Bending elastic moduli of lipid bilayers : modulation by solutes

(1)

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Bending elastic moduli of lipid bilayers : modulation by

solutes

H.P. Duwe, J. Kaes, E. Sackmann

To cite this version:

(2)

Bending

elastic moduli of

_{lipid bilayers :}

modulation

_by

solutes

H. P.

_Duwe,

J. Kaes and E. Sackmann

Physik-Department, Biophysics Group,

Technische Universität München, D-8046

_Garching,

F.R.G.

(Reçu

le 12

_septembre

_1989,révisé le 5 _janvier1990,

accepté

le 31 janvier

1990)

Abstract. 2014 We

present

high

precision

measurements of the

_bending

elastic moduli for

_bilayers

of

a

_variety

of different

_lipids

and of modifications of the flexural

rigidity by

solutes. The

measurements are based on the Fourier

_analysis

of

_thermally

excited membrane undulations

(vesicle shape fluctuations) using

a

_recently

_{developed dynamic image processing}

method.

Measurements of the

_bending

modulus as a function of the undulation wave vector

_provide

information on the limitation of the excitations

_by

the constraint of finite membrane area

_(surface

tension

effects)

and

by

transient lateral tensions

arising

in each

monolayer by

restricted diffusion

at

_high

wavevectors. Measurements of the autocorrelation function of the undulation

_amplitudes

provide

a further test of the theoretical models. Studies of the effect of solutes show that cholesterol increases the

bending

modulus of

_{dimyristoylphosphatidylcholine}

from

Kc

= 1.1

10-12

_ergs_to4.2 _x

10-12

_ergs

(at

30 mole

%). Incorporation

of about 2 mole % of _a

short

_{bipolar lipid}

reduces

_Kc

to the order of kT. A

_scaling

law between the

_projected

radius of

gyration,

_Rg,

of these

_hyperelastic

vesicles and the surface area, A

(or

number of

lipid

molecules

N)

of

_{R2g ~}

A 0.94 ± 0.02 was established.

Classification

Physics

Abstracts 87.22F

1. Introduction.

Lipid bilayer

vesicles fascinate

_physicists

because of their

_peculiar

elastic and

_dynamic

properties

which are dominated

_by

a _{very soft}

_bending

elastic modulus on the order of 10 kT

[1-4]. They appeal

to

_{biophysicists}

since

_(despite

their

_{simple structure) they}

exhibit

_typical

mechanical and

_rheological

features of cell membranes. The

_bending

elastic modulus may be lowered

_by

solutes towards the _{thermal energy in order}to

_approach

the limit of random surfaces. New

_{fascinating phenomena}

arise in mixed

_bilayers

where the

_coupling

of lateral

phase separation

and curvature effects lead to a manifold of metastable vesicle

_{shapes [8].}

_By

using dynamic image processing

methods,

it has become

_possible

to determine the mean square

amplitudes

and the correlation times of the thermal membrane excitations as a

function of the wavevector in the

_{optical regime,}

thus

_{allowing high precision}

measurements

of the

_bending

elastic modulus. In the _presentwork we

_present

first a more detailed

description

of our

experimental procedure

of evaluation of the membrane undulations

_[1].

Secondly,

we summarize our recent measurements of the

_bending

elastic moduli of various

lipid bilayers

and demonstrate that the method is well suited to

_distinguish

vesicles

_composed

of different numbers of

bilayers.

In a third

_part

we

_present

measurements of the

(3)

autocorrelation function of the membrane excitations and show that the

_equilibrium

and the

dynamic

models of the vesicle fluctuations

[6,

7]

agree well with

experimental

data.

_Finally,

we

_present

measurements of the radius of

gyration

of vesicles with

bending

elastic moduli of

the order of kT as function of the total surface area and establish a

_scaling

law with broken

exponent.

2. Theoretical

_background

of the method.

It is

_{generally accepted}

now that the thermal undulations of vesicles and also biconcave

_(and

moderately swollen) erythrocytes

can be described in terms of the

_{quasi-spherical}

model first worked out

_by

Helfrich

_[4]

and Schneider

_[5]

and

_{subsequently improved by}

Peterson

[6]

and Milner and Safran

_[7]

to include

_spontaneous

curvature effects and to account for the

suppression

of

_{long wavelength}

modes

_[1]

_]

_by

the constraint of constant area. The thermal

excitations of

_{quasi-spherical}

vesicles are described in terms of a

spherical

harmonics

expansion (Yfm ( v, Q ) )

of the middle surface

_separating

the two

_monolayers.

The latter is determined

_by

the radius

_{Br( v,}

_{’P) (v : polar angle,}

Q azimuthal

angle)

which is

where _rois the so called

_equivalent

radius. It is

_equal

to the radius of a

_sphere

of the same

volume,

V,

as the vesicle. For

_{symmetric bilayers,}

where

_spontaneous

curvature effects can be

ignored,

the undulations are determined

_by

the elastic energy.

where H is the mean curvature : H =

_{1 /r1}

+

_{1 /r2}

_{(r 1,}

r2 :

principal

radii of

curvature),

Kc

is the

_bending

elastic modulus and A is the surface area of the vesicle. y is a

_Lagrange

multiplier

which accounts for the constraint of constant area.

Physically

it

corresponds

to the lateral tension which arises for those excitational modes that

_require

a

_larger

excess area than available. Provided the excitation of the

_spherical

harmonic modes follows the

_{equipartition}

theorem,

the normalized mean square

amplitudes

of the

_spherical

harmonic mode charac-terized

_by

the

_angular

momentum

_(f)

and

_magnetic

_quantum

number

_(m)

are

_{given by}

where y =

_{y ro 2/ Kc.}

As

_pointed

out first

_by

Brochard and Lennon

_[9],

the

damping

of the membrane

_bending

undulations is determined

_by

the

_hydrodynamic

flow of the enclosed and

surrounding

fluid and for that reason the modes are

_completely

_overdamped.

The

(temporal)

autocorrelation function of the

_{amplitude alm ( t )}

are

_exponentials

_[6,

_7]

(4)

and q is the

_viscosity

of water.

It is

_important

to note that

_{equipartition}

is

possible

only

if the flow within the

_bilayer

is fast

enough

to accomodate for local

_density

fluctuations

_(in

each

_monolayer)

associated with the undulations. A break-down of the

_{equipartition}

is

_suggested

for membranes

containing

cholesterol

_(cf.

Sect.

_4).

Experimentally,

the excitations of the vesicle have to be

_analysed

in terms of the fluctuations of the contour observed in the

_phase

contrast

_microscope.

This is

_possible

provided

1)

the vesicle is

_{quasi-spherical}

so that the

_{image plane}

goes

through

its center of mass

2)

the time over which the fluctuations are observed is

_{large compared}

to _{the response}

(= relaxation)

time of the excitations.

Let the Fourier transform of the momentaneous contour at time t be

where ço is the

longitudinal angle

and

v (Q, t )

is determined

by

the deviation of the vesicle

surface from the

_sphere

at it =

_{TT’ /2.}

_By

_{using equation (1)}

one obtains for the Fourier

component

_{Vq (t)}

of

_{equation (7) [1, 3]}

The sum starts at

f

_{= q}but the minimum of

f

must be

_larger

than two. The mean _square

amplitudes

of the contour fluctuations are then

where

_{P p, q}

_(cos

_{(’TT’ /2))}

are the values of the

Legendre Polynominals

in the

equatorial plane.

Thus the mean _square

_amplitudes

of the fluctuation of the contour are

directly

related to the

mean _square

amplitudes

of the

spherical

harmonics

given

in

_{equation (3).}

The sum in

equation (9)

converges

rapidly

with

_{increasing f}

and has been calculated for the

_quantum

numbers

_{2 _ q _}

10 _upto

f

=

lmaX

= 34

[3].

3.

_{Expérimental procedure.}

3.1 PREPARATION OF THIN WALLED VESICLES. - Giant vesicles with diameters

_larger

than 10 _umwere

prepared by

the

following procedure.

The

_lipid

or

_lipid

mixture was dissolved in

2 : 1

_{chloroform/methanol (1}

mM

_solution).

20

_u1

of this solution

(containing

about

2 x

10- 8

mole or

10- 5

_gof

_lipid)

was distributed as a thin film on the surface of a

microscope

cover

_glass.

The solvent was

_evaporated

_by

_placing

the substrate in a vacuum chamber for a

minimum of one hour. The cover

_glass

was inserted in a

_sample

cell

(where

it served as the

bottom

_window)

and was fixed with silicon _grease.The

_sample

cell was filled with distilled

water or a 100 mM solution of mannit and the

_top

closed

_(and

_sealed)

with a second cover

glass (the

top

window).

The whole

_sample

cell was inserted into a thermostated

measuring

chamber made of V2A steel. The

_temperature

could be varied between 0 and 50 °

(5)

Peltier elements cooled

_by

water. The

_temperature

in the

_sample

chamber was measured with a Pt 100

_temperature

sensor.

3.2 EXPERIMENTAL PROCEDURE USED FOR FOURIER ANALYSIS OF VESICLE CONTOUR FLUCTUATIONS. - The vesicles were observed and evaluated with an inverted Zeiss Axiomat

microscope.

It was mounted

(on

air

_cushions)

on a

_{heavy ground plate}

which was

_suspended

from the

laboratory

ceiling

to reduce vibrations caused

_{by walking.}

A schematic

view

of the

optical

set-up

is shown in

_figure

1. One

_advantage

of the Axiomat is that the

_position

of the

image plane

of the

_{(infinity adapted) objective}

can be choosen

_freely.

In _{this way}a normal

bright

field

_objective

can be used for

_phase

contrast

_{microscopy by placing}

a

_{phase plate}

in the

image plane

far away from the

_objective.

For the

present

work,

a

_bright

field air

_objective

of

magnification

50 x

(Zeiss)

was used. Since the

_working

distance of the

_objective

was

_only

300 _ktm,the vesicles near the bottom window of the

_sample

cell had to be observed

_by

an

inverted

_{microscope. Images}

of the vesicles selected were taken with a

_{charge coupled}

device

(CCD)

camera

_(frame

transfer camera,

HR600M,

Aqua-TV

company,

Kempten, FRG)

onto

which a section of 40 x 40

itm 2

of the

_{image plane}

was

_projected.

The camera is connected to

an

_{image processing}

_system

_(MAXVIDEO,

DataCube

_BOSTON,

_USA)

with VME bus and

OS-9

_operating

_system.

With this

_system

the

_images

from the video camera are

_digitized

512 x 512

_pixels

and 256 gray levels.

A fast

_algorithm

was

_developed

which allowed us to evaluate the

_(time

_dependent)

contour

of a vesicle at a rate _{of 5-10 per second and}to store the data in the

_computer

memory. The

procedure

is illustrated in

_figure

2. Since the

_phase

shift of the

_light

is _{maximal if it passes}

tangential

to the vesicle

surface,

the transient contour

_(that

is the

_equator

of

_{quasi-spherical}

vesicle) corresponds

to a

_sharp

minimum of the

_brightness

of the

_image.

First a

_starting

point

(coordinate

a%,

yo)

of the contour is selected

by

the cursor. Each

pixel

has 8

neighbours

which

define 8 directions

_(numbered

0 to 7 in

_{Fig. 2).}

As second

_step

the

_adjacent

contour

_point

is searched as follows : the

intensity

over three bands

comprising

46

pixels

and

laying

in three

different directions of the

_eight

are summed. The

_position

of the contour

_{(xl, y1) adjoining}

the

starting point (xo, yo)

lies in the direction of minimum total

_intensity

and is

_{given by}

(xo

+

1,

yo +

1).

The three test directions are determined

_by

the direction of the

_previous

_step.

The above

_procedure

is

_repeated

until one returns to the

_position

of the

_{starting point.}

The number of 46

_pixels

is of

arbitrary

source. It has been found

_optimal

to account for the fact that the width of the

_{intensity valley}

is 5 to 6

_{pixels (cf. Fig. 2).}

After

_having

determined

_(and

stored)

the

_positions

of the contour the center of mass of the vesicle is determined

according

to

where K is the number of

_pixels

of the contour and is

_typically

K = 500-1 000.

Now,

the

polar

coordinates

_{( Q i, Ri)}

of the contour are determined with the center of mass as

_origin.

The Fourier

_{decomposition}

of the contour is

_{performed using}

a Fast Fourier Transform

procedure [10].

For this

_procedure

the contour line must be divided into 2N

_equidistant

_points.

Since the contour line is determined

_by

500-1 000

_pixels

a new set of

29 = 512 points

(Ro,511 )

is determined from the

_original

set

_{( Ri, x )}

_{by averaging}

the values

lying

within

segments

of

_{opening angle}

2

_Tr/128.

The Fourier transformation

yields complex

Fourier coefficients

(6)

Fig.

1.

_{- a)}

Schematic view of _apparatusbuilt around a inverted Zeiss Aximat

_microscope

used for

observation of vesicle. Since the focus of the

objective

is set at

_infinity

its

image plane

can be

positioned

at will

_{by using}

the lenses

_LI

and

_L2

and the mirror. Therefore a normal

_bright

field

_{objective (Hère :}

Zeiss 50 x , Air, Pol, numerical _aperture0.95,

working

distance 300

um)

can be used for

_phase

contrast

microscopy by placing

a

_{(ring-like) phase plate}

in the

image plane.

The

sample

could be illuminated with

(7)

Fig.

2. - Schematic

representation

of

algorithm

used to determine the transient contour of the vesicles. The

_{starting point}

at

_{(xo, yo)}

is selected

_{interactively}

the first time. Then the

_intensity

over three bands

comprising

46

_pixels

and

lying

in three different directions

_(of

the

eight possible)

are summed to find the direction of minimum

_intensity.

An

_example

of the

_depth

of the

_{intensity valley}

is shown at the left lower

corner.

As the next

_step,

the mean _square

_amplitude

is determined as follows :

Since the excitation modes are

_{statistically independent,}

the time average of the contour is determined

_{by averaging}

over the Fourier coefficients of an

_arbitrary

number of M-contours

where n is an

_eligible

number of that contour from which the

_procedure

is started. It must be

n .

M/2.

The mean _square

_amplitudes

of

_{mode q}

are

_{finally given by}

The

_bending

modulus is

_directly

obtained from

_{equation (9).}

_{The average vesicle radius is}

given by

where an,

o,

bn, o

are the

averaged

Fourier coefficients

for q

= 0. It should be noted that in the

(8)

cq

at time t and the contour obtained

_{by averaging}

over a

_time,

At,

which is

considerably

longer

than the correlation time of the

_{longest wavelength}

mode. In this _wayit was

_possible

to

analyse

also vesicles of

_slightly

oval

form,

which

change shape

very

slowly,

as

_{predicted by}

Millner and Safran

_[7].

The values of

_Kc

obtained for

_slightly

oval vesicles _agreewell with those

_exhibiting

a

_{spherical shape, showing}

_{that the}

_{quasispherical}

model

_applies

also to

slightly

oval vesicles.

4.

_{Expérimental}

results.

4.1 BENDING MODULI OF SYNTHETIC AND NATUREL LIPIDS AND THE EFFECT OF SOLUTES.

-Figure

3 shows values of the

_bending

modulus of DMPC

_bilayers

as a function of the wave

vector q

characterizing

the undulations of the contour

_{(cf. Eq. (7))}

measured at 30 ° C. The

Kc-value

of the lowest order mode

_(q

=

2 )

is

clearly larger

than that of the

higher

modes

(q

= 3 to

10).

The latter _agreewithin

experimental

error

_although

a shallow minimum appears to exist

_{around q}

= 5. The average value of

Kc

obtained for

3 _ q 10

is

Kc =

( 1.15 ± 0.15 ) x 10-12

erg. It is essential to

_analyse

a

_{sufficiently large}

number of

_images

in order to

_probe

all excitational states of the vesicle.

_By

_attributing

the

_apparent

incrase of

Kc

at q

3 to the influence of the surface

tension,

we determined values

of 00FF

for the

f

= 2 and f = 3 modes. These values were used below in order to correct the correlation times in table II and are

_given

there. In the

present

experiments,

we

_analyzed

at least 100

images

in order to determine

_K,

from the lowest order mode

_(f

=

2).

The average value of

K,,

decreases with the number of contours

_analysed.

That is the true

_bending

elastic modulus

is

given by

the

_asymptotic

values of

_Kc.

Figure

4 shows measurements of the

_bending

_modulus,

_Kc,

as a function of the wave vector q for DMPC

containing

20 mole % and 30 mole % cholesterol. A

qualitatively

different

q-dependence

as in

figure

3 is observed. As in the case _{of pure DMPC,}the

_Kc-value

for the lowest order mode

(q

=

2 )

is

larger

than

for q

= 3.

However,

_{at q}

>

3, Kc

increases

again

with the wave vector.

_Below,

this behaviour is

_interpreted

in terms of a reduction of the

high

order modes

_by

the reduced flow of the

_bilayer.

Fig.

3. -

Bending

elastic

_{modulus, Kc, of (most probably) single}

walled vesicle of DMPC as function of

wave

_{number q}

_{characterizing}

_{the undulations of the vesicle}_contour._Vesicle

_{radius ro}

= 9.91

f.Lm ;

(9)

Fig.

4.

_{- Bending}

elastic modûli of

_single

walled vesicle of DMPC

_{containing (a)}

20 mole % and

_(b)

30 mole % cholesterol,

respectively. Measuring

temperature 30 ° C. Case of

_figure

4a : 176

_images

taken

at time intervals of 0.2 s were

_{analysed ;}

vesicle radius was 9.88 ilm. Case of

figure

4b : 178

images

taken at time intervals of 0.25 s were

_{analysed ;}

vesicle radius 12.19 um.

Figure

5 shows a _{summary of the}_average

_bending

moduli of vesicles of DMPC.

_Altogether

26 vesicles were

_analysed.

The ordinate of the

_{histogram gives}

the number of cells

_yielding

the

same

10 -12

ergs are

_{distinguished.}

The

_larger

_Kc-values

are

multiples

of the minimum value. This leads to _{the conclusion that these groups}

_correspond

to

vesicles with shells

_composed

of one, two,

three,

and four

_bilayers.

Figure

7 summarizes the values of the

_bending

moduli of

_{lipid bilayers}

determined in our

laboratory.

The two more exotic

_lipids

1,2

di

_{(5CI-16: O)-PC}

and

_{galactosyldiglyceride}

are

included in this

_figure.

For

_comparison,

we also

_give

the value of

_Kc

of

_erythrocytes

measured

by

direct Fourier

_analysis

of the membrane excitations

_by

reflection interference contrast

microscopy

[11].

Each

_{point corresponds}

to the measurement of one vesicle. The most

remarkable results are :

1)

In several cases the measured

K,-values clearly

form groups. The differences between

(10)

Fig.

5.

_{- Summary}

of average values of

Kc

measurëd for DMPC vesicles at 30 ° C. The ordinate of the

histogram gives

the numbers of vesicles

exhibiting

the same

_bending

elastic modulus.

Fig.

6. - Plot of

auto-corrélation

_{function Vq(t) v q (0)}

of

amplitudes

of contour fluctuations for the three lower order modes. The data

_points

are fitted

_{by single exponential}

functions

_using

a least square

fitting procedure.

group of

Kc-values corresponds

to a vesicle

composed

of a fixed number of

bilayers.

The lowest values are attributed to

_single

shell vesicles. This demonstrates the

_high

degree

of

accuracy of the

present

technique.

The average values of the

bending

moduli of the

single

shell vesicle are

_given

in table 1.

(11)

Fig.

7.

_{- a) Summary}

of

_bending

elastic moduli,

Kc,

of vesicles of various

_synthetic

and

natural lipids

and of DMPC-cholesterol mixtures. On the

right

side the value of red blood cell membrane is shown. The

_lipid

C5-PC is a

_{phosphatidylcholine}

_(PC)

with a branched chain. G-DG

₍₌

galactosyldiglyceride)

is a

_plant

_{cell lipid}

with a

_high

content of C =

_C-double

bonds. Each

point

was obtained

by

evaluation

of one

_single

vesicle. Note that the values for DMPC, DMPC + 20 % cholesterol egg PC and C5-PC

form well

_separated

_{groups. The differences between the}average values of

_Kc

of

adjacent

groups are

equal

to the lowest value of

_K,.

Thus each group

corresponds

to vesicles with the same number of

(12)

Table I.

_{Summary of}

_averagevalues

_of

_{bending stiffness of bilayers}

studied in _presentwork.

Most

_remarkably

is the low value

_{of Kc}

(:::::: 5

kT ) for

the

_{galactosyldiglyceride}

and the low value

for

erythrocytes.

3)

Most

_remarkably

_{is the very low}

_bending

modulus of the

_{digalactosyldiglyceride,}

that is about

_{Kc ~}

3

_{k B}

T. This is

_partially

due to the low area

_{packing density}

of this

_{lipid. Judged}

from

_{accompanying monolayer}

studies

_(at

the air water

_{interface) performed}

in this

laboratory (H.-P.

Duwe doctoral

_thesis)

the area per molecule of this

lipid

is

_by

about 30 %

larger (100

Â/molecule)

than for DMPC

₍₆₅

_{Âl/molecule).}

4)

Another remarkable

_aspect

is the rather low

_bending

modulus of the red blood cell membrane. The

_{lipid/protein bilayer}

of this

_composite

membrane contains about 50 % cholesterol and 50 % of its total mass is

_composed

of

_integral

_proteins.

One would thus

_expect

a tenfold

_higher

value of

_Kc.

In a

_forthcoming

_paper

_(K.

Zeman and E.

Sackmann,

to be

published)

we will

_provide

evidence that the

bending

excitations may be determined

_by

the

spectrin/actin

network

_coupled

to the

_bilayer.

4.2 CORRELATION TIMES OF MEMBRANE EXCITATIONS.

_{- According}

to

_{équation (4)}

the

dynamics

of each

_spherical

harmonic mode is determined

_by

an

_exponential

auto-correlation

function. Since the mean square

amplitude

of each contour excitation of wave

_{vector q}

is the sum over the mean square amplitudes

_{1 al,.}

2,

the auto-correlation

_{function (Vq(t) Vq(O))}

is

also a sum over

_{exponentials.}

_However,

it can be well

_{approximated by}

a

simple exponential

with the

_decay

constant

_(reciprocal

correlation

time)

_Wq

= W l for the

following

reasons.

_First,

the lowest order term of each sum

is q

= f

(cf.

_Eq.

_(8)).

Second,

the correlation time w 1 increases

approximately

with the third power

of f and the

_{amplitudes ai m}

of the sums in

equation (8)

decrease with the fourth power of

f.

The auto-correlation

functions,

Aq ( T ),

of the three lowest order modes

_(q

=

2,

3 and

4)

of the contour fluctuations were

directly

determined from the

_amplitudes

_Vq(t)

_according

to

where

N is the number of

_images

evaluated for the calculation of the correlation function. The auto-correlation functions thus determined are

_plotted

in

_figure

6 for the three lowest order modes

_(q

=

2,

3,

4).

The

_experimental

data _canbe well-fitted

by

a

single exponential

function. The oscillations of the

_{experimental plots}

of

_Aq(t)

are due to

_experimental

errors

(13)

times of the three lowest order modes are summarized in table II and

compared

with valùes

(denoted

as theoretical in Tab.

II)

calculated from

equation (5)

with the measured values of

Kc

and the average vesicle radius ro.

For ÿ -

0 one observes a

_systematic

_discrepancy

between calculated and

_experimental

values. Good

agreement

is,

however,

observed for a value of

y = 2013

2.3. This value has indeed been obtained from the static

_{experiment by fitting}

the

apparent

value of

_Kc

for the lowest order mode

_(q

=

2 )

to the values obtained for

q > 3. We thus conclude that both the

dynamic

and the

equilibrium

model of the vesicle

fluctuation

[7]

agree well with

experimental

data.

Table II.

_{- Comparison}

_{of experimental}

with theoretical relaxation times

₍₂

_{’TT T l}w

1)

for

modes q = 2 to _q= 4 as calculated

_for

_{y =}0 and _{y =}+ 2.3. Vesicle radius

_Ro

= 9.91 _ktm,

Kc

= 1.1 _x

10 -12

_{erg ;}water

viscosity

₁₇= 1.002 mPs..

4.3 FREEZING OF UNDULATIONS AT THE

_{La-P/3,-PHASE}

TRANSITION.

_{- Figure}

8 shows the

temperature

dependence

of the mean _square

_amplitudes

of the second and third order modes

in the

neighbourhood

of the fluid-to-solid

_{(La-P /3’-)}

transition of a DMPC

bilayer

vesicle.

Clearly,

the

_higher

order modes freeze at lower

_temperature

than the second order excitation. Thus at

_decreasing

_temperature

the mean square

amplitude

of the

f

= 2 mode starts to

decrease

_already

at 24 ° C whereas that of the f = 3 mode remains

essentially

constant to

T = 23 ° C before reduction sets in. This is

_largely

a _{consequence of the reduction of the}

vesicle area and the

_{corresponding}

increase of the influence of the surface tension

( y ) .

The

_{gradual freezing}

of the

_{long wavelength}

excitations well above the transition

Fig.

8. - Plot of mean _square

_amplitudes

of 2nd and 3rd order modes of DMPC

_bilayer

vesicle as a

function of _temperatureupon

approaching

the fluid-to-solid

phase

transition. The temperature varies

linearly

between 24 ° C at

_picture

number n = 0 to 22 ° C at

picture

number n = 600. The

(chain) melting

(14)

temperature

( Tm

=

23.8 ° C)

may be the

origin

of the

precritical

behaviour of the

phase

transition.

4.4 HYPERELASTIC BILAYER. - An

_example

of a drastic decrease of the membrane

_bending

modulus

_by

solutes is shown in

_figure

9. If a small amount

_(about

2-5 mole

%)

of a

bipolar

amphiphile (called

bola

_lipid)

is added to

_DMPC,

the membrane undulations are

_drastically

enhanced. As can be seen in

figure

9a,

the contour is smeared which is a consequence of the very

strong

excitations of undulations in the sub-micrometer

_regime.

The

_amplitudes

of these modes are

_considerably

larger

than the half width of the contour

_{intensity profile}

and a

_large

number of these are

simultaneously

excited which cannot be resolved.

In addition to these fast fluctuations of the

membrane,

one observes slow

shape changes

of

the vesicle which can also be seen in

_figure

9a

_(compare

the first 3

_pictures

with the last

_3).

Long

time observations have

_clearly

shown that these slow

_{shape changes}

are

_periodic.

A

rough

estimation shows that for a vesicle of about 10 pm

radius,

excitations of 5 ktm

wavelength (corresponding

to f

=

5)

exhibit

amplitudes

of about 1 _pm.This

yields

an

approximate

value of

_Kc

==== 0.5

kB T.

The very small

bending

stiffness and the

_{large amplitudes}

of the undulations show that the DMPC-bola vesicles exhibit a very small

persistence length [14].

Therefore _{it may be}

_expected

that these vesicles exhibit

_typical

features of random surfaces

_{(two-dimensional polymers)}

[14].

In order to

_explore

this

_aspect,

we determined the radius of

_gyration

of vesicles of various size as a function of surface area at a

_temperature

of 27

° C,

that

is,

well above the

chain-melting

transition. For this purpose the coordinates of 25

_points

which were

equidis-tantly

distributed around the contour line of the vesicle were determined

_{by image processing.}

Then the radius of

_gyration

of the two-dimensional vesicle

_image

was obtained

_according

to

where

_Rij

is the distance between two

_points

on the vesicle contour. We

_repeated

this

procedure

for 10 different

_pictures

of each vesicle evaluated. With these values of

Rg,

an

_averaged

radius of

gyration,

_Rg,

was calculated.

_{Simultaneously}

the surface area,

A,

was measured as follows : The vesicles were cooled below the

_{La-P p-transition}

where

they

assume a

_{spherical shape}

which allowed the determination of A from the vesicle radius. The

area,

A,

at the

temperature

of the

_{Rg-measurement}

(27 ° C)

was obtained

_{by assuming}

that the

bola-containing

vesicles exhibit

_{approximately}

the same

_change

in area at the

_phase

transition

(AA /A -

20

_%)

and the same thermal

_expansion

coefficient

_{( a}

= 7.22 _x

10 - 3

_per

degree)

_as

normal DMPC vesicles. The DMPC-bola vesicles had an average excess area of 30 % relative

to the surface of a

_sphere

of radius

_Rg.

The relative error of our measurements was 5 % for

Rg and 9 % for A.

In

_figure

9b the radius of

_gyration,

_Rg,

is

_plotted

as a function of the vesicle area A in a

(15)

Fig.

9.

_{- a)}

Transient

_shapes

of vesicles of DMPC

_containing

2 mole % of the

bipolar lipid

denoted as

bola

lipid.

The

images

were recorded from top left to bottom

_right.

The time intervals between two

images

is about 10 s. The _temperaturewas 27 ° C.

b)

Double

logarithmic plot

of

_projected

radius of

gyration

_Rg

of DMPC vesicle

containing

2 mole % of the

_bipolar

bola

_lipid

as a function of the vesicle

area A.

is confirmed

_by

a coefficient of correlation of 0.99. From the

_slope

of the

_straight

line,

one

obtains

Thus

_Rg

scales as

NO,94:t 0.02

where N is the number of

_lipid

molecules.

5.

_Concluding

discussion.

The

present

Fourier

_analysis

of the

_shape

fluctuations of vesicles allows the determination of membrane

_bending

elastic moduli to an _accuracyof 10 %. In

_particular,

shells

_composed

of

different numbers of

_bilayers

_maybe

_{well-distinguished.}

The detailed

_study

of DMPC showed that the

_{quasispherical}

model can describe the static

(16)

certainly

breaks down at

_{wavelengths comparable}

to the

_bilayer

thickness

_{(--- 10 nm)}

where the undulations are

_expected

to be associated with lateral tension and with shear deformations in the direction of the membrane normal

_[12].

The

_present

measurements of

_{cholesterol-containing lipid bilayers provide}

evidence that in this

_particular

case, the

_bending

mode

concept

breaks down even in the

_{optical wevelengths}

regime.

There are, of course, several reasons for the

(anomalous)

monotonous increase of

Kc

with f :

(1)

the decrease of thc correlation time of the undulations to values on the order of the camera

_integration

time

_(0.02

_s),

₍₂₎

the shear associated with the mutual shift of the

_opposing

monolayers,

and

(3)

the lateral _pressure

_arising

in both

_monolayers

if the

_lipid

flow is too slow

to accommodate for the local

_density

fluctuations. At the

_{present stage}

of the

_technique,

it is

not

_possible

to

_{distinguish unambiguously}

_amongthese

_{possibilities. Concerning}

the first

effect,

we should like to

_point

out that we did not observe the same monotonic increase in

Kc

for the multilamellar

_vesicles,

even in the case of vesicles

_composed

of four

_bilayers

for which

_K,

is increased

_by

a factor of four. We thus favour the last effect for two reasons :

_first,

the lateral

_{compressibility}

modulus is

_strongly

increased

_by

cholesterol

_(by

a factor of four at

30 mole %

_cholesterol,

_[13])

and

_second,

the lateral

_mobility

is

_{substantially}

reduced

_(by

a

factor of about two at 30 %

_{cholesterol).}

Finally

it is worth

_{mentioning (1)}

that in mixed

_systems,

one _mayalso have a

_coupling

between undulations and

_{phase separation}

_[8,

_14,

_15]

and

₍₂₎

that

_strong

evidence has been

provided

that the fluid DMPC-cholesterol mixture exhibits

_{phase separation}

between 10 and 30 mole % of cholesterol

_[16].

Thus

_{phase separation}

may be another reason for the

anomalous behaviour of the

_Kc-versus-f

_plot

of the DMPC-cholesterol mixture.

The

_remarkably

low value of

_Kc

observed for the

_{digalactosid diglyceride}

is

_surprising

in view of the fact that the area

_{compressibility}

modulus

_reported

for this

_lipid

is similar to other

common

_lipids

_[18].

The correlation between

_bending

and area

_{compressibility}

modulus

suggested by micropipette experiments

[17]

holds well for other cases such as the cholesterol-DMPC

system.

On the other

side,

the low

_Kc-value

for the G-DG is

_expected

from theoretical considerations of Szleifer et al.

_[19]

who showed that

_Kc

decreases

_drastically

with

_increasing

surface area _permolecule. As mentioned above the area per molecule for the

galactolipid

is

by

about 30 %

_larger

than for

_DMPC,

and a

_considerably

lower

_Kc-value

is thus

_expected.

Since the

_bending

stiffness

_depends

also

_critically

on the

_bilayer

thickness

_[19,

_20],

extensive

studies of

_bilayers

of various thicknesses and

_packing

densities are

_required

in order to

_clarify

this

point.

Our value of

_{Kc =}

_{( 1.15}

+

_{0.8 )}

x

10-12

_ergfor _egglecithin is about a factor of two

_larger

than the value

_{reported by}

Faucon et al.

[21].

Since the above value was the lowest observed

by

us for this

_lipid,

we assume that it

_corresponds

to a

single

shelled vesicles. We do not have

an

explanation

for this

_discrepancy.

On the other

_side,

the value of the surface tension

(y =

1.5 x

10 - 5

erg/cm2)

_{reported by}

Faucon et al. agrees well with our data. Thus from

y = 2.3

for ro = 9.9 _pm,we obtain _{y =}1.5 x

10- 5

erg/cm2.

Recently,

another

_technique

for the measurement of

_{bilayer bending}

stiffnesses was

reported

by

Bo and

_Waugh

_[22].

The value obtained

by

these authors for

stearoyl-oleyl-phosphaticlylcholine (Kc

= 2 x

10-12

_ergs)

is somewhat

_larger

than for DMPC.

In contrast to

_cholesterol,

_manysmall molecules lead to a reduction of the

_bending

modulus. One

example

is

_pentanol

which reduces the

_bending

stiffness of DMPC

_bilayers

quite drastically

as shown

_{by Safinya}

et al.

_[20].

This has been attributed to a

_thinning

of the

bilayer.

A correlation between

_Kc

and the thickness of the

_bilayer

has indeed been established for the case of

_{SDS-cosurfactant-water}

_mixtures

_[20].

(17)

bola

_lipid

on the behaviour of DMPC

vesicles. With

_respect

to these systems the

_following

question

arises : At

_large

excess areas

_(of

some 10

_%)

vesicles _may

_undergo

a

_blebbing

transition

_leading

to the formation of tethers or chains of small vesicles which emanate from the

_giant

vesicles. Such a

_blebbing

transition has been

_{reported by}

Evans et al. for

stearoyl-oleyl

phosphatidylcholine

vesicles

_(E.

_Evans,

_{private communication).}

In _contrast,the

_giant

DMPC-bola vesicles can assume

_large

excess areas without

blebbing.

One

likely

_explanation

is that the

_{bipolar lipid impedes}

the mutual

_sliding

of the two

_monolayers

which is

_required

for the formation of blebs.

On the other

side,

DMPC vesicles with bola

_lipid

are unstable and

_{eventually expel}

small vesicles. These vesicles are

completely

and

irreversibly

separated

from the

giant

mother

vesicle and swim _away.After this _{process the mother vesicle}

_eventually

exhibits normal

shape

fluctuations characteristic for DMPC. Therefore we conclude that the

_expelled

vesicles are

enriched with the bola

_lipid.

Our

_scaling

law

_{Ri oc}

A o.9a + 0.02

establishes a fractional

dimensionality

of the vesicle

surface. A fractional

_{dimensionality}

of

_hyperelastic

surfaces with

_{Kc :5}

kT was

_predicted

on

the basis of Monte Carlo simulation studies

_by

Leibler et al.

_[14]

for 2-dimensional vesicles and

_by

Baumgàrtner

(private communication)

for the 3-dimensional case. The latter author

found an

_exponent

of 0.8. The

_discrepancy

of the latter result with our data is most

_probably

a consequence of the differences in the

boundary

conditions. For vesicles the volume V is

essentially

constant while in the Monte Carlo studies V is considered as variable. Another

reason for the

_discrepancy

may be attributed to the fact that we can measure

_only

the

projected

radius of

_gyration.

A very

interesting phenomenon

exhibited in

_figure

9a is the slow and reversible transition between a more

prolate

and a more oblate average

_shape.

Such slow and reversible transitions

have been observed for several DMPC-bola vesicles. The response time of these slow

_shape

changes

may vary between one and some 20 minutes. These ultraslow

_shape

fluctuations may

thus

_correspond

to the Goldstone-like modes

_{predicted by}

Peterson

_[6]

and Milner and Safran

[7].

As follows from

_figure

5 the

_lamellarity

of

_giant

vesicles may be determined

by

Kc-measurements.

In

_principle

the

_lamellarity

of vesicles can also be detetmined

_{by optical}

contrast measurements

_[23].

However this method

_{requires high-precision}

measurements of absolute contrasts if vesicles from different

_preparations

have to be

_compared

as in our

studies.

Moreover,

small enclosed vesicles

present

in most cases may introduce

large

errors.

For these reasons this

_technique

was not followed further after some initial

attempts.

In contrast, the undulation

_{analysis by}

the

_present

_technique

allows absolute measurements of the

_{lamellarity, provided}

the value of

_Kc

is known.

The effect of the shear associated with the

_{monolayer-monolayer slip depends}

on their

coupling strength.

Lateral diffusion measurements on

_{supported bilayers [18] provide}

evidence for such

strong

coupling

effects. The

_{impeded monolayer slip certainly}

comes into

play

in the 100 nm

_regime

as shown

_by

the

_following

consideration :

According

to

equation (5)

and table

_II,

the relaxation time of the undulations is of

order f

is

Trel = 10 x

10- 3 s

for a vesicle

_{of ro}

= 10 _ktm,whereas the undulations

wavelengths A

scale as

A = 2

_1Tro/f.

On the other

side,

the

_{reciprocal jump frequency}

of the

_lipid

lateral Brownian motion is ~

107 s-1.

Thus the

monolayer-monolayer shearing

is

expected

to become essential

for t - 1 500

or for undulation

wavelengths

of 50 nm.

Very

recently

we

_performed

coherent

_quasielastic

neutron

_scattering

_experiments

of oriented multilamellar DMPC

systems

(at

a

hydration

of 20 %

_{water) by}

the

Spin-Echo

(18)

line width _wlmof the

_dynamic

surface

_roughness)

scales as

q 2.5

to

q 3,

that

is,

similar as in the

optical wavelength

domain. However the value of

_{Kc corresponding}

to the measured line width would be an order of

_magnitude

smaller than

expected.

These short

wavelength

excitations may thus be of different

origin.

Acknowledgements.

This

_manuscript

was

_partially

written

_during

a visit at the

_University

of British Columbia in Vancouver and one of the authors

_(E.

_S.)

is most

_grateful

to E. A. Evans for his

_hospitality

and for _many

_enlightening

discussions. We also

_{greatly profited}

from

_helpful

discussions with W.

_Helfrich,

S.

_Leibler,

R.

_Lipowski,

S. T. Milner. Last but not

_least,

we would like to thank H.

_Engelhardt

who

_pioneered

the

_present

_{dynamic image processing technique.}

The bola

lipid

was a

_{gift by}

Prof.

_Furhop

from the Freie Universitât Berlin. The work was

_financially

supported by

the Deutsche

_{Forschungsgemeinschaft}

_(Project

SA

_246/16-2

and SFB 266

_(Bl))

and

_by

₍₁₉₇₅₎

1035.

[10]

NIEDERDRENK _K.,« Die endliche Fourier- und Walsh Transformation mit einer

Einführung

in die

Bildverarbeitung

»,

Vieweg Verlag Braunschweig,

2nd ed.

57.

[24]

PFEIFFER W., HENKEL _Th.,SACKMANN E., KNOLL W. and RICHTER D.,

Europhys.

Lett. 8