartoon of vesicle in a pipette

32

3420 3440 3460 3480 3500

1

2

3

4

5

6

0

Ap [µm2]

τ [m

N/m

]

Aspiration dataLinear fit

Ap,0

Figure 11: Determination of KA. Tension versus relative area increase.

Lanosterol

Cholesterol

Ergosterol

Figure 12: Structures of cholesterol, lanosterol, and ergosterol. In the biosynthetic pathway,the methyl groups on lanosterol’s α-face are shed giving rise to cholesterol. Ergosterol differsstructurally from cholesterol in that it has two additional double bonds as well as a methylgroup on the side chain.

33

Figure 13: Generic phase diagram for binary phase diagrams involving cholesterol and sat-urated and mono-unsaturated lipids.

0.20

0.22

0.24

0.26

0.28

0.30

0.32

0.34

0.36

0

5 10 15 20 25 30

POPCPOPC/cholPOPC/lanoPOPC/erg

Ka [

J/m2 ]

mol% sterol

45

50

55

60

65

70

75

0 5 10 15 20 25 30mol% sterol

M1 [

103

s-1]

(a) (b)

Figure 14: Plot of (a) the apparent area expansion modulus, Ka and (b) the first moment ofthe 2H-NMR spectrum, M1, as a function of sterol content. Ka is determined by micropipetteaspiration and M1 by 2H-NMR. The extent to which these sterols increase Ka and M1 followsthe sequence cholesterol > lanosterol > ergosterol for all measured sterol concentrations.

34

0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38

45 50 55 60 65 70 75 30

40

50

60

70

80

90

100

45 50 55 60 65 70 75

κ [k

BT]

M1 [103 s-1] M1 [103 s-1]

Ka [

J/m2 ]

POPCPOPC/cholPOPC/lanoPOPC/erg

(a) (b)

Figure 15: Plot of (a) the area expansion modulus, Ka, and (b) the bending rigidity, κ, as afunction of the acyl chain order measured by M1. Both mechanical moduli exhibits a uniquefunctional dependence on M1 independently of sterol structure and concentration.

Figure 16: The bending rigidity of DMPC and DPPC versus T − Tm

35

Figure 17: Comparison between the experimental decrease of the bending elasticity κ as afunction of the peptide area coverage.

36