Bending stiffness of lipid bilayers. V. Comparison of two formulations

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Bending stiffness of lipid bilayers. V. Comparison of two formulations

Thoms Fischer

To cite this version:

Thoms Fischer. Bending stiffness of lipid bilayers. V. Comparison of two formulations. Journal de

Physique II, EDP Sciences, 1993, 3 (12), pp.1795-1805. �10.1051/jp2:1993230�. �jpa-00247939�

Classification Physics ^Abstiacts

87.20 62.20D

Bending stiffness of lipid bilayers. V. Comparison of two formulations

Thomas M. Fischer

Institut ffir

Physiologie,

Medizinische

Einrichtungen

der Rheinisch-Westffilischen Technischen

Hochschule Aachen, D-52057 Aachen, Germany

(Receii>ed 30 July J993, rec.eii>ed in final

form

23 August J993,

accepted

30 August J993)

Abstract. Two formulations of the intrinsic bending elasticity of

lipid bilayers

_are compared which _are based _on IA) the _mean and the Gaussian curvature and (B) the _mean and the deviatoric

curvature. It is shown that ii) formulation A contains

implicitly

^the elasticity ^{associated with} the

deviatoric curvature _even if the

topology

of the membrane does not change, iii) the deviatoric

elasticity

is

responsible

for the

change

in elastic energy that _comes with _a

change

in membrane topology, (iii) the elastic constant associated with the Gaussian _{curvature is} negative, and (iv) the

spontaneous-curvature ^model may be extended by _a curvature-independent term which becomes relevant when the membrane surface _area changes. Formulation B is then (for the special _case of

symmetric

bilayers) refined in that the _curvatures of the _two monolayers are accounted for

separately.

This leads _{to a} curvature

independent

term and _a correction of the elastic constants of the

bilayer

which both depend on the value of the monolayer-curvature. A continuum mechanical

model for the

bending

elasticity of _a

lipid monolayer

is

developed. By comparison

of this model _to

experimemal

data ii) the elastic constant associated with the deviatoric _curvature is estimated and iii) the decrease in experimentally determined elastic constants with

increasing lipid

unsaturation is traced back to the change in _area compressibility and _a trend in the spontaneous curvature of the

monolayers. Finally

it is shown that upon a change ⁱⁿ temperature ^the change ⁱⁿ elastic energy of _a

lipid

vesicle due _to the curvature-independent term is much

larger

than the concomitant change in elastic energy due _to the shape change ^but only a small fraction of the total change ^{in internal} energy.

1. Introduction.

this

_{is the last paper of}

a series.

Papers

number I and IV dealt with the distinction of

bilayer- couple bending

and

single-layer bending elasticity

of

lipid

vesicle and red cell membranes

[1, 2]. Bilayer-couple bending

arises from the 2D

isotropic elasticity

of the individual

layers

and

(heir

fixed distance.

Single-layer bending

denotes the intrinsic

bending

stiffness of each

layer.

tapers

number II and III

presented

two concepts ^for an altemative formulation of

single4ayer

bending elasticity.

In number II the consequences of _a separate account of the _curvatures of the two

monolayers

_were ^studied

[3].

^In number III it _was

suggested

to _use the deviatoric instead Qf the Gaussian _curvature

[4].

The _treatment of these _two concepts was

incomplete

in that _:

(a)

the second concept was not

compared

in _a

complete

fashion _to the traditional formulation based _on the Gaussian curvature, and

(b)

the second concept was not

incorporated

in the treatment of the first _one.

The present paper removes these flaws. Point

(a)

is treated in sections 2,I and 2.2. In section 2.3 the alternative and the traditional formulation are

applied

to

budding

of

giant lipid

vesicles in the

spontaneous-curvature

^{model. Point (b) is treated in section 3. I. In section 3.2} a

continuum mechanical model is

developed

that describes the

bending elasticity

of _a

monolayer. By comparison

^with

experimental

^data elastic constants in the alternative formulation _are estimated. In section 3.3 the consequences

arising

for the

bilayer~couple

model from _a

change

in membrane surface _{area are} considered.

2.

Consequences

from the use of the deviatoric instead of the Gaussian curvature.

2,I ZERO SPONTANEOUS CURVATURE. In this section _we consider the intrinsic

(here

called

single layer) bending

resistance of _a

bilayer.

We

adopt

the traditional strategy

by lumping together

the

properties

of the two

monolayers

into _a

conceptual single layer.

First _we consider the _case of _zero spontaneous ^{curvature. The} energy

density

_{e,j can} be written

[4, 3]

e~j=28,B~+2B~i~, ₍₁₎

where

B,

and

B~

are the elastic constants of the

bilayer

in

isotropic

and deviatoric

bending, respectively.

F denotes the local _{mean curvature.}

I,

the deviator, is defined _as follows

~

cj c~

~ ~~~

2 ^~

where _c-i and c~ are the local

principal

curvatures. The _curvatures c-j, c~, F, and

/. _are those of the neutral

plane

of the

bilayer.

Since any deviation from the unstrained

(plane) configuration

of the

bilayer

would increase its elastic energy both B~ and B~ must be

positive.

We compare

equation (I)

to the energy

density

_e~ introduced

by

Helfrich

[5]

for the _case of

zero spontaneous curvature :

e~ = 2 k~

?~

₊

i~

_{cj c~}

(3)

Here _k~ and

i~

denote the elastic constants associated with the _mean and the Gaussian

curvature.

Obviously

there is _no restriction _on the

sign

of the elastic _constants since the

product

ii c~ may ^assume

positive

_or

negative

values. Both notations

(Eqs.

(I) and

(3))

are identical

when

k~=B,+B~, ⁽⁴⁾

and

k~ ₌ 2

B~. (5)

From

equations (I)

to

(5)

we conclude _:

(a)

For _constant

topology

^of a

laterally

uniform

(for

an

explanation

_see next

item)

vesicle

membrane it is sufficient to consider either B _or /.. The Helfrich notation _uses fi, but _note that k~ ^as determined from

fitting

to

experiments

_covers both types ^of

bending

stiffness

(isotropic

and

deviatoric).

If, in

equation (3),

F _was

replaced by I, equation (4)

would ~till

apply.

(b)

_k~ as defined in

equation (3)

is

negative. Experimental

and theoretical

findings

as to the

sign

of k~ ^{have been} controversial

[6, 7].

If

alternatively

? had been chosen instead of

? in the first term of

equation (3)

the

corresponding

_k~ would have been

positive.

This shows that k~ ^is a

phenomenological

_parameter and not an elastic _constant in the strict _sense. Please note that _to omit the Gaussian _curvature term in energy

minimizing procedures

_k~ must be

laterally

uniform _; I-e- if _a

monolayer

consists of _{two or more}

lipid species

their concentration must in

general

be

laterally

uniform.

(c)

A

straightforward

determination of B~

(or k~)

ⁱⁿ a

bilayer

vesicle would

require

a

change

in its genus

[8],

_e-g- the fission of _a budd. Such _a

change is,

however, not

possible

in

equilibrium

due _to the

hydrophobic

interaction of the

hydrocarbon

chains with _water.

(d)

The

drop

in energy

by (4 ^ark~ predicted by equation (3)

and

(5)

_upon fission of _a budd

is due _to the relaxation of deviatoric

bending

moments in the neck

region.

This _was not

recognized

as

long

as k~ was

interpreted

as the elastic constant in

isotropic bending

alone

[9].

One could say that the _use of the Gaussian curvature has hidden the existence of _a deviatoric

elasticity although

it is

implicitly

contained in the Helfrich formulation.

2.2 NON-zERo SPONTANEOUS CURVATURE. Second _we treat the _case that the spontaneous

curvature assumes a non-zero value. This spontaneous curvature can also be

decomposed

in an

isotropic

and a deviatoric contribution. As

suggested [4]

_we retain the old name spontaneous

curvature for the

isotropic

contribution and call the deviatoric one _« spontaneous warp ».

Accounting

for both

quantities

the energy

density (stored

in

single layer bending)

of the

bilayer

reads

[4, 3]

_:

e~j =

2

B~(fi f~j)~

+ 2

B~(?

_{it )~}

,

(6)

where

f~j

denotes the spontaneous curvature in

single layer bending

^{and it the} spontaneous warp. Due _to the rotational

mobility

of

phospholipids,

the spontaneous warp is free to orient with respect to I. For this _reason ? and dare

by

definition

positive [4],

it does _not carry an

index because there is _no deviatoric contribution in

bilayer-couple bending

[4]. B~ ^and

B~

are

positive

for the _{same reason as} above.

Helfrich

[5]

accounted for the spontaneous curvature _as follows

Co ²

eH = 2 k~ C + k~ _{Cl 1'2}

(7)

2

where co was called spontaneous curvature.

Equation (7)

^{is the} basis of the so-called

Spontaneous-curvature ^model.

Equation (6) (for

the

special

_{case :} it

=

0)

and

equation (7)

_are

identical

_except for _{a term}

independent

of _curvatures when besides

equations (4)

and

(5)

the

following

holds _:

B~

co ~

2

f~j

~ ~

(8)

Adding

the constant term and

using

b

=

B~/B, equation (6)

(for it 0) _can be written

altematively

_:

~ ~ Co ²

~

_{~ ~~} ^{C'o 2} _~

~~~ _~ ^~ 2 ^{~ ~ ~} ~~ ~

~ 2

It is

confusing

that both

f~j

^and _co are called spontaneous curvature. It has been shown I that

f~j

corresponds

to what _one would

intuitively

call the spontaneous curvature. We conclude that c-o is _a

phenomenological

_parameter. For b

=

0

equation (8)

reduces _to

co/f~j

₌ ^2. With

increasing

b the ratio

co/f~j

^becomes

larger reflecting

the fact that

equation (7) implicitly

accounts for the elastic contribution of deviatoric

bending.

In section 3.2 the value of b will be

estimated.

When different

shapes

with the _same surface _area and volume _are

compared

with respect to their elastic energy the constant term in

equation (9) drops

out. This _means minimum energy

shapes predicted by equation (6)

_or

equation (7)

would be identical.

If, however,

the surface

area is

changing

the constant term in

equation (9)

must not be omitted.

2.3 APPLICATION TO MODEL CALCULATIONS. We consider

budding promoted by

_a

spontaneous curvature

(in single layer bending)

when the surface _area of an initial

sphere (radius R)

^{is increased} at constant volume and _a constant

positive

value for _c-o. Based _on the first _term of

equation (7)

the elastic energy decreases with

increasing

number of budds

[10].

From

equation (6),

however, one would expect ^{that above} a threshold value for b the energy would increase becau~e _more elastic energy has to be

paid

for the formation of the necks than is

released in the

spherical portions

of the membrane.

As _an

example

we pre~ent

approximate

calculations

following

a case studied in detail

by

Miao et al.

[10].

To _ease

comparison

with their data _we scale

lengths by

_co. In their

figure

6

they

start with _a

spherical

vesicle of radius 4.0?16.

They

find for

distinguished

values of the

surface _area the vesicle

shape

to consist of a number of

spherical

budds

(with

radius

2/co)

and the mother vesicle

(spherical

as

well),

all connected

by microscopic

necks.

We estimate the elastic energy at these

distinguished

values of surface _area. To this end _we

assume

(I)

F

=

0 in the neck

region.

Then

according

to

equation (7)

the contribution of the necks to the elastic energy is _zero. We further

neglect

^their surface _area and calculate _as if the mother vesicle and the budds _were closed

spheres.

Then

according

to

equation (7)

the energy stored in the budds is _{zero as} well.

Energy

is

only

stored in the mother vesicle. The values calculated

according

to the first term on the

right

hand side of

equation (7)

_are shown in the

second column of table I.

Comparison

with

figure

6 of Miao _et al.

[10]

shows that with

increasing

number of budds _our estimate becomes

increasingly larger

than their

precise

^values.

The values in the third and the fourth column of table I _are calculated

according

to

equation (9).

The values for _no budd _are

larger

than in the second column due _to the two additional _terms. But _note that the decrease in energy with

increasing

number of budds is much smaller for b

= 0.2 than in the second column. For b

= 0.3 the

energies

increase.

Table I.

Comparison of

elastic

enei-gies

calculated

appro.<imately (for

details _see te,rt)

according

to

equation (9)

and the

first

_term

of equation (7).

The

suifiace

_area in each line is

constant and chosen such that the uiother >,esicle _as well _as the budds _are

spherical.

The

»oluuie

of

all

shapes

considered is the _saute.

E/2

ark~

^E/2

ark~

E/2 ark~

Number of _{acc. to} first _{term acc. to}

Eq. (9)

_acc. to

Eq. (9)

of

Eq. (7)

b

=

0.2 b

=

0.3

0 4.09 6.65 8.02

3.42 6.52 8.14

2 2.76 6.37 8.25

3 2.10 6.21 8.35

Alternatively

we use

equation (6)

to calculate the elastic

energies. f~j

is calculated from c~ and b

according

to

equation (8).

In this formulation energy is stored in the budds and necks

as well. Under the _same

simplifying geometric assumptions

as above the first _{term on} the

right

hand side of

equation (6) gives

the contributions of the

spherical portions

whereas the second term that of the necks. The surface _area of _a neck is in

good approximation inversely

correlated

to the square of its _curvature.

According

to

experimental

evidence

[4]

we assume

it

=

0. The contribution of each neck to the elastic energy is then

approximately

constant

irrespective

of the size of the neck. If _we take this contribution _as 8

arB~

we get ^the same values for the total energy as in columns 3 and 4 of table I. The choice of 8 _was the constant of

proportionality corresponds

to a surface _area of the neck of 4

art(,

where r~ is half the neck diameter.

3.

Bending

_energy of _a

bilayer by accounting

for the two

monolayers separately.

3. I FORMULATION. In this section _we consider

again single layer bending

but this time _we account for the two

monolayers separately.

We treat the _case of _a

symmetric bilayer

where the spontaneous curvatures of the individual

monolayers

^{have the} same absolute value (different from

zero)

and have

opposite sign [I II-

The spontaneous curvature of the

bilayer

is _zero. The

spontaneous warp of the

bilayer,

_on the other

hand,

has the same value _as the spontaneous warp of the

monolayers.

This is due _to the rotational

mobility

of

phospholipids [4].

Besides the individual spontaneous curvatures the

analysis

accounts for the

slight

difference

in actual curvature between the two

monolayers.

From the _sum of the _two

monolayer

contributions the energy

density (in single layer bending)

of the

bilayer

is obtained

by expansion

in _a

Taylor

series and collection of _terms up ^{to a} power of 2 in 6. The

isotropic

contribution, i~

reads

[3]

:

i~

₌ ² B~

(i~

+ fi~ ₊ 2

16

_(P~

+

i~)

₊ ^~

6~(c(

₊

c()

₊ 6^~cj

c~(c(

_{+ cj c~ +}

cl )) (10)

4 2

where b,

I,

_cj, and c~ are as above the curvatures of the neutral

plane

of the

bilayer.

6 denotes the distance of the neutral

plane

of each

monolayer

from that of the

bilayer. /

_{denotes the}

spontaneous curvature of the _outer

monolayer. i~

and fi~ are

quantities describing

the

bilayer.

fi~ ^is

positive

since it is derived from the

properties

of the

single layer.

To

distinguish

the formulation in

equation (10)

from that in

equation (6)

_a tilde is added _to the

respective symbols.

/

_is

_by

_definition

a local

quantity

since it

depends

on the local

composition

of the

lipid monolayer.

Under _most circumstances the thermal energy

is, however,

sufficient to randomize

the distribution of different

lipid species

_on the surface of the

monolayer

so that

/

_{is then}

essentially

uniform.

The first _term in the

curly

brackets of

equation (10)

is _a consequence of the

opposite

spontaneous curvatures of the two

monolayers.

It relates _to the elastic energy stored in the

bilayer

^{when it is}

plane,

I-e- when _no extemal

bending

moments are

acting

_on it.

The second and the third term

give

the increase in energy ^{when the}

bilayer

deviates from

planarity

due to the action of extemal

bending

moments. The second term is the standard _one.

It

corresponds

to the first term _on the

right

hand side of

equation

(I).

The third term relates to the difference in the spontaneous curvatures and the difference in

the actual curvatures of the two

monolayers [3].

This term becomes relevant when

216

_is

_comparable

_to

_{unity. Using}

_{the estimates of}

I _presented

_{in section 3.2} and

taking

6 ₌ 1nm

II

_{we can}

appreciate

that for the

lipids

listed in table II 0 _w

216

~ l/3.

Table II. Calculation

jaccoiding

to

Eq. (25)) oj

the spontaneous

uionolayer-curvatm.e

/,fi.oni

the elastic constafit k~ in

single layer bending [16],

the

isotropic,modulus

K

[16],

and the thick.~Jess d

of

the

bilayei [32] for four different

_pure

lipids (DMPC

=

diniyiistoyl- phosphatidj~fcho!ire,

^SOPC ₌

steaioyf-ofeoyf-phasphatidyfcholine,

DGDG

=

digalactosyl- diacylglyceiol,

DAPC

=

diaiachidonyl-phosphatidylcholine)

and _a

(I/I) lipid

uii~<tune w~ith Cholesterol

(= Chol)

finder the

assumption

f ₌ ⁰

for

DMPC. C

= C denotes the number"

of

carbon-carbon double-bonds _per

lipid

molecule.

~~~~~

~

"

~

~

)

^~

f

^~~ ~C

ill

erg

dyn/cm

A ~

~2

~~

DA4PC 0 0.56 145 35.I 1.50

SOPC 0.90 190 39.8 1.44 57.4

SOPC/Chol 2.46 640 44.9 0.92 7.7

DGDG 6 0.44 160 38.7 0.88 6.2

DAPC 8 0.44 135 47.3 0.70 5.9

The last _{two terms} contain

higher

order contributions which become relevant at radii of

curvature

comparable

_to the thickness of the

bilayer [12].

The deviatoric contribution

i~

to the energy

density

stored in the

bilayer

has _as yet not been

given accounting

for the curvatures of the _two

monolayers individually.

Bv _an

analogous

calculation _as

presented by

^Fischer

[3]

for

isotropic bending

we obtain _:

~a ~

~

~a(

(~ ~f )~ ~f~ ~j^C~

~'

₊ ~~(C~ + ~~~ ~^~ ~l C2(C~ + ~l C2 ⁺

~~))

(~

0 does not carry a tilde since the

spontaneouq

_warps ⁱⁿ

equations (6)

and I I _are identical

[13].

fi~ ^is

again

a

positive quantity.

AII _terms in the

curly

^{brackets of}

equation

II except ^{for the} first _one become releN.ant

only

at small radii of curvature.

3.2 COMPARISON _To EXPERIMBNTALLY DETERMINED ELASTIC CONSTANTS.

Adding

equations (10)

and II) we obtain the total energy

density

stored in

single layer bending

:

?~j = 2

fi,(1

_{+ 2}

_f6

_{B~ +}

2(ii~

₊ 2 fi~

16 i~

_{+ 2 ii~}

(f~

+ O

(6~

c~

))

,

(12)

where _we have taken it

=

0

according

to

experimental

evidence

[41. Except

for the

higher

order _terms,

equations (12)

^and

(I)

~are identical when

/

=

0. This _means in this _case

B~ = B~ and B~ =

fi~.

If

equation

ii is used when f #

0,

_B~ and B~ ^{become effective elastic}

constants

[14]. Equation (12)

shows that these~are the smaller the _more

negati;,e

I. _Lipids

_may exist _so that the effective elastic constants and thus the

bending

_energy become

negative.

Then the

plane configuration

of the

bilayer

would not be the _one of lowest elastic energy Giant vesicles would

probably

not form

spontaneously

in this _case

[15].

Experiments [16]

with vesicles

prepared (via

spontaneous

formation)

from various

lipid species

showed that k~

[17]

decreases with

increasing degree

of unsaturation of the

hydrocarbon

chains. The 2D

isotropic

modulus which _was measured in the _same

experiments

decreased _as well. It _was, however, noted

[3, 16]

that besides the influence of K there is _an

additional influence _on the trend of _k~~ ~To trace it back _{on a} trend in

f

we rewrite

equation (12)

_:

i~j

=2(B,+h~+48,16)?~-?(fi~+?fi~f6)cjc~+?fi,(f~+O(6~c.~)). (13) Comparison

with

equation (3

shows that the

expression

within brackets in the first term _on the

right

hand side of

equation (13) corresponds

to k~. It is _our strategy to express

h,

and fi~

by

K. To this end _a

lipid monolayer

^{is modelled} as a thin

plate

of a

linearly

elastic 3D material with

in-plane

(~,

y) isotropy.

The material is assumed to resist volumetric

changes

with _a 3D

isotropic

modulus

k,

transverse shear strains with _a 3D shear modulus

G,

and in-

plane

shear strains with _a 3D shear modulus g. From the symmetry

properties

of the material it follows

ii 8]

that 5 elastic constants are needed for its

description.

The

respective

stress-strain relations _{are :}

«,, = aj u~, +

(ai

2 _a~)_u~, + a~ u~- (14)

«,, =

(aj

2

a~)

_u,~ + aj u,~ + a~ u--

(15)

«_ = a~ u~~ ⁺ a~ u,~ ⁺ a~ u--

(16)

«~. = a~ 2 _u~~

(17)

«~. = a~ 2 u,-

(18)

and «,~, = a~ 2 u,,

(19)

where u,~ and

«,~ (I, j

_{= ~r, y,}

=)

are the

components

^{of the strain and} stress tensor in linear

elasticity. Trivially

it follows a~ = G and a~ = g. Deviatoric

bending

leads _to

in-plane

shear strains which _are not uniform in thickness direction of the

plate [4].

The

respective

distribution is characterized

by

a

rotary-reflection

_symmetry of order 4 around _an axis normal to the

plate.

Such strains _are resisted

by

a

lipid monolayer

because

they change

the

shape

oi individual

lipid

molecules.

In-plane

shear strains which _are uniform normal to a

lipid monolayer

_are, of _course, not resisted

elastically

because the

lipids

_can flow past each other. We account for these

properties by restricting

ourselves in the

following

_to

boundary

conditions which _are characterized

by

_an absence of

in-plane

shear strain that is uniform in thickness direction.

To express aj, a~, and a~

by G,

_g, and k _we

require

that the stress-strain relations

(Eqs. (14- l9))

must be identical to those of _an

isotropic

3D material

(characterized by

_a shear modulus

~1 and _an

isotropic

modulus K) under the

following

two

boundary

conditions. The first _one is

unilateral

compression

_{(u== #} 0 all other u,~ =

0).

With k

= K and G

= ~1 ^we obtain

by

comparison

_{: a~}

=

k ~

G and a~ =

k ₊ ~

G. The second

boundary

condition is

isotropic

3 3

compression.

With k

= K we obtain aj =

k ₊ G + g. With these coefficients _we obtain for 3

the free energy

~f)

_per volume _:

f

=

~

(u~~ + u,~ +

u,,)~

₊ ^~

_(u,~

_{+ u,,, )~ +} 4

u-(u,-

_{u~, u,,, +}

2 6

+

12(u),

₊

u(, ))

+

~

(u,,

_{u~,, )?} + 4

u), (20)

2

K is calculated

subjecting

the

plate

to

isotropic in-plane

stress

(«,,

= «,~ # 0, all other

«~~ =

0). Using equations (14), (15),

and

(16)

_we obtain

(fi,,

_{+ u,,}

)

which is

equal

to the relative

change

in surface _area of the

plate.

From the definition of K

[161

_we obtain _:

K 18 kG

~~

(21)

§~3k+4G

'

where _ui is the thickness of the

plate

which in turn

corresponds

to the thickness of _a

lipid

JOURNAL DE PHYSIQUE il -T ~ N'12 DECEMBER 19W 6'

monolayer.

To obtain

fi,

_and

_fi~

we follow the standard treatment

[19]

and calculate the

bending

_{energy per} surface _area

(e~)

^{stored in} a thin

plate. Using equations (14-16,

19) _we obtain

e~ =

~

_{~ ~~}

~~

+

~~

+ ~ ~~~ ~~~ ^~

+ 4 ~~~

j

,

(22)

4 3 k ₊ 4 G _jx- jy- ³ j_j2

j~2

d<

dy

where ( denotes the

displacement

of the neutral fiber normal to the

plane.

In

principal

_axes the mixed derivative

vanishes,

the other derivati~es _are

replaced by (? f)

and

(I

~l).

The energy

density

^of a

monolayer

^is

given by equation

(6) with B~. B~, ^and

f,j replaced by fi,/2, h~/2,

and f.

Comparison

with

equation (22)

results in

fi~ 3 k~

I 23k+4G'~~ ⁽²³⁾

and

~~

=

gni~ (24)

From

equations (21)

and

(23

_we obtain fi~ ₌

Kni~/12. b~

as determined in

equation (24)

cannot be

expressed by

^K.

By analogy

to 3D

elasticity

_{we assume}

ii~

=

bB,,

where b is _{a constant.} The

expression

for k~ as obtained from

equation (13)

is then

k~ =

~~~ (l ₊ b ₊

id _{), (25)}

where _we have used d

= 2 _1?1

=

4 6, d

being

the

bilayer

^{thickness. Table} II shows the numbers calculated for the

expression

+ b ₊

Id

_from the

experimental

data of Evans and Rawicz

[16].

To determine the unknown parameter ^b we assume

f

=

0 for the

lipid

with saturated

hydrocarbon

chains

[20].

We get ^b ₌ ^{0.5 and values for} f which decrease with

increa~ing

unsaturation

(Tab.

II).

The trend in the values of f ^{is in}

qualitative

_agreement with the observed _structures of

lipids

in water upon

increasing

the

degree

of unsaturation

and/or

_upon addition of cholesterol

[20].

Independent

determinations of

f

_are available from the radius of ~tress-free

cylinders

of the

inverted

hexagonal phase [21].

For

phospholipids

^with two monounsaturated

hydrocarbon

chains this radius has been measured to be _on the order of 3 _nm

[21]. Taking

the _mean

curvature of the

cylinders

as an estimate for

f

_we obtain I/f

=

6 _nm, which is

comparable

to the value

given

in table II for

polyunsaturated hydrocarbon

chains. To

explain

this

discrepancy

we note that

(al

the methods _to determine

f

are different and

(b)

the

headgroup

size of the

lipids

used

[21, 16]

differs

markedly.

The main

uncertainty

ⁱⁿ our determination of b is the

large

variation in the value of k~

reported by

different laboratories. We used the values of Evans and Rawicz

(16]

because

their

study

is the

only

one in which K and _{L~ were} determined _at the _same

preparation

of vesicles. We conclude that it is

likely

that b is

appreciably

greater than _zero. This _means the

conclusions reached in section 2 have _a realistic foundation.

From data _on microemulsions

Farago (7]

concluded _:

k~/k~1-

2.

By comparison

of the coefficients of

equations (3)

and

(13)

we obtain _:

~~ ~ ~ _~

~~

(26)

~C I ₊ h ₊

fd

Figure

shows

i~/k~

as a function of the

monolayer

curvature for b

=

0.5 and d

=

4 _nm. The

strong

dependence

on

f

indicates that k~/k~ ^is not a well

behaving

indicator of the ratio of deviatoric and

isotropic elasticity.

The values of f in table II

correspond

_to values of

i~/k~

^{between 0.67 and 0.39.}

i~/k~

= -2 _as found

by Farago [7] corresponds

to

lll

=

2 _{nm a} value which is _not unreasonable for the surfactant used in the

study

of this author.

ic/kc

-1 -2 -8 8 S Z I

lli(nm)

-7 -6 -5 ,S -3 -Z ~l 2 3 S 5

id

-1

Fig,

I.

-k~/k~

(according to Eq. (26) for b 0.5 and d 4nm) _{as a} function of the monolayer

curvature

I.

3.3

CONSEQUENCES

FOR THE BILAYER-COUPLE MODEL. -The so-called

bilayer-couple

model

[9]

accounts for

single layer bending by using equation (3)

and for

bilayer-couple bending by using

_a

boundary

condition _on the difference in surface _area of the _two leaflets.

However, when the surface _area of the membrane

changes,

the _constant term in

equation (13) predicts

a contribution _to the elastic energy not contained in

equation (3).

In _contrast to the.

spontaneous-curvature model the _constant term in

equation (13)

has _a

straightforward physical interpretation

_: it _accounts for the energy stored in the balance of

bending

moments of the _two

monolayers

with

opposite

spontaneous curvature.

To get an idea _on the

magnitude

of the increase in this energy

during

a thermal increase in surface _{area we} compare it _to

a

typical change

in elastic energy

occurring

ⁱⁿ

parallel

due _to the

shape change,

_{on one}

hand,

and _to the energy calculated from the heat

capacity

of the membrane, _on the other.

Typical changes

in elastic energy are obtained

by comparing

the

experimental

data of Khs and Sackmann

[22],

in

particular

the

change

from

picture (3)

to

(5)

in their

figure

3 with

figures

4 and 5 of Seifert _et al.

[23].

We get ^1.6

ark~

^{for the} increment in elastic energy due to the

change

in curvature. With _an increase in surface _area of 60 ~cm2

[22]

and from

equation (13)

we obtain for the additional energy stored in the balance of

opposite

bending

moments 2 k~

f~

₆₀

~m~/(I

₊ b ₊

Id). _Using

_{the data in table II}

we get ^2.5 x

10~

_k~

and 5 _x

10~

_{k~ for SOPC and}

DAPC, respectively.

This is several orders of

magnitude

_greater

than the increase in elastic energy due to the

shape change.

It is,

however, only

_a small fraction of the

change

in total internal energy of the membrane. For _a

rough

estimate _{we use} the heat

capacity

of

glycerin,

_a temperature

change

of 3.5 °C

[22],

and _an average membrane surface

area of 2 800

~m2 [22]

and get an increase in energy of 10~ k~.

4.

Summary.

Two formulations of the elastic energy

(of

a

bilayer)

stored in

single-layer bending

were

compared

_: the traditional formulation which is based _on the _mean and the Gaussian _curvature and _an alternative formulation which is based _on the _mean and the deviatoric curvature. A

further distinction is that the traditional formulation does _{not account} for the

monolayers individually,

in contrast to the alternative formulation.

The

advantage

of the traditional formulation is that _as

long

_as the

topology

remains

unchanged

and in the absence of lateral

segregation

of

lipid species

it

requires

one stiffness parameter ^and one type ^of curvature less than the alternative formulation. Calculations _are therefore

simpler.

The

advantage

of the alternative formulation is that it allows _a

straightfor-

ward

interpretation

of its ~tiffness parameters ^and spontaneous curvatures. Their use facilitates

physical thinking.

An

example

would be _two budds connected

by

a

microscopic

neck

having essentially

the _same energy ^as the budds after fission.

To describe _a

symmetric bilayer

the alternative formulation _uses

h~

^and

B~,

^{the elastic}

constants for

isotropic

and,

respectively,

deviatoric

bending

and f, the spontaneous

monolayer-curvature.

The traditional formulation _uses the elastic _constants k~ and k~.

B~, B~ ^and f are

independent

of each other whereas both k~ and k~

depend

on

B,,

B~ ^and f.

At first

sight

it appears that the altemative formulation needs _one parameter more than the traditional formulation.

However,

for _a

complete description

of the elastic

properties

of a

symmetric bilayer

we have _to add in both formulations the 2D

isotropic

modulus

(K)

to account for

isotropic

tension and the distance

(2 6)

of the neutral surfaces of the

monolayers

to account for

bilayer-couple bending.

Since

ii~

can be

expressed by

K and 6 the total number of parameters ^{is the} same in both formulations.

Acknowledgments.

I thank

Dip]. Phys.

J.

Hektor,

RWTH Aachen, for

reading

the

manuscript

_; ^{Dr. U.} Seifert, KFA

JUlich, Germany,

for

helpful

discussions _; and Mr. P. Steffens for

typing.

References

[1] Fischer T. M., Biophy.I. J. 63 (1992) 1328.

[2] Fischer T. M., Biophys. J. 65 (1993) 687.

[31 ^{Fischer T.} M., J. Phys, if Fiance 2 (1992) 327.

[4] Fischer T. M., J. Phys, ii Fiance 2 (1992) 337.

[5] Helfrich W., Z. Natuifi(».sch. 28c (1973) 693.

[6] Harbich W., Ser~uss R. M, and Helfrich W., Z. Nat~lifiorsch. 33a (1978) 1013 Helfrich W., Liq. Cry.it. 5 (1989) 164?

Gompper G. and Zschokke S., R.

Lipowsky,

D. Richter and K. Kremer Eds., Springer

Proceedings

in

Physics

66, The structure and conformation of

amphiphilic

membranes

(Springer-Verlag,

Berlin, 1992).

[7]

Farago

B., R.

Lipowsky,

D. Richter and K. Kremer Eds.,

Springer Proceedings

in

Physics

66, The

structure and conformation of

amphiphilic

membranes

(Springer-Verlag,

Berlin, 1992).

[8] This is _a correction of what I suggested ^{earlier, reference} [4].

[9] Wortis M., Seifert U., Berndl K., Fourcade B., Miao L., Rao M. and Zia R. K. P., D. Beysens, N. Boccara and G. Forgacs Eds.,

Proceedings

of the workshop on

Dynamical

phenomena at

interfaces, surfaces and membranes (Les Houches, 1991) (Nova Science, Commack, 1991).

[10] Miao L., Fourcade B., Rao M., Wortis M. and Zia R. K. P.,

Phys.

Ret,. A 43 (1991) 6843.

II Curvatures _are defined _as positive in _a spherical vesicle for both the inner and the outer monolayer.

[12] These _{terms are a}

by-product

of _our

analysis.

For _a better _account of the energy density at large

curvatures see : Kozlov M. M, and Winterhalter M., J Phys. if Fiance 1 (1991) 1077.

[13] For the _{same reason} equation ill) is very similar _to the

corresponding expression

for isotropic

bending

in _an

asymmetric bilayer

(see Ref. [3],

Eq.

(7)).

[14] There is _no dependence of these effective _{constants on} the

shape

of the vesicle _as I

conjectured

earlier, reference [3].

[15] A way ^to produce bilayers ^from such

lipids

_may be to start from spontaneously ^{formed vesicles} and to reduce f by _a chemical modification of its

lipids

_{or a} change in the ionic environment while the membrane is under

isotropic

tension. Upon release of this tension such membranes would be unstable and bend until

higher

order _terms would limit the _curvature and/or until the _excess

surface _area would be used up and isotropic tension would stabilize again the vesicle shape.

This exotic _case has been treated

extensively

in reference [3].

[16] Evans E, and Rawicz W., Phys. Rev. Lett. 64 (1990) 2094.

[17] It has been shown that bilayer-couple bending does _not appreciably contribute to the total bending stiffness in thermally excited

shape

fluctuations [I]. For this _reason the values determined by Evans and Rawicz [16] correspond to k~.

[18] Sneddon I. N. and Berry D. S., S.

FlUgge

Ed., Handbuch der

Physik

VI, Elastizitfit und Plastizitht

(Springer-Verlag,

Berlin, 1958).

[19] Landau L. D, and Lifshitz E. M., Theory of Elasticity (Pergamon Press, London, 1959).

[20] Israelachvili J. N., Marcelja S, and Horn R. G., Q~lait. Ret>. Biophys, 13 (1980) 121.

[21] Rand R. P., Fuller N. L., Gruner S. M, and

Parsegian

V. A., Biochenii.iti=v 29 (1990) 76.

[22] K§s J. and Sackmann E., Biophy.I. J. 60 (1991) 825.

[23] Seifert U., Bemdl K. and

Lipowsky

R.,

Phys.

Ret,. A 44 (1991) 1182.