Membrane-induced interactions between inclusions

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Membrane-induced interactions between inclusions

N. Dan, A. Berman, P. Pincus, S. Safran

To cite this version:

N. Dan, A. Berman, P. Pincus, S. Safran. Membrane-induced interactions between inclusions. Journal

de Physique II, EDP Sciences, 1994, 4 (10), pp.1713-1725. �10.1051/jp2:1994227�. �jpa-00248072�

Classification Fhv.ni,I Ab.in.act.<

87. lo 87.20

Membrane-induced interactions between inclusions

N. Dan

(')~

A. Berman

(2),

P. Pincu~

(3)

and S. A. Safran

(')

(')

Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot76100, Israel

j2) Department ^{of Chemical and Nuclear} Engineering,

University

of California. Santa Barbara CA 93106-5080, U-S-A-

f)

Departments of Materials and Physics, University of California, Santa Barbara CA 93106- 5080, U.S.A.

(Reieii,ed /5 Mai, /994, a(iepted 8./uly /994)

Abstract. We investigate theoretically the effect of embedded inclusions _on membrane

structure, and the corresponding membrane-induced interactions between inclusions. We find that

the membrane thickness, which _is perturbed from its equilibrium value by the

coupling

to the embedded inclusions, decays non-monotonically ^{with di,tance from} the inclusion boundary. As _a result, the membrane-induced interactions between inclusions vary non

monotonically

with

spacing.

Tfie penodicity of the perturbation profile, _as well _as the strength and range of the induced interactions, _are proportional to the ratio of the amphiphile bending modulus and compre,sibility.

In systems where the inclu~ions impose a thickness-matching constraint, the induced interaction~

are attractive. However, the pre~ence of _an energy barrier _{at a} finite spacing _may hinder aggregation. In ~ystem; where the inclusion~ impose _a specific contact-angle, the interaction energy is minimal _{at a} finite inclusion

spacing.

1. Introduction.

The _structure and

properties

of model and

biological

membranes have been

extensively

studied

jl-8].

Simple

model membranes _are

composed

of _a

single amphiphile [9]

or of mixtures of surfactants and co-surfactants

[10]. Biological

membranes, however, are far _more

complex

and contain various

inhomogeneities

in the form of

large

^and

rigid

inclusions, e.g., embedded

proteins

and cholesterols. The interactions and

phase

behavior of these embedded inclusions

play

_an essential role in the functional

specialization

of membranes

[1, 2]. Understanding

the

forces

acting

on embedded inclusions is necessary, ^not

only

for the

study

^of

biological

systems, ^{but for successful} utilization of artificial

bilayers.

A strong

coupling

between the

hydrophobic

core of the membrane and

hydrophobic regions

in the embedded inclusions leads to local

perturbation

of the

bilayer

structure (see

Fig. I).

Direct interactions between inclusions due to van der Waals and electrostatic forces _are well understood

[I1, 12]

electrostatic interactions _are

repulsive

between like inclusions,

decaying

exponentially

with the

spacing.

Van der Waals interactions _are attractive, their

strength

decreasing

(for

cylindrical objects)

with distance to the 3/2 _power. Inclusions embedded in

bilayers,

however, _are also

subject

to indirect forces which _are

membrane-induced,

and arise

(a)

uo

^"

/

~ ^---

r

equilibrium (unperturbed)

~~~

T

Ii,(

U r

r

a = tan 8

(c)

2~

Fig. I. Cylindrical inclusions embedded

in a

bilayer.

(a) For the stretch

boundary

condition, the inclusion _{impose~ a} thickness-matching constraint _on the bilayer. _ujj is the induced monolayer thickne~s at the inclusion

boundary.

(b) For the fixed slope boundary condition, the inclusion imposes _a finite

contact-angle with the monolayer, 0. jc) A top ^{view of} an array of cylindrical inclusion~ of radiu~

I,j. The average ~pacing between adjacent inclusions _is 2 L.

from the

inclusion-imposed perturbation

of the

bilayer

structure.

Long

_range, membrane- induced interactions _are due to

suppres~ion

of

long wavelength bilayer

fluctuations, and have been discussed

by

Goulian et al.

[7]. Short-range,

induced interactions _are due to the local

deformation of membrane _structure in the

vicinity

of the inclusion

[1-8].

Most models

discussing

the inclusion-induced local deformation of _a membrane considered

two contributions _to the

bilayer

_energy

[1-5]

_; ^molecular

expansion/compression,

which is

imposed by

the inclusion-induced

boundary

condition, and _an interfacial tension term, which

accounts for the

change

in overall surface _area. The membrane-induced interactions between

inclusions _were

found,

in these systems, to be

always

attractive, their

strength decreasing exponentially

with

spacing.

Recent studies

[6-8]

have shown, however, that

accounting

for the

bendifig

of the membrane at the inclusion

boundary significantly

affects the

perturbation profile [6, 8]

and membrane- induced interactions

[7, 8].

Goulian _et al.

[7]

examined both

long

and

short-range

membrane- induced interactions in systems ^{where the}

bilayer

_energy is dominated

by

the

bending

stiffness.

The

expansion/compression

and interfacial energy were assumed _to be

negligible.

In the limit of _zero

temperature,

^where

long

_range interactions _are

suppressed,

the

short-range

interaction

energy between conical inclusions _was found to scale

inver~ely

with the fourth power of the

spacing

between inclusions.

The spontaneous curvature

[13]

of _an

amphiphilic monolayer

characterizes the

tendency

of the

monolayer

head-tail interface _{to curve to or} from the water

phase,

and the

magnitude

of this

interface curvature. An

amphiphile

with _zero spontaneous curvature forms _a

monolayer

that has _no

preferential

curvature. Most

amphiphiles, however,

have _a finite spontaneous

curvature I

].

In _a

bilayer~

the tendencies of the _two

monolayers

to curve balance~ so that the

bilayer adopts

_a

locally

flat

configuration. Therefore,

the

energetic

contribution of the

amphiphile

spontanous curvature has

previously

been

neglected

in discussions of inclusions

embedded in

bilayers. Recently~

Dan et al.

[8]

have shown, for _a one-dimensional system, ^that the inclusion

decouples

the two

monolayers.

This

implies

that spontaneous curvature of the

amphiphile

dominates the

perturbation profile

and membrane-induced interactions between

inclusions. The

perturbation profile

_was found _to

oscillate, thereby allowing

the local

curvature of the

monolayer

to match the spontaneous curvature of the

amphiphile

and

greatly

reducing

the

energetic penalty

due to the inclusion-membrane

coupling.

The type ^of membrane-induced interactions between inclusions

(namely,

attractive _or

repulsive)

was

shown _to

depend

on the

magnitude

of the spontaneous curvature and the inclusion-induced

boundary

condition. In the _case of conical inclusions

[14]~

the interactions between

closely packed, large

inclusions _were found to scale

inversely

^{with their}

spacing.

The differences between the latter

prediction

^{and the Goulian} et al. [7]

analysis

is

primarily

due to a difference in

degrees

^{of freedom Goulian} et al. con~ider _a

pair

of conical inclusions that _are finite in size and _can tilt

freely

with respect to each other. The tilt

adjustment,

which

was not allowed in the Dan et al. model, lowers the

magnitude

of the local

bilayer

deformation and thus reduces the range of membrane-induced interactions.

In this paper we calculate the

shape

^of the

perturbation profile

and the membrane-induced interactions for _a two-dimensional array of

cylindrical

inclusions

(Fig. lc),

embedded in _a

bilayer composed

of

amphiphiles

with _zero spontaneous curvature. This

configuration,

unlike the one-dimensional model

employed by

Dan et al.

[8]

_or the two-inclusion model of Goulian

eta'.

[7],

enables ;imulation of _an efisemble of finite sized inclusions. The

energetic

contributions of

expansion/compressioi~ iiterfacial

_{tension and the}

bending

_energy are taken

into _account. We show that the

perturbation profile

oscillates _{as a} function of distance from the

inclusion

boundary.

The

periodicity

of the oscillations and the

decay length depend

on the ratio of

bending

modulus _to molecular

compressibility,

and _on the

spacing

and radius of the inclusions. The membrane-induced interaction~ between the embedded inclusions _are therefore

non-monotonic, and may be attractive _or

repulsive, depending

_on the type ^of

inclusion-bilayer coupling.

2. The model.

Consider _a membrane

composed

of _a

single

type ^of

amphiphile.

The spontaneous curvature of the

amphiphile [13]

is taken _to be _{zero. so} that there is _no

preferential

curvature of such _a

monolayer

at the interface between water and oil. The membrane contains _a uniform array of embedded~

rigid cylindrical

inclusions (see

Fig.

lc). The strong~

hydrophobic coupling

between the

hydrophobic

core of the

bilayer

and _a

hydrophobic region

in the inclusion leads to

a local deformation of the membrane structure. The inclusion-induced deformations _can be

divided into _two classes. The first type ^{is induced}

by regularly-shaped~ strongly hydrophobic

inclusions. The thicknes~ of the membrane in the

vicinity

of the inclusion is constrained _to match the inclusion dimensions,

thereby minimizing

_exposure of the

hydrophobic

^inclusion to

water

(Fig.

la). We _use the term stietified

couplifi,q

^for the~e

perturbations

to membrane

thickness. The second type ^of

perturbation applies

to

irregularly shaped

inclusions, or those which form _a

specific

bond with the membrane, _so that the contact

angle

at the inclusion

boundary

is finite

(Fig.

b) and

specified by

the local interaction. The term

sloped couplifi~g

is

used to describe these systems. ^We assume that the inclusion-induced deformation is

symmetrical.

The two

monolayers composing

the

bilayer

_are, then,

similarly perturbed,

and

thus,

equivalent.

The free _energy

(per

molecule) of _a

monolayer

with no ~pontaneous curvature

is

composed

of contributions from the

bending

and

stretching ji.e., expan~ion/compression)

deformations, and _can be written _as

[8, 13]

flu, ~) =fain, ~j+Kji)(v2n)2 ₍₁₎

where

fn(ii, II

is the free energy of _a flat

monolayer,

which is _a function of the _area per molecule,

I,

the interfacial tension, _y, and the local thickness of the

monolayer,

u(r ) jr is the radial coordinate from the center of the inclusion). The

unperturbed

membrane

structure is flat

[12],

determined

by

minimization of

j'n(u, 3)

and characterized

by

an

equilibrium

area per molecule,

2~,

and _an

equilibrium

thickness, _ii~.

Assuming incompres- sibility

of the

amphiphiles,

the volume per

amphiphile

molecule~ _{v, can} be used _to relate the

thickness and surface

density

of the

perturbed monolayer through

the constraint

d~ 2 l12

U _I iii ₊

(2)

dr

K is the

bending modulus,

_per molecule

[13],

of the

monolayer,

^{which is} related to the

bending

modulus per unit _area

by

a factor of the _area per molecule.

To capture ^the

physics

of _{an aiiav} of inclusions, _we consider _a

single

inclusion surrounded

by

its

Wigner-Seitz

cell I.e. the cell formed

by

the

perpendicular

to the bisector~ of the _vectors

between the inclusion and all its nearest

neighbors).

We idealize this cell

by

_a circle whose radius is

equal

to the average

spacing

between inclusions. Within this cell~ we solve for the

membrane thickness and the energy as a function of the cell size. The calculation is

simplified by

the

cylindrical

symmetry ^{of this idealized}

problem.

The local

monolayer

curvature is

given by V~u,

and is reduced

by

the circular symmetry to

V~u

=

~

(i

^{~" All}

^energies

^are

^given

r Jr %r

in units of

k~

T~ where

k~

^is the Boltzmann coefficient and T is the temperature.

For

simplicity,

we consider

only

small

perturbations

of the

monolayer thickness,

since

large

deformations _are inhibited

by

the finite

extensibility

of the

amphiphilic

tails. We define

A(I)

as the

change

in

ii(I)

relative _to the

unperturbed

thickness of the membrane,

~°~ ll(I') _ll~

A(I>

= (3)

ll~

and _we limit ourselves to the _case where J _« I. The

energetic penalty,

_per molecule, due to the

perturbation

of

monolayer

structure can be written

b

f

-

~~ ~°~

_~~ ~

~(

₊

K(2~ )(V~u

_)~

=

Ed'+ Ku( (V2~)~ j4)

~

~~-

~ ~

-w

where B

m

~$

_[%~

_f~~/%i~]

~ ~

defines the lateral

compressibility,

or _«

~tretching

modulus _»,

2 -~

of the

amphiphile

interface and K

=

K(i~)

is the

bending

modulus

(per

molecule) of the

unperturbed monolayer.

The membrane-induced interaction energy, per inclusion, is

equal

to the

perturbation

_energy of _a circular

region

of radius _I

= L ₊ i-n, where

i-ii is the radius of the inclusion and L defines half the

spacing

between

adjacent

inclusions boundaries (see

Fig- c).

The total free energy per

inclusion, F, relative to the free energy of the membrane in the

unperturbed,

flat, state is

L+in

_ll~

, , ~ ,

F

=

di ? _ri

[Ed-

+

Kup(V-A)-] j5)

U

>~>

In order to focus _on the effects of the membrane deformations _on the interactions between

embedded inclusions _we omit the effects of the direct

(electrostatic,

_van der

Waals)

interactions. These interactions _are most sensitive _to the

properties

of the

inclusions,

while the membrane mediated interactions _are modified

by changing

the

properties

of the

bilayer.

The

equilibrium monolayer perturbation profile

minimizes this membrane-induced interac-

tion energy. The

Euler-Lagrange equation corresponding

to the minimization of F is

V~A

=

~ A

=

~

~

j6)

Ku$ p~

4

ll~

K l14

where _p

m

defines the characteristic correlation

length

of the

perturbation.

In _a B

system ^where the

stretching

_energy is weak

compared

to the

bending modulus,

_p is

large.

Conversely,

_{a strong}

stretching

modulus and

relatively

low

bending

modulus results in _a small

value of _p. Since the

bending

modulus of most

amphiphilic monolayers

is of order

k~

T

[I I],

the value of _p is

primarily

set

by

the

magnitude

of the

stretching

modulus in _one component ^chainlike

amphiphile monolayers,

the

stretching

modulus is

large (roughly proportional

to the molecular

weight

of the tail

chain)

and _p is therefore small. In multi-

component systems ^that contain either solubilized oil _or co-surfactants, the

stretching penalty

can be relaxed

by

redistribution of the various components

[15],

and _p is therefore

large.

Cell membranes _{are a}

typical example

of the latter.

The solution of the

Euler-Lagrange equation

is _{a sum} of Bessel functions of the first and second kinds. Due _to the radial symmetry,

only

the _zero order Bessel functions _are

kept

and the scaled thickness is

given by

A(I)

=

Aj Jojz>

+

A~JO(z'j

_{+ A~}

Yo(=j

₊

A~ Yo(z'> (7j

,~j ^i/2

fi

_~~/2

where _z

=

'

r and z'

=

'

r.

P P

The _constants

A~ ^are determined

by

four

boundary

conditions (see

Appendix).

We consider both types ^of

inclusion-bilayer couplings

^{discussed above.} In the first _case,

stretching

constraints _set A(r

=

ro)

₌

Jo,

^where

Jo

defines the size of the

hydrophobic

inclusion

region.

In the other _case, the

slope

at the inclusion

boundary

is _{set, so} that ~~

is defined

by

the

~l' in

contact

angle

(see

Fig.

lb).

3. Results and discussion.

The

perturbation profile (Eq. (7))

is _set

by

the

competition

between _an

exponential

term, which determines the

decay length,

and _a

periodic

term which determines the

periodicity

of the

oscillations. Both the

decay length

and the

periodicity depend

on the ratio between the

monolayer bending

and

stretching

moduli, _p.

In

figure

2, the thickness

profile

of three _« stretched _» membranes is

plotted,

_{as a} function of distance from the inclusion

boundary.

In all three _cases, the

monolayer regains

the flat

layer

thickness

(defined by

A

=

0)

when the distance from the inclusion

boundary

is of order lo _p.

This

scaling

is due to the balance between the

bending

and

stretching energies

the

penalty

for

stretching

is

roughly proportional

to the _area of the

perturbed region,

while the

bending penalty

(which increases with curvature) is

inversely proportional

to the

decay length.

Therefore, in systems ^{where B dominates and} _p ^is small, the

decay length

^{is also} small, while in systems

where the

bending

modulus is

dominant,

_p and the

decay length

_are

large.

jj

"~'p"

^0.I

j[

~'p"

^I,0

~p"10.0

~ / _'

~/ _"'

,, ^{), ~/}

^l'

" I' _/ -'

, j _'

/~ ^' . ~ ,'

~w

0-5

_'

I "'

~/

i

" "m'

~fl

',

. 0 2 4 6

j,

,~

'

,~i₍

o

0

(r-r~)/p

Fig. 2. The scaled th(cknes~ change, ~ of the monolayers where the interaction with the inclu~ion obeys the ~tretch boundary condition, _as calculated from equation (7). _{p is} proportional to the ratio of the

bending

and ~tretching moduli. _i~j and _{iii> in} all three _ca~e~. The profiles were calculated for L W p, so that

increa~ing

the

spacing

did _not affect the profile. The inset show~ _an

enlargement

of the

region _near the first _minimum.

The oscillations in the

perturbation profile

_are

primarily

manifested in the first minimum, where the thickness of the

monolayer

decreases below the flat

layer

value. The

periodicity

of the oscillations is

proportional

to the

position

of the first minimum. As shown in the inset, the location of the

minimum,

and hence, the

periodicity,

_are

approximately proportional

to p.

Since the oscillations _are

~trongly damped,

their effect _on the

bilayer

thickness is

relatively

small, and may be difficult _{to measure}

directly.

However,

they significantly

affect the type ^and

strength

of the membrane-induced interactions between inclusions.

The

equilibrium

membrane-induced interaction energy, per inclusion, is

simply

related to the thickness

profile

as discussed in the

Appendix,

where we show that for the _case of the

stretching boundary condition,

2

wKu$

_d

F ~

=

~A(I)

^{I (V-~)}

⁽⁸⁾

v dr

,

~,~,

with _a similar

expression given

in the

Appendix

for the

slope boundary

condition.

In

figure

3 the interaction energy of stretched membranes, ^scaled

by

the value of the free energy at infinite

separation,

F (L _- cc

w F

~,

is shown _{as a} function of the reduced inclusion

spacing L/p.

We see that, _as

expected [1-5],

the lowest _energy state is one where the inclusions

are

aggregated IL

=

0)

and the _area of

perturbed bilayer

is minimal. At distances where L is of the order of 10 _{p or more,} the

perturbed

_regions of

adjacent

inclusions do _not

overlap

(see

3

-.-.p= ^0.I

!'j _p= 1,0

'

~p"10

0j

j

~

j

I

(

j

_~o

p~

^,

~~'

i

0

0 2 4 6 8

L/p

F(g. 3.-The interaction energy, F, normalized by the value of the interaction energy at infinite separation F~, _{m a} function of the reduced spacing L/p for the three _case, where the ~tretch boundary condition applie; _as in di,cussed _in

figure

2. The value of the energy at large inclusion

,pacings roughly

qcale~ _a~

I/p~.

Fig.

2) and F does _not vary with

spacing.

Thus,

F~

defines the

energetic penalty

for

inserting

_a

single

inclusion into the

bilayer.

The oscillations in the

bilayer profile

_are manifested in the presence of _an energy

peak,

at a

spacing

of

Lip,

and _a shallow

secondary

minimum at

L=3p.

In the limit of p ^w the

height

of the barrier is

negligible,

while for _{p «} the barrier

position

is _near the inclusion

boundary.

In these limits, therefore,

aggregation

is unhindered. However, in systems where _p is finite the location of the barrier and its

height

_may be sufficient _to prevent

aggregation,

so that _a metastable

ordering

characterized

by

the

secondary

minimum may

occur.

The membrane-induced interaction energy for the _case of the

sloped boundary

^{condition is}

plotted

in

figure

4.

Contrary

to the ~ystem ^{with stretch}

boundary

conditions, where

aggregation

is

preferable, aggregation

is

prohibited

in these systems ^the

slope boundary

condition _cannot be satisfied when the inclusions are

closely packed

unless the local curvature (and thus, the

interaction

energy)

is infinite. One

might

expect ^{the minimal} energy ^{state to} occur, then, _at

large spacings

where the

perturbed regions

do not

overlap.

However, _we see that the energy

minimum is obtained at finite

spacings.

This minimum (liLe the energy barrier in the systems with stretch

boundary

conditions) is due to the balance between the

bending

and

stretching energies.

Thus systems ^{where the inclusions attach to the membrane with} a characteristic

slope

are

expected

to order with _a finite

spacing proportional

to p. In the Goulian et al. model

[7]

which does not include the

stretching

_{energy, p}

- cc and this minimum is therefore absent.

A better

understanding

of the

relationship

between system parameters ^{and the induced}

interaction energy may be

gained by considering

characteristic limits. These _are defined

by

the relative

magnitudes

^of the three

length

scales in the ~ystem Jo, L, and the

perturbation decay

length,

_p.

~~~~P"

-...p= I,q

'

-p=10

~~

Uw

0 2 4 6 8

L/p

Fig.

4. The interaction energy, F, for the _case of the

slope boundary

condition, normalized

by

the value of the interaction energy ^at infinite separation

F~,

a~ a function of the reduced spacing L/p, _i-n and _a

= for all three _cases.

The first _case is _one where the

spacing

between inclusions is small

compared

to the

bilayer

correlation

length (L

« p

).

For the

stretching boundary

condition we find that the interaction energy ^varies with Au, p and _{i~j as}

F( ~-0) _~~(L(i~+~ ). ₍₉₎

P

~

2

This is _an attractive interaction between inclusions. We _see that, if the inclusion

size,

i~, is much

larger

than the

spacing,

the membrane-induced interaction energy between

inclusions increases

linearly

with L, in agreement ^{with the}

predictions

of Dan _et al.

[8, 14].

Increasing

the distance between inclusions increases,

linearly,

the _area of

perturbed membrane,

and thus, the

energetic penalty.

However, if the inclusions _are

«

point-like

_»

(i~

« L), the

perturbed

area and the interaction energy both increase with L~. The

strength

of the interactions scales _as

p~~,

_{so that small}

_changes

_{in the}

_bending

or

stretching

_energy

significantly

affect the interaction energy.

For the

slope boundary

condition, the free energy in the limit of small

spacings

between inclusions scales _as

F ~

-

0) ^~~

^"~

⁽¹⁰⁾

P L 2

ro

It is

interesting

to note that the

magnitude

of the energy is

independent

of _p, in _{contrast to} the

case of the

stretching boundary

condition. The interaction energy is

repulsive

and

diverges

when L

- 0 since the

slope boundary

condition _cannot be satisfied at close

packing.

In the limit that L _« r~, the interaction _energy

diverges

_as

io/L,

while in the limit of small inclusions

where

Lwr~,

the interaction

decays

_as

(io/L)~.

The softer power ^law interaction of

/L~

obtained

by

Goulian et al.

[7]

for two inclusions with

slope boundary

conditions is due to

tilting

of the

inclusions,

which _{was not} allowed in _our system. ^In an inclusion array such _as the

one we discuss,

tilting

would either break the symmetry ^{of the} system, or lead to overall

bending

of the

bilayer

(and the formation of

equilibrium vesicles).

We have therefore limited

our discussion _to the _case where there is _no

macroscopic

bend of the

bilayer

with _an array of inclusions.

The second limit of interest is _one where the inclusions _are

widely spaced

and the

regions

of

perturbed bilayer

do not

overlap.

At infinite

separation,

this energy defines, therefore, the minimal

inclusion-bilayer coupling

_energy below which the inclusions would be

expelled

from

the membrane. We

distinguish

between _{two cases; one} where the inclusion

size,

i-o, is small

compared

to the

bilayer

correlation

length,

_p, and _one where i~ ^w p. In the limit of

large spacing

and small inclusions

compared

with _p

(L

w p, and _i-o « pi, the interaction energy scales, _up to terms of order

p/L,

as

F~Fj(1+4e ^(sin~~+.

+.

(11)

~~

P where, for the

stretching boundary

condition

fi2

Fj~

^° j12)

~

10)~

p~ In-

P

and, for the

slope boundary

condition

1-o ^a 2

Fj (13)

P

When both the

spacing

between inclusions and the inclusion size _are

large compared

with

p(L

_{w p,} and _{i-u w}

p),

to

leading

order in

p/L

and

pfi.u.

F~F~(I

^{+e ~~}

(2(sin~~±cos~~

+.

)j ⁽¹⁴⁾

P P

where for the stretch

boundary

conditions, the minus

sign applies

and

Aji~

F~

_~~

⁽¹⁵⁾

P'

For the

slope boundary

condition, the

plus sign applies

^and

~ 2

F~ ~,^" II 6)

P

i hus, for

large spacings,

the membrane-induced interactions between inclusions

decays

^like

an

oscillating exponential

in all _cases. In the limit of infinite

separation,

the deformation

energy,

F~,

induced

by

an isolated inclusion is

given by Fj

or

F~;

^these

expressions

determine the minimal inclusion-membrane

coupling

_{energy necessary} to prevent ^inclusion

expulsion.

Note that, _as in the

opposite

limit

(Eqs. (9), (10),

where _p is

large),

the

dependence

of F _{on p} is weaker for systems ^with a

slope boundary

condition than those with _a stretch _one.

4~ Conclusions.

In this paper we

investigated

the

properties

of membranes

containing

embedded inclusions.

We find that the

bilayer perturbation profile

does not

decay monotonically

with distance from

the inclusion

boundary.

The

physical origin

of the oscillations in the

profile

is the

tendency

to

reduce the local curvature, and hence, the

bending

_energy.

The

inclusion-imposed perturbation

of the

bilayer

structure induces

indirect,

membrane-

mediated interactions between embedded inclusions. Previous studies _{on zero} spontaneous

curvature membranes

containing

embedded inclusions

[1-5] neglected

the

bending

stiffness

and

predicted monotonically

attractive interactions for membranes which stretch _to match the

inclusion size. More

recently [7]

it has been shown that the interaction energy between

inclusions for the case of the

slope boundary

^{condition is}

repulsive (for

values of the saddle-

splay bending

modulus,

K~0)

ⁱⁿ

bilayers

where the

bending

_energy dominates and the

stretching

_energy is

negligible.

Our

analysis

shows that the

magnitude

of the interactions and their range scale with _p, the ratio of

bending

to

stretching energies.

Therefore, small oscillations in the deformation

profile,

which may be undetected

experimentally,

could induce

significant

membrane induced

interactions between inclusions.

Accounting

for the

bending

_energy does not

change

^the

minimal energy ^state, which is

aggregation,

for the _case of the stretch

boundary

condition.

However, _{an energy} barrier appears, which may induce _a metastable _state where the inclusions

are orderd _{at a} finite

spacing.

For the _case of the

slope boundary

condition,

accounting

for both

stretching

and

bending

shows that the membrane-induced interaction energy is minimal _{at a} finite

spacing

of

approximately

2 _p.

Biological

membranes _are

composed

of

lipid

mixtures and contain _a

large

number of

impurities.

The inclusion

imposed

membrane deformation could therefore be relaxed

through demixing

in the

vicinity

of the inclusion.

Although

such _a process reduces the

stretching penalty,

there is _an

energetic

cost due to decreased

mixing

entropy ^and

possible

direct

interactions between the various membrane components

[15].

These contributions _can be

scaled, however, into effective

stretching

and

bending

moduli, _so that _our

analysis

still

applies.

Acknowledgments,

We would like _to thank J. Israelachvili for his

help

and S. Gruner and S. Keller for useful discussions.

Acknowledgment

is made _to the Israel

Academy

of Arts and Sciences 122/91/2,

the donors of the Petroleum Research fund, administered

by

the ACS, for support ^{of this}

research. PP

acknowledges

the support ^{of the National} Science Foundation, grant ^DMR-93-

0 II 99, and the Materials Research

Laboratory

at

UCSB, supported by

NSF grant DMR-91-

23048. AB is

grateful

for the support ^{of the National Science Foundation Graduate}

Fellowship

and of the Materials Research

Laboratory

at UCSB.

Appendix

A.

The

Euler-Lagrange equation

is

(Eq. j6))

V~A

=

~

~

(A.I)

P~

and is of 4th order. A is

given

therefore,

by (Eq.

(7))

~(r)

=

AjJjj(=) +A~JO(±')+ Ai Yo(=) +A~ Yo(=') jA.2)

j

Ill _j~~

where ₌

=

'

j. and -,

,/2

P p

~"

Four

boundary

conditions define ~. The first condition is that of symmetry at the

midpoint

between

adjacent

inclusions

l~~

₌ ^0.

^(A.3

11-

,

=L+,,,

The second

boundary

condition is defined

by

the

coupling

between the membrane and the

inclusion _:

~ii-

i~j)

₌

do jA.4a)

/

a jA.4b)

I , =,

for the stretch and

slope boundary

conditions,

respectively.

The _two

remaining boundary

conditions _are the natural _ones, and _are

given by

) _~~~

_Ii'

=

L ₊ r~~) =

o jA.5

and,

v2~

_jr

= i-ui o iA.6a

I ~~Aji

=

ioi

=

o jA.6b ₎

for the stretch and

slope boundary

conditions,

respectively.

The coefficients A~ ^can be written _as functions of the reduced parameters

ii

=

,/2

^{~ ~ °}

_(A.7

P

ii

=

,1

^{~ ~~'}

_(A.

8

pi =

,~

^~°

_(A.9)

P

pi

='~

"

(A.10)

For the stretch

boundary

conditions, these coefficients _are

Aj

₌

~°~~~~~~

(A.I la) 2lJjjiPii Yiifii Jiifii _{YoiP ii)}

A~

=

~°~~

~~~~

(A. I16>

2(JjjiPii Yiifii -Jiifii _YiiPii)

~

~°~'i~'i

(A.ilc)

~~2lJjifjiYniPii-JoiPiiYiifii)

~

~ ? (J

if

~

Yj

ill ^~il

ip

~

i

Y

iii

^{~~' ~}

The coefficients for the

slope boundary

condition _are

~

i~~~

p£VYjjfjj

2~~~(J]lpllY]lill~J]lfllYllp(1) ^{~~ ~~~~}

~i/~ _~ ~y

jt

A~=

~~~

' ~

(A.12b)

2

(Jj ~pii

Yj

iii] _Jj iii]

_Yj

ip~i)

~

i^3/~ _p «J

iii j

~

2~~(JilPil Yjliil -Jjlfil _YilPil) ^{~~ ~~~~}

~~ 2~?(J ip~illtii~~Jlil~i

_y

_ip~i) ^~~''~~~

Once the

perturbation profile

A is known, direct

integration

of

equation (5) yields

the membrane-induced interaction energy between inclusions. However, a

general expression

for

F can be obtained

by integrating 6f by

parts ^twice.

Defining

F

=

2

wiu~/u

_we find

L +,~, ^L +,,~

F ₌ di _r

jBA~

₊

Ku$ (V~A)?j

= di i-J

jB~

₊

Ku$ (V~A)j

₊

,) ,>

+

Ku~

r

) ^V~~))~

^~'°

^Ku(

_Ar

^{~(~~~ ~}

^{~ "~}

_(A.13)

1' ,n ^I _,n

The first term

(in

_square

brackets)

in this

expression

vanishes because A satisfies the Euler-

Lagrange equation.

The _{next term}

(in curly

brackets) vanishes

by

the choice of

boundary

conditions. For the

stretching boundary

condition, the last term

(in curly brackets)

vanishes at the

endpoint

L ₊

i-ii

by

the choice of

boundary

conditions, so that the free energy is

given by

F

= 2 wK

~

~A(r)

^r

~

V~A)~

(A.14)

di'

, =,,>

For the

slope boundary conditions,

similar considerations

yield

F

= 2 wK

~ _{r ~~~~ V~A)~}

(A.15)

v dr

, ~,~,

Thus,

the free energy per inclusion unit cell

(equivalent

to the interaction free _energy in _our model) is

simply

related to the thickness

profile

at the

boundary

_{I = i-u.}

References

[lJ Bloom M.. Evans E., Mourltsen O. G.. Q. Ret,. Biaphj.,I. 24 (199i) 293.

[2] Abney J. R., Owicki J. C.,

Progress

in Protein-Lipid Interactions, Watts and de Pont Eds. jEisevier, 19851.

[3] Marbelja S., Biophy.I At la 455 II 976) 1.

[4] Owicki J. C., Mcconnell H. M., Plot.. Na'l. At-ad- St-i- USA 76 II 979) 4750.

[5] Fattal D. R., Ben-Shaul A.,

Biophyi.,1.

6511993) 1795.

j6] Huang H. W., Biophj's. J so j19861 lo61.

j7] Goulian M.. Bruinsma R., Pincus P., Ew.oph_vs. Le" 22 (1993) 145.

j8] Dan N., Pincus P.. Safran S. A., Lan~gmiiir 9 (1993) 2768.

j9] See, for

example

^Kahlweit M.,

Strey

R., Busse G., J. Phj,s. Cfiem. 94 (1990) 3881.

[io] See, for example _; Kaler E. W., Herrington K. L., Miller D. D., Zasadzinski J. A., NATO ASI Seiies C 369 I991) 57i.

[))j Israelachvili J., Intermolecular and Surface Forces, 2nd Ed. (Academic Press, London, 1992).

[12] Safran S. A., Statistical

Thermodynamics

of Surfaces, Interfaces, and Membranes (Addison- Wesley, Reading. MA, 1994j.

[13] Helfrich W., Z. Natuifioic.h 28c ii 973) 693.

[14] Dan N.,

Unpublished.

[lsj Dan N., Safran S. A., Ew.oph>'s. Le't. 21 (1993) 975.

Jt)URN~L DE PHi S'QUE T 4 N'II) t)(Tt)HER >L'<'4