Adhesion in egg lecithin multilayer systems produced by cooling

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Adhesion in egg lecithin multilayer systems produced by

cooling

W. Harbich, W. Helfrich

To cite this version:

Adhesion in

_egg

lecithin

_multilayer

_systems

_produced

_by

cooling

W. Harbich

_(*)

and W. Helfrich

Freie Universität Berlin, Fachbereich

_Physik,

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

(Reçu

le 3 août _1989, révisé le Il _janvier 1990,

accepté

le 16 janvier

1990)

Abstract. 2014

Slightly hydrated

egg lecithin was

_{aligned mostly parallel}

to the

_{glass plates}

of the

sample

cells

_{by squeezing}

it to a thickness of 20 03BCm. The rest of the cell was filled with water and the

lipid

was left to swell at elevated _temperatures to

_bilayer

mean

_spacings

between 3 and 17 nm.

Cooling by

a few °C gave rise to induced adhesion in the

semicylindrical

border

_{(and myelin}

cylinders)

where the mean

_spacing

was

_larger

than in the bulk. From the contact

_roundings

of membranes

_adhering

to stacks of other membranes we obtained the lateral tension

_(dyn

cm-

1)

_inducing

the adhesion as a function of the

equilibrium

mean

_spacing.

An extensive

analysis

of the data in terms of van der Waals attraction and

undulatory

repulsion

leads to serious contradictions.

They

suggest the intervention of a new,

submicroscopic roughness

of unstressed

lecithin membranes

absorbing plenty

of membrane area. Classification

Physics

Abstracts 62.20D - 68.15

1. Introduction.

It seems to be a

_{general experience}

that egg and other lecithins in a

_large

excess of water form

giant

vesicles and that the vesicles do not stick to each other when

_they

come in contact. The absence of mutual adhesion between lecithin membranes _may

_surprise

in view of the van der

Waals attraction and adhesion

_energies

measured

_by

several authors

_[1-4].

On the other

_hand,

unstressed fluid membranes

_perform

_strong

thermal undulations which are associated with a

repulsive

force.

_{Theory predicts}

this kind of steric interaction to be

_possibly

_strong

_enough

to

overcome van der Waals attraction

_[5-7].

The adhesive behaviour of _egg lecithin membranes has been

_investigated

in our _group for

many years, the method of observation

being light microscopy.

Very

early

it was checked that also in salt solution vesicles emerge,

though

very

slowly

at 0.5 M

_NaCl,

and do not adhere to

each other

_[8].

This was first evidence

_{against separation by}

electrostatic

_repulsion

which would have to

_originate

from

_impurities

because of the zwitterionic nature of lecithin.

Adhesion was observed later on when membranes

_happened

to be stretched

during

the

swelling

of lecithin in water

_[9-11].

A

_rounding

of the membrane in the immediate

_vicinity

of the adhesive contact was used to

_compute

the lateral tension.

_Being

controlled

by

the

(*)

Present address : Bundesanstalt für

_{Materialprüfung, Abteilung}

6-2, D-1000 Berlin 45, F.R.G.

membrane’s

bending

rigidity

and lateral

tension,

contact

_rounding

is

_optically

resolvable

_only

at

_extremely

low tensions. The contact

_angles

made

_by

the contact area and the

_asymptotic

direction of the free membrane were found to be

_practically

constant in the _range of measured tensions

_extending

from ca.

10- 4

to

_{10- 6 dyn cm-1.}

_They

were ca. 40° for

_symmetric

adhesion,

i.e. two

_single

membranes under

_equal

_tension,

and ca. 70° for adhesion of a

_single

membrane to a bundle of several others. In another

_study,

thin unilamellar tubes with radii of

1 um or less were seen to adhere to the

_glass

slides of the

_sample

cell and there to themselves while wider tubes did neither

_[12].

_{All these instances of induced adhesion may be}

_regarded

as

indirect evidence for the decisive role of steric interaction in membrane

_{separation. They}

show van der Waals attraction to dominate whenever lateral tension or

_geometric

constraints

reduce the

_amplitude

of the fluctuations.

In work basic to the

present

one, we studied the

_{orderly swelling}

of

_{slightly hydrated}

egg lecithin in contact with excess water

_{[13, 14]. Orderly swelling}

takes

_place

_{if the lecithin spans} the thickness of the

_sample

cell

_(which

should not exceed 40

ktm)

and the membranes are

aligned mostly parallel

to the

slides,

forming

a well but not

_perfectly

ordered

_{planar phase}

with a

_{semicylindrical}

border to the water. It starts with the formation

_{of myelin cylinders}

and

water channels

_reaching

from

_top

to buttom of the cell

_which,

in

effect,

increase the

_length

of the border. We measured over

_days

or weeks the

_gradual

increase of mean membrane

spacing, covering

the _range of 1 to 15 nm in the

_{planar phase.}

There were no

_signs

in our

samples

of an

_{equilibrium spacing,}

located

_by

others in

_{egg-lecithin multilayer}

_systems

at

2.75 nm

_{[1, 2],}

or of a

_{phase separation.}

For the maximum mean

_spacing

we could estimate

the pressure between

bilayers

in the

_{planar phase, using}

the

_{bending rigidity}

of the

_bilayers

and the radius of very thin tubes

protruding

from the

semicylindrical

border into the free

water. _{The pressure conformed with the}

_{predicted strength}

of the undulation forces within

experimental

and theoretical uncertainties. We inferred from the result that electrostatic

forces,

even if

_they

would have been

_predominant

at

_{15 nm,}

were

_{certainly negligible}

near

3 nm. Like vesicle formation in salt solution the

_experiments

_suggest

that stacked _egg lecithin

membranes can be driven

_apart

_by

steric interaction alone.

It is

agreed today

that undulations

_play

a

_significant

role in the

interplay

of forces between fluid membranes

_[6,

15,

16].

Undulation forces have been measured and found to be of the

predicted strength

in

_multilayer

_systems

_{of very flexible membranes where}

_{they clearly}

dominate

_[17,

_18].

Whether unstressed lecithin membranes adhere to or

_separate

from each

other is still considered a controversial issue. It is therefore of interest to

_investigate

the interaction of these membranes

_by

novel

_{experimental techniques.}

In the

_following

we

_report

on adhesion induced

_by

lateral tension in

_multilayer

systems

of known mean membrane

_spacing.

The tension was

produced by cooling

the egg lecithin

samples

after

_{orderly swelling}

at elevated

_{temperatures.}

Adhesion was inferred from a

separation

of the

_multilayer

_system

into domains of

_higher

and lower lecithin

_density.

This

« phase separation

», as it will be called for

short,

was seen almost

_exclusively

in the

semicylinders

and

_cylinders

where the mean membrane

_spacing

is

_larger

than in the

_planar

phase.

Unless the lecithin

_density

was

_{extremely high, single}

membranes became visible after some

_{cooling, making}

contact

_angles

_again

of ca. 70° with extended domains of adhesion.

Lateral tensions between

4 x 10- 5

and 2 x

10- 3

dyn cm-1

1 were obtained from contact

roundings

and other estimates.

_They

seemed to saturate

_quickly

with

_decreasing

_temperature,

approaching

but not

_exceeding

the

_strength

needed to induce adhesion in the

_{planar phase.}

The data are

_compatible

with an inverse

square relationship,

a - 1

_/z2eq,

between the lateral tension or and the

_equilibrium

mean

_spacing

_zeq

of the membranes in domains of adhesion.

To

_interpret

the data we _propose a model for the

restructuring

of the

multilayer

system

experiment,

and

_permits

an estimate of intermembrane

repulsion

in the absence of tension

which conforms with the theoretical

_strength

of undulation forces. The

_scaling

law

o- ~ 1/Zeq

is showri to be

_part

of a

simplified

model of induced adhesion which also

predicts

ga ’- U, ga

being

the adhesion energy, and constant contact

_angles.

The

_validity

of the

_scaling

law seems to confirm that the steric interaction is due to undulations.

However,

the tensions associated with the

_{equilibrium spacings}

of adhesion are smaller

_by

a factor of at least 100 than appears

permissible theoretically.

A similar and

_probably

related contradiction was

inferred

_previously

from the

_largeness

of the contact

_angles

of induced adhesion

_[11]

_and

will be discussed

_again.

Now as

then,

it appears necessary to

_postulate

an

_optically

unresolved

roughness

of the membranes which absorbs much more area than do the undulations and

possibly

enhances steric interaction.

2.

_{Experimental.}

The

_samples

were

_prepared

as

_{previously [13, 14].}

Pure egg lecithin

(research grade,

99

%)

was used as

_purchased

from Serva

_{(Heidelberg).}

Water was twice distilled. After

hydrating

the lecithin

₍₁₀

to 20 vol.%

_H20),

we

deposited

it on the bottom

_glass

of the

_sample

cell and

pressed

it flat which the cover

_glass

to a final thickness of

_typically

20 _um. The

_squeezing

aligned

the

_{bilayers mostly parallel}

to the

_glasses.

Water sucked in

_{subsequently by capillarity}

filled the rest of the cell.

The

_samples

were

_placed

under a Leitz Ortholux

_microscope

and observed

_directly

or on a

monitor

_equipped

with a recorder.

_Photographs

were made with a camera attached to the

microscope.

The standard mode of observation was

_phase

contrast which is well suited to

bring

out

_single

membranes. The numerical

_aperture

was in

_general

0.75 and the

_depth

of

field about ± 1 _um. All

_pictures

were taken of the central

_plane

of the

_sample

cells.

In contrast to our earlier

work,

we controlled and varied the

_temperature

with a

thermostat.

_Swelling

was done at elevated

_{temperatures,}

_typically

55

_{° C,}

where it took about half as

_long

as at room

_temperature.

After

_reaching

the desired lecithin

_density

in the

_planar

phase

of the

_multilayer

_system,

we lowered the

temperature

to induce adhesion. At the usual

heating

and

_cooling

rates of 0.5 ° C

min-1

the

_temperature

was known with an _{accuracy of}

± 1 ° C while

_being

varied.

The

_stages

of

_swelling

of the

_slightly

_hydrated

_egg lecithin were the same as at room

temperature.

Swelling began

with the

_{rapid generation}

of

_{myelin cylinders growing}

from the border of the

_{planar phase}

into the free water.

_(We

did not

_study

here the less

_frequent

water

channels

_penetrating

the

_{planar phase.)}

The

_cylinders

consisted of concentric

_bilayers,

spanned

the thickness of the

_sample

cell,

and tented to fold into

_patterns

of

_parallel,

equidistant

sections that were in contact with each other. The

_cylinders

_may be

_regarded

as an

extension of the

_{semicylindrical}

border of the

_{planar phase.}

After their

_growth

slowed down

to become

_{quasi imperceptible they}

started

_developing

visible water cores.

_Cylinders

and

border

_began

to widen

_noticeably,

their cross sections

_turning

from

_nearly

circular to

elliptical. Subsequently,

some of the

cylinders began

to

_{expand, forming secondary planar}

phase.

The lecithin

_density

of the latter tended to be

_slightly

lower than that of the

_primary

planar phase.

The width of border and

_cylinders

had

_roughly

doubled at this

_point.

It

_tripled

in

_places

before the

_growth

of

_cylinders

in

_length

and width ceased

_{completely. Secondary}

planar phase

continued to

_develop

and

_occasionally

the border of the

_{planar phase}

and the

cylinders

became circular

_{again. Finally,}

the observations were discontinued at a lecithin

density

of ca. 20 vol. % in the

_{planar phase.}

The lecithin

_density

was evaluated on the basis of a few

_{representative}

measurements of its

the

_{planar phase.}

Since the decrease of

the

_density

with time varied

_considerably

_{among and} within

samples,

we

simultaneously

watched the structural features and the halo effect of the curved

_regions

and related them to the measured densities. Fluctuations in the curved

_regions

were used for the same _purpose,

_being

visible at lecithin densities of 40 vol.% and less. The

density

in the middle of the curved

_region

was taken to be that of the

_{adjacent planar phase}

multiplied by

the ratio of the

_principal

axes of the

_{(nearly) elliptical cylinder. Quantitative}

data for the

_{planar phase}

are listed in table 1.

Table 1. - The lecithin

_density

in the

_{planar phase}

as a

_function

_{of swelling}

time

_for

the

_typical

sample

thickness

_of

20 um. The values at

20 °C,

i. e. room

_temperature,

are taken

_from

reference [13].

A and B

_simply

mark

_swelling

_{times ;}

C and D denote the

_inception

_of

the

growth

o f

secondary

planar phase

and the cessation

_{of cylinder}

_growth,

_{respectively ;}

E marks a

concentration.

3. Results.

3.1 PHASE SEPARATION. -

Cooling by

a few °C or more after

_swelling

at elevated

temperatures

gave rise to deformations of the

_multilayer

_system

which were in

_general

restricted to the curved

regions,

unless the

_temperature

was lowered very

much,

e.g.

by

several tens of

_{degrees. They}

were identical or similar in the

semicylindrical

border of the

planar phase

and in the

_myelin

_cylinders,

in accordance with the fact that

_cylinders

and

_planar

phase always

remained connected. All the structural

_changes

indicate a

_separation,

to be called «

_{phase separation}

_», of the

_multilayer

_system

into domains of

_higher

and lower lecithin

density. They

followed the

_temperature

without noticeable time

_lag

and were reversible within very wide

limits,

again

without

_delay.

_However,

_{complete healing}

often

_required

a

temperature

up to 5 ° C

_higher

than that of

_swelling.

If a

_sample

was not

_reheated,

the

deformations vanished

_by

themselves in a

_day

or two,

provided

the

temperature

had not been lowered

_excessively.

The fluctuations seen in the curved

_regions

were

usually

not much

affected

_{by cooling.}

The deformation

_patterns

observed after

_{cooling dependend}

on the lecithin

_density

and the

temperature

change.

The effects of

_cooling

on the

_{myelin cylinders}

in a

_sample

of

_{high density}

are shown in

figures

1 to 3. After

_swelling

for a

_day

at 55 ° C down to a

_density

of ca. 45 vol.% in the

_{planar phase}

and ca. 40 vol. % in the center

_plane

of the

_cylinders,

the

_temperature

was

lowered to 45 ° C. In all three

_pictures

one sees

_{bright stripes running through}

the

_cylinders

at

oblique angles. They

are faint in

figure

1 taken at 51

_{° C,}

but become wider and

_brighter

with

decreasing

temperature

as

_displayed

for 48 and 45 ° C

_{by figures}

2 and 3. The

_stripes

are

Fig.

1.

_{- Myelin}

_cylinders

in a

_{sample being}

cooled after

swelling

at 55 °

C, viewed in

_phase

contrast

microscopy.

The three curved

cylinders span

the

_sample

cell from _top to bottom and are seen in a

longitudinal

section, the thin white center lines

being

their water cores. Lecithin

_density

in

_{planar phase}

(not shown)

ca. 45 vol. %,

_sample

thickness ca. 20 _um. The bar represents 20 lim. The instantaneous

temperature is 51 ° C.

Fig.

3. - Same

_experiment

as in

figure

1, but the instantaneous _temperature is 45 ° C.

surroundings.

Since the

_bright

lines in the middle of the

_cylinders

_represent

water cores

_[13,

14],

the

_stripes

may be

regarded

as domains of low lecithin

density. Changing

the

focus,

we

had the

_impression

that

_they

were ribbons of some width in the direction of

viewing.

Their

surroundings

look as before

_{cooling, including}

the fluctuations. A

_possible

_arrangement

of the

bilayers

in and outside a

_stripe

is sketched in

_figure

4. As a

_{cylinder comprises}

about 1 000

bilayers,

the few dark strokes

_subdividing

the

_{bright stripes}

cannot be

_single

membranes.

Fig.

4. - Sketch of

presumable

arrangement of membranes in an

_{oblique stripe.}

Note the increased membrane

spacing

within the

stripe.

Deformation

_patterns

such as those of

_figures

1 to 3 were seen in many

samples,

both in

cylinders

and

border,

and used to look

_equal

over

_large

areas. Fluctuations were of a

_speckled

type

at these

_high

lecithin densities.

_They

were

_barely

visible in the

_example

shown but well

developed

or not discernible in others of

_quite

similar

_composition.

Phase

_separation

was

found up to a

_density

of ca. 55 vol.% where curved and

planar regions

differed little in

concentration and

_temperature

decreases of a few 10 ° C were needed to

_produce

defor-mations.

_However,

the

_strong

halo effect

_typical

_{of these very}

_high

densities could have

The second series of

_pictures

exhibits a small

_part

of the

semicylindrical

border of a

_sample

whose lecithin

_density

was ca. 40 vol.% in the

planar phase. Swelling

had taken

_place

at 65 ° C

and the

_cooling

rate was 1 ° C

min-1.

Figures

5 to 7 show the

_sample

at

_roughly

_55,

40 and 25

_{° C,}

_{respectively.}

The short white strokes

_parallel

to the border

occurring

in

_figure

5 are

probably

water

_pockets.

_{They join only rarely}

to form structures

_resembling

the

_oblique

white

stripes

seen in the first series. The border widens

_considerably

as the

_temperature

continues to

be lowered. Its dark areas in

_figures

6 and 7 are domains of

_{relatively high}

lecithin

density.

They

are

largely parallel

to the direction of the border and

_multiply

connected

_by

short dark

lines. The lines

represent

bundles of membranes of variable thickness.

_Especially

in

_figure

7 some of them seem to be

_{single bilayers}

since

_they

have the same minimal

strength.

Also,

they

make contact

_angles

of 70° with the dense domains as is

typical

of

single

_{egg lecithin}

membranes

_adhering

to

_stacks,

i.e. bundles of ten or more

_{bilayers [11]. Being}

in

_equilibrium

with

_single

membranes,

the dense domains of the

_multilayer

system

must be

_subjected

to

adhesion.

Fig.

5.

_{Fig. 6.}

Fig.

5. -

Semicylindrical

border of

planar phase.

The

_multilayer

_system is on lower

_right,

free water

(and

an accidental

_{single membrane)}

on upper left. The

sample

is

being

cooled after

_swelling

at 65 ° C. The

sample

thickness is 18 um, the lecithin

density

in the

planar phase

ca. 40 vol.%. The bar _represents 20 jjbm. The instantaneous temperature is ca. 55 ° C.

Fig.

6. - Same

experiment

as in

figure

5. The instantaneous _temperature is ca. 40 ° C.

Many

of the lines in

figure

7

_display

a contact

_rounding

where

_they

_merge with a dark area.

Although

barely resolved,

the

_rounding

will be used below to evaluate the lateral tension.

Figure

7 also exhibits structural

_changes

in the

_{planar phase}

which started to be visible at

30 ° C. Deformations were in

_general

not

_fully

reversible after

_temperature

_drops

of this size. Fluctuations

_looking

like

_speckles

were well visible before

_cooling

and in the state of

_figure

5. The

_single

membranes and thin bundles seen at lower

_temperatures

also fluctuated

_strongly.

The fluctuations seemed to be reduced after

_cooling

in the dense

_domains,

but this may have

been an

_optical

illusion.

The next two

_photographs

show the border

_region

of a

_sample

swollen at 55 ° C. The lecithin

density

was between 35 and 40 vol.% in the

_{planar phase}

and 20 % less than that in the center

plane

of the border before

_cooling.

The initial state is seen in

figure

8,

while

figure

9

displays

the same section after

lowering

the

_temperature

to 45 ° C. After

cooling,

the border has lost its uniform appearance and

clearly

increased its width. One sees bundles of membranes

traversing

water-rich domains or water

_pockets

and

_merging

with each other or with broad domains of adhesion.

_Many

of the bundles seem to be bent where

_they

merge, but it is

hardly

possible

to

_quantify

the curvatures. Fluctuations

_appeared

_strong

before and after

_cooling.

The

_experiment

was not continued to lower

_{temperatures.}

Fig. 8.

Fig. 9.

Fig.

8. -

Semicylindrical

border of a

_sample

swollen to a lecithin

_density

of 35 to 40 vol.%.

_Multilayer

system is on lower

right,

free water on _upper left.

Sample

thickness 20 ktm. The bar represents 20 um.

The

_picture

shows the initial state after

_swelling

at 55 ° C.

Fig.

9. - Same

_experiment

as in

_figure

8. The

picture

shows the

_sample

after

_cooling

to 45 ° C.

The

_sample

shown in

_figure

10 was

_only

12-14 um thick. It had swelled for 3

days

at 50 ° C

down to a lecithin

_density

of 20-25 vol.% in the

_{planar phase.}

Then it was cooled to 40 ° C

where the

_picture

was taken. Phase

separation

is _very

_conspicuous

in much of the

_border,

but

the outermost and innermost

_parts

of the

_semicylinder

were little affected

_{by cooling}

in this

Fig.

10. -

Semicylindrical

border

_{of planar phase}

of

_a’sample

swollen at 50 0 C to a lecithin

_density

of 20

to 25 vol.%.

_Multilayer

_system is on the side of the bar

_representing

20 jim. The

sample

thickness is 12

to 14 _um. The

picture

shows the

_sample

after

being

cooled to 40 ° C.

display

well visible contact

_roundings

where

_they

merge with domains of adhesion. In

addition,

they

are

_{slightly bulged}

over their entire

_length

towards the water side of the border. This bias is also noticeable in

_figure

_{7 ;}

we will return to it in the discussion. The lowest lecithin densities in further

_experiments

were 18 vol.% in the

_{planar phase}

_and,

before

cooling,

8 vol.% in the curved

_regions.

At these low

densities,

little

_{cooling produced}

many

single

membranes,

but the overall appearance of the

samples

was rather

_irregular.

3.2 LATERAL TENSION. - Membranes in adhesion make rounded contact

_angles

because of their

_{bending rigidity.}

The

_gradual

turn of the visible contour of the membrane from some

other direction into that of the contact area is

_governed

_by

an

_angular

correlation

_length

which

also

_depends

on lateral tension. If

_optically

_resolvable,

this

_length

can be used to calculate the

tension. We

_repeat

here its derivation

_[9-11]

_J

as the method will be

_applied

not

_only

to

membranes but also to bundles.

_{Denoting by cp (s )}

the local

_angle

made

by

the free membrane and its

_asymptotic

direction and

_{by s}

the arc

_length,

we _may write for the energy per unit area

of membrane.

where K is the

_bending

_rigidity

_{and ai}

the lateral tension of the free membrane.

_(The

asymptotic

direction is

_{only approximately}

known because of other deformations of the

membrane.)

The _{formula presupposes}

_negligible

curvature in the direction of

_viewing

which

seems reasonable whenever the

_angular

correlation

_length

is much smaller than the

_sample

thickness. The

_{Euler-Lagrange}

calculus leads

_immediately

to the one-dimensional membrane

shape equation

The last

_{equation gives}

the lateral tension if the correlation

_length

and the

_{bending rigidity}

are

known. The solution of the differential

equation

(2)

has been

_plotted

elsewhere

[ 11 ].

It is

practically

an

exponential

for ~ 30° ,

but turns more

_sharply

into the direction of the contact

area at

_larger

_angles.

Equations (2)

should be

_equally

valid for bundles of membranes

_adhering

to other bundles

since both K and o- have to be

_{multiplied by}

tbe number of’membranes if more than one is

involved.

However,

the contact

_angles

are smaller with bundles than with

_single

membranes. The

_{bending rigidity of egg}

lecithin membranes has been measured

_by

several authors

[19-23],

the values

_ranging

from 0.5 to 2.3 x

10-12

_erg, all obtained at room

_temperature.

We use

here 1 x

10-12

erg, instead of 2 x

10-12

_erg as

_previously,

since we

_suspect

that the smaller

and

_larger

values are

_typical

of

_big

vesicles and thin

_tubes,

_{respectively [23].}

The decrease of

the

_rigidity

with

_temperature

is

_probably

less than a factor of two in the

_temperature

range of

our

experiments [24].

Angular

correlations

_lengths

almost down to

_1/4

_ktm were read or estimated from the

deformation

_patterns,

_{corresponding}

to an _{upper limit of almost 2} x

10-3

_dyn

cm-1

for the

lateral tensions. Tensions at this virtual limit of

_optical

_resolution,

which was attained

only

with

bundles,

were checked in a

_simple

but

_approximate

way

by

establishing

lower and upper bounds.

_Regarding

one

_quarter

of the width of the thinnest

bright stripes

seen in

figure

1,

namely 1/4

um, as an upper limit to the

angular

correlation

_length,

one finds from

(2)

crf > 1.6 x

10- 3 dyn em-1.

The other bound to the tension stems from the observation that fluctuations were not

_{markedly suppressed by cooling.}

For an

_estimate,

we need the tension crb of the membranes where

they

are bound to one another. The two tensions of a membrane are related

_by

where gi

is the contact

_angle

the free membrane makes with the stack of membranes to which it adheres. It is assumed here that the deformation of the stack

_by

the

_adhering

membrane is

negligible.

For the

_typical

contact

_{angle gi}

=

70° ,

one has Ub =

( 1 /3)

_ai. The elastic energy of

an undulation mode contains a and K in the combination

crb q 2+

K q 4

_{where q}

is the

component

of the wave vector

_parallel

to the membrane

[9].

(We disregard

the

_{perpendicular}

component

giving

rise to

_compression

and dilation of the

_stack.)

One may

expect

the first

term not to exceed the second for the shortest resolvable

_{optical wavelength À}

as

_long

as there

is no

_{significant suppression.}

This leads to

_{0-b «-- K (21r /,k )2}

_and,

after the

_plausible

substitution À = 2 _um, to _{Ub ’--} 1 x

10- 3 dyn cm-1

1

and Ur

3 x

10- 3 dyn cm-1. Evidently,

the bounds are not far

_apart

and conform with the estimate for or obtained from contact

rounding practically

within the

_{large experimental inaccuracy.}

Looking

at

_{samples exhibiting}

well-resolved contact

_roundings,

we found the

_angular

correlation

_length

not to be

_entirely

uniform over the

_sample.

This _may have been due to local variations in lateral tension and to

_{optical perturbations,}

_e.g. the halo effect characteristic of

phase

contrast

_microscopy.

These difficulties were

_{partially compensated by}

the

large

number of contact

_roundings

available for

_analysis

in each

_sample.

In a few cases we evaluated the

_angular

correlation

_length

at different

temperatures.

It

appeared

to increase

_slightly

on the _way to room

_temperature,

but the

_change

was not

significant

in view of the

_general

uncertainties. This has the remarkable

_implication

that the lateral

_tensions,

once

measurable,

did not

_change

much when the

_temperature

continued to

be lowered. The width of the curved

_regions

often increased

_dramatically

with

_temperature

as

shown

_{by figures}

5 to 7.

However,

the sum of the widths of the dense domains as

opposed

to

the total width of the border

_(or

half the

_cylinder)

seemed to

_change

little. It is

_particularly

large enough

to be

_assessable,

_indicating

a lecithin

density nearly equal

to that of the

_planar

phase.

A

_plot

of the lateral

_{tension ai}

versus the mean membrane

spacing

z in the

_planar

phase

is shown in

_figure

_11,

each

_{point representing}

a different

_sample.

We assumed a

constant membrane thickness of 3.6 nm

_[1,

_2]

when

_converting

volume densities into

_spacings.

The data _agree

_fairly

well with an inverse square

_{law, ai -}

( 1 /z )2,

_especially

if z is

replaced

by

z - 2 nm to account at least

_crudely

for the effect of rather

_impenetrable

_{hydration layers}

[4].

The

_{quadratic dependence}

will be seen to _{agree with} a theoretical model.

_However,

it

remains to be verified in further

_experiments

because of the

_{large experimental}

errors.

Fig. 11. Doubly logarithmic plot

of the lateral tension _o-f vs. the mean membrane

_spacing

in the

planar

phase.

The solid dots are the raw data,

increasing

the membrane thickness

_by

2 nm results in the

open dots.

4. Discussion.

4.1 INTRODUCTORY REMARKS ON INDUCED ADHESION. - The

_experiments

have shown that

cooling

causes deformations in the curved

_regions

of the

_multilayer

_system.

The structural

changes

are characterized

_by

the formation of domains of increased water content and

_by

the

generation

of lateral tensions.

_Assuming

a constant

_bilayer

thickness of 3.6 nm, we

_compute

from the measured lecithin densities mean

_spacings

that range from 3 to 17 nm in the

_planar

phase

and to 41 nm in the curved

_regions

before

_cooling.

Phase

_separation

in a

_multilayer

system

does not

_{necessarily imply}

induced adhesion. A coexistence of two lamellar

_phases

of different finite

_spacings

but

_equal

intermembrane net

_repulsion

is in

_principle

also

_possible

[2].

However,

we think that in the

_present

_{study phase separation}

has

_always

been due to

adhesion. The coexistence of

single

membranes with domains of

_high

lecithin

_density

_may be

regarded

as a direct

_proof

of adhesion.

_Ideally,

the

_system

should

_split

into a stack of

adjacent bilayers

leads to the formation of water

_{compartments.}

_Single

membranes

_(or

_very thin

_bundles)

were detected for mean membrane

_spacings

down to t = 4.5 nm in the

_planar

phase.

A coexistence of two lamellar

_phases

both with

_spacings

between 5 and 3 nm cannot be

excluded on the basis of our

_microscopic

observations.

_However,

it is _very

_unlikely

as there is no

_plausible

mechanism to

_explain

it.

Lateral tensions induce adhesion

_{by reducing}

fluctuations and thus steric

_repulsion,

as will

be discussed below in terms of undulations. This can result in a total interaction of the membranes which is still

repulsive

at small

spacings

but turns attractive at some distance. A

schematic

_plot

_{of the total interaction energy} per unit area, gtotal

( a’b z),

as a function of mean

spacing

at fixed _0-b is

_given

for such a case in

_figure

12. The energy minimum marks the

equilibrium

value of the mean

_spacing,

_zeq,

which

_according

to the

_experiments

and

theoretical

_predictions

is a

_decreasing

function of lateral tension.

_(Any

stable deviations of

f from

_zeq

would,

of course,

require

pressure differences for a

_complete

balance of

forces.)

For the reasons

given

above,

the function

_{f (of) plotted}

in

_figure

11 should be

_practically

identical to the function

_Zeq(Tr)

which

_by

use of

₍₃₎

can be converted into

_{zeq ( b ).}

Fig.

12. - Schematic

_plot

of the total interaction per unit area vs. the mean

_spacing

of two membranes

(or

a

_stack).

The full line _represents the interaction at some fixed lateral tension while the dashed line

depicts

the interaction in the absence of tension.

The minimum

_{equilibrium spacing}

of induced adhesion as observed in our

_experiments

is

3 nm. We may infer that the membrane

_separation

in egg lecithin

_multilayer

_systems

would

collapse

from

_infinity,

i.e. from the dissociated state to 3 nm or less if the membranes were

flat rather than

_fluctuating.

The obvious conclusion that electrostatic

_repulsion

has been

insignificant

in our

_samples

can be drawn with confidence

_only

when we discuss forces and

their

_dependences

on membrane

_spacing.

_Beforehand,

we _propose a model to

_explain

how the lateral tension comes about and

_why

it levels off when

_cooling

continuous after the

appearance of induced adhesion.

4.2 RESTRUCTURING OF MULTILAYER SYSTEMS AND SELF-LIMITATION OF LATERAL

TEN-SIONS. - Lecithin membranes shrink in area when the

_temperature

is lowered. This would lead to _very

_large

lateral tensions if the

_system

retained its structure. An estimate of those

theoretical _{tensions may be based} on the thermal area

_{expansity a}

= 2.4 x

10- 31°C

and the

stretching

elastic modulus _{K =} 140

_dyn

_cm-1,

both measured

_by

Kwok and Evans

_[25].

for

expansivities

of water

_(1.3

_{x 10-4/OC)}

_{and egg lecithin}

_(probably

also much less than a

_[26]),

one has for the tension

_{produced by cooling}

the

_{simple expression}

Inserting

the above numbers and Ar==-10"C for the

_change

in

temperature

yields

3.4

_dyn

cm-1

which is several orders of

_magnitude

above the actual tensions and at the limit of membrane

_rupture

_{[25]. Evidently,}

the

multilayer

system

is

capable

of

_{restructuring}

itself,

thus

_avoiding

the

_build-up

of

_high

lateral tensions.

To

_develop

a model of

_{restructuring,}

we

_begin

with the unrealistic

_example

of a

_long

_myelin

cylinder

that is disconnected from the

_{planar phase.}

When the membranes shrink in area each

of the unilamellar

_cylinders

can conserve its enclosed volume

_{by becoming}

shorter and wider

simultaneously.

For an estimate of the

_{geometric changes}

one _may choose a convenient cross

section of the

_cylinders,

_e.g. two semicircles of fixed radius r connected

_by

two

_straight

lines of

variable

_length

w.

_Omitting

the effect of

_cylinder

ends and

_starting

from

circles,

i.e.

w =

0,

one has the first-order

_equations

and

linking

the relative

_changes

of area

_A,

_length

_L,

and volume V. With à V = 0 one obtains

and

where the radius r and the width w are different for each membrane. Because of

DA /A

= a AT one finds with AT= - 10 ° C and the

_{quoted a}

the ratios

_{DA /A}

=

-

0.024,

ALIL

= -

0.05,

and

w/r

= 0.08. As a

rule,

the

widening

of the curved

regions

and

the

_shortening

of the

cylinders

have been a few times

larger

than the estimates

_suggest.

The

amplification

should be due to the connectedness of the

_cylinders

to the

_{planar phase,}

as is shown next.

Shrinkage

of the membranes in the

_{planar phase}

releases water which also has to be accommodated

_by

the

_{myelin cylinders}

and in the

_{semicylindrical}

border. The

capacity

of the curved

_regions

for additional water is small between the innermost but

_large

between the

outermost

_membranes,

_{being proportional}

to the individual radius r for the

_simple

_geometry

introduced above. The

_{planar phase}

can

_adjust

to the nonuniform

_{capacity by}

transverse

dilation and

_compression

of the

_multilayer

_system.

In the center

_plane

the membrane

_spacing

has to increase so that no water is

squeezed

out

_despite

membrane contraction. The dilation has to decrease

_linearly

on the way to

_top

and bottom of the

_sample

cell,

changing halfway

into

_compression

as the

_sample

thickness is fixed.

_Accordingly,

the increment Af of the mean

where D is the

_sample

thickness and 0 r D

_/2.

The increase of the membrane

_thickness,

at

fixed bulk

_density,

is omitted in this

_equation.

It results in a factor 2 on the

_right-hand

side when membranes and water

_layers

are

_equally

_thick,

but becomes less

_important

with more

water.

If the mean membrane

_spacing

in the

multilayer

system

is nonuniform and the

intermem-brane forces decrease with

_spacing,

the membranes will

_experience

forces. In mechanical

equilibrium

the net _{forces per unit} area must be balanced

_by

pressure differences between successive water

_{layers. Recalling}

that

_especially

after

_cooling

the membrane

_spacing

is

definitely larger

in the curved

_regions

than in the

_{planar phase}

and

_dismissing,

for the

moment, the

_possibility

of

_adhesion,

we

_expect

membrane interaction to be absent or reduced

in these

_regions.

As a _consequence, all or

_part

of the pressure difference

àp

= pin -Pout between the water

_layers

on the inner and outer sides of a membrane must be balanced

_by

other means in the curved

_regions.

The additional force can

_only

be

_brought

about

_by

a lateral

tension whose

_largest

_possible

value is

for

_cylindrical

curvature

_{1 /r.}

_Existing

also in the

_{planar phase,}

the tension _may reduce the intermembrane forces. We will retum

_immediately

to this

_{important point.}

The presence of adhesion

requires

minor modifications

which,

in a sense,

simplify

the

model.

_Large

_spacings

are now restricted to the small areas where a

_single

membrane runs

between water

_{compartments.}

Intermembrane forces

_certainly

vanish for these

_pieces,

permitting a

or, more

_{precisely, uf}

to assume its maximum value

_(5).

_However,

the free

membrane is

_likely

to form the characteristic contact

_{angle 03C8 ( =}

70° for egg

lecithin)

where it

hits the stack of outer

_membranes,

which is assumed to be like the

_{planar phase.}

It follows that the radius of

_cylindrical

curvature should be

_{r/cos 03C8}

instead of

_{1 /r.}

In

_compensation

of has to

satisfy

instead of

_(5).

Because of

_(3),

_however,

the tension _(Tb of the membrane where it is bound should be

_independent

of

cos gi

and,

like o, in

_{(5), obey}

Since the contour

_length

over which a membrane

stays

single

is so

_small,

_usually

less than the

sample

thickness,

the curvature of the free membrane need not be

_cylindrical.

This

_might

explain

the outward

_bulging

of

_single

membranes mentioned above.

_Spherical

instead of

cylindrical

curvature would lower the tension

_{(7) by}

half. We

_disregard

the effect of

_bulging

as

these curvatures are difficult to determine and do not seem to exceed

_{cos t/J Ir.}

We also refrain

from

_discussing

the

_adjustment

Ai of the mean

_spacing

in the curved

_regions

of adhesion

which will differ from that in the

_{planar phase}

because of a different balance of forces. Our model of

_{restructuring}

_predicts,

to first

order,

the

_justments

of the mean

spacings

in

the

_{planar phase}

and thus the lateral tensions to b

_proportional

to the decrease of

temperature.

However,

the

_initially

linear rise will level off when the tensions become

_large

enough

to reduce the steric

_repulsion

between membranes. The tensions cannot _{surpass the}

value at which the intermembrane force vanishes for the mean

_spacing

in the

_planar

_phase.

To be more

_{precise, they}

can reach this

point only

asymptotically

because it is the

repulsive

net

force that causes the tensions. The self-limitation of the lateral tension agrees with the

the curved

regions.

The

_apparent

absence of a

_dependence

of o- on r i.e. on the

position

of the

membrane in the

_multilayer

_system, seems to be _{another consequence of self-limitation.} 4.3 ESTIMATES AND SCALING LAWS. - For the further

_discussion,

membrane interactions need to be considered in some detail. The energy of direct interaction per unit area,

gdir, between a

_pair

of flat - membranes of uniform

_spacing

_may be divided into three

_parts

The first

contribution,

_{ghyd (Z),}

is the energy of the

hydration

force. It is

repulsive

and

drops

exponentially

with z, the characteristic

length being

ca. 0.2 nm

_{[1, 2].}

It will be

_neglected

in

our estimates or

simply replaced by

a hard core

_{hydration layer}

of 1 nm on each interface so

that the minimum membrane

_spacing

is 2 nm.

_(Marra

and Israelachvili

_[4]

found the minimum

of gdir to be near 2.4 nm for

_synthetic

lecithins in the fluid

_state.)

The second term of

_(8),

gvdw’

signifies

van der Waals interaction which is

_always

attractive between

_equal

membranes while the other direct interactions are

_repulsive.

It

_obeys

in the

_{half-space approximation}

for small z, H

being

the Hamaker constant. The _range of

spacing

where

₍₉₎

is a

_good

approximation

may be a few times the membrane thickness of ca.

3.6 nm because of the effect of the very

large

dielectric constant of water on the

zero-frequency

term

_[27].

To take account of the finite membrane thickness

_b,

_{equation (9)}

is often

replaced by

the ansatz

resulting

from the

_integration

of molecular interactions. This energy goes with

1 /z4

for

large

z, as it

_should,

but does not reflect the enhanced range of

(9).

Marra and Israelachvili

[4]

who measured for lecithins a

_particularly

small value of the Hamaker constant

_{(H =}

1.3 x

10- 14

_erg instead of the more

_typical

7 x

10- 14 erg)

also located the

_plane

of

_origin

of the van

der Waals interaction

₍₉₎

ca. 0.5 nm in front of the theoretical

_position

calculated for flat

interfaces,

which is

_equivalent

to an increase of the membrane thickness

_by

1 nm. To

improve

agreement

of theoretical and actual van der Waals interaction Attard et al.

_{[28] recently}

treated the

_region

of the

_polar

heads as a third

_layer

besides

_lipid

and water. The last term of

(8),

ges,

represents

electrostatic

_repulsion

between

_{charged bilayers.}

For fixed surface

_charge

density

it varies as In

_{(z/zo )}

at small

_{enough spacings (ideal}

_gas

_{approximation ;}

zo is a reference

_spacing)

and as

1 /z

for

_{larger spacings}

_[12,

_29].

Electrostatic

_repulsion

generally drops

with smaller powers of

1 /z

than van der Waals attraction and

_undulatory

repulsion (see below),

but

_eventually

it is cut off

_{by Debye-Hückel screening.}

As the

_polar

head of the lecithin molecule is zwitterionic in _pure water, any

appreciable

surface

_charge

would have to be due to ionic

_impurities

in the membrane or the water.

We discard the

_possibility

of electrostatic interaction in our

_samples

for the

_following

reasons. No

signs

of an electrostatic effect were noticed in the

_study

of

symmetric

adhesion

[ 11 ]

which covered a _{wider range of} tensions than do our data. In

particular,

the contact

angles

did not

_depend

on the

_origin

_{of the egg lecithin and} were the same when salt solution

was used instead

_{of pure}

water, as could be checked _up to 0.1 M NaCl.

_Moreover,

the age of

the

_sample

made little or no différence

_during

the one or two weeks needed for an

typically

near 70°. Just the same

_angle

is characteristic of the

_present

_experiments.

Its small

scatter, now as then

rarely

more than ±

_{10° ,}

is remarkable as the adhesion _energy should be

very sensitive to electric

_charges

on the membranes in view of the

_{large Debye length}

of twice

distilled water and the slow decrease of

_{ges at}

_spacings

below this

_length.

Without reference to

any other

work,

it may be

argued

that electrostatic forces too weak to

_prevent

induced

adhesion must be

_negligible

at the smallest

_spacings

around 5 nm, unless the

_{Debye length}

is very much smaller in the

sample

cells than the 0.1 to 1 _um of twice-distilled water.

The

_only

steric interaction of fluid membranes that is known to date arises from the mutual

hindrance of their undulations. For a stack of membranes

_regarded

as a smectic

_{liquid crystal,}

the energy of

undulatory

interaction per unit area of membrane was calculated to be

_[5]

Here k is Boltzmann’s constant, T the

temperature,

K the

bending rigidity

of the

membrane,

and z the mean

_spacing

between membranes.

_{Equation (11)}

is based on the

_hypothesis

that the direct interaction is that of a hard-core

_potential

at the membrane interfaces. The numerical factor in

_{equation (11)}

has been confirmed

_[17,

_18]

within

_experimental

accuracy

(ca.

20

_{%) by X-ray}

studies of

_multilayer

_systems

of _very flexible membranes

_{( K ~- kT).}

The direct interaction of membranes can in

_general

not be

_approximated

_by

a hard-core

potential.

So the

question

arises if the total interaction may be

expressed,

to a reasonable

approximation, by putting

z = _z and

adding

direct and

undulatory

interactions. Whether or

not

_{superposition}

is

_acceptable

as an

_{approximation}

_{depends mostly}

on the power

( 1/z )n

to which an attractive direct interaction is

_proportional

in the range of

spacings

swept

by

the undulations.

_Lipowsky

and Leibler

_[6]

have shown that

_{superposition}

must not be used

to calculate

_{equilibrium spacings}

whenever n >

_2,

i.e. when direct attraction falls off more

rapidly

than

_{undulatory repulsion. Assuming}

a direct interaction

_consisting

of

hydration

forces and the van der Waals

_{potential (10)}

with its

_{1 /z4}

tail,

and

_employing

a renormalization group

method,

they predicted

for a

_pair

of membranes a continuous

_unbinding

transition if the Hamaker constant is lowered to a critical value. The same transition would be

discontinuous,

the

_{spacing jumping}

from a finite value to

_infinity,

if

_{superposition}

were

applicable.

In order to decide whether to

_{expect spontaneous}

adhesion or

separation

and afterwards to

deal with the effect of lateral

tension,

we consider in the

following

the

simplified

case of « ideal

competition »

between direct and

_undulatory

interactions,

expressing

the former

solely by

the

_{half-space approximation (9)}

of van der Waals attraction. Since this attraction

varies as

1 /z 2,

we _may

_hope

its

_{superposition}

with

_undulatory

interaction to be

_(marginally)

correct.

_{Adopting superposition,}

we write for the total interaction in the absence of tension

Here

_Hc

is a critical Hamaker constant

_denoting

in the case of ideal

_competition

the limit

between total

_separation

_{(H Hc)}

and total

_collapse

_{(H>- Hc)}

of the membranes.

_Apart

from a numerical factor not too far from

_unity

we _may

_expect

for the

_stack,

because of

₍₉₎

and

(11),

It may be

argued

that

_He

should be somewhat smaller than

_{given by}

this

_equation

because z

distribution function of the

_{spacing. Inserting}

the measured

_rigidities

into

_{(13), calculating}

the values of

_Hc,

and

_comparing

them to the measured Hamaker constants, one ends up on the

borderline,

unable to

_predict

whether unstressed egg lecithin membranes should adhere to or

separate

from each other

_[11].

Likewise,

the criterion obtained

_{by Lipowsky}

and Leibler

[6]

in their renormalization _group calculation

_permits

no decision. This is even more so if the

theoretical uncertainties of the two methods are taken into consideration.

The effect of lateral tension on undulation forces is treated as elsewhere

[9,

10] by

means of an

_{independent-membrane-piece approximation.}

The

_approach

starts from the mean-square

undulation

_amplitude

_of,

say, a

_{hexagonal piece}

of

single

membrane of area

_S,

which is

_independent

of (T for or « K 7T

2/S.

The

_plaquette

size S is chosen such that the

undulations

_just

fill the interval between

_adjacent

membranes. For a stack of mean

spacing

z we _may

_put

as was done for the unstressed membrane between

rigid plates

[5].

Eliminating

(u2)

between

(14)

and

_{(15) yields}

The undulation _pressure could now be calculated _very

_directly,

but with a _poor numerical

factor,

by treating

each

_piece

of size S as an

_{independent particle}

of a one-dimensional ideal gas in an interval of width 2 z.

We

_prefer

to

_multiply

the undulation forces

_deriving

from

(11)

by

S(O)/S(u)

to obtain those forces in the _presence of tension. The effect of o- on the ratio becomes

_important

when

the

_exponent

in

₍₁₆₎

_{approaches unity. Any}

further increase of the

_exponent

_{by raising}

z2

or o- leads to a

_steep

_drop

of

_S(o)/S(o)

and a

corresponding

decrease of

undulatory

interaction. If the

_exponent

is

_exactly

one, i.e. for

one has

_{S(O)IS(o,) - 0.58.}

Note that this

_relationship

and the ratio

_{S(0) /S(a )}

at small

or in

general

do not contain the

_{bending rigidity. Incidentally,}

a

_relationship

like

₍₁₇₎

may also

be obtained

_{by equating}

guna of

(11)

and the total

_stretching

energy per unit area of a

freely

undulating

membrane

_{[9, 11 ],}

The

_{regular quadratic}

term

_containing

the

stretching

elastic modulus K s = 140

dyn cm-1

1

[25]

is

_negligible

at the tensions in

_question.

The numerical factor comes out 12 times

_larger

than

3/2

7T..

We are interested in adhesion induced

_by

lateral tension which takes

_place

for

o H «

_H,.

The associated

_{equilibrium spacing}

_may be derived from a balance of forces.

Following

the indicated

_procedure

and

_{replacing u by}

_Ob, we write the undulation force _per