Interactions between membrane Inclusions on Fluctuating Membranes

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Interactions between membrane Inclusions on Fluctuating Membranes

Jeong-Man Park, T. Lubensky

To cite this version:

Jeong-Man Park, T. Lubensky. Interactions between membrane Inclusions on Fluctuating Membranes.

Journal de Physique I, EDP Sciences, 1996, 6 (9), pp.1217-1235. �10.1051/jp1:1996125�. �jpa-00247242�

Interactions between Membrane Inclusions

_on

Fluctuating

Membranes

Jeong-Man

Park

(*) ^{and T.C.} Lubensky

Department

of

Physics, University

of

Pennsylvania, Philadelphia,

PA 19104, USA

(Received

29 December 1995, revised 24

April

1996,

accepted

13

May1996)

PACS.87.22.Bt Membrane and subcellular

physics

and structure

PACS.82.65.Dp Thermodynamics

of surfaces and interfaces

PACS.34.20.-b Interatomic and intermolecular

potentials

and forces,

potential

_energy surfaces for collisions

Abstract. We model membrane

proteins

_as

anisotropic objects

characterized by

symmetric-

traceless tensors and determine trie

coupling

between these order-pararneters and membrane

curvature. We consider the interactions 1) between transmembrane

proteins

that respect _up-

down

(reflection)

symmetry of

bilayer

membranes and that have circular _or non-circular _cross- sectional

areas in the

tangent-plane

^of membranes,

2)

between transmembrane

proteins

that

break reflection symmetry ^{and have circular} or non-circular cross-sectional _areas, and

3)

be-

tween non-transmembrane

proteins. Using

_a field theoretic

approach,

_we find

non-entropic

1/R~

interactions between

reflection-symmetry-breaking

transmembrane

proteins

^{with circu-}

lar cross-sectional _area and entropic

1/R~

interactions between transmembrane proteins ^with circular cross-section that do not break

up-down

symmetry in agreement ^with previous cal-

culations. We also find anisotropic

1/R~

interactions between reflection-symmetry-conserving transmembrane

proteins

with non-circular cross-section, anisotropic

1/R~

interactions between

reflection-symmetry-breaking

transmembrane

proteins

with non-circular cross-section, and _non- entropic

1/R~ many-partiale

interactions among non-transmembrane

proteins.

For

large

R, these interactions _are

considerably langer

than Van der Waals interactions _or screened electro- static interactions and

might provide

the dominant force

inducing aggregation

of the membrane

proteins.

1. Introduction

Recently,

tue structure and

properties

^of model and

biological

membranes bave been studied

extensively- Biological

^membranes

^play

a central role _m botu tue structure and function of cells. Biomembranes divide

living

tissue into dilferent

compartments

or cells and act _as cell boundanes.

Tuey

determme tue nature of ail communication between tue inside and tue outside of cells. Tuis communication _can take

place

_ma tue actual _passage of ions _or molecules between two

compartments

or ma conformational

changes

induced in membrane

components.

Model

bilayer lipid

membranes in aqueous environments exuibit many of tue attributes of trie

biological

membranes. For

example,

these membranes _can form vesicles _{or more}

complex

structures that divide _space into

separate compartments,

^{which like cells} can fuse _or divide-

(*)

Author

for.correspondence (e-mail: jeongillubensky.physics.upenn.edu)

©

Les

Éditions

_de

_Physique

₁₉₉₆

However,

there _are many

properties

of biomembranes that cannot be mimicked

by lipid bilayers.

Energy-driven transport

of ions _across membranes and

receptor-mediated

events _are

only

_a few of trie

myriad

of membrane-associated functions that

lipid bilayers

_are

incapable

of

performing

on tueir _own- Sucu _{processes are} mediated

by proteins

tuat _are attacued to _or dissolved in

biological

membranes

[1-5].

Tuus in order to make

lipid bilayers

_more realistic models of

biomembranes,

it is necessary to introduce into tuem membrane inclusions sucu _as

proteins.

Membrane

proteins

_are classified _as

integral proteins

_or

peripueral proteins according

to uow

tigutly tuey

_are associated witu membranes.

Integral _proteins

_{are so}

tigutly

bound to membrane

lipids by uydropuobic

forces tuat

tuey

_can be freed

only

under

denaturing

conditions.

Peripueral proteins

associate witu _a membrane

by binding

at tue membrane

surface; tuey

_can be

non-destructively

dissociated from tue membrane

by relatively

mild

procedures.

Some

integral proteins,

known _as transmembrane

proteins,

_span tue

membrane,

wuereas otuers _are attacued to _a

specific

surface of _a membrane. For

brevity,

_we refer tue latter

proteins

_as non-transmembrane

proteins

to

distinguisu

tuese from transmembrane

proteins- However,

_no

proteins

_are ^known to be

completely

buried in _a membrane

[2j.

Interactions between membrane

proteins

is

expected

to be controlled

by lipid aflinity,

direct interactions sucu _as electrostatic and Van der Waals

interactions,

and indirect interactions mediated

by

tue membrane [6j- Tue latter interaction _anses from

tuermally-driven

undula- tions of tue membrane and is

analogous

to tue Casimir force between

conducting plates.

Since tue

degree

to wuicu fluctuations of _a membrane _are restricted

depends

_on distance between

membrane

proteins,

its free _{energy also}

depends

_on tuis

distance, decreasing

witu

decreasing separation-

Tuis

implies

_an attractive

force,

wuicu leads to _a

tendency

for membrane

proteins

to

aggregate.

Tuis indirect interaction between membrane inclusions _was first calculated

by Goulian, Bruinsma,

and Pincus

(îj-

^Before

presenting

^tue interaction models introduced in tuis

paper m

detail,

in Section 2 _we will review its

results, indicating

wuere

tuey

dilfer from and extend tuose of Goulian _et ai- In Section

3,

we introduce _a

puenomenological

model

(Model I)

for

protein

interactions. In tuis

model, proteins

_are cuaracterized

by symmetric-traceless

ten- sors

depending

on tue cross-section

suapes

of

proteins

_on tue

membrane,

and interactions

are described

by symmetry-allowed couplings

between tuese tensor

order-parameters

and tue curvature tensor of tue

fluctuating

membrane. In Section

4,

_we introduce anotuer

puenomeno- logical

model

(Model II),

in wuicu tuere is _an interaction between _a membrane

protein

and membrane

lipids

at its

perimeter.

At tue

penmeter, lipids

tend to

align

witu tue direction of tue

protein

at _a certain

angle depending

on wuetuer _or not

proteins

break tue

up-down

symmetry of tue

bilayer

membrane. Tue interaction _is described

by

tue fluctuation of tue normal _vector of tue membrane around tuis

preferred

direction. Tue interaction between _non-

transmembrane

proteins

_is described in Section 5. Non-transmembrane

proteins

_are

proteins

with

preferred

center-of-mass

positions

not at trie _center of trie

bilayer

membrane. We

expand

trie

potential

_{energy m} terms of trie deviation from this

preferred position

and include other

couphngs considering

trie

symmetry

to calculate tue interaction between non-transmembrane

proteins- Finally,

_a discussion _{is given} in Section 6.

2. Review and

Summary

2.1. REVIEW _{OF THE} PREvIous WORK. Trie interaction between membrane inclusions

witu circular cross-section _on tue membrane bas been calculated

by

Gouhan et ai.

[7j. Tuey

use tue Helfricu-Canuam Hamiltonian

[8, 9j,

7i0

_"

/ _d~UVj(~H~

+

kK), (2.1)

to describe fluctuations of tue

membrane,

wuere _~ and K _are tue

bending

and tue Gaussian

rigidities

^and H and K _are tue _mean and tue Gaussian curvatures,

respectively-

Tue surface tension is not taken into account since it

elfectively

vanisues

(10,11].

Wituin inclusions witu circular

cross-section,tue

constants _~ and K _are assumed to dilfer from tuose of tue

surrounding

membranes. For _some

inclusions,

sucu _as

proteins,

^{tue circular}

regions

are assumed _to be

rigid

witu _~

= -k ₌ oo. Tuis _case is tue

strong-couphng regime.

On tue otuer

uand,

for

regions

witu _excess concentrations

oilipids,

_~ and R wituin circular

regions

are assumed to bave values

close to those oi trie

surrounding

membrane. This _case is trie

perturbative regime.

In both tue

strong-coupling

and

perturbative

_regimes, ^reierence

I?l

^{finds tuat tuere is} an

entropic 1/R~

interaction,

^wuich is

proportional

to tue

temperature

^and to tue square oi tue _area oi tue circular

region. Also,

at low

temperatures,

there is _a

non-entropic 1/R~

interaction between

proteins

which varies with trie _square oi

angle

oi contact between membrane and

proteins.

Recently, Golestanian, Goulian,

and Kardar (12] extended trie calculation in reierence I?l ^{to trie} interaction between two rods embedded in _a

fluctdating

membrane.

They

find _an

anisotropic

1/R~

interaction between two rods-

2.2. SUMMARY OF THE PRESENT WORK. In ibis _{paper, we} introduce three models for

tue interaction between membrane inclusions sucu _as transmembra~ie

proteins

and non-trans- membrane

proteins. First,

we

present

Model I.

Here, proteins

are cuaracterized

by symmetric-

traceless tensor

order-parameters,

^{and trie}

couph~ig

between tuese

order-parameters

and _mem- brane curvature is determined

by symmetry

^and power

counting.

Ii

proteins

don't

respect up-down symmetry,

we allow for

couplings

that break

up-down symmetry. Otherwise,

we re-

quire up-down symmetry.

^For

proteins

with circular cross-sectional _{area on} trie membrane and

preserving up-down symmetry,

we find trie

leading distance-dependent

iree _energy,

where trie

coupling

constants are denoted in terms oi trie

bending

and trie Gaussian

rigidity dilferences, ô~(x)

and

ôk(x),

between

proteins

and

surrounding

membrane. This

corresponds

to tue

perturbative

_regime oi reierence

[7],

^{wuose results} we

reproduce.

For

proteins

witu circular cross-sectional _area,

up-down-symmetry-breaking couplings

do not affect tue lead-

ing

contribution to tue iree _energy, and tue

leading distance-dependent

iree energy remains

identical to

equation (2-2)-

We also calculate interactions between

proteins

witu non-circular cross-sectional _area. Wuen

up-down symmetry

is

conserved,

_we find _an interaction energy

~2

F =

-kBT

~ ~ ~

((q4Q4

+

q2Q()(q4Q4

+

q2QÎ

+

d2QÎ)

_cos

4(91

+

92)

64~ _~ R

+2(Q4Q4

+

Q2QÎ

~

à2QÎ)~2QÎ

COS~

2àl

C°S~

2à2

+

(~2QÎ

~

I~2QÎ)Q2QÎ~ (2.~)

wuere A is tue cross-sectional area oi

proteins, Q2

and

Q4

_are

magnitudes

oi 2nd-rank and 4tu-rank tensor order

parameters measuring

orientational anisotropy,

respectively,

and _{9~ are} tue

angles

oi tue directions oi

proteins

measured witu

respect

to tue

separation

vector between

proteins (See Fig. 1)-

^Tuis anisotropic

1/R~

interaction contains

anisotropic ^cos~ 291cos~ 292

interaction in addition to

anisotropic cos4(91

+

92)

interaction also round in tue recent inde-

pendent

^work

^by

Golestanian et ai.

[12].

^{However, wuen}

up-down symmetry

is

broken,

tuere is _an

anisotropic 1/R~

interaction:

~2 d2Q2

F =

~

cos2(91+ 92). (2A)

16~KR

Fig.

l. Proteins make

angles

_9~ measured with respect to the separation vector R. The distance between

proteins

R is taken to be much

larger

than

protein

_size.

Tuus tue

leading

term _m tue iree _energy ialls off witu

separation

_as

1/R~

ratuer tuan

1/R~.

Next,

_we introduce Model II in wuicu _we

impose

a certain

boundary

condition at tue

perime-

ter oi

proteins

witu tue circular cross-sectional _area- For

proteins

witu

up-down symmetry,

_we find

6

~2

F =

-kBTj. _{(2 5)}

~ R

Tuis iree _energy looks similar to tuat oi reierence I?l in tue

strong-coupling regime.

Wuen

proteins

break

up-down _symmetry,

_we find

"

~

~~~~Î~4

⁺

~~~~ °Î

₊

_O(

~ ~~ ^'

wuere _a~ is tue _contact

angle

between tue direction oi1-tu

protein

and tue unit normal oi tue membrane

(See Fig. 2).

In tue limit T _~

0,

tuis iree _energy becomes tue result in reference [7j for tue

low-temperature

_regime.

Finally,

_we introduce _a

ueigut-displacement

model in wuicu

protein positions

normal to tue membrane _{can vary.} In tuis

model,

_we find turee- and

four-body

interactions _m addition _to

a

two-body

interaction. Tuese turee- and

four-body

interactions also fall off _as

1/R~

and _are tue _same order of

magnitude

as

two-body

interaction.

Consequently, by introducing

turee models to descnbe tue interaction between membrane inclusions sucu _as

proteins,

we recover all tue results

m reference

[7j. Furtuermore,

_we obtain

anisotropic

interactions between

proteins

witu non-circular cross-sectional _area-

Also,

_{we ex-}

tend tue calculation to tue

up-down symmetry breaking proteins

witu non-circular cross-section and find

anisotropic 1/R~

interaction between tuem.

Moreover, using

_a

ueigut-displacement

model,

we find turee- and

four-body 1/R~

interaction in addition _to

two-body 1/R~

interaction.

OE' tli

Fig.

2- Contact

angle

_a~ is measured between the direction of1-th

protein

and the unit normal of the membrane at

protein's perimeter.

3. Mortel 1

For _a fluid membrane free of membrane

proteins,

tue energy of membrane conformations _can be described

by

trie Helfrich-Canuam Hamiltonian

[8,9j,

7io

=

/ _d~u@(~H~

+

KK), (3.1)

2

expressed

in terms of trie local _mean and Gaussian curvatures. We will work at

length

scales

large compared

with trie membrane thickness but small

compared

with trie membrane's _per- sistence

lengtu. Tuus,

_{we can}

parameterize

tue membrane _m trie

Monge

_gauge ^R ₌

ix, h(x))

wuere _x

=

(ui, u2).

In terms of R and tue unit normal vector of tue membrane

N,

tue metric

tensor gap is

given by ôaR ôpR

and tue curvature tensor

Kap

is

given by

N

DaDpR,

wuere

Da

is tue covanant denvative

along

_ua direction on tue membrane. In tue

Monge

_gauge,

H =

g°~Kap

=

i7~h

₊

O(h~), ^(3.2)

K = det

g"~Kp~

₌

^i7~hi7~h ôaôphôaôph

+

O(h~). (3-3)

Wuen tue

topology

^{of tue membrane is}

^fixed,

^{tue Gaussian curvature term can be}

^dropped,

and tue

leading

term in

7io

in _an

expansion

in derivatives of h is

7io

"

)~ / d~xi7~hi7~h. (3.4)

Now let _us consider tue

coupling

between membrane

proteins

and membranes.

3.1. PROTEINS WITH CIRCULAR CRoss-SECTION. Membrane

proteins

can bave

arbitrary

suapes;

_{as a} result tueir

tangent-plane

cross-sections _can be any

suape.

Now _we will

compute

tue undulation mediated force between

proteins separated by

_a distance

larger

tuan tue _size of

proteins.

For

simplicity,

let _us first consider membrane

proteins

tuat bave _a circular _cross-

sectionaÎ _{area on} tue membrane. Tuese

proteins

_may be described

by

_a scalar

density,

_p, wuicu may be

interpreted

_as tue distribution function of

proteins describing

tue

positions

^of

proteins

and tue

configurations

^of

^protein's

^{amino acid} sequence

Plx)

=

~ )filx ^{xi), 13.5)}

wuere _x

=

(u~, u~)

is _a

point

on tue membrane and tue _{sum is over} all

proteins.

^Tue

specific

form of

f~(x x~) depends

_on tue

specific

conformation of1-tu

protein

at tue

position

_x~- It

vanisues outside tue

protein

_cross section:

riz) _'X'

_{< ap}

j3 6) f~i~~

_"

o, 'X'

>

apl

wuere ap =

/fi

_{is tue radius of tue}

_protein

_{wuere A is its} cross-sectional _area. We _assume all

proteins

_are identical _so tuat

tuey

_are all described

by

tue _same function

f([x x~[).

For

membrane

proteins,

_ap is of order

10~À-

_If

we model tue

protein

as a uniform

cylinder,

tue distribution function of

protein

will be

f(x)

= inside tue

projected

_area and

f(z)

= 0 outside- In

general, proteins

bave non-uniform

folding

of trie amino acid

chain,

and

f(z)

will bave small

deviations from

unity

mside trie circular cross-sectional _region D- In tuis _{case, we use}

/d~z f(x)

= A

(3.7)

D as tue definition of A.

Wuen

proteins

do not break

up-down symmetry,

tue relevant

coupling

between _p and tue

ueigut

fluctuation field of tue membrane _is

~'~~ Î ~~~~°~~~~~~~~

~

~~~~~~~~~~

"

~ / _{~~~[°Î(ÎX XII)fiÎiÙÎ}

~

~Î(ÎX XII)~ÎiÙÎÎ, _(~.~)

~

D~

wuere

D~,1

=

1,2;

denote circular _regions

occupied by

membrane

proteins.

Tue

couphng

constants _o and _~ describe

couplings

between tue

density

inside

protein's

_cross section and tue curvature of _a membrane. Tuus tuese _can be related to tue

bending

and Gaussian

rigidities:

a

jjx)

=

ô~jx)

₊

ôiix)

,

+~jjx)

₌

-ôijx)

,

j3.9)

wuere ô~ and ôk _can be

interpreted

_as tue

changes

m tue

bending

and tue Gaussian

rigidities

due to tue existence of

proteins

on tue membrane. In tue

Monge

_gauge, to lowest order in h

K]

₌

^-i7~h

,

KfÂ(

=

ôaôbhôaôbh, (3.10)

and tue relevant

couphng

becomes

~iint

"

/ d~X(Op(X)(À7~h)~

+

~p(X)ôaôbhôaôbh]. (3.Il)

Tue free _{energy is given}

by

exp[-flfj

=

/(Dh] ^{exp[- fl~} / _{d~x(i7~h)~ fl} / d~z(ap(x)(i7~h)~

₊

_{~p(x)à ôbhô ôbhj}

2 2 ^~ ~

(3.12)

We _{can use} tue cumulant

expansion

to calculate tuis form of tue free _{energy. We write}

e~~~~~~°)

=

lexp[- ^fl / d~x(op(x)(i7~h)~

₊

p(x)ôaôbhôaôbhjj)

(313)

2

wuere

()o

^{denotes tue} ensemble average over tue fluid membrane Hamiltonian

only

and

~~~'

"

/lDhl ~XPl~jfl~ / _d~Xlv~h)~l

13.14)

Tue cumulant

expansion gives

le~)o

=

11+V+)V~+.

)o

= exP

lV)o

+

)llv~)o ^IV)]]

⁺

°lV~)j ^13.15)

Plugging equation (3.13)

into tue cumulant

expansion equation (3.15)

and

keeping

terms up to order

h~,

_we find tue free _energy

pif Fo)

"

fl d~x[ap(x)(i7~h)~

₊

p(x)ôaôbhôaôbhj1

2

o

~

~ Î~ Î ~~~~°~~~~~~~~~~

~

~~~~~~~~~~~~~~~~~Î~

~- ^fl / d~x[ap(x)(i7~h)~

+

~p(x)ôaôbhôaôbh]

~

(3.16)

2 2

o

Tuis _can be

expanded

in terms of tue

ueigut

correlation function

Ghh lx y)

and its denvatives.

Tue

ueigut

correlation function in tue real _space is

Ghhlx-Y)

=

lhlx)hlY))o

Î _(ÎÎ2 ^Î~Î2

_{~~~ ~~~~}

~~~~~~~~~~~°

Î _(ÎÎ2 ^{~ÎÎÎ4~~ 61fl~~~}

_~~

_{~~' ~~'~~~}

wuere _we used

(h(p)h(q))o

=

(2~)~ô(p

+

q)/fl~p~

in momentum space and _{x y}

= R-

Tuen, taking

derivatives _we find

(ô~ôbh(x)fôjh(y))o

"

ô~ôbfôjGhh lx y)

4~i~R2

^{~~~~~~~ ~ ~~~~~~ ~ ~~~~~~~}

-2(É~Ébô~j

+

É~lÎ~ôbj

+

É~Éj

ôbi +

ÉbÉ~ô~j

+

ÉbÉjô~~

+

É~Éjô~b) +8É~ÉbÉ~Éjj

~i~R2~~~~~~~'

~~'~~~

wuere

É

=

R/R,

R

=

(R(.

^We

proceed

to calculate tue terms m

equation (3.16).

We _are

only

interested in terms tuat

depend

_on tue distance between membrane

proteins.

We _can,

tuerefore, drop

tue first and tue last terms in tue RHS of

equation (3.16)

smce

tuey

do not

depend

on distance:

ld~xp(x)ô~ôbhô~%hl= Î d~xp(x)i7~Ghh(x y) _[y=x

o

Î

Î ~~~~~~~ Î _(ÎÎ2

i~ ~ ^~~ ~~~~~~~ Î

(ÎÎ2 i~

2~ÎÎfl~~~~~~~ ~~°~~~~'

~~'~~~

~~~ ~~~~~~~~~

/ d2zpjx)(V~h)~

=

Î~ ^l~P~~~'

^{~~ ~~~}

o

wuere N is tue number of

proteins-

In

equation (3.19)

_we introduced tue cut-off for tue

ueigut

fluctuation field A

r~J

llap

wuere ap ^{is tue} radius of tue

protein.

Tue second _term

gives

contribution to tue

distance-dependent

free _energy:

fl2 d~xd~yp(x)p(y) [4o~ô~ôbi7~Ghh lx y)ô~ôbi7~Ghh lx y)

8

+2~~ô~ôbfôjGhh(x y)ô~ôbfôjGhh(x y)] (3.21)

Î ~~~~~~~~~~~~~~ 4~fl~R2

)2

Î~°~~~~"~~~~~" ^~~~

^~

~~~~~° ~~~~~°

~~~Î

'

wuere _we

kept only

tue

leading distance-dependent

terms and R

= x y.

Tuus,

tue

leading distance-dependent

free _{energy is}

given by

fl~

"

/ _d~X /

d~YP(X)U(ÎX YÎ)P(Y), (3.22)

~~~~~

(a~

₊

~~)

j3_23)

~~~~

_~~~

(~~)~(x-y(~

For two

proteins separated by

_a distance

R,

tue

leading

R

dependence

is

pJ~

=

1

/ _d2x / _{d2y [a~}

+

~2] flx) jjy) 13.24)

2~2~2R~

Di D~

Relating

tue

couplings

a and _~ witu tue variations of tue

bending rigidity

and tue Gaussian

rigidity

_{as m}

equation (3.9),

_{we recover} tue result of Goulian et ai. _I?1

F ₌

~~ljl~~ /~ ^d~z /~ ^d~y iô~iz)ôiiy)

+

ô~iy)ôiix)1 13.25)

~ ~

If membrane

proteins

break

up-down bilayer symmetry,

tuere is anotuer

possible

relevant

coupling,

7i;nt

₌

jÀ / _d~zp(x)K]

=

~À ~j / _{d~x f([x x~[)K]. (3.26)}

2

~

D,

However,

tuis term does not contribute to

protein-protein

interactions since tue distance-

dependent

contribution vanisues _as

follows,

/ d~xd~YPlx)PIY)lv~hlx)V~hlY))o

=

/ d~zd~YPlx)PIY)V~G~~ix y) 13.27)

=

) / _d~x / d~Yflf)flY)ôlx

_Y)

= 0.

3.2. PROTEINS WITH NON-CIRCULAR CRoss-SECTIONS. SO far _we

bave,

for

simplicity,

considered

protein-protein

interactions wuen

proteins

bave circular _cross section.

However,

m

general proteins

bave

asymmetric

conformations

giving

_use to non-circular foot prints on

tue membrane surface.

Tuey

_can tuen be cuaracterized

by symmetric-traceless

tensor order-

parameters

sucu _as

Ô~~,

Ô~~~~> and so on:

Qab j~) ~

^l

_{Qab jj~}

~

Qabcd j~) ~

^l

_{Qabcd j j~}

~

j~ ~~)

~

và

^{" ~}

~

ô

^{" ~}

wuere x~ denotes tue

position

of

v-tu protein

^and

Q(~

^and

Q(~~~

are tue

symmetric-traceless

tensors constructed from tue cuaracteristic direction vector of

v-tu protein

_on tue membrane.

Wuen

up-down _symmetry

is not

broken,

tue relevant

coupling

between inclusions and curvature is

7i;nt

₌

/ d~xÉ~~~~(x)K~bK~d, (3.29)

2 wuere

(abcdj

_X

Qabcdj

~

Qabj )Qcdj

_{~ d}

Qacj )Qbdj j~~~)

" Q4 X Q2 ^{X X} 2 X X

Tue

coupling

constants q4, q2 and

d2

describe

couplings

between tue

protein

tensor order

parameters

and tue curvature of _a membrane. Results for membranes witu circular _cross- sections _can be obtained

by cuoosmg

Si~~~

=

aô~~ô~~

₊

~ô~~ô~~, 13.31)

ratuer tuan

insisting

tue order

parameters

be

symmetric

and traceless. Witu tuis

coupling,

we

proceed

_as before

using

tue cumulant

expansion.

For two

proteins separated by

_a distance vector R from _one to tue

otuer,

tue free _energy becomes

In terms of tue tensor

T~bj

introduced

before,

tue final form for tue free _energy writes _as

flF

₌

d~x d~yÉ~~~~(x)U~bcd

_vki

lx y[)É"~~ (y), (3.33)

4 '

wuere

j~~~~ j§)T~dki là)

₊

Tabki(É)Tcdv là))

(3.34) Uabcd,iJki (ÎX YÎ)

8~~~~_IX YÎ~

For two identical

proteins separated by

R

= xi x2, tue

leading distance-dependent

free energy is found _to be

~2 sabcdj )sqklj~

y

=

-kBT

^{~~ ~} [Tab~j

(É)T~dki (É)

₊

_Tabki (R)T~dv(R)]. (3.35)

64~2~2R~

Now tue free energy is

anisotropic, depending

on tue direction of tue

separation

vector R and tue orientation of

proteins

^described

by S~~~~(x~)-

Q~~~~ is _a 4tu-rank

symmetric-traceless

tensor, ^wuicu can be

expressed

_as

Qabcd

~

Q

~

e~e~e~e~

(e~e~ô~~

+

e~e~ô~~

₊

e~e~ô~~

₊

e~e~ô~~

₊

e~e~ô~~

₊

e~e~ô~~)

6

14~~~~~~~

~ ~~~~~~ _~

~~~~~~~' _~~'~~~

and

Q~~

^is a 2nd-rank

symmetric-traceless tensor;

Q~~

"

Q2 ^{e~e~ ~ô~~}

,

(3.37)

2

wuere

el

= cos9 and e~

= sm 9 cuaractenze tue direction of

protein

witu 9 measured witu _re-

spect

to tue

separation

vector R and

Q2

and

Q4

are

magnitudes

of 2-fold and 4-fold

anisotropy, respectively- Tuen,

tue free _energy becomes

Again,

for

proteins breaking up-down symmetry,

_we bave tue additional relevant

coupling

7i;nt

=

/d~xÉ~~(x)Kab. ^(3.39)

In _{contrast to} tue _case of circular _cross

section,

tuis

couphng

leads to a

qualitative change

in tue

protein-protein

interaction.

Proceeding

_as

above,

_we find tue

leading

distance

dependence

of

protein

interaction _is

1/R~:

~à~ fl2 j / _~~~ / ~~YÉ~~(X)É~~(Y)(ôaôbll(X)ôiôjll(Y))0 fl2

=

/ _d~x / d~ys~~(x)É~J(y)ôaôbfôjGhh(x y)

4

"

/ _d~~ /

d~YÉ~~l~)UCb,u(lX Yl)É~~IY), 13.40)

wuere

Uab,~j

llx _Yl)

=

))~ ^))~

13.41)

Tuis interaction is also

anisotropic, depending

_on R and

S~~(x~).

For

spuerical

_cross

section,

smce

S~~

r~J ôab and ô~~ô~JT~b~j "

o,

tue contribution to tue interaction vanisues _as before. For

ellipsoidal

_cross

section, S~~

₌

dio~~

wuere

dl

_is tue

coupling

constant and tue free _energy becomes

~2 d2Q2

~

I6~ÎRÎ

^{~°~ ~~~~ ~ ~~~'}

^~~'~~~

Tue minimum energy

configurations

_are at Hi + _H2

"

0,

_~.

Consequently, by introducing symmetric-traceless

tensors _as tue

order-parameters

for aniso-

tropic proteins

and

by determining

tue relevant

couplings by _symmetry,

_{we were} able to red- erive tue results for tue circular _cross section

by

Gouhan _{et ai-} I?i

Furtuermore,

_we ob- tained

anisotropic

interactions between

proteins

wuicu bave tue non-circular _cross section.

Tuis

anisotropic

interaction bas tue

leading

distance

dependence 1/R~

and

1/R~ depending

_on

up-down symmetry breaking-

4. Mortel II

In tue

previous section,

we introduced _a

coupling

between membrane

proteins

and tue

ueigut

fluctuation field of trie membrane

by considering symmetry

and _power

counting-

Since trie order parameter for

proteins

_in tue

coupling equation (3.8)

_can be

interpreted

_as tue distribution function of

proteins,

tue

puysical implication

of tuis

coupling

_can be tuat tue

bending

and

Gaussian

rigidities

mside tue

protein

_cross section differ

sligutly

from tuose of tue

surrounding

membrane. Tuus tuis

coupling

_can be

tuougut

of _as

perturbative. However,

if

proteins

are

infinitely rigid

witu _~

= -k ₌ oo inside tue

protein

_cross

section, perturbation tueory

fails. In tuis _{case, we can} derive tue

protein-protein

interaction

by considering

tue

puenomenological

interaction between membrane

proteins

and membrane

lipids

at tue

perimeter

of tue

proteins.

First,

let _us consider

proteins

whicu bave circular _cross sections and do not break

up-down symmetry.

Tuese

proteins

can be modelled _as

inversion-symmetric

turee-dimensional

ellipsoids

of revolution

(or cyhnder)

witu _a

major

axis

pointing along

a unit vector _{m m} turee-dimensions.

Tueir orientational order _can be cuaractenzed

by

tue

symmetric-traceless

tensor

Qv

"

(m~mj

[j/3).

We _assume tue axis _m

prefers

to

align along

tue membrane normal N. A

simple

interaction

favoring

tuis

alignment

is

7i;nt

" a

£ /

^~~

_{Q° N~Nj. (4.1)}

2

~~

2qp

zz

In tue

Monge

_gauge,

Q~JN~Nj

₌

(m N)~

=

-ô~~(m~ N~)(mb Nb)

+ constant,

(4.2)

3

wuere _a, b _{run over}

1,

2

only

and

N~

₌

-ôah

to lowest order in h. Now _{we can}

Taylor-expand

tue unit normal of tue membrane at tue perimeter ^{from tue center} of tue

protein

to lowest non-trivial order _in ap

N~(r)(~~

=

N~(R~

+

apb(1 (b

_mi

)~)~/~)

₌

N~(R~)

+

apb i7N~

₊

,

(4.3)

wuere b is tue unit vector from tue center of tue

protein

to its

perimeter

and

N~(R~)

is tue average of

N~ jr) along

tue

penmeter

_c~.

Dropping

tue constant term, ^tue

coupling

becomes

1i;n~ ₌ a

~ / £ _{imaiR~) NaiR~) apb~ô~Nai2}

~ ~~ ^p

~2

=

a~j ih~(R~)ih~(R~)+ ~ôbN~ôbN~

,

(4A)

2

zz

wuere

fli~(R~)

=

m~(R~) N~(R~).

Tue free _energy is

e~~~

=

/[Dhj[Dih~(R~)je~~~°~~~'" ^(4.5)

=

(constant) /[Dh] ^{exp[- fl~} / _{d~x(i7~h)~ flaA ~j} ôbN~(R~)ôbN~(R~)],

2 2~

zz

wuere

7io

=

)K Jd~x(i7~h)~

and tue

integration

over

fli~

is trivial and

gives

constant con-

tribution. Since tue

coupling

bas _a

quadratic

form, _{we can} evaluate tuis usmg tue Hubbard- Stratonovicu transformation.

Altuougu

it is

notuing

more tuan

completing

tue _{square, we} will find tuis

technique

to be very useful-

By introducing

^tue

auxiliary

fields

W(v)

and

defining V(v)

_as

VilJ~)

=

ôiNilJ~),

_V21J~)

=

/ôiN2lJ~)

=

/ô2NilP), _AIR)

=

ô2N21J~), 14.6)

we bave

e~~~

=

/lDhllDwl/L)1exPl-fl7io j _{~ Wl/L) Wl/L) +1~ Wl/L)}

Vl/L)1

=

e~~~°

/lDWl/L)1exPl- j _~ ^Î/L) Wl/L)1lexPli

l~vl/L)

Vl/L)1)o, 14.7)

wuere r

=

floA/~. Using

tue cumulant

expansion again,

to tue lowest order _we obtain

e~~~~~~°~

=

/[DW(/t)] ^exp[- [ W(/t) V(/t)W(v) V(v))oÎ

"

/ÎDW(11)1exPÎ- ^{Wa Ill) (Va (11)~i} (v))owb Iv)] (4.8)

zz,v

For two

proteins separated by

_a distance

R,

_we find

e~~~~~~°~

=

/lDwli)llDW12)lexPl-j ~ Wal/L)lval/L)Vblv))owblv)1

=

jdetjvaj~t)l&lV))°)~~~~

8

j

^~~/~

(4.9)

=

(constant) 1 3(~)

'

wuere A

r~J

llap

is _a cut-off for tue

ueigut fluctuation,

and ap ^is tue radius of tue

protein

introduced in Section 2.1-

Tuus,

tue free _energy bas _an

R-dependence

_as

F =

-kBT]

=

-kBT ~~, _(4.10)

in accord witu tue

previous

calculation

by

Goulian et ai. [7]. ^{In tue above}

equation,

we used

a cut-off for tue

ueigut fluctuation,

A

=

21ap [13].

For

proteins

^{tuat break} tue

up-down _symmetry,

tue unit normal of tue membrane at tue

perimeter

of tue

protein

^is not forced to be

parallel

to tue direction of tue

protein. Instead,

tue unit normal is forced to bave _a fixed

angle

_a~ witu tue direction of tue

/t-tu protein.

Tuus tue

preferred

unit normal at tue

perimeter

is

N01~)lc

"

~ ~

~"~

j4 Il)

'

fi~

"

Tue

coupling

between trie

protein

and trie membrane at trie

perimeter

of tue

protein

_is

7iint

_{" Ci}

£ / £ _{(N N0)~.}

(4.12)

~ cp ~~P

"'

N

a)

Z

b)

Fig.

3.

a)

The unit vector _m denotes the direction of the

protein

and N denotes the unit normal

vector of the membrane at the

perimeter

of

protein (Side view). b)

b is the unit _vector from the center

of the

protein

to its pefimeter

(Top view)-

We

Taylor expand

to find to lowest order

~'~~ ~jÎ~21~p~~~~

~~~~~~ ~~~ ~°"~~~~~

a

~j[ih~(/t)ih~(/t)

+

a)ôbN~ôbN~ apa~ô~N~]. ^(4.13)

2

v

Tue free _{energy is now}

e~~~

=

/[Dh][Dùi~(/t)]exp[-~fl~ /d~x(i7~h)~

₂

-fia ~j[ih~(/t)ih~(/t)

+

a)ôbN~ôbN~ apa~ô~N~jj ^(4.14)

2

zz

~~~~~~~

Î~~Î~~~~~~~~~

ÎÎ~~~~~~~~~~ Î~~~~~l

Re-defining V(/t)

_as

VI

Ill)

"

ôlNl(/L) ^~~ _V2(/L)

"

ôlN2(/L)

_"

ô2Nl(/L)>

_1~3(/L)

62~Î2(/L) ~~, _(~.là)

ap p

tue free _energy becomes

e~V

=

/[Dhj

^exp

-

fl~ / _d~x(i7~h)~ ^~°~ _{~j V(/t)} ^(/t)j

2 2~

~

=

/[Dhj[DW(/t)]

^exp

(- ^fl~ / _{d~x(i7~h)~ ~j W(/t) W(/t) +1~j W(/t)} ^(/t)j

~ ~~

v v

"

/ll~wlll)1eXPl~j ~ _Will) 'Wlll)lleXPli ~ Will) 'Vlll)1)o, (4.16)

v v

wuere r

=

flaAlir-

For two

proteins separated by

_a distance

R,

we find

e~~~

=

/lDwll)llDW12)1exPli É

WlJ~) lvlJ~))o

2

à ~l _{llwl/1) Vl/1)Wlv) Vlv))o lwl/1)} Vl/1))oiwlv) Vlv))o)1

~

jÎÎÎjÎ-jwùw-iÂw

=

idetù)-~/2e~l~~~~~, _14.17)

wuere

W =

(Wi(1),W2(1),W3(1),Wi(2),W2(2),W3(2)), _(4-18)

=

(à,0,à,à,0,%), _(4.19)

ap ap ap ap

3 0 1 -À 0 -À

0 2 0 0 -2À 0

~

111~ _{À À} Î

~~'~~~

0 -2À 0 0 2 0

-À 0 3À 0 3

witu =

8/R~A~

and A

=

21ap

is tue membrane cutoff. Tuus tue

R-dependence

of tue free energy

6ecomes

~~~

_{~~~~ ~~}

~A2

~~~

~~Î~~

~

~

R~A2 ~

~°~

~

°~~~~~

~A2

^{~ ~}

Î~Î~~ ^{~Î~~~ ~~'~~~}

Dur final form for tue free _energy is

~

~BT ~~

_~

^{~~IÀ~ a(}

⁺

^~2

~ R4

2

~ ~~

Tuis

gives

tue

previous

result

equation (4.10)

for ay = 0 wuicu

corresponds

to tue

strong-

couphng regime

m reference

I?l.

In tue limit T _~

0,

tuis gives tue result for tue low

temperature

Fig.

4. Non-transmembrane proteins have

preferred

center-of-mass positions non ai trie center of trie

bilayer. (

denotes the

position

of the

protein

and h is trie membrane

height

fluctuation field. The

potential

_energy has _a minimum at the

non-vanishing

value of ( h

= ro.

regime

in reference

[î]. Tuus,

_m tuis

puenomenological model,

_we obtain tue

general

interaction between tue

up-down symmetry breaking proteins

at finite

temperature

T.

Tuis

calculation,

wuicu focuses _on tue

change

in free _energy

brougut

about

by

tue addition of

inclusions,

does not show

explicitly

uow tuese inclusions

modify

tue

suape

of tue membrane at

large

distances from tue inclusions. Careful treatment of tue minimum energy

configuration

of

h,

about wuicu _we calculated Gaussian

fluctuations, yields

tue _same

large

distance distortion

(h

r~J

cosnH)

_as calculated

by

Goulian et ai. We beheve tuis result to be true for _a free membrane witu _no

imposed boundary

conditions. If the membrane is forced to be flat

by

_an

aligning field,

tuese

long-range

^{forces will become}

short-range

and surface tension will bave _a similar effect. It is not _so clear wuat will

uappen

_{on a} vesicle of

spuerical topology

witu _no

Laplace

_pressure. Tuis

question

^is

currently

under

investigation.

5.

Height-Displacement

Model

Non-transmembrane

proteins

_are

exposed

to a

specific

surface of _a membrane.

Tuus, tuey

bave

preferred

center-of-mass

positions

not at tue center of tue

bilayer (See Fig. 4)-

We consider tue interaction between tuese

proteins by introducing

tue

potential

_energy

il(( h)

wuere

(

is tue

position

of tue

protein

and h is tue membrane

ueigut

fluctuation field. For

integral proteins, il(( h)

bas _a minimum at tue

non-vanisuing

value of

(

h

= ro. We _can

expand il(( h)

in terms of tue deviation from tuis

preferred

value _ro

~~~

_~~

~~~°~

~

~~~

~~ ~°~~

~Î~ _Î((-h)=ro

^~ ' ~~ ~~

and if

proteins

are

tigutly bound, @

~

~ ~~_~ » 1.

Considering

tue

symmetry,

we introduce

tue

couplings

^~

7i;nt

₌

~ / _{d~x[ki (m}

+ i7h)~ + k2

(( h)i7~h

+

k3 (( h)m~mbô~ôbh

+

il(( h)], (5.2)

2

~ D~

wuere _m is tue

preferred

direction of tue

protein. By minimizing

_{over m~,} we find

m~ =

-ô~h

+

~~

(( h)ôbhô~ôbh

+

O(h~) (5.3)

ki

Substituting

tuis result into tue

couphng,

_we obtain _m lowest order

7i;nt

=

~j / _{d~z[k2(( h)i7~h}

+

k3(( h)ô~hôbhô~ôbh

2

~ D~

~~~~°~~Î~~~

^~~

°~~~Î~Î((-h)=ro~

~~'~~

Minimizing

_over

(( h) gives

tue

preferred position

of tue

protein

_as

62~y

^-1

(

h

= ro

k2 (j _i7~h

+

O(h~) (5-5)

à(

((-h)=ro Tuus _we obtain tue

coupling

~'~~ ~ ^D ~~~~~~~°~~~

~

~~~°~~~~~~~~~~~ Î~~

~Î~ _((-h)=ro

^~

~~~~~~~

~~ ~~

Tue first and tue last terms look similar to tue _ones in tue

puenomenological

model.

However,

tue non-linear second term is allowed because tue

up-down symmetry

is broken

by

tue

preferred position

of tue

protein-

In tue low

temperature hmit,

_{we assume}

lipids

_{are so}

tigutly

bound

62~y

to tue

proteins

tuat

~)

_{is mucu}

_bigger

_{tuan k2 and}

we

drop

tue last term in

(

(-h)=ro~

equation (5.6).

In tuis

limit,

tue Hamiltonian becomes

7i _{= ~}

/ _d~z(i7~h)~

+

~j / d~z[k2roi7~h

₊

k3roô~hôbhô~ôbhj- (5.7)

2 2

~

v »

Now

by minimizing

tuis Hamiltonian _over

h,

_we obtain tue low

temperature

^{limit for} tue

protein

interactions. From tue minimum

condition,

(

= 0

=

~V~h

₊

~j _{k2roV~ô(r ry)}

v

+

~j k3ro(26aô(r ry)ôbhôaôbh

+

ôahôbhôaôbô(r ry)), (5.8)

v

we obtain tue

equilibrium ueigut

for tue membrane

h(r)

₌

~j _G(r-ry) ) /d~r'G(r-r') ~j ôaG(r'-ry)ôbG(r' -rv)ôaôbG(r' -r>). (5.9)

~ro

M M>v,À

Tuus tue interaction becomes

1i =

)~ / _d2z (~ ^{v2Gir rv)}

⁺

_([~ ^{i ôaGir rv)ô~Gir rv)ôaô~Gir >)j}

v v,v,>

~

t

j~ £ ^~~~°

^~

_ô(ry

ru

~,~

~

-~

i [- ~j° ^{[~ ôaGirv} rv)ô~Girv r>)ôaô~Girv r«) làio)

vv,>,«

For two

proteins separated by

_a distance

R,

_we obtain tue

leading

distance

dependence

of tue interaction

7i =

18k3roô~G(R)ôbG(R)ô~ôbG(R)

~~

k2ro

^~

R~Rb(ô~bR~ 2R~Rb)

3ro

~

_~~

=

~~° ~

~~jj~° (5.Il)

~~l

Above we used

G(R)

=

-(k2ro/4~~)

In

R~.

_We

can

interpret

trie

parameters

in

equation (5-11)

as trie _area of

proteins

^{and trie} contact

angle

between

proteins

and

lipids.

Wuen tuere _are several

proteins,

from

equation (5.10),

we find tuat turee- and

four-body

interactions exist in addition to

two-body

interaction. For turee

proteins separated by Ryv

which is trie vector from trie v-th

protein

to trie

/t-tu protein,

_we find

turee-body

interaction to be

~~~~~~

_-

-i~k~ro iii

^~

^(~

RilRi«

~ ~~llv'l~~~ ^~~~~~~~

'

~~~~~

where

£'

_means all _{/t, v, a are}

different,

in addition to

two-body

interaction between each _pair of

proteins given by equation (5.Il). Similarly,

_we find

four-body

interaction to be

~~~~°~~

~~~°~ÎÎÎ~~ (

R(~j~R(~

~~~ ~~~ ~~~~

~i~~~ ~~~l'

v,v, ,a

(5.13)

where

£'

_{means all /t, v,À,}

a are different. Note that these three- and

four-body

interactions

are also

1/R~

interaction which is trie _same order _as

two-body

interaction.

6. Discussion

We model

biological

membrane _{as a} continuons

bilayer

^of

^lipid

^{molecules in which} varions

membrane inclusions sucu _as

proteins

are embedded. Such model membranes with inclusions also bave

potential applications

for

target drug delivery,

^nano-scale _pumps, functionized inter-

faces,

and cuemical _reactors. In tuis _{paper, we}

study

uow tue membrane contributes to tue interactions between inclusions.

Also,

it is

interesting

to understand uow inclusions affect tue

properties

sucu _as

rigidity

_or

suape

of model membranes.

Tue interaction between membrane inclusions sucu _as

proteins

witu circular cross-sectional

area was first calculated

by

Goulian et ai-

Using

three

models,

wuich _we refer to _as Model

I,

Model

II,

and _a

ueigut-displacement models,

_{we recover} ail tue results

by

Goulian et ai- The

interaction in

equation (2.2)

is a

temperature-dependent

interaction between two circular inclu-

sions tuat falls off witu distance as

1/R~. Assuming

two inclusions are

identical,

tue force will

be attractive if ô~ and ôR bave tue

opposite sign

and

repulsive

otuerwise- For inclusions witu

up-down symmetry,

^tue interaction is attractive and falls off _as

1/R~ again-

Tue

magnitude

is set

by kBT

and is

independent

of tue

rigidity

_~. Wuen inclusions break

up-down symmetry,

in addition to tue attractive interaction set

by kBT,

we find _a

repulsive

interaction

proportional

to tue square of tue contact

angle

ⁱⁿ

equation (2.6). Tuus,

for

up-down asymmetric inclusions,

tuere _are

competing

attractive and

repulsive

interactions and we

migut

bave _an

mteresting

transition between

aggregation

and

mixing

of inclusions wuen

kBT

_r~J

~o~-

_Botu Model I

(for

soft

inclusions)

and Model II

(for

hard

inclusions) predict potentials

tuat fall off witu distance

as

R~~.

Tuis interaction is attractive for hard inclusions. For soft inclusions tue

sign

of tue interaction

depends

_on tue relative

sign

of ô~ and ôk and is attractive if

tuey

bave

opposite signs-

^{Reasonable models}

predict

ô~ôR _< 0 _so tuat tue

prediction

of models I and II _can be

viewed _as

being

consistent.

Furtuermore,

_we calculate tue interaction between

proteins

witu tue non-circular _cross- sectional _area and find

anisotropic 1/R~

and

1/R~

interactions

depending

on wuetuer _up- down

symmetry

is broken _or not. In

equation (2.3),

_we find tue interaction between

proteins

witu non-circular

cross-sectional

_area wuen

up-down symmetry

is

conserved,

and tue free _en- ergy

again

^{falls off} as

1/R~

but tue

magnitude depends

_on tue orientations of inclusions.