Eur. Phys. J. E 3, 259–271 (2000)

THE EUROPEAN
PHYSICAL JOURNAL E
EDP Sciencesc Società Italiana di FisicaSpringer-Verlag 2000

Adhesion-induced lateral phase separation in membranes
S. Komura1,2,a,b and D. Andelman2
1
2

Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv
69978, Tel Aviv, Israel
Received 12 January 2000 and Received in ﬁnal form 15 May 2000
Abstract. Adhesion between membranes is studied using a phenomenological model, where the intermembrane distance is coupled to the concentration of sticker molecules on the membranes. The model
applies to both adhesion of two ﬂexible membranes and to adhesion of one ﬂexible membrane onto a second
membrane supported on a solid substrate. We mainly consider the case where the sticker molecules form
bridges and adhere directly to both membranes. The calculated mean-ﬁeld phase diagrams show an upward
shift of the transition temperature indicating that the lateral phase separation in the membrane is enhanced
due to the coupling eﬀect. Hence the possibility of adhesion-induced lateral phase separation is predicted.
For a particular choice of the parameters, the model exhibits a tricritical behavior. We also discuss thenon-monotonous shape of the inter-membrane distance occurring when the lateral phase separation takes
place. The inter-membrane distance relaxes to the bulk values with two symmetric overshoots. Adhesion
mediated by other types of stickers is also considered.
PACS. 87.16.-b Subcellular structure and processes – 68.10.-m Fluid surfaces and ﬂuid-ﬂuid interfaces –82.70.-y Disperse systems

1 Introduction
Adhesion of membranes and vesicles has attracted considerable experimental and theoretical interest because of
its prime importance to many bio-cellular processes [1,2].
Theoretical treatments of membranes composed of single
component lipid bilayers have revealed that generic interactions such as van der Waals, electrostatic or hydrationinteractions govern the adhesive properties of interacting
membranes. It is also worthwhile to mention that related
phenomena are found in unbinding transition of nearly ﬂatmembranes [3] or adhesion of vesicles to surfaces [4].
In addition to general non-speciﬁc interactions mentioned above, it is known from the works of Bell andcoworkers [5,6] as well as others [7], that highly speciﬁc
molecular interactions play an essential role in biologicaladhesion. This interaction acts between complementary
pairs of proteins such as ligand and receptor, or antibody
and antigen. A well-studied example of such coupled systems is the biotin-avidin complex. The avidin molecule
has four biotin binding sites, two on each side, and forms
a ﬁve-molecule biotin-avidin-biotin complex. The resulting speciﬁc interaction is highly local and short-ranged.
aOn leave from Faculty of Computer Science and Systems
Engineering, Kyushu Institute of Technology, Iizuka 820-8502,
Japan.
b
Present address: Department of Chemistry, Tokyo
Metropolitan University, Tokyo 192-0397, Japan.

Measurements by surface force apparatus [8] or atomic
force microscopy [9,10] have shown that the force required to break a biotin-avidin bond is about 170 pN. In
related experiments measuring chemical equilibrium constants [11], it was found that the biotin-avidin binding
energy is about 30–35 kB T which is larger than thermal
ﬂuctuations. Other coupled systems are those of selectins
and their sugar ligands where the bond is much weaker,of the order of 5kB T [12,13].
More recently several models taking into account thermal ﬂuctuations in membrane adhesion have been proposed. Zuckerman and Bruinsma [12,13] used a statisticalmechanics model which is mapped onto a two-dimensional
Coulomb plasma with attractive interactions. They predicted an enhancement of the membrane adhesion due
to thermal ﬂuctuations. In another work, Lipowsky considered the adhesion of lipid membranes which includes
anchored stickers, i.e., anchored molecules with adhesive
segments [14,15]. It was shown that ﬂexible membranes
can adhere if the sticker concentration exceeds a certain
threshold. If the multi-component membranes, including
lipids and sticker molecules, undergo a phase separation,
the adhesion is dominated by the sticker-rich domains.
Further studies in this direction using mean-ﬁeld theory
and Monte Carlo simulations [16] obtained a phase separation which is driven both by attractive intra-membranesticker interactions and ﬂuctuation-induced interactions
between stickers.

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The problem of multi-component membrane adhesion
is intimately related to that of formation of domains (a lateral phase separation). This has been observed by several
experiments. For example, the biotin-avidin interaction
occurring during vesicle-vesicle adhesion was investigatedby a micropipette technique [17]. The adhesion between
one avidin-coated vesicle and a second biotinylated vesicle
is followed by an accumulation of biotin-avidin complex in
the contact zone. This accumulation of cross-bridges between the two vesicles is found to be a diﬀusion-controlled
process.
Adhesion-induced phase separation has been observedby Albersdörfer et al. and results from the interplay
between long-range repulsive and short-range attractive interactions [18,19]. The membrane includes repellermolecules in the form of lipopolymers (modiﬁed DOPE
lipid with a polyethyleneoxide headgroup), mimicking
glycocalix in real biological systems. The other component is a receptor molecule in the form of biotinylated
lipids (DOPE-X-biotin). This lipopolymer is responsible
for longer-range repulsive interaction, while the shortrange attractive interaction is introduced by adding streptavidin to the extra-cellular solution. The streptavidin
acts as a connector between the biotinylated lipids on thetwo membranes. A technique of reﬂection interference contrast microscopy [20] was used to observe domain formation on a vesicle adhering to a membrane supported ona solid substrate. The lateral phase separation on both
membranes leads to the formation of domains of tight adhesion separated by domains of loose adhesion [18,19].
In a related work, adhesion between cationic vesicles
and anionic supported membranes revealed that electrostatic interactions induce lateral charge segregation on
the membrane [21,22]. This phase separation leads topatches of tight inter-membrane contact and decoupled“blisters”. Furthermore, adhesion of membranes including self-recognizing homophilic molecules and lipopolymers has been investigated [23]. It was found that the initial weak adhesion is followed by slower aggregation into
tightly bound domains coexisting with domains of weakadhesion. The result has been interpreted in terms of adouble-well inter-membrane interaction potential due tothe presence of the lipopolymers. Let us emphasize that in
all the above-mentioned experiments, it was reported thatadhesion molecules aggregate spontaneously and form domains of tight adhesion.
It is generally believed that multi-component biomembranes in physiological conditions are close to their critical point, and membrane functions are partially governed
through phase separation processes. Moreover, concentration ﬂuctuations in the vicinity of the critical point may
aﬀect biophysical properties of membranes and can be ofimportance in regulating membrane processes in a robust
way. Recently this conjecture was supported by an experiment of an insoluble Langmuir monolayer at the air/waterinterface [24,25]. The monolayer was prepared in two different steps. The ﬁrst mimics the composition of the inner leaﬂet of a cell biomembrane, while the second mimics
the outer leaﬂet. In both cases, by using ﬂuorescence mi-

(a)

(b)

(c)

Fig. 1. Schematic representation of various types of adhesion between two membranes. The two membranes are represented by black lines. (a) Bolaform-sticker adhesion: bridges
consist of a single type of sticker molecules which are anchored
to one membrane (ﬁlled circle) and stick to the other membrane by another sticky part of the molecules (open circle).
(b) Homophilic-sticker adhesion: bridges consist of two identical stickers which are bound together by their respective sticky
end segments. (c) Lock-and-key adhesion: bridges consist of
two diﬀerent stickers forming a ligand-receptor type bond.

croscopy technique, it was found that the Langmuir monolayer is close to its corresponding critical point of demixing.
So far, the interplay between lateral phase separationand membrane adhesion has not been considered theoretically in detail except in references [14–16]. The workin references [21,22] deals only with the speciﬁc case of
oppositely charged membranes. In this paper we provide
a general phenomenological approach for the adhesion ofmulti-component membranes. Using a mean-ﬁeld theory,
we investigate how the lateral phase separation within
the membrane is aﬀected by the adhesion of membranes.
Like in references [14–16], we consider adhesion mediatedby sticker molecules. Sticker molecules are polymers or
macromolecules anchored to one membrane and interacting with the other membrane by another sticky part ofthe molecule. They can form bridges between two adjacentmembranes (so-called trans-interaction) [14], and play anessential role in the adhesion of cell membranes in biological systems.
We distinguish three types of adhesion dependingon the structure of bridges as represented in Figure 1.
i) “Bolaform-sticker” adhesion where each bridge moleculeconsists of a single sticker having two sticky ends (Fig.
1(a)). One sticker end is anchored to one membrane while
the other end is adhering directly to the second membrane.
ii) “Homophilic-sticker” adhesion where the bridges areformed by two stickers of the same type (Fig. 1(b)). Eachsticker is anchored on one of the membranes, while their
free ends bind together to form the bridge. iii) “Lock-andkey” adhesion where the bridges consist of two diﬀerent
stickers forming a ligand-receptor type bond (Fig. 1(c)).
This case represents an asymmetric adhesion due to thelack of symmetry between the ligand and receptor. In thepresent work, we mainly discuss the symmetric bolaformsticker adhesion (case i) above) using a model where theequilibrium spacing between two membranes is coupled to
the local concentration of stickers. Even in the latter symmetric case, a certain asymmetry can be obtained by controlling separately the sticker chemical potentials on the

S. Komura and D. Andelman: Adhesion-induced lateral phase separation in membranes

ψ1(r)two membranes. An important consequence of our modelis that the lateral phase separation is enhanced.
This paper is organized as follows. In the next section, we explain our phenomenological model of bolaformsticker adhesion. The mean-ﬁeld phase diagrams are given
in Section 3. The inter-membrane distance between two
coexisting domains is calculated in Section 4. Finally discussion is provided in Section 5 where the other types ofadhesion mentioned above are considered.

1
l 1(r)

ψ2 (r’)2l2 (r’)

2 Bolaform-sticker adhesionr

In this section, we treat the case where the adhesion ismediated by a single type of sticker molecules which areanchored irreversibly to one membrane and stick to theother membrane by another sticky part of the molecules
as in Figure 1(a). The anchor segments consist of a hydrophobic segment and penetrate into the hydrophobic interior of the lipid bilayer. The sticky segments, on the other
hand, adhere directly to another membrane having somepotential of sticking [15]. As mentioned above, we callthis a “bolaform-sticker”. Consider two interacting membranes labeled by i = 1, 2 consisting of lipid molecules andbolaform-stickers as schematically shown in Figure 2. Let
the sticker concentration in each membrane be denoted by
ψi (r), where r = (x, y) is a two-dimensional planar vector
and 0 ≤ ψi (r) ≤ 1. Note that the average concentrationsof stickers on the two membranes, ψ1  and ψ2 , do nothave to be the same.
When the adhesion molecules are very ﬂexible, they
can bend back to form arches on a single membrane. In
order to avoid such a situation in experiments and in themodel, the bending rigidity of the sticker molecules should
be suﬃciently large. Hence we assume that for stiﬀ enoughstickers all the bonds are inter-membrane ones connecting the two separate membranes as considered in references [14,15].
The interaction between two stickers on the same
membrane is called cis-interaction and can be repulsive
or attractive. Here we discuss the case in which this interaction is attractive. Then, below a certain critical temperature, the multi-component membrane undergoes a ﬁrstorder phase transition and stickers form lateral domains.
As shown in Figure 3, a sticker-poor phase coexists witha sticker-rich phase in the two-phase region of the phase
diagram. The sticker critical concentration ψc and the critical temperature Tc are assumed to be the same for thetwo planar membranes. We deﬁne the concentration difference φi (r) for each of the membranes with respect tothe critical concentration by
φi (r) = ψi (r) − ψc261

(i = 1, 2),(2.1)

where φi satisﬁes −1 ≤ φi ≤ 1.
The total free energy of the two coupled membranes
can be written as a sum of several terms detailed below.
The ﬁrst contribution describes the lateral phase separation of each membrane. Motivated by recent experiments
on Langmuir monolayers [24,25] demonstrating that ther’Fig. 2. Two adhering membranes. The reference x-y plane isshown as a dashed line. The height of two membranes measured
from this plane is denoted as 1 and 2 , respectively. The sticker
concentration on each membrane is denoted by ψ1 and ψ2 ,respectively.

Τ

Tc

A

B

A+B

ψc

ψ

Fig. 3. Schematic phase diagram for a single membrane containing sticker molecules. The concentration of the sticker
molecule is ψ. The critical concentration and the critical temperature is denoted by ψc and Tc , respectively. Within the coexistence curve the membrane separates into sticker-rich andsticker-poor region (A+B coexistence). The membrane is in a
one-phase outside the coexistence curve.

inner and outer leaﬂets of biomembranes are close to theircritical point, we employ a phenomenological GinzburgLandau free energy which is an expansion in powers of
the order parameters {φi }. Hence we have



1
1 
1
1
c(∇φi )2 + tφ2i + φ4i − µi φi .
F1 =
d2 r
2 i=1,22
2
4(2.2)
This expansion for the free energy can be justiﬁed close
to a critical point where the φi ’s are small enough.
The parameter c representing the line tension acting
at the domain boundary, and the reduced temperature

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t = (T − Tc )/Tc are taken to be the same for the two
membranes. On the other hand, the chemical potentialµi , coupled to the membrane sticker concentration φi , candiﬀer between the two membranes since the sticker concentrations on the two membranes do not have to be thesame. We recall that each bolaform-sticker is modeled with
one of its ends anchored irreversibly to one membrane,
while the second sticky end is attracted by the second
membrane. The sticker concentration is associated withthe anchored end of the stickers. The normalization factor 1/2 in (2.2) is introduced in order to write down thefree energy per single membrane. The coeﬃcient of the
fourth-order term can generally be set as a positive constant without loss of generality.
It is convenient to introduce the following new variables for the average and the diﬀerence between the twoconcentrations:ψ2 + ψ1
φ2 + φ1=− ψc ,
22
ψ2 − ψ1
φ2 − φ1
=
,
φ− =
2
2
φ+ =

(2.3)

where −1 ≤ φ+ ≤ 1 and −1 ≤ φ− ≤ 1. In terms of thesenew variables, (2.2) can be written as


 1
1 
d2 r c (∇φ+ )2 + (∇φ− )2 + t(φ2+ + φ2− )
F1 =
2
2

1 4
2 2
4
+ (φ+ + 6φ+ φ− + φ− ) − µ+ φ+ − µ− φ− , (2.4)
4whereµ2 + µ1µ2 − µ1
,
µ− =
.
(2.5)
2
2
The chemical potential µ− associated with the order parameter φ− is non-zero when the symmetry between thetwo membranes is explicitly broken. Namely, the two interacting membranes have diﬀerent average concentrations ofstickers.
Next we consider the out-of-plane deformation energyof the two membranes. As depicted in Figure 2, the membrane shape is parameterized by their heights 1 (r), 2 (r  ),above the x-y reference plane. Working in the Monge representation it is implicitly assumed that the membranes
remain ﬂat on average and have no overhangs. This approach can be also useful to treat adhesion of vesicles intheir contact zone. When the vesicle is large enough, it will
be roughly ﬂat close to the contact region, and the entire
vesicle can be thought of as a reservoir for the stickers.
Returning to the deformation energy, it can be written
as the sum of the bending energy and the surface tensionof each of the two membranes separately, as well as theinteracting potential energy between them [26,27]:



1
1 
1
κ(∇2 i )2 + σ(∇i )2
F2 =
d2 r
2 i=1,2
2
2
µ+ =



+

d2 r v(1 − 2 ; ψ1 , ψ2 ),(2.6)

where κ is the bending rigidity, σ is the mechanical surface tension acting on the membranes, and v is the potential energy per unit area representing the inter-membraneinteractions. For simplicity, κ and σ are assumed to beequal for the two membranes and do not vary as afunction of the sticker concentration ψi . The potentialv(1 − 2 ; ψ1 , ψ2 ) can be generally assumed to be a function of the local relative height coordinate 1 − 2 and the
sticker concentration ψi . The former assumption is the socalled Derjaguin approximation [28]. The dependence on
the sticker concentration ψi will be considered later.
We now make a change of variables and transform tothe center of mass and relative coordinates given, respectively, by2 + 12 − 1
L=
,
=
.
(2.7)
2
2
Only terms which depend on  can be considered in thecase where the center of mass is stationary, hence L is aconstant of motion. Then (2.6) can be written as [30]



1
1
F2 = d2 rκ(∇2 )2 + σ(∇)2 + v(; ψ1 , ψ2 ) .
2
2
(2.8)
In the high-temperature phase, the stickers are homogeneously distributed, and each of the membrane isin a one-phase region on the phase diagram. We assume
that even in the absence of sticker molecules, such membranes are bound to each other due to the balance between the short-range repulsive (e.g., hydration interaction) and longer-range attractive interactions (e.g., vander Waals interaction). Hence, we do not consider the interesting problem of the unbinding transition [14–16]. Although the membranes are always bound together, theirequilibrium distance  depends on the sticker concentration. Let us consider the potential v(; ψ1 , ψ2 ) in (2.8) for
φ+ = 0. Note that φ+ = 0 means that (ψ1 + ψ2 )/2 = ψc ,namely, the average sticker concentration on the two membranes is at its critical value ψc . The inter-membrane potential v(; φ+ = 0) is assumed to have a single minimum
at a certain inter-membrane distance  = 0 for φ+ = 0.
This gives the equilibrium distance between the two boundmembranes. The deviation of the inter-membrane distancefrom 0 is deﬁned by the dimensionless quantity δ(r) givenby(r) − 0
δ(r) =.
(2.9)
0For small deviations from the minimum of the potential,v(; φ+ = 0) can be expanded to second order. This is
known as the harmonic approximation and gives
1v(; φ+ = 0) ≈ v(0 ) + v  (0 )( − 0 )2
2
1= v(0 ) + V δ 2 ,(2.10)
2
where v  (0 ) is the second derivative of v with respect to 
evaluated at  = 0 , V ≡ v  (0 )20 , and v  () = 0 at  = 0[31]. Using (2.9) and (2.10), (2.8) can be written as



1
1
1 2
2
2 2
2
K(∇ δ) + Σ(∇δ) + V δ , (2.11)
F2 ≈ d r222

S. Komura and D. Andelman: Adhesion-induced lateral phase separation in membranes

LOOSE
TIGHT
1
l0
2

potentials, i.e., (1 − ψ)vf () + ψvr (). By expanding vf ()and vr (), we get a coupling term which is proportional toψ. In the case of adhesion between two membranes, we
add the contributions from both of the membranes andobtain the coupling energy as given in (2.12). The same
argument can be repeated for any arbitrary adhesion potentials, vf () and vr (), provided each of them has a single
well-deﬁned minimum at some distance .
The total free energy considered in our model is thesum of (2.4), (2.11), and (2.12):
F = F1 + F 2 + F 3 .

Fig. 4. Schematic representation of two adhering membranes
undergoing a lateral phase separation in the case of thebolaform-sticker adhesion. Coexistence between tight (T) andloose (L) membrane domains is shown. The inter-membrane
distance is smaller than 0 for the tight phase, whereas it larger
than 0 for the loose phase.

with K ≡ κ20 and Σ ≡ σ20 . This is the expression of thedeformation energy within the harmonic approximationand it served as a starting point to many calculations onmembrane adhesion [4,27,29].
Now we will include the eﬀect of the adhesion on thephase separation and suggest a lowest-order coupling between the composition φi (r) and the inter-membrane distance δ(r). When the membranes are quenched into a twophase region of the phase diagram a sticker-poor phase coexists with a sticker-rich phase. As shown in Figure 4, thiscan lead to diﬀerent inter-membrane distance for the different membrane domains. Since the sticky segments of thebridges adhere directly onto the two membranes, the coupling is proportional to the sum of the local sticker concentrations of the two membranes. This can be phenomenologically represented by the following coupling term:


α
2
F3 =
d r (ψ1 + ψ2 ) = α d2 r φ+ δ + · · · , (2.12)
20where the coupling constant α is positive preferring
smaller separation δ < 0 in regions where the average concentration φ+ is positive (or ψ1 + ψ2 > 2ψc ). In the last
expression of (2.12), we have neglected the linear terms
in φ+ and δ, which merely shift the chemical potential or
minimum of the potential, respectively. Depending on thevalue of φ+ , this coupling term not only introduces a shiftof the minimum of the potential v but also changes theminimum value of the potential. Note also that (2.12) issymmetric with respect to the exchange of the two membranes 1 ↔ 2.
The above linear coupling energy can also be understood in the following way. Let us ﬁrst consider a single ﬂexible membrane with sticker molecules adhering to
a ﬂat substrate. Suppose vr () and vf () are the potentials for sticker-rich and sticker-free membrane, respectively. Following the same discussion as in (2.10), eachof the potential is parabolic around a diﬀerent separation:
vr ≈ a( − r )2 and vf ≈ a( − f )2 . The eﬀective potential can be obtained by a linear combination of these two263(2.13)

Here it is convenient to convert to Fourier space. The
Fourier transform of any function f (r) is deﬁned as˜(2.14)
f (q) = d2 r f (r)eiq ·r ,where q is the two-dimensional in-plane wave vector. The
total free energy can be expressed as



1  1
(t + cq 2 ) |φ̃+ (q)|2 + |φ̃− (q)|2
F =
A q 2

1
+ (V + Σq 2 + Kq 4 )|δ̃(q)|2 + αφ̃+ (q)δ̃(−q)
2


1
2
+ d r (φ4+ + 6φ2+ φ2− + φ4− )4
−µ+ φ+ − µ− φ− ,(2.15)
where A is the area of the membranes projected on the x-yplane. For convenience the free energy (2.15) is written as
a combination of real space and Fourier space terms.
Within the mean-ﬁeld level, the free energy in termsof φ+ and φ− is obtained by functionally minimizing F
with respect to δ̃(q). Then we ﬁndδ̃(q) = −αφ̃+ (q).
V + Σq 2 + Kq 4

(2.16)

Hence the inter-membrane distance δ = ( − 0 )/0 is fully
determined by the value of φ+ . By inserting (2.16) into
(2.15), the resulting free energy depends only on φ+ and
φ− , and becomes


1  1
1
F =
Γ̃+ (q)|φ̃+ (q)|2 + (t + cq 2 )|φ̃− (q)|2
A q 2
2


1
+ d2 r (φ4+ + 6φ2+ φ2− + φ4− )4
−µ+ φ+ − µ− φ− ,(2.17)whereΓ̃+ (q) = t + cq 2 −α2.
V + Σq 2 + Kq 4

(2.18)

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The European Physical Journal E

If we expand the last term in (2.18) for small q, we obtainΓ̃+ (q) ≈ (t − γ) + c +

α2 Σ
V2

q2 ,

(2.19)

φ− = ± −t − 3φ2+ .

with

α
.
(2.20)V
The parameter γ is an important parameter characterizing
the coupling strength. The ﬁrst two terms in (2.19) implies
an upward shift of the transition temperature, as will be
discussed in detail in the next section. We also ﬁnd thatthe presence of the coupling (α = 0) increases the line
tension c provided the mechanical surface tension Σ isnon-zero; c → c + α2 Σ/V 2 .
γ≡

In this section, we calculate the mean-ﬁeld phase diagrams for bolaform-sticker adhesion using the free energy
explained in the previous section. In order to study thebulk properties of the system, we set q = 0 and study thehomogeneous solutions, φi ’s and δ being constants. From
(2.16), the inter-membrane distance which minimizes thefree energy is given by
δ=−

(3.4)

2

3 Phase diagrams

αφ+
.
V

(3.1)

Since α is positive, δ is negative (smaller inter-membrane
distance) for positive φ+ , and δ is positive (larger intermembrane distance) for negative φ+ . By substituting backthis δ into the free energy f per unit area for homogeneous(constant) φ+ and φ− , we obtainf=where the two membranes have the same concentrations,
φ1 = φ2 . Likewise, two “asymmetric phases” are obtained
for t + 3φ2+ < 0 with1
1(t − γ) φ2+ + tφ2−
2
2
1
+ (φ4+ + 6φ2+ φ2− + φ4− ) − µ+ φ+ − µ− φ− ,4Notice that this free energy is continuous at t = −3φ2+ .
This free energy f1 can now be minimized with respect toφ+ . The resulting equation of state is written as
for t + 3φ2+ > 0 ,(t − γ) φ+ + φ3+ ,
µ+ =
(3.6)
− (2t + γ) φ+ − 8φ3+ , for t + 3φ2+ < 0 .
The phase diagram can now be calculated and the twophase region is obtained by the Maxwell construction. The
phase diagram for µ− = 0 is illustrated in Figure 5.
For t > 0, only the symmetric phase with φ− = 0
can appear since t + 3φ2+ > 0. Two symmetric phases with
diﬀerent φ+ can coexist when t < γ. The coexistence curveis simply given by√(3.7)
φ+ = ± −t + γ,
and the associated critical point is located at(t, φ+ , µ+ )c = (γ, 0, 0).

(3.2)

where γ is deﬁned in (2.20). Notice that γ is never negativeand vanishes only when α = 0. Therefore, when µ+ =µ− = 0, the ﬁeld φ+ will order before φ− , and the phasewith φ+ = 0 and φ− = 0 is expected [32]. Although thephase behavior of this free energy can be examined ingeneral, we concentrate here only on two particular cutsin the parameter space, i.e., µ− = 0 and µ+ = 0. In
these cases one can clearly see the eﬀect of adhesion on
the lateral phase separation.
3.1 The case µ− = 0
When µ− = 0 the two membranes have the same chemical potential µ1 = µ2 . Since the chemical potential µ+ is
coupled to φ+ , f can be minimized ﬁrst with respect toφ− . A “symmetric phase” is obtained for t + 3φ2+ > 0 with
φ− = 0,In the asymmetric phase the two membranes have diﬀerent
concentrations φ1 = φ2 . After inserting these expressions
into (3.2) with µ− = 0, the free energy becomes

1
1


(t − γ) φ2+ + φ4+ − µ+ φ+ ,


2
4



for t + 3φ2+ > 0 ,
(3.5)
f1 =

1 2 1

2
4

−
t
(2t
+
γ)
φ
−
−
2φ
−
µ
φ
,
+ +

+
+

2

 4
2
for t + 3φ+ < 0 .

(3.3)

(3.8)

We stress that the critical temperature is increased fromtc = 0 to tc = γ = α2 /V due to the coupling between
the composition φ+ and the inter-membrane distance δ asgiven in (2.12). In other words, the phase separation isenhanced by the adhesion of membranes. As presented in
Figure 4, the two coexisting values of φ+ given by (3.7)
lead to diﬀerent inter-membrane distances δ according to(3.1). Since α > 0, δ is negative ( < 0 ) in the sticker-richdomain, and this phase is called the “tight phase” (T). On
the other hand, δ is positive ( > 0 ) in the sticker-poor
domain and this phase is called the “loose phase” (L).
However, for each of the coexisting tight and loose phases,
φ− = 0, which means that the sticker concentration is thesame in the two membranes, φ1 = φ2 .
For −γ/2 < t < 0, the asymmetric phase with φ− = 0
is always unstable, and the tight and loose phases coexist
according to (3.1) and (3.7). For t < −γ/2, the asymmetric phase can be locally stable but it is only metastable.
Namely, its free energy is higher than that of the symmetric phase. Hence the coexistence between the tight andloose phases given by (3.1) and (3.7) preempts the asymmetric phase. The limit of metastability of the asymmetricS. Komura and D. Andelman: Adhesion-induced lateral phase separation in membranes
1.5

265

existence region. The tight and loose phases coexist for
t < γ = α2 /V according to (3.1) and (3.7).

(a)1

3.2 The case µ+ = 0

TL

Next we consider the case of µ+ = 0 but with µ− = 0.
This means that the chemical potentials of the two membranes have the same magnitude but opposite sign, i.e.,µ1 = −µ2 . This is a special case of the more general situation where the symmetry between the two membranes is
explicitly broken. Now f in (3.2) can be minimized withrespect to φ+ ﬁrst. As long as t + 3φ2− > γ, the only solution is(3.10)
φ+ = 0.

t/γ

0.5

L+T

0

-0.5

-1
-1.5

-1

-0.5

0

φ + /γ 1/2

0.5

1

1.5

1.5

This is called the “middle phase” (M) where the intermembrane distance is exactly 0 (or δ = 0). Again notethat φ+ = 0 means that ψ1 + ψ2 = 2ψc . For t + 3φ2− < γ,
we have the tight (or loose) phase with(b)

φ+ = ± −t + γ − 3φ2− ,

1

0.5

t/γ

L

T

0

-0.5

-1
-1.5

-1

-0.5

0

µ + /γ

0.5

1

1.5

3/2

Fig. 5. The phase diagrams for the bolaform-sticker adhesion
when µ− = 0 as a function of (a) rescaled average composition
φ+ /γ 1/2 and temperature t/γ, and (b) rescaled average chemical potential µ+ /γ 3/2 and temperature t/γ. The continuousline is a ﬁrst-order phase transition line, whereas the dottedline is the limit of metastability of the asymmetric phase withφ− = 0. The metastable region of the asymmetric phase is indicated by a thick line in (b). Open circle (◦) indicates a criticalpoint. Below the critical temperature, there is a coexistence
region between the loose (L) and the tight (T) phases as denoted by L+T. The phase diagram is symmetric with respect
to both φ+ → −φ+ and µ+ → −µ+ .

phase is obtained by calculating the second derivative of
the second equation of (3.5) with respect to φ+ . This leadsto
−2t − γφ+ = ±,(3.9)
24
which is also shown as a dotted line inside the L+T coexisting region of Figure 5(a).
In summary, for µ− = 0, the asymmetric phase φ− = 0
does not exist as a stable phase for any temperature.
At most it is metastable and occurs within the L+T co-(3.11)

where  deviates from 0 (or δ = 0) according to (3.1).
Since µ+ = 0, both the tight and the loose phases are energetically degenerated and they coexist. By substituting
φ+ back into (3.2) with µ+ = 0, the free energy becomes

1 2
1


tφ− + φ4− − µ− φ− ,


2
4




for t + 3φ2− > γ ,


1
1
f2 =
(3.12)
2− (t − γ) + (−2t + 3γ) φ2−



4
2


−2φ4− − µ− φ− ,





for t + 3φ2− < γ .
After minimizing with respect to φ− , the equation of stateis given as
for t + 3φ2− > γ , tφ− + φ3− ,µ− =(3.13)
(−2t + 3γ) φ− − 8φ3− , for t + 3φ2− < γ .
The calculated phase diagrams for µ+ = 0 are shown in
Figure 6. The phase diagram is symmetric about φ− = 0
and µ− = 0 as a consequence of the φ2+ φ2− coupling term,
and lack of any odd terms in φ+ in the free energy. For
t > γ there is a one-phase region of the middle phasewith φ+ = 0 since t + 3φ2− > γ. For 5γ/6 < t < γ, thesystem undergoes a second-order phase transition between
the middle phase (φ+ = 0) and the tight (or loose) phase(φ+ = 0). The analytical expressions of the second-orderphase transition lines are−t + γ,(3.14)
φ− = ±
3and

2t + γµ− = ±
3

−t + γ,3(3.15)

266

The European Physical Journal E
1.5

(a)
1
T
0.5

t/γ

M1

M2

(L)

0

-0.5

-1
-1.5

T+M2

M 1+T

-1

-0.5

0

φ − /γ

1/2

0.5

1

1.5

1.5

(b)
1

t/γ

0.5

0

M1

M2

T
(L)

-0.5

-1
-1.5

-1

-0.5

0

µ − /γ

3/2

0.5

1

1.5

Fig. 6. The phase diagrams for the bolaform-sticker adhesion
when µ+ = 0 as a function of (a) rescaled composition diﬀerence φ− /γ 1/2 and temperature t/γ, and (b) rescaled chemical potential diﬀerence µ− /γ 3/2 and temperature t/γ. The
continuous line is a ﬁrst-order phase transition line, whereas
the dashed line is a second-order one. The loose, tight andmiddle phases are denoted as L, T, and M, respectively. The
two tricritical points are indicated by a ﬁlled circle (•). Belowthem there are two regions of coexistence of the middle phase(φ+ = 0) and the tight phase (φ+ = 0) denoted as M1 +T and
T+M2 . The phase diagram is symmetric with respect to bothφ− → −φ− and µ → −µ− . Because µ+ = 0 there is a degeneracy between the tight (T) and the loose (L) phases on thephase diagram.

in Figure 6, respectively. On the second-order phase transition line, φ+ goes continuously to zero.
For t < 5γ/6, the transition changes to ﬁrst order. Thishas been numerically determined by the Maxwell construction. The point which connects the ﬁrst- and secondorder phase transition lines is a tricritical point [32]. In

our model, it is located at


√
1 1/2
5
4 2 3/2
γ, ± √ γ , ±γ.
(t, φ− , µ− )tcp =627
3 2(3.16)
The ﬁrst-order phase transition corresponds to the coexistence of the middle phase with φ+ = 0 and the tight (orloose) phase with φ+ = 0. The obtained two-phase coexistence region is indicated by “M+T” in Figure 6(a). Within
the present Ginzburg-Landau expansion, the tight phasepersists even if we go to low temperatures. Because of thedegeneracy between tight and loose phases, the ﬁrst-orderline near the tricritical point actually corresponds to coexistence of three phases: tight, loose, and middle phases.
In continuation to the discussion of the previous subsection (µ− = 0), we see that the phase separation is
also enhanced for the µ+ = 0 parameter space. It occurs
at higher temperatures, since the tricritical temperaturettcp = 5γ/6 = 5α2 /6V is positive for α = 0. It is important to notice that in the middle phase with φ+ = 0, theinter-membrane distance is 0 since δ = 0. On the other
hand, in the tight (loose) phase with φ+ > 0 (φ+ < 0),according to (3.1) and (3.7),  < 0 ( > 0 ). The coexisting membrane domains between tight and middle phases,or between loose and middle phases is schematically represented in Figure 7.
We end this section by commenting on the generalcase when both µ+ and µ− are non-zero. When µ+ becomes non-zero, the degeneracy between the tight andthe loose phases is lifted. In such a case, instead of thethree-phase coexistence for µ+ = 0, there is a coexistence
between either tight and middle phases, or between loose
and middle phases as shown in Figure 7(a) and (b), respectively. Notice that the tricritical point exists only whenµ+ = 0. In a more general phase diagram drawn in the(t, µ+ , µ− ) space, three second-order lines meet at the tricritical point. In the three-dimensional parameter space,these second-order lines lie on the perimeter of two-phasecoexistence planes between either tight and middle phases(T+M) or between loose and middle phases (L+M).

4 Non-monotonous membrane proﬁleOne of our assumptions was that the inter-membranes potential v(; φ+ = 0) has a single minimum at  = 0 whenφ+ = 0. In the absence of thermal ﬂuctuations, two homogeneous membranes are bound with inter-membrane
distance 0 for φ+ = 0. In this section, we calculate the
proﬁle of the inter-membrane distance between two membranes which are quenched below the phase separation
temperature.
We ﬁrst expand the potential v(; φ+ = 0) up to thefourth-order terms in δ;11v(; φ+ = 0) ≈ v(0 ) + V δ 2 + U δ 4 ,
2
4

(4.1)

where V ≡ v  (0 )20 as before and U ≡ v (4) (0 )40 > 0.

S. Komura and D. Andelman: Adhesion-induced lateral phase separation in membranes(a)

We now minimize F with respect to the concentrationsφ̃+ (q) and φ̃− (q) and obtain

MIDDLEφ̃+ (q) = −

TIGHT
1
l0
2

(b)
LOOSE

l0

V <

Both V and U are positive constants because of the convexity of v at its minimum. Suppose that each of the membranes is in its high-temperature phase (t > 0). Then the
fourth-order φi terms in the Ginzburg-Landau expansion(2.4) can be neglected since the second-order terms are
positive. The resulting free energy with µ+ = µ− = 0 is

d r
2


1 
c (∇φ+ )2 + (∇φ− )22

1+ t(φ2+ + φ2− ) + αφ+ δ
2
1
1
1
1
+ K(∇2 δ)2 + Σ(∇δ)2 + V δ 2 + U δ 4
2
2
2
4
=






1  1
(t + cq 2 ) |φ̃+ (q)|2 + |φ̃− (q)|2A q 2+αφ̃+ (q)δ̃(−q)
 
1
1
2
4
2
+ (V + Σq + Kq )|δ̃(q)| + d2 r U δ 4 . (4.2)
2
4

(4.3)

Σe ≡ Σ +

α2 c
,
t2

Ue ≡ U.

(4.5)

We see that for t > 0 the coupling always increases the mechanical tension Σe > Σ, but reduces the rigidity Ke < K
and the potential strength Ve < V .
Let us consider the strong coupling case when Ve < 0
but still having Ke > 0, namely,

2

Fig. 7. Schematic drawing of the inter-membrane distanceof two adhering membranes with coexisting domains between:(a) tight (T) and middle (M) phases, and; (b) loose (L) andmiddle (M) phases, as applies from Figure 6 for bolaformsticker adhesion. The inter-membrane distance is exactly 0
for the middle phase, whereas it is smaller (larger) than 0 for
the tight (loose) phase.

φ̃− (q) = 0.

By inserting these equations into (4.2) and expanding forsmall q, the free energy can be written as



1
1
1
1
F = d2 r Ke (∇2 δ)2 + Σe (∇δ)2 + Ve δ 2 + Ue δ 4 ,
2
2
2
4
(4.4)
with
Ke ≡ K −MIDDLEF =αδ̃(q),
t + cq 2

α2 c2
,
t3
2
α
Ve ≡ V −
,
t

1



267

α2
<K
t

t
c

2

.

(4.6)

For t > 0, although no phase separation occurs in theabsence of coupling (α = 0), it occurs for non-zero α. The
minimum free energy of the membranes is given by solving
the Euler-Lagrange equation obtained by minimizing (4.4)
with respect to the inter-membrane distance δ:Ke ∇4 δ − Σe ∇2 δ + Ve δ + Ue δ 3 = 0.

(4.7)

The two uniform (bulk) solutions of (4.7) areδ0 = ± −Ve /Ue .

(4.8)

We assume a one-dimensional proﬁle δ(x) describing
the inter-membrane distance along the x-direction. A typical proﬁle determined by a numerical solution of the EulerLagrange equation (4.7) using a relaxational method isshown in Figure 8. It is convenient to rescale the variables
1/3−1/4 1/12
Ue x, respectively,
δ and x as ζ = Ue δ and u = Keyielding the following one-dimensional proﬁle equation:
d4 ζ
−
du4

Σe
1/2 1/6
Ke Ue

d2 ζ
+
du2

Ve
1/3
Ue

ζ + ζ 3 = 0.

(4.9)

Only two independent combinations of the four parameters Ke Σe , Ve and Ue exist. In Figure 8 they are set to
1/2 1/6
1/3
be Σe /(Ke Ue ) = 0.1 and Ve /Ue = −1, respectively.
The proﬁle has a large slope at the interface x = u = 0,
but relaxes to the bulk values ±δ0 at x = ±∞ in a nonmonotonic fashion with two symmetric overshoots, having
a height greater than δ0 . These overshoots are suppressedby increasing Σe or by increasing the coupling strength α.
The maximum value of δ at the overshoot scales as |Ve |1/2
as can be seen from (4.8). The overshoot of the proﬁle is268

The European Physical Journal E
1.2

the adhesion of a single ﬂexible membrane onto a supported membrane, the supported membrane cannot haveany shape ﬂuctuations. Therefore, the inter-membrane
distance proﬁle calculated in this section can be regarded
as a distance of the ﬂexible membrane from the substratewith respect to its equilibrium distance 0 .

0.8

ζ

0.4

0.0

5 Discussion-0.4

5.1 Main ﬁndings

-0.8

-1.2
-10

-5

0

5

10

u
Fig. 8. Inter-membrane distance proﬁle at the domain bound1/3
ary. The variables are rescaled as ζ = Ue δ and u =−1/4 1/12
Ke
Ue x. The values of the two independent parameters
1/2 1/6
1/3
are chosen to be Σe /(Ke Ue ) = 0.1 and Ve /Ue = −1.

1

TIGHT

LOOSE

2
interfaceFig. 9. Schematic drawing of the domain boundary and intermembrane distance of two adhering membranes. Two symmetric overshoots in the interface region between the loose andtight domains are shown, in agreement with the model andFigure 8. They result from the combined eﬀect of membrane
curvature and lateral tension.

followed by a damped oscillation which minimizes the curvature energy. This behavior is similar to the non-linear
response of membranes to local pinning sites [33–35] ormembranes adhering to a geometrically structured substrate [29] and is a result of the 4th-order derivative inthe proﬁle equation. The oscillatory decay has been also
predicted for the membrane proﬁle between two inclusions
such as proteins [36–39].
The conﬁguration of the phase separated membranes
corresponding to the above inter-membrane distance δ
is schematically represented in Figure 9. In the case of

In this paper, the interplay between adhesion and lateral phase separation of multi-component membranes isinvestigated. We consider the “bolaform-sticker” adhesion
where adhesive bridges are formed by a single sticker having two sticky segments and adhere directly onto the two
membranes, as shown in Figure 1(a). We proposed a phenomenological free energy consisting of three parts: i) the
free energy describing the lateral phase separation of stickers on each membrane (see (2.2)); ii) the deformation energy of the two membranes, which is the sum of the bending energy, the surface tension, and the potential energy(see (2.6)); and, iii) the coupling energy between the intermembrane distance and the average concentration of stickers on both membranes (see (2.12)). The diﬀerence of thechemical potentials between the two membranes is also
taken into account because the sticker concentrations do
not have to be the same.
We calculate the phase diagrams describing the bulk
properties for two particular choices of the chemical potentials, i.e., µ− = 0 (µ1 = µ2 ) and µ+ = 0 (µ1 = −µ2 ).
In the case of µ− = 0, the critical temperature increases
depending on the coupling strength and the potential
strength (see (3.8)). Hence the lateral phase separationis enhanced due to the adhesion. This is one of the mainconsequences of our model. When the phase separation
takes place, the inter-membrane distance is smaller for thedomains rich in the sticker molecules (“tight phase”), andlarger for the domains poor in the stickers (“loose phase”).
In the case of µ+ = 0, our model exhibits a tricritical behavior. The upward shift of the tricritical temperature also
indicates the enhancement of the lateral phase separation.
We ﬁnd that the line tension for the lateral phase separation increases because of the coupling eﬀect as long as
the mechanical surface tension is non-zero. We have also
calculated the inter-membrane distance proﬁle between
the two membranes which are quenched below their phaseseparation temperature. Because the membrane shape isgoverned by the bending rigidity, the inter-membrane distance proﬁle relaxes to the bulk values in a non-monotonic
way with two symmetric overshoots.
5.2 Membrane adhesion on solid surfaces andsupported membranes
So far, we have mainly discussed the adhesion of two membranes. Our model also applies to the case where a single
ﬂexible membrane with sticker molecules adheres to a ﬂat

S. Komura and D. Andelman: Adhesion-induced lateral phase separation in membranes

substrate or a supported membrane [20]. Let us discussthese two cases separately. For a ﬂat substrate without
any supported membrane on it, the contributions fromthe second membrane (say i = 2) can be dropped fromthe model. The coupling term (2.12) simply reduces to
αφ1 δ because the stickers are assumed to adhere directly
to the substrate. The second case is that of a supported
membrane with sticker molecules. Unlike the case of two
ﬂuctuating membranes discussed in Section 4, the supported membrane does not have any shape ﬂuctuations.
However, even in such situations, there is an enhancementof the phase separation due to the coupling eﬀect and the
upward shift of the critical temperature is given by (3.8).
Another related situation is the case where a membrane
is composed of two diﬀerent lipids and the membrane isput close to a ﬂat substrate. If the two lipids feel diﬀerent hydration force and prefer diﬀerent distances from thesubstrate, the phase separation between the two components will be enhanced by the adhesion for the same reason
described in this paper.

5.3 Relation to other models
There exists an analogy between the phase behavior of
our membrane system with that of metamagnets (magnets which undergo ﬁst-order phase transitions in an increasing magnetic ﬁeld) or 3 He-4 He mixtures described by
the BEG (Blume-Emery-Griﬃth) spin-one model [40,41].
Moreover, the phase diagrams for µ+ = 0 are analogous
to those describing the phase separation of two-component
mixtures in ﬂuid bilayers which also exhibits tricritical behavior [42] and other related amphiphilic systems [43–47].
However, in the former case of two-component bilayers
the concentration diﬀerence between the two leaﬂets ofthe membrane is linearly coupled to the curvature of thebilayer and the diﬀerence in the chemical potential is nottaken into account.
In our paper, we did not address the problem of theunbinding transition. We rather assumed that the membranes are always bound together, even in the absenceof any sticker molecules. This assumption is partiallymotivated by the experimental study of reference [23]where suspended membrane (part of the giant vesicle)
was claimed to be bound to the supporting membraneeven in the absence of sticker molecules. In this case, theinter-membrane distance  stays ﬁnite and it is permissible to expand the free energy around the minimum. Hencethe phase separation consists of loosely and tightly boundpatches. The interplay between unbinding transition andphase separation of multi-component membranes has been
considered in other theoretical works [14–16]. The adhesion there is only brought about by sticker molecules, andthe phase separation is induced both by attractive interactions and ﬂuctuation-induced interactions between the
stickers. Although their model treats a diﬀerent aspectof the more general problem, the ﬂuctuation eﬀect yields
similar consequences compared to ours. We assumed thatthe cis-interaction in (2.2) is attractive, and tracing over269

the inter-membrane distance δ yields a term proportional
to γφ2+ . Since this term does not depend on the sign of
φ+ , it has a similar eﬀect as ﬂuctuations although ourtreatment is restricted to the mean-ﬁeld level.
It is worthwhile to comment here the diﬀerence between the present study and that of reference [23]. In theirpaper, it is found that the adhesion between the membranes including homophilic recognition molecules and repeller molecules is controlled by lateral phase separation.
The multiple competing states of adhesion is attributed
to the double-well inter-membrane interaction potential
generated by the competition of two forces; attraction between homophilic molecules and the repulsion between
repeller molecules. By changing the repeller concentration, the double-minimum potential causes the ﬁrst-ordertransition between a state with inter-membrane spacing
set by the thickness of the repeller molecules to a statewith a spacing set by the bare potential (van der Waals
plus hydration interactions). In our work, the eﬀect of repeller molecules is not taken into account and the minimum of the potential depends on the sticker concentration through the coupling term (2.12). When the stickers are phase separated and two diﬀerent values of the
sticker concentration coexist, the inter-membrane potential has double-minimum. However the physical origin of
this double-minimum potential is diﬀerent from that inreference [23] because it is not due to the presence of repeller molecules.

5.4 Other types of sticker molecules
As mentioned in the introduction, “homophilic-sticker”
adhesion occurs when the adhesive bridges are formed by
two stickers of the same type bound together by their two
sticky segments (see Fig. 1(b)). Suppose ψi (i = 1, 2) denotes the sticker concentration on each membrane. Thenthe inter-membrane distance depends on the product ofeach sticker concentration expressing the probability tohave two stickers —one on each membrane— at the same
position. Using (2.1) this coupling term can be written interms of φi asψ1 ψ2 = φ1 φ2 + ψc (φ1 + φ2 ) + ψc2 .

(5.1)

An interesting remark can be made for homophilic-sticker
adhesion. The resulting phase separation within eachmembrane leads to three diﬀerent values for the intermembrane distance. The inter-membrane distance between domains rich in stickers on both membranes (richrich), as well as between rich-poor domains, and poorpoor domains can be diﬀerent [14,15]. Notice that thesethree diﬀerent inter-membrane distances correspond to
the tight, middle, and loose phases in our model.
A third case is that of “lock-and-key” adhesion due
to the formation of chemical bonds between lock-and-key
types of stickers, e.g., ligands and receptors (see Fig. 1(c)).
Suppose that both types of stickers are distributed on thetwo membranes and ψi now represents the local concentration, say, of the lock molecules. First let us assume

270

The European Physical Journal Ethat the membranes are saturated with sticker molecules(no lipid). Then, 1 − ψi (r) represents the concentration of
key molecules. Since domains rich in lock (key) molecules
on one membrane adhere with domains rich in key (lock)molecules on the other membrane, the coupling term in
the free energy F3 will be a coupling between the intermembrane distance andψ1 (1 − ψ2 ) + (1 − ψ1 )ψ2 =−2φ1 φ2 + (1 − 2ψc )(φ1 + φ2 ) + 2ψc (1 − ψc ).

(5.2)

This term is symmetric with respect to the exchange oftwo membranes. Both in (5.1) and (5.2), we see that thelowest-order term in the concentration (except the constant term) is proportional to φ+ . If there is a linear
coupling between the inter-membrane distance and φ+ inthese cases, we expect an upward shift in the transition
temperature and the phase separation will be enhanced asargued above. Due to the presence of higher-order termsin (5.1) and (5.2), however, the phase behavior will be
more complex.
Let us now take into account the presence of lipids
in the lock-and-key adhesion. If a single type of sticker
is present on each membrane, namely, lock molecules onmembrane 1 and key molecules on membrane 2, we canregard ψ1 and ψ2 as the concentrations of lock and key
molecules embedded in the lipid membrane, respectively.
Then 1 − ψ1 and 1 − ψ2 describe the concentration of
the second component (lipid) on each membrane, respectively. In this case, the inter-membrane distance dependson ψ1 ψ2 as in (5.1). When both lock and key molecules
are present on both membranes, one has to start witha three-component mixture for each of the membranes.
The generic lattice model to study the behavior of ternary
membranes of monolayers is the BEG spin-one model [40,41,48]. Here one has to include the coupling between thetwo membranes. If we denote the concentration of lock andkey molecules on each membrane as ψiL and ψiK (and hence
the concentration of the dilution lipid is 1 − ψiL − ψiK ), theinter-membrane distance now depends on ψ1L ψ2K + ψ1K ψ2L
which is similar to (5.2). More detailed calculations for thehomophilic stickers and lock-and-key stickers and their inﬂuence on membrane adhesion are left for future studies.

We have greatly beneﬁted from the discussions and correspondence with R. Lipowsky, R. Netz, J. Rädler, E. Sackmann,
and T. Weikl. SK would like to thank the Ministry of Education, Science and Culture, Japan for providing ﬁnancial
support during his visit to Israel. Support from the exchange
program between the Japan Society for the Promotion of Science (JSPS) and the Israel Ministry of Science and Technology is also gratefully acknowledged. DA acknowledges partial
support from the Israel Science Foundation founded by the
Israel Academy of Sciences and Humanities, Centers of Excellence Program and the US-Israel Binational Science Foundation (BSF) under grant number 98-00429.

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