Bending rigidities of some biological model membranes as obtained from the Fourier analysis of contour sections

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Bending rigidities of some biological model membranes

as obtained from the Fourier analysis of contour sections

M. Mutz, W. Helfrich

To cite this version:

Bending rigidities

of some

_biological

model membranes as

obtained from

the

Fourier

_analysis

of

contour

sections

M. Mutz and W. Helfrich

Fachbereich

Physik,

Freie Universität Berlin, Arnimallee 14, D-1000 Berlin 33, F.R.G.

(Reçu

le 3 octobre 1989, révisé le 12 _janvier 1990,

accepté

le 18 janvier

1990)

Abstract. 2014 We have measured the

bending rigidites

kc

of egg lecithin

(EYPC),

dimyristoyl-phosphatidyl-ethanolamine (DMPE)

and

_{digalactosyl-diacylglycerol (DGDG)}

membranes in the fluid state,

subjecting

the fluctuations of

large nearly planar bilayer

sections to

_computerized

Fourier

analysis.

For EYPC we found

_kc

= 0.8

10-12

_erg which is less than half the value

obtained earlier for egg and other lecithins from the

bending

fluctuations of tubular vesicles but in the range of more recent measurements on

_spherical

lecithin vesicles. For DMPE we found

kc

= 0.7 _x

10-12

_erg at T = 60 °C, while we

_{derived kc}

= 1.7 x

10-12

erg from the

bending

fluctuations of tubular vesicles of

_{dilauroyl-phosphatildylethanolamine (DLPE)}

at T = 46 °C. As

in the case of lecithins the different results seem to be

_mostly

due to the different methods.

_Very

low values

of kc = (0.12 2014 0.27)

x

10-12

erg were measured for DGDG. The scatter was less but

also

_large

for the other materials. Classification

Physics

Abstracts 87.20E

Introduction.

The

_{bending rigidities kc}

of the fluid

_bilayers

of some lecithins havé been measured

_by

various

methods. The first

_value,

_kc

= 2.3 _x

10 -12

erg, was obtained for _egg lecithin

_(EYPC)

membranes from the fluctuations of the

_angle

made

_by

the two ends of

_long

tubular vesicles

[1].

Later on, the same method was

_applied,

_apart

from

EYPC,

to

_{dimyristoyl-lecithin}

(DMPC), dipalmitoyl-lecithin

and

_{distearoyl-lecithin, yielding bending rigidities}

in the range

of

_{(1.8 - 2.4 )}

x

10-12

_erg

[2].

_Studying

fluctuation

_amplitudes

and relaxation times of tubular and

_spherical

_vesicles,

Schneider et al. found

_{kc = (1 - 2) x 10- 12}

_erg for the

_{bending rigidity}

of the EYPC

_bilayer

_{[3, 4].}

The variations in

_rigidity

_among the vesicles were + 30 % or

_larger

in most of those

_studies,

_being

at least as

_large

as the

_typical

statistical error of the value

computed

for a

_single

vesicle.

_Very

_recently,

Bo and

_Waugh

determined the radii of

(invisibly)

thin vesicular tethers and the forces

_acting

on them for

_{stearoyl-oleoyl-lecithin}

(SOPC)

vesicles

_{manipulated by}

a

_{micropipette,}

which

permitted

them to

_calculate,

with a

rather

large

_{uncertainty, kc}

= 1.8 x

10 -12

_erg

_[5].

Distinctly

lower values of the

_bending

rigidities

of lecithin membranes have been

_reported

_by

other authors. From a

_computerized

Fourier

analysis

of the contours of

_{fluctuating spherical}

vesicles

_{Engelhardt et}

al. and later

Duwe et al. derived 1.1 x

10-12

_erg for DMPC

_[6-8].

Bivas

et al. ;

also

_{studying spherical}

vesicles,

expanded

the radial fluctuations in

Legendre polynomials [9].

Their

_preliminary

experiments

on EYPC were continued

by

Faucon et al. who corrected the mean _square

fluctuation

_amplitudes

for lateral

_tension,

white noise and video

_sampling

_time,

_arriving

finally

at a _{very low value of the}

_{bending rigidity}

of egg lecithin

_bilayers,

kc =

(0.4 -

0.5 )

x

10 -12

erg

[10].

Theoretical estimates of the

_{bending rigidity}

may be obtained from the formula

Containing

the

_bilayer

thickness b and the area

stretching

modulus

A,

it is based on the

assumption

that the constituent

_monolayers

behave like

_homogeneous

sheets of thickness

b /2

with a neutral surface in the middle of both of them

[ 11 ].

Insertion of b = 4 nm and

A = 140

dyn

cm-1,

the area

_stretching

modulus measured

_by

Evans and coworkers for the EYPC and DMPC membranes

_{[12, 13],}

leads

_{to kc}

= 0.5 x

10 -12

_{erg. Similar} values can be

obtained

_by

other model calculations

_[14],

_including

the mean-field statistical mechanics of

fluctuating hydrocarbon

chains

_[15].

The aim of the

_present

work was to

_develop

a further method of

_measuring

the membrane

bending rigidity. Nearly straight

sections of

_fluctuating

membrane contours are recorded and

Fourier

_{analysed. They belong}

to extended

_{single bilayers}

which seem to turn in semicircles from the

_top

to the bottom of the

_sample

cells. The materials

_investigated

are EYPC

and,

for

the first

time,

two other

_{electrically neutral lipids abounding}

in

_biological

membranes,

namely

dimyristoyl-phosphytidylethanolamine (DMPE)

and

_{digalactosyl-diacylglycerol (DGDG).}

Preliminary

data on DGDG have been used in

_discussing

the

_unbinding

transition found

_by

us

with DGDG if the

_experiment

was done in 0.1 M NaCI solution instead

_{of pure}

water

_{[16, 17].}

Specifically,

the _enormous

_spread

of the transition

_temperature

was attributed to the

_large

scatter of the measured

_rigidity.

In order to check wether the

_rigidities

of PE membranes

depend

on the

_experimental

method,

as seems true of

lecithins,

we also

_analyse

the

angular

fluctuations of tubular vesicles of another

_cephalin,

_{dilauroyl-phosphatidylethanolamine}

(DLPE).

Theory.

Since the

_cylindrical

curvatures of the membranes

_investigated

are _very

_weak,

it is

_probably

sufficient in

_calculating

_{the mean-square fluctuation}

_amplitudes

to consider a square

piece

of membrane of area A

_{lying essentially}

in the

_{(x, y ) plane}

_{[ 18, 19].}

The instantaneous

_shape

of the membrane may be

represented by

the scalar field

_{u (x, y) describing}

the local

displacement

of the membrane in z direction. The

_displacement

can be Fourier

_expanded

in

terms of real waves

where q =

(qX’

qy)

= 2 03C0

(m, n ),

with m and n

being integers.

The

prime

at

_E

indicates that

A 1 /2

the summation has to be restricted to a

_half-plane

so that

_pairs

of

_opposite

wave vectors are

counted

_only

once. The curvature elastic energy per unit area of fluid _{membrane may be}

written as

where cl

and c2

are the

_principal

curvatures. The

_spontaneous

curvature _co vanishes if the

the

_integral

of Gaussian curvature

_{depends only}

on the _genus of the surface. For the

weakly

deformed

_membrane,

_{i.e. 1 Vu}

₁

«

_1,

we _{may express the} sum of curvatures

_by

The curvature _{elastic energy associated with} a

_pair

of modes is then

_readily

seen to be

In the presence of lateral

tension a,

a further contribution to the deformational _energy results from the fact that the thermal undulations absorb membrane area. The extra area _per

unit basal area

being 1

(VU)2

for

_ou

₁

«

_1,

one obtains as absorbed area _per

_pair

of modes

2

The addition of deformational energy

is taken into account in

_calculating

the mean-square fluctuation

amplitudes

from the

equipartition

theorem. The mean _energy of deformation _per

_pair

of modes

being

kB

T,

one has because of

₍₅₎

to

₍₇₎

Let the x axis be in the focal

plane

of the

microscope

and

the y

axis coincide with the

direction of

_viewing.

We then see a cut

_throught

the membrane in the x, z

_plane

and

_along

the x axis. To calculate the mean _squares of the fluctuation Fourier

_amplitudes

_Aqx

and

Bqx

of this

line,

we have to sum over al

_qy

at _{fixed qx.}

_Replacing

the sum

_by

an

_integral,

and

_{using (8),}

we find

For low lateral

_{tensions o- « q x 2k,}

this takes the

_simple

form

The fluctuations of the square have been derived on the tacit

asumption

of

_periodic

boundary

conditions without discontinuities of u and Vu. As in the

experiments

we see

only

a

section of the total

membrane,

we have to check if we can still use

₍₁₁₎

to calculate

k,,.

For this purpose, we examine the contributions of all the basic modes of the total

membrane to the Fourier

_amplitudes

of the membrane section seen

_{at y}

= 0 in a window of

width L. The total membrane area A is now

_imagined

to be infinite. If the visible section of membrane contour is

_{expressed by}

For x =

L/2

to x = +

_L/2,

the

_{amplitude An}

is

_{given by}

Insertion of

₍₂₎

in

_{(13) yields}

The

_integral

of cos

_{(kn . x).}

sin

_{(qx .}

x)

cancels in the

_given

interval and has therefore been

omitted in

_{(14). Replacing}

the sum over the basic modes

by

a double

integral

and

_integrating

over q y

as in

₍₁₁₎

results in

An

_{analogous expression}

with sines

_replacing

cosines is obtained for

_{(B2n) .}

_Figure

1 shows how the basic modes contribute to

_(Afl)

of the window mode n = 5. The main effect cornes

from the interval

_kn

_{= 2 7T / L : q x : kn + 2 7T /L.}

The lower qx modes contribute 22

%,

the

higher

modes less than 3 % as much as the main

peak.

In accordance with the

_analysis

of

Fig.

1. - The

_integrand

of

_{equation (15)}

for n = 5,

representing

the contributions of the basic modes to

the mean _square

_amplitude

(A2n)

of the window mode as a function of the wave vector _component

experimental

contours, the effect of the

_{longest wavelength}

modes is reduced

_{by subtracting}

a

symmetric parabola

from cos

_{(qx .}

_x).

The

_parabola

is taken to coincide with the

_endpoints

of the cosine and determined

_by

the method _{of least squares. The} subtraction of the

_parabolas

results in the curve

plottèd

in

_figure

2. The effect of the

longest wavelength

modes is seen to

be

markedly

reduced and the total contribution of all side

_peaks

is now

_only

13 % of

that

of

the main

_peak.

In the case of

_{(B n 2) ,}

a

_straight

line

_coinciding

with the

_endpoints

of the

sin ( qx . x )

is subtracted from the

_sine,

_again

in

_agreement

with the

_analysis

of the

experimental

data. After this

_subtraction,

which is even more necessary than the

other,

the

side

_peaks

contribute to

_{(B n 2>}

is

_only

ca. 8.5 % as much as the main

peak.

Side

peak

contributions are about

_{equally important}

for n = 10 _as for n =

5,

both with

(A2n)

and

(B2n).

The modified

_{integral (15)}

turns out to be

_practically

identical

_{to (A2qx)}

as

given

by

(11)

for the same wave vector,

being

smaller

_{by only}

1 and 2 % for

_(A2n)

_{and (Bn2) ,}

_{respectively.}

Fig.

2. - The

_integrand

of

_{equation (15)}

for n = 5, modified

by subtracting

the

parabola.

Materials and methods..

Dilauroyl-phosphatidylethanolamine

(DLPE),

dimyristoyl-phosphatidylethanolamine

(DMPE), egg-yolk-phosphatidylcholine (EYPC)

and

_{digalactosyldiacylglycerol (DGDG)}

extracted from wheat flour were obtained from

_{Sigma (Munich).}

The

_glycolipid

_DGDG,

again

extracted from wheat

floor,

was also

_purchased

from Serva

_{(Heidelberg).}

The materials were used without further

_{purification.}

The

_purity

of DGDG claimed

_by

the

distributor was 95 %

(Sigma)

and 99 + %

_(Serva).

We checked the

_{purity by thin-layer}

chromatography

on silica

gel

60F. The runs were

_perfomed

in a mixture of

_chloroform,

methanol,

and water

_{(65 :}

65 :

_2.5).

DLPE,

DMPE and DGDG showed a

_single

_spot

when

developed

with iodine vapour. In view of the

sensitivity

of the

_{procedure, impurity}

concentrations

_larger

than 1 % can be excluded. We dissolved the

_lipids

in chloroforme

(10 mg/ml)

and stored them below - 20 ° C.

A small amount

_(some

_u1)

of the chloroform solution was

_spread

on a

_glass

slide and left

overnight

to

_evaporate

under vacuum.

_Subsequently

40

_u,1

of ion

_depleted

water from an ion

exchanger

(Seradest)

of

_pH

5.5-6.0 were added. A cover

_slip

was

_put

on

_top

and the

approximately

40 _ktm

_high

cell was sealed to

_prevent

_evaporation.

All the

_lipid

was close to

the bottom slide at the

_beginning

of

_swelling.

All four

_lipids

swelled

_{spontaneously, forming}

DGDG are fluid at room

_temperature

_{[20, 21].}

When heated in excess _water, DLPE and

DMPE

_undergo

their main transitions at 44 ° C and 53

_{° C,}

_{respectively,}

so that the

experiments

had to be

_performed

above these

_temperatures

_[23].

After several hours of

swelling

we started to look for extended membrane contours with

_straight

sections and tubular vesicles. The former

_belonged

to

_huge

sheets

_{apparently going semicylindrically}

from the bottom to the

_top

of the cell.

They

were seen in the middle

_plane

of some of the

_samples.

The

_samples

were observed with a

_phase

contrast

_{microscope (Leitz),}

the numerical

aperture

of the

_objective

(PH

40x) being

0.75. The

_object

_stage

could be heated to the

desired

_temperatures

_electrically

or

_{by connecting}

it to an external thermostated water bath

(Haake).

Membranes are seen in the focal

_plane

where

_they

are

_parallel

to the

_optical

axis of the

_microscope.

The

_depth

of field was ± 1 _pm.

An

_{image analysis}

_system

_{(Imaging-Technology}

_Inc.,

ITI)

with videocamera

_(Grundig)

and PDP-11

_computer

_(DEC)

were attached to the

_microscope.

It allowed us to store an

_image

in a frame

buffer,

_composed

of 512 x 512

_pixels

with a 6-bit resolution for the _grey levels and 1

pixel corresponding

to about 135 nm. The

sampling

time of the video

images

was 40 ms. The

contour of the membrane was then determined

_by

a

special finding algorithm.

After

_being

oriented

_{horizontally,}

the

_length

of the

_straight

section of membrane contour

within a frame was about 70 ktm. In many cases the membranes were not

_quite

horizontal and

slightly

curved. We eliminated

_gradient

and curvature

_{by subtracting}

a

_{generalized parabola}

consisting

of a

_{symmetric parabola}

and a uniform

_slope.

Its ends

_coinciding

with those of the

contour, the

_parabola

was determined for every contour _{with the method of least squares.}

This

_procedure

has the

_advantage

of

_reducing

the contribution of

_{long wavelength}

modes of the total membrane to the window

mode,

as was shown in the theoretical calculation of the

mean-square

amplitudes.

We used a fast Fourier

algorithm

to calculate the window mode

amplitudes

from the

_adjusted

contours, after

_testing

it

_by

simulation. The number of

_pictures

entering

into each calculation of the mean _square

amplitudes

was in

_general

200 and never less than 100. The time intervals of more than 10 sec between

_pictures

were

_longer

than the

relaxation times of the slowest modes.

Results.

An

_example

of a DGDG membrane

_{showing typical}

fluctuations is

_displayed

in

_figure

3. We

selected unilamellar structures

_{by looking}

for contours of lowest

_optical

contrast.

_Single

membranes _gave the lowest

_{bending rigidities,}

but in the cases of EYPC and DGDG the

spread

between maximum and minimum values was still a factor of two. We

_{analysed only}

membranes which were

_{fluctuating freely}

and not attached to other membranes or

vesicles,

which diminished

_strongly

their number. In the

_experiments

with DMPE

_{particular problems}

arose from the fact that to

_keep

the

_bilayers

in the fluid

_phase

the measurements were

performed

at T = 60 ° C. At this

_high

_temperature

convection currents in the

_samples

sometimes caused a distortion of the membrane contour.

A

_typical

result of the

_{computerized analysis}

for a DGDG membrane is shown in

_figure

4. The

_{bending rigidity}

_kc

was calculated

_by

means of

_{equation (11)}

from the average of the mean square Fourier

amplitudes

_(A2n)

and

_{(B n 2) .}

The calculated

_{bending rigidity}

k,,

forms a

_plateau

_up to the 12th mode and then starts to decrease.

_However,

as indicated

by

apparently higher

values of

_kc,

the first two modes are

_considerably

weakened because of

geometric

constraints

_{(see below).}

_{The average of the values in the}

_plateau

was taken to be the definitive value of the

_{bending rigidity.}

Table 1 summarizes the results for the three different

_lipids

_investigated

with this method

Fig.

3. -

Typical

fluctuations of extended DGDG membrane

(the

bar represents 10

_um).

Fig.

4. - Results of the

computerized

Fourier

_analysis

for a DGDG membrane.

Average

( Cn2) = 1/2

n

((A n2) + (B2n))

of mean-square

function

_{amplitudes (left)}

and

_rigidities

calculated

_according

2 n n

to

_{(11) (right).}

The dotted line _represents the

_plateau

from where the final value

_{of kc}

is calculated.

deviations

_{of kc}

for the individual

_bilayer

were less than 15 % for all

_lipids.

However,

the standard deviations of the individual values from the mean value for one material were 32 %

for

_DGDG,

25 % for

_EYPC,

and 14 % for DMPE. We also measured the

_apparent

_bending

rigidities

of a few multilamellar contours and obtained

_multiples

of

_kc.

DGDG membranes sometimes showed _very

_strong

undulations which could not be

Table 1.

_{- Averages}

and standard deviations

_of

the

final

values

of

the

_bending

_rigidities

as

obtained from

the undulations

_of

extended membranes.

Fig.

5. - DGDG membrane contour smeared in part

by

very strong undulations

(the

bar _represents

10 um).

There was no notable difference between the DGDG from

_Sigma

and that from Serva. We did not _{find any tubular vesicles of DGDG and DMPE that} were suitable for

measuring

the distribution function of the

_angle

made

_by

their ends. On the other

hand,

the

_swelling

of

DLPE,

which differs from DMPE

_{only by}

two carbons less in the saturated

_hydrocarbon

chains,

produced

some suitable tubes but no extended membranes. The measurement of the

angular

distribution function and the calculation of the

_{bending rigidity}

from its width have been described elsewhere

_{[1, 2].}

The results of our

experiments,

three with unilamellar and

two with multilamellar

tubes,

are listed in table II. Note that the average

bending rigidity

obtained with unilamellar DLPE

_tubes,

_kc

= 1.7 x

10 -12

_{erg, is} more than two times

_larger

than that obtained with extended DMPE membranes.

Discussion.

We have measured the

_{bending rigidity}

_kc

for three different

_types

of

_lipids.

Besides the often studied _egg

_lecithin,

we

_{determined kc}

also for two

_{synthetic cephalins}

_(DLPE

and

DMPE)

,

and a natural

_{galactolipid (DGDG).}

Our new method is similar to the

_expansion

in

_spherical

Table II.

_{- Bending rigidities of}

individual DLPE membranes and groups

of

such membranes at T = 46

0

C as obtained

_from

the distribution

of angular fluctuations of

tubular vesicles. The

three values with the lowest

_{kc belong}

to unilamellar tubes.

Instead of

_{complete spherical}

vesicles we studied

_{nearly planar}

membrane sections which _may

be

_regarded

as

_part

of a

_huge

vesicle.

Contours with

_long

_{practically straight}

sections should be characteristic of

_vanishing

lateral

tensions,

especially

in the

_frequent

case of several of them

_running

_parallel.

Tensions tend to

be associated with circular contours which are often

grouped

in onion-like structures, as has first been noted with EYPC

[24].

The

_{long wavelength}

fluctuations of our extended

membranes were not

_{fully developed.}

Rather than to lateral tension this seemed to be due to

the constraints

_{imposed by}

the

top

and bottom of the

_sample

cell where the membranes

forming semicylinders

in

between,

are in contact with other membranes or the

_glass

slides.

We think that the absence of lateral

_tension,

which can be

_positive

or

_negative

in the case of

quasi spherical

vesicles,

as well as the

_large

radius of the

_{semicylinders}

of ca. 20 pm, are among the

advantages

of the new method.

_Also,

a

_possible

deformation of the membranes

_by

gravity

would not be serious and could be checked

_{by carefully measuring}

the

_height

of the membrane « equator » where one sees the contour.

_Many

of the extended membranes are

enclosed

_by

others and thus shielded from

_{impurities dissolving}

from the

_glass

slides.

_Finally,

the extended membranes remain in

_place

for

_{long periods,}

while

_spherical

vesicles must be free to float

_vertically

and

_{horizontally.}

A

_disadvantage

of the new method is the fact that it

mixes modes of different wave _vectors,

_fixing

_qx but

_integrating

over

_qy,

while the wave

vectors or, more

_{precisely, angular}

momenta can be

_separated

in the case of

_{spheres [9, 10].}

However,

because of the

_{1 /q 4}

_dependence

of mean _square

_amplitudes

most of the contributions

_{to (Aqx2 and (B2qx)}

should come from basic modes with wave vectors not much

larger

than _{q xe} An additional

_mixing

of modes stems from the

_{subsequent integration}

over qx as was shown above.

The admixture of other wave vectors also mixes relaxation times and makes it more difficult

to deal with the effect of the video

_integration

time which is 40 ms

_{[6, 10].}

The

_long

integration

time reduces in effect the

_amplitudes

of the

_high

wave vector modes.

However,

in

our

experiments

the reduction was in

general

more than

compensated by

white noise.

Limiting

ourselves

to the

plateau

modes,

we

disregard

both effects. Estimates of the

relaxation times

_[25]

and the direct observation of fluctuations

suggest

that relaxation can be

neglected

up to n = 8 for EYPC and n = 15 for DGDG. Faucon et al.

[10]

corrected for lateral

tension,

white

_noise,

and video

integration

time,

finding kc _ (0.4 -

0.5 )

x

10-12

_erg

for egg lecithin. We doubt that the omission of these effects

explains why

we obtained a

_larger

value. The

_{large spread}

of the

_{bending rigidity}

of DGDG membranes may be the reason for

We have used two methods to measure the

bending rigidities

of

_cephalines

_(PE’s),

obtaining kc

= 0.7 x

10 -12

_erg for DMPE from the undulations of extended membranes and

kc =

1.7

x

10 -12

erg for DLPE from the

angular

fluctuations of tubular vesicles. The

difference between the two results

by

factor of two to three is

_suprising

as DMPE differs from DLPE

_only

_by

two additional

_CH2

groups in the

hydrocarbon

chains. The

_bilayer

thickness b

of DLPE is kown to be

41.0 Â

[26]

and from other data

_[27]

one can estimate b =

42.5 Â

for

DMPE. The

_simple

model

_{underlying equation (1) predicts}

the

_{bending rigidity}

to _vary as the

cube

_pf

the

_bilayer

thickness b if the

_stretching

modulus A is assumed to be

_proportional

to b. On the basis of this model the

_{bending rigidity}

should be little

_larger,

not much

smaller,

for DMPE than for DLPE. The difference is also

_unlikely

to result from the effect of a

higher

temperature

(AT

= 14

K)

on chain

flexibility

and

_bilayer

thickness.

Rather,

it seems that the

bending rigidity depends

on the method

by

which it is measured. The same

_discrepancy

between methods appears to exist in the case of EYPC. Servuss et al.

_{[1] and}

Beblick et al.

_[2]

obtained

_{kc = ( 1.8 -}

_{2.3 )}

x 10-12

erg from the

bending

fluctuations of tubular

vesicles,

while

we derive

_kc

= 0.8 x

10 -12

_erg from the undulations of extended membranes.

Evans and coworkers

_[13],

who measured the membrane

_stretching

modulus to be 140

_dyn

cm-1

1

for EYPC and

_DMPC,

found À = 190

dyn

cm-

1 for DGDG. Their data on

mixtures of

_{dipalmitoyl-oleoyl-PE}

and SOPC

_suggest

À = 240

dyn

cm -1

for

the pure PE. The

thicknesses of all the common

biological

model membranes are of the order of 40

Â.

Experimental

values for PE’s have been

_quoted

above ;

42 Â [21]

]

and

_{36 À [22]}

have been

reported

for DGPD and

_EYPC,

_{respectively.}

_Accordingly,

one

_might

infer on the basis of

equation (1)

that the

_{bending rigidities}

should all fall between 0.5 and 1 x

10-12

erg. The

considerable deviations from this narrow _{range of values} are difficult to understand.

_Equally

puzzling

is the

_dependence

of the results on the

_experimental

method

(and

the wide

spread

of

the

_rigidities

measured with the same

_method).

It seems difficult to

_imagine

that the

_large

uncertainties result from an extreme

_sensitivity

to

_impurities.

All the membranes

_investigated

in the

_present

work

_separate

_by

themselves in

_pure

water

but

_display

mutual adhesion when under lateral tension. Detailed studies of the tension-induced adhesion

_{of egg}

lecithin

_{bilayers [28,}

_29]

_point

to a

submicroscopic roughness

of these

membranes much

larger

than that

_{produced by}

the usual thermal undulations. One may

speculate

that such a

_roughness,

if it also exists in the

_bilayers

of the other

_{lipids, might}

be the

origin

of some of the inconsistencies.

Conclusion.

The

_present

work shows

_again

that the

_{bending rigidities}

of

_electrically

neutral

_biological

model membranes are rather elusive

_quantities.

There remains the

_challenge

of

determining

the true values which may be a function of wave vector and

_geometry.

So

_far,

most of the

experiments

have been done with the natural

materials,

especially

egg lecithin. Both egg

lecithin and wheat flour

_{digalactosyl-diacylglycerol}

are fluid at all

_practical

_temperatures

and,

moreover, swell very

readily

to

_{give large}

unilamellar structures.

_{However, they}

are

likely

to differ from batch to batch. Small variation in

_composition

may

produce large

differences in

behavior,

if the latter is

_subject

to defects or other

singularities

in the

_bilayer.

To eliminate extrinsic sources of

_potential

scatter _{it will be necessary in the} future to do more measurements of the

_{bending rigidities}

of

_{one-component}

_lipids.

A

_{practical problem}

which remains to be addressed has to do with

_optical

observation. The

finite

_depth

of field of ± 1 _{pm may} cause a shift of the membran contour seen under the

microscope

to

_{positions deviating}

from the

_planar

cross section. This is because the membrane

produces

an additional noise which may result in a too small

bending

rigidity

if it is not taken into account.

_Clearly,

the effect is the less

important

the

larger

the

_{investigated objects}

and fluctuation

_wavelengths

are.

Acknowledgment.

This work was

_{supported by}

the Deutsche

_{Forschungsgemeinschaft through}

SFB 312. W. H.

thanks the Institute for Theoretical

_Physics,

UCSB,

Santa

_Barbara,

CA

93106,

USA for their

hospitality

when the final version of the

_manuscript

was written.

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