PHYSICAL REVIEW E 86, 021916 (2012)

Concentration fluctuations and phase transitions in coupled modulated bilayers
Yuichi Hirose and Shigeyuki Komura*
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, JapanDavid Andelman
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel(Received 28 March 2012; published 16 August 2012)
We consider the formation of ﬁnite-size domains in lipid bilayers consisting of saturated and hybrid lipids.
First, we describe a monolayer model that includes a coupling between a compositional scalar ﬁeld and a
two-dimensional vectorial order parameter. Such a coupling yields an effective two-dimensional microemulsion
free energy for the lipid monolayer, and its characteristic length of compositional modulations can be consideredas the origin of ﬁnite-size domains in biological membranes. Next, we consider a coupled bilayer composed oftwo modulated monolayers and discuss the static and dynamic properties of concentration ﬂuctuations above
the transition temperature. We also investigate the micro-phase separation below the transition temperature andcompare the micro-phase separated structures with statics and dynamics of concentration ﬂuctuations above thetransition.
DOI: 10.1103/PhysRevE.86.021916

PACS number(s): 87.16.D−, 64.60.−i, 47.57.jb

I. INTRODUCTION

Biomembranes are two-dimensional (2D) ﬂuids that separate the inner and outer environment of organelles in biologicalcells. Naturally occurring biomembranes consist typicallyof numerous lipid species, sterols, sugars, and membraneproteins. According to the “lipid raft” hypothesis [1], someof the lipid components and/or proteins are incorporated intoﬁnite-size domains, which play an important role on cellularfunctions such as signal transduction processes. Recent experiments suggest that lipid rafts are nothing but dynamicalmolecular assemblies of 20 nm in size with ﬁnite lifetimes in
the order of 10–20 ms [2].
Being motivated by the raft hypothesis, a large numberof investigations [3–14] have been conducted to reveal theproperties of artiﬁcial membranes consisting of lipid mixturesand cholesterol. Below the miscibility transition temperature, formation of micron-size domains were observed using
ﬂuorescent microscopy [3–6]. In some cases, rather than amacroscopic phase separation, domains with distinct size inthe micrometer range have been reported [7–9]. For example,
various types of modulated (stripe or hexagonal) patterns
have been found for multicomponent lipid and cholesterol
mixtures [10]. Above the miscibility transition temperature,even multicomponent membranes do not phase separate, andtheir concentration ﬂuctuations around the homogeneous state
can be investigated [11–13], in particular, close to the criticalpoint, Tc . Furthermore, it is interesting to note that criticalconcentration ﬂuctuations have been observed in membranes
extracted from living cells [14].
One of the main reasons that initiated the notion of lipid
rafts in biomembranes is the existence of ﬁnite-size domains
rather than domains resulting from a macroscopic phaseseparation [15]. Assuming that membranes are in equilibrium,
the same question can be phrased in terms of whether thebiomembrane state is above or below the miscibility transition*

komura@tmu.ac.jp1539-3755/2012/86(2)/021916(13)temperature. In the high-temperature one-phase region, theonly relevant length scale is the correlation length associatedwith concentration ﬂuctuations, and it diverges at the criticalpoint, Tc [16]. Below this temperature, there should be aphysical mechanism suppressing domain coarsening in order
to explain the existence of ﬁnite-size domains in equilibrium.
Yet another characteristic feature of biomembranes isthat the lipid composition of the two leaﬂets (monolayers)constituting the bilayer is not the same [17]. Moreover,such asymmetric monolayers are not independent but arecoupled to one another. This was conﬁrmed experimentallyby investigating the phase separation of bilayers with different
monolayer lipid compositions [18] or seen in simulations [19].
One mechanism that leads to the coupling between the two
leaﬂets is the lipid chain interdigitation occurring at the
midplane of the bilayer [20–22], which may affect the domain
size in asymmetric bilayers.
One possibility to account for such ﬁnite-size domains
in lipid mixtures is to consider the special role played by“hybrid lipids” such as POPC or SOPC. These lipids have
one saturated hydrocarbon chain and another unsaturated
one and are major components of biological membranes. It
was suggested that hybrid lipids act as a line-active agent
at boundaries of 2D domains of immiscible lipids [23–26].
Hence, a mixture containing hybrid lipids can be regarded
as a “2D microemulsion” [13,23–27]. For microemulsions it
is known that there is another length scale, in addition to thecorrelation length mentioned above, related to the size of waterand oil microdomains [28].
Using the microemulsion analogy, we suggest that anadditional length scale can be responsible for the ﬁnite-size
structures in biomembranes. We propose a model for bilayers
consisting of two coupled monolayers that are both in a 2D
microemulsion state, manifesting modulated phases. First,
we describe a monolayer model that includes a coupling
between the lipid composition and a 2D vectorial orderparameter. The model accounts for the line active nature of
hybrid lipids. Next, we consider a bilayer in which two such

021916-1©2012 American Physical Society

YUICHI HIROSE, SHIGEYUKI KOMURA, AND DAVID ANDELMAN

monolayers are coupled through an interleaﬂet interaction.
For the coupled bilayer, we discuss the static and dynamic
properties of concentration ﬂuctuations above Tc and investigate their micro-phase separation below Tc [29]. Intermediate
structures arise when two competing structures have different
characteristic length scales. One of our important conclusions
is that the micro-phase-separated structures below Tc reﬂect
the static and dynamic properties of concentration ﬂuctuations
above Tc .
This paper is organized as follows. In Sec. II, after describing how the microemulsion state is obtained for monolayers, amodel for coupled modulated bilayers is presented. In Sec. III,
we show the results of static and dynamic structure factors
for the coupled bilayers above Tc . In Sec. IV, we describe
some results for competing micro-phase separation in coupledbilayers below Tc . Finally, in Sec. V, further discussion andﬁnal remarks are presented.
II. MODEL
A. Modulated lipid monolayers

We consider a 2D microemulsion formation in a monolayer
consisting of two types of lipids: a saturated lipid (denoted
by “S”) and a hybrid one (“H”). As shown schematically in
Fig. 1(a), the saturated lipid has two saturated chains, whereas
the hybrid lipid has one saturated chain and an unsaturated
one. It is well known [3–6] that the resulting liquid-ordered(Lo ) and liquid-disordered (Ld ) phases are rich in saturated
and hybrid lipids, respectively, as is depicted in Fig. 1(c). In
the experiments mentioned previously, cholesterol is usually
added as a third component and is known to affect the area
per headgroup of lipids [30,31]. However, since cholesterol
has a strong preference for the Lo phase and affects mainly thesaturated lipid, we do not consider cholesterol explicitly in ourmodel and neglect its presence hereafter.
The local area fraction of saturated and hybrid lipids
are deﬁned as φS (r) and φH (r), respectively, where r =(x,y) is a 2D vector. Under the incompressibility condition, φS + φH = 1, the only relevant order parameter isthe difference between the two lipid compositions, φ =φS − φH . The phenomenological free energy that describes
the 2D phase separation between S and H lipids is given
by a Ginzburg-Landau expansion in terms of the orderparameter φ,



σ
τ
1
(∇φ)2 + φ 2 + φ 4 − μφ .
Fs [φ] = d 2 r
(1)
2
2
4
Here σ > 0 is related to the line tension between monolayer
domains, τ ∝ T − Tc is the reduced temperature with respect
to the critical point Tc , and μ is the chemical potential that
regulates the average φ value in the monolayer. Without lossof generality, the coefﬁcient of the quartic term is set to be apositive constant by an appropriate rescaling of the position
variable, r.
Next, we discuss the role of the hybrid lipid and its
effect on the phase separation. We deﬁne a 2D lateral vectorb pointing from the unsaturated tail of the hybrid lipid
toward its saturated one, as depicted in Fig. 1(a). We thenintroduce a coarse-grained 2D vectorial order-parameter ﬁeld

PHYSICAL REVIEW E 86, 021916 (2012)m(r) = (mx (r),my (r)), as shown in Fig. 1(b). This vector is thespatial average of the b vectors over areas large as comparedwith molecular size but still small enough as comparedwith macroscopic scales. The phenomenological free energy
associated with the vectorial ﬁeld m (and due to the hybrid
lipids) can be written as an expansion up to quadratic orderin m [32],



K
a
(∇ · m)2 + m2 .
(2)Fh [m] = d 2 r
22
The coefﬁcient K is the 2D elastic constant, while a istaken to be positive so that m = 0 is the stable homogeneousstate. Although an additional term (∇ × m)2 is allowed bysymmetry, it is not included in Eq. (2) because this term does
not modify our results. The reason why the term (∇ × m)2 isirrelevant after the minimization is due mainly to the form of
the coupling term discussed in the next.
At the Lo /Ld interface, hybrid lipids orient their saturated
and unsaturated chains toward the Lo and Ld phases, respectively, thereby reducing the chain mismatch [25,26] as shownin Fig. 1(c). Within our phenomenological approach, the roleof hybrid lipids can be represented by a coupling between(a)

H lipid

S lipidbsaturated
(b)

(c)

m

m

unsaturated

m

Lo

Ld

FIG. 1. (a) Saturated lipids (S) with two saturated hydrocarbon
chains, and hybrid lipids (H) with one saturated and anotherunsaturated chain. For each hybrid lipid, we deﬁne a 2D vector b
pointing from its unsaturated tail toward the saturated one. (b) The
vector m denotes a coarse-grained 2D vectorial order parameter overthe microscopic b vectors. (c) A binary lipid monolayer consistingof saturated and hybrid lipids. At the interface between the Lo andLd phases, hybrid lipids orient their saturated chains toward the Lophase. The distribution of the m(r) vector ﬁeld is shown by arrows ofvariable length.

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CONCENTRATION FLUCTUATIONS AND PHASE . . .

PHYSICAL REVIEW E 86, 021916 (2012)

the lateral variation of φ and the vectorial ﬁeld m. To lowest
orders, the coupling term in the free energy is given by(3)
Fc [φ,m] = − d 2 r m · (∇φ),
where  is a positive coupling constant because the vector mtends to orient toward the Lo domains. The total monolayer freeenergy is given here by the sum of the three terms introducedin Eqs. (1)–(3): Fm = Fs + Fh + Fc . Notice that our model
is valid in the weak segregation limit (close to Tc ), because
slow spatial variation of the order parameters is intrinsicallyassumed. We further note that a similar free-energy functional
using a vectorial order parameter was proposed for 3D
microemulsions [33].
The total free energy can be conveniently expressed using
the 2D Fourier transform of φφ(q) = d 2 r φ(r)e−iq·r ,(4)

Taking the inverse Fourier transform, Eq. (7) can be
expressed in position space asτ 2 1 42
2 2
2Fm [φ] = d r 2B(∇ φ) −2A(∇φ) + φ + φ − μφ .
24(10)This is the “2D microemulsion” free energy for a monolayercomposed of a binary lipid mixture. When A > 0, the negative
gradient-squared term favors spatial modulations, while thepositive Laplacian squared term with B > 0 suppresses modulations. As mentioned above, Yamamoto et al. [25,26] showedthat the effective line tension between domains becomes
negative for membranes consisting of saturated and hybrid
lipids.
More generally, models based on equations similar toEq. (10) have been used successfully in the past to describe
modulated phases [34] arising in a variety of different
biophysical and chemical systems such as Langmuir ﬁlms [35],lipid membranes [36,37], and diblock copolymers [38].

where q = (qx ,qy ) is a 2D wave vector and, similarly, m(q) is
the Fourier transform of m(r). Minimizing Fm with respect tom, we obtain its optimum value asm(q) =iqφ(q),a + Kq 2(5)

where q = |q|. By substituting back Eq. (5) into the monolayer
free energy Fm , the minimized free energy is expressed as
d 2q 1
2q 2
2
φ(q)φ(−q)τ+σ
qFm [φ] =−(2π )2 2
a + Kq 2

1+ d 2 r φ 4 − μφ ,(6)4where the Fourier transform has been used only for the secondorder terms in φ. Expanding the effective binary interaction
for small q, we obtain



d 2q
τ
4
2
2Bq
φ(q)φ(−q)
Fm [φ] ≈−2Aq+(2π )22

1+ d 2 r φ 4 − μφ ,(7)4where two new parameters are deﬁned


K 2
1 2
B≡
, A≡
−σ .
4a 2
4 a

(8)

When the coupling constant  is small enough, i.e.,  2 /a < σ(or A < 0), the minimum of the Gaussian term (ﬁrst integral)in Eq. (7) occurs at q ∗ = 0 and is a signature of a macroscopic
phase separation (for τ < 0). On the other hand, when  islarge enough, i.e.,  2 /a > σ (or A > 0), the minimum occurs
at a nonzero wave number q ∗ = 0 given byAa(1 − aσ/  2 )
∗
=
,
(9)
q =
2B
2K
indicating a potential micro-phase separation with q ∗ > 0
modulation.

B. Coupled lipid bilayers

We consider two coupled modulated monolayers forming
a bilayer as shown in Fig. 2. Each monolayer is a binary
mixture of saturated and hybrid lipids. We deﬁne two local
order parameters for the two monolayers, φ = φS − φH andψ = ψS − ψH , depicted in Fig. 2. The coarse-grained freeenergy functional is then written as [29]τφ
1
Fb [φ,ψ] = d 2 r 2B(∇ 2 φ)2 − 2A(∇φ)2 + φ 2 + φ 4
24− μφ φ + 2D(∇ 2 ψ)2 − 2C(∇ψ)2τψ
1+ ψ 2 + ψ 4 − μψ ψ − φψ .
2
4

(11)

The ﬁrst ﬁve terms depend only on φ and its derivatives
and describe the upper monolayer in Fig. 2 and its possible
modulations. These are the same terms as in Eq. (10)for the single monolayer case. Similarly, the latter ﬁve ψterms describe the lower monolayer, where τφ and τψ are
the two reduced temperatures, while μφ and μψ are thecorresponding chemical potentials. The last term, φψ, represents the coupling between the two leaﬂets with a coupling
constant .
We comment here on the physical origin of the proposed
coupling term, −φψ, in Eq. (11). Note that this quadraticterm is invariant under exchange of φ ↔ ψ. When  > 0,φψFIG. 2. Schematic illustration of two coupled modulated monolayers forming a bilayer membrane. Each monolayer is composedof a binary S/H lipid mixture, which can have a lateral composition
modulation. The relative composition of S and H lipids in the upperand lower leaﬂets are deﬁned by φ and ψ, respectively. The lipid tails
interact across the bilayer midplane.

021916-3

YUICHI HIROSE, SHIGEYUKI KOMURA, AND DAVID ANDELMAN

this term can be obtained from a term such as +(φ − ψ)2 ,which represents a local energy penalty when the upper andlower monolayers have different compositions [20,21]. For
mixed lipid bilayers, such a coupling may result from theconformational conﬁnement of the lipid chains and, hence,
would have an entropic origin [20]. By estimating the degreeof lipid chain interdigitation, the order of magnitude of thecoupling parameter  can be evaluated [22]. In general, can also be negative depending on the speciﬁc couplingmechanism [39]. However, the sign of  does not affect our
results in any essential way because the free energy Eq. (11) isinvariant under the combined exchange of either φ → −φ and → − or ψ → −ψ and  → −. Since such an effectonly changes the sign of the chemical potential (either μφ orμψ ), it is sufﬁcient to consider the  > 0 case.
The other possible higher-order coupling terms which areallowed by symmetry are φ 2 ψ 2 , φψ 3 , and φ 3 ψ. However,these terms do not affect the properties of concentration
ﬂuctuations in any essential way and will not be consideredhereafter.

PHYSICAL REVIEW E 86, 021916 (2012)and similarly for Sψψ and Sφψ ,2gφ (q),4gφ (q)gψ (q) − 2,Sφψ (q) = Sψφ (q) =4gφ (q)gψ (q) − 2Sψψ (q) =A. Static structure factor

The spatially varying φ(r) and ψ(r) can be written as φ(r) =φ0 + δφ(r) and ψ(r) = ψ0 + δψ(r), respectively, where φ0 =φ and ψ0 = ψ are the spatially averaged monolayercompositions, and δφ and δψ describe the deviations fromtheir average values. In thermal equilibrium, φ0 and ψ0 satisfy
τφ φ0 + φ03 − μφ − ψ0 = 0,τψ ψ0 + ψ03 − μψ − φ0 = 0.

(12)

Expanding the free energy, Eq. (11), and retaining the quadraticorder terms in δφ and δψ, we obtain the Gaussian free energy2FG [δφ,δψ] = d r 2B(∇ 2 δφ)2 − 2A(∇δφ)2+

φ(δφ)2 + 2D(∇ 2 δψ)2 − 2C(∇δψ)2ψ
+ (δψ)2 − (δφ)(δψ) ,
(13)
2
2

where the notations φ = τφ + 3φ02 and ψ = τψ + 3ψ02 as
well as Eq. (12) are used.
The static partial structure factor is Sφφ (q) =δφ(q)δφ(−q) , and because of the radial symmetry,Sφφ (q) = Sφφ (q), where q = |q|. It is given by2gψ (q)Sφφ (q) =,4gφ (q)gψ (q) − 2

(14)

(16)

and
gφ (q) = 2Bq 4 − 2Aq 2 +gψ (q) = 2Dq 4 − 2Cq 2 +

φ

2

,

ψ

2

(17)
.

Since the structure factors diverge at Tc , we see that the
coupling parameter  effectively shifts the critical temperature
to lower values.
B. Decoupled leaflets ( = 0)

When the two leaﬂets are decoupled,  = 0, it is sufﬁcient
to present results for only one of the two monolayers, say, theφ one. From Eq. (14), the decoupled structure factor is simply
given byIII. CONCENTRATION FLUCTUATIONS ABOVE Tc

Using the bilayer free energy, Eq. (11), we obtain the staticand dynamic structure factors, which describe the properties ofconcentration ﬂuctuations for two coupled monolayers in theirrespective one-phase region (above Tc ). We shall closely follow
the formulation of Ref. [40] in which the coupled macro-phaseseparation was discussed for bilayers.

(15)

Sφφ (q) =1.
2gφ (q)(18)

The correlation function Gφφ (r) = δφ(r)δφ(0) = Gφφ (r),where r = |r|, is obtained by use of the inverse 2D Fouriertransform of Eq. (18) (see Appendix A for the derivation)
ξφ λ φr(1) 2π r
Gφφ (r) =
,
(19)
+i
Re H0
32π B
λφ
ξφ
where H0(1) (z) is the zeroth-order Hankel function of theﬁrst kind and “Re” denotes the real part. This correlation
function contains two length scales; the ﬁrst is the modulation
periodicity
 1/4
B
2
λφ

=
(20)
2π
1
− γφ
φ
and the second is the correlation length 1/4
2
B

ξφ =
,
(21)
1 + γφφwhere γφ = −A/ φ B.
When −1 < γφ < 1, both λφ and ξφ are ﬁnite, and thecorresponding phase is called the structured-disordered phase.
In Fig. 3, we plot Eq. (19) for certain parameter values. The
correlation function has exponentially decaying oscillations as
a function of the distance r, similar to 3D microemulsions[28].
√
The peak of the structure factor Sφφ occurs at q ∗ = A/2B >0 for −1 < γφ < 0, whereas q ∗ = 0 for 0  γφ < 1. In theformer case, the peak height is given by
Sφφ (qφ∗ ) =

1
−
A2 /B
φ

(22)

and diverges at φ∗ = A2 /B. The line at γφ = 0 is the Lifshitz
line. The modulation periodicity λφ diverges for γφ → 1, andthe monolayer transforms into a disordered phase for γφ > 1.

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CONCENTRATION FLUCTUATIONS AND PHASE . . .

PHYSICAL REVIEW E 86, 021916 (2012)(a)

Gφφ (r)/(ξφλφ/32πB)

1

OD

SD

DD

0 OS

SS

DS

SO

DO

0

1

1

γψ

0.5

-1

0

OO

-1

0

10

r

0
20

C. Coupled leaflets (  = 0)

When the two monolayers are coupled ( = 0), thecorresponding phase diagram can be obtained by analyzing
the poles in the complex plane of Eqs. (14)–(16). One example
is shown in Fig. 4(b). The boundary between the OO and SS
phases (solid line) is determined by the condition that all thestructure factors diverge. In the DD phase, all the poles are pure
imaginary, whereas in the SS phase the poles are complex, andat least one real pole exists in the OO phase.
In Fig. 4(b) we see that the asymmetric phases such as DO
and SD are suppressed as compared with Fig. 4(a). This isbecause one of the leaﬂets induces modulations in the otherleaﬂet due to the interleaﬂet coupling, . In fact, all theasymmetric phases disappear for any ﬁnite value of . In
addition, the OO and SS phases are expanded as compared
to Fig. 4(a). As mentioned before, increasing the coupling effectively lowers the bilayer temperature. The Lifshitz lines(dotted lines) are obtained by plotting the main peak positionsof Eqs. (14) and (15). Due to the coupling effect, they are tilted
as compared to the  = 0 case where the lines occur at γφ = 0and γψ = 0.
In Figs. 5(a) and 5(b), we plot the structure factorsof the decoupled and coupled bilayers, respectively. As an
illustration of the coupling effect with  = 0.3, we consider

DD

(b)

FIG. 3. The rescaled (dimensionless) real-space correlation function, Gφφ , of a single monolayer in the structured-disordered phase as
a function of r. The parameters are B/ φ = 1 and γφ = −0.9, and thetwo resulting characteristic lengths are λφ /2π 1.45 and ξφ 6.32.

The line of γφ = 1 is called the disorder line and is not a phasetransition line. On the other hand, the correlation length ξφdiverges for γφ → −1 (hence, this is the critical point), andthe ordered phase appears for γφ < −1 [28].
As shown in Fig. 4(a), the phase diagram of a decoupledbilayer is easily obtained by combining the phase sequencesin terms of the two independent parameters γφ = −A/ φ Band γψ = −C/ ψ D. The various phases are expressed by
the binary combination of ordered (O), structured-disordered(S), and disordered (D) phases for each of the leaﬂets. The twodisorder lines are shown in the ﬁgure by dashed lines, while
the Lifshitz lines occur at γφ = 0 and γψ = 0 (not shown).

γφ

1
SS

γψ 0
-1

OO

-1

0

γφ

1

FIG. 4. (a) Phase diagram of a decoupledmodulated bilayer( =0) plotted in the plane of γφ = −A/ φ B and γψ = −C/ ψ D.
The phases are labeled using a binary combination of two lettersrepresenting phases in the φ and ψ monolayers. “O,” “S,” and “D”
denote the ordered, structured-disordered, and disordered phases,
respectively. Solid and dashed lines are the phase transition anddisorder lines, respectively. (b) Phase diagram of a coupled modulated
bilayer ( = 1) for φ = ψ = 2, B = D = 0.5. In addition, the two
dotted lines are the Lifshitz lines.

two different characteristic wavelengths, qφ∗ = qψ∗ . The heightsof the two peaks are set to be equal by requiring φ = ψ andA2 /B =√C 2 /D = 1 [see Eq. (22)]. The peak height of Sφφ atqφ∗ = 1/ 2 is increased [as in Fig. 5(b)] due to the coupling√
effect, whereas that of Sψψ at qψ∗ = 3/ 2 is almost unchanged
compared with the decoupled case. We also plot Sφψ given by
Eq. (16) which represents the cross correlation of ﬂuctuations
between the two monolayers. This quantity is proportional tothe coupling constant  and its peak
√ position is essentiallydetermined by that of Sφφ at qφ∗ = 1/ 2.
∗
for the coupled
The peak position of Sφψ denoted as qφψcase ( = 0.3) is obtained numerically and plotted in Fig. 6(a)as a function of qφ∗ and qψ∗ , which are, respectively, the peak
positions of Sφφ and Sψψ for the decoupled case. We ﬁnd that∗for the coupled case, the value of qφψis almost equal to thatof the smaller of (qφ∗ ,qψ∗ ). In Fig. 6(b), the peak height of Sφψ
∗at qφψ
is plotted as a function of the coupling parameter and the ratio of the two characteristic wave numbers qψ∗ /qφ∗

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YUICHI HIROSE, SHIGEYUKI KOMURA, AND DAVID ANDELMAN

3(a)

PHYSICAL REVIEW E 86, 021916 (2012)

(a)

Sφφ
S ψψ

2

q∗φψ=2

S(q)

2

qψ∗

1.5

1

1

1
0.5

0 -1
10
3

10

0

q

(b)

10

0
0

1

2

qφ∗

(b) 3

Sφφ
Sψψ
Sφψ

0.1

1

5

10

0.4

0.6

0.8

0.5

2.5
∗

2 qψ

S(q)

2

1

2

√

1

1.5
0 -1
10

10

0

q

10

1

1
0

FIG. 5. Bilayer structure factors Sφφ , Sψψ , and Sφψ as a functionof the wave number q√for φ = ψ √= 1.5, B = A = 1, D = 0.0123,
C = 0.1111 (qφ∗ = 1/ 2,qψ∗ = 3/ 2). (a) The decoupled case,  =0; (b) the coupled case with  = 0.3.

√
for the speciﬁc case of qφ∗ = 1/ 2, as the parameters are ﬁxed
to B = A = 1 and C 2 /D = 1. The peak height increases as is increased because the temperature of the coupled bilayer
is effectively lowered. To see this clearly, we have plotted inFig. 6(b) the phase-transition line (dashed line) separating thestructured-disordered and ordered√ phases. The peak height ofSφψ also increases as qψ∗ /qφ∗ = 2qψ∗ approaches unity, whichis the case where Sφφ and Sψψ completely overlap.

D. Dynamic structure factors

The dynamical ﬂuctuations in composition, δφ(r,t) andδψ(r,t), depend on time t and are now considered for coupledmodulated bilayers. We assume that both δφ and δψ areconserved for each monolayer. This means that the exchangeof lipids between the two monolayers is negligible. Such anassumption is justiﬁed when the typical time scale for lipid
ﬂip-ﬂop motion is considerably large as compared to that forthe domain formation (see Sec. V). Hence, the time evolutionof δφ and δψ are given by the following diffusive equations

0.2

Λ

1

∗
FIG. 6. (a) Contour plot of the peak position qφψ
of the structure√√
factor Sφψ as function of qφ∗ = A/2B and qψ∗ = C/2D. The
2parameters are  = 0.3, φ = ψ = 1.5, and A /B = C 2 /D. The
dotted line represents qφ∗ = qψ∗ . (b) Contour plot of the peak height√∗of the structure factor Sφψ (qφψ) as a function of  and 2qψ∗ . For√
the choice of parameters B = A = 1 and C 2 /D = 1, qφ∗ = 1/ 2, the√two characteristic wave numbers satisfy qψ∗ /qφ∗ = 2qψ∗ . The dashedline is the phase-transition line.

for each monolayer:∂δφ(r,t)δFG= Lφ ∇ 2+ νφ (r,t),∂t
δ(δφ)∂δψ(r,t)
δFG= Lψ ∇ 2+ νψ (r,t).
∂t
δ(δψ)

(23)

Here Lφ and Lψ are the kinetic coefﬁcients taken to be constants, and νi (r,t) represent Gaussian white noise, satisfying
νi (r,t) = 0 andνi (r,t)νj (r ,t  ) = −δij Li ∇ 2 δ(r − r )δ(t − t  ),(24)

where i,j = φ,ψ and · · · now indicates the average overspace and time. In the above equations, we have also neglected
any hydrodynamic effects. In general, one has to take into
account the ﬂuid nature of both the bilayer membrane and thesurrounding bulk solvent [41].

021916-6

CONCENTRATION FLUCTUATIONS AND PHASE . . .

PHYSICAL REVIEW E 86, 021916 (2012)

TABLE I. Expressions for αij (q) and βij (q).
αij (q)
2Lφ q 2 (ωψ2 − ω+2 )ω−2 − ω+2

βφφ =2Lψ q 2 (ωφ2 − ω+2 )αψψ =ω−2 − ω+2
αφψ =

2

(a)

βij (q)
2Lφ q 2 (ω−2 − ωψ2 )ω−2 − ω+2

2Lψ q 2 (ω−2 − ωφ2 )βψψ =ω−2 − ω+2

2Lφ Lψ q 4 ωφψ
ω−2 − ω+2βφψ = −

1/ω

αφφ =

10

2Lφ Lψ q 4 ωφψ
ω−2 − ω+210


δφ(q,ω) =d 2 r dt δφ(r,t)e−i(q·r−ωt) ,

0

-2

-1

The Fourier transform of ﬂuctuations in both space and time
is

10

10

(25)

10

1/ω

and a similar Fourier transform is used for δψ andνi . The dynamic structure factors such as Sφφ (q,ω) =δφ(q,ω)δφ(−q,−ω) are given by
αij (q)βij (q)+ 2,
Sij (q,ω) = 2
ω + [ω+ (q)]2
ω + [ω− (q)]2(26)

where the explicit expressions of αij (q) and βij (q) are given
in Table I and the characteristic frequencies ω± (q) are[ω± (q)]2 =∓ωφ2 (q) − ωψ2 (q)2+ 4Lφ Lψ q 4 2 ωφψ(q) ,ωφ2 (q) = 4L2φ q 4 [gφ (q)]2 + Lφ Lψ q 4 2 ,(28)

ωψ2 (q) = 4L2ψ q 4 [gψ (q)]2 + Lφ Lψ q 4 2 ,(29)

ωφψ (q) = 2q 2 [Lφ gφ (q) + Lψ gψ (q)],(30)

and gφ and gψ are deﬁned in Eq. (17). In the more special caseof Lφ = Lψ = L, ω± reduce to a simpler form,(gφ (q) − gψ (q))2 + 2 .
(31)
The intermediate structure factors Sij (q,t) depend explicitly
on time and are obtained by taking the inverse Fouriertransform in t:dω
Sij (q,t) =(32)
Sij (q,ω)e−iωt ,2π
and from Eq. (26), we obtain
αij (q) −ω+ (q)tβij (q) −ω− (q)tee+.
2ω+ (q)2ω− (q)

0

10

q

(b)

1

1/ω+
1/ω−

0

-2

-1

2

10

2

10

(27)

ω± (q) = Lq 2 gφ (q) + gψ (q) ∓

10

10

1 2
ω (q) + ωψ2 (q)
2 φ

1/ωφ
1/ωψ

10

0

q

10

1

FIG. 7. Decay time of concentration ﬂuctuations as a functionof the wave number q for φ = ψ√= 1.5, B = A =√1, D = 0.0123,
C = 0.1111 (resulting in qφ∗ = 1/ 2 and qψ∗ = 3/ 2), and L = 1.
(a) The decoupled case,  = 0; (b) the coupled case with  = 0.3.

1/ω± (q). However, when the two monolayers are decoupled( = 0), the Sφφ and Sψψ structure factors decay with a single
exponential characterized by a decay rate ωφ and ωψ [seeEqs. (28) and (29)], respectively.
For the decoupled case ( = 0), the decay times 1/ωφand 1/ωψ are plotted as a function of q in Fig. 7(a), withthe same parameters as those in Fig. 5. The plots show ashoulder√reﬂecting the characteristic
structure at wave numbers√qφ∗ = 1/ 2 and qψ∗ = 3/ 2. Notice that the larger the initiallength scale, the longer the decay time. For the coupled casewith  = 0.3, we plot 1/ω± in Fig. 7(b). Due to the coupling
effect, the two decay times split into a larger and smaller one,
1/ω+ > 1/ω− . The larger one, 1/ω+ , exhibits two shoulders,
while the smaller one, 1/ω− , has a shoulder between the two
characteristic wave numbers. The coupling affects the decay
time of the structure corresponding to the smaller wave number(larger length), similar to the effect seen in Fig. 5 for the static
structure factor.

(33)

IV. COUPLED MODULATED PHASES WITH DIFFERENTPERIODICITIES qφ∗  = qψ∗Hence, the decay of concentration ﬂuctuations is describedby a sum of two exponentials with two characteristic times

In our previous paper [29], we investigated the phasebehavior of coupled modulated bilayers by using the freeSij (q,t) =021916-7

YUICHI HIROSE, SHIGEYUKI KOMURA, AND DAVID ANDELMAN

energy, Eq. (11), below Tc . When the two monolayers havethe same preferred periodicity, qφ∗ = qψ∗ , we obtained themean-ﬁeld phase diagram exhibiting various combinationsof modulated structures such as stripe (S) and hexagonal(H) phases. In some cases, the periodic structure in one of
the monolayers induces a similar modulation in the secondmonolayer. Moreover, the region of the induced modulated
phase expands as the coupling parameter  becomes larger.
However, when the preferred periodicities in the two
leaﬂets are different, qφ∗ = qψ∗ , it is difﬁcult to obtain thephase diagram because the free-energy densities cannot be
obtained analytically. We then have to rely on numericalsimulations [29,42] to solve the time evolution of the two
coupled order parameters as explained below.
For the dynamics of coupled modulated bilayers below Tc ,we use the following time evolution equations [29]δFb∂φ(r,t)= Lφ ∇ 2
,
∂t
δφ

δFb
∂ψ(r,t)= Lψ ∇ 2
,
∂t
δψ

(34)

where the bilayer free energy, Fb , is given by Eq. (11). As in
Eq. (23), we have assumed that both φ and ψ are conservedfor each monolayer and neglect any hydrodynamic effects.
Below Tc the effect of thermal ﬂuctuations is less importantand the noise terms have been omitted in the above equations.

φψ

PHYSICAL REVIEW E 86, 021916 (2012)

Hereafter, the kinetic coefﬁcients Lφ and Lψ are set to unity
for simplicity.
We solve numerically the above 2D equations using periodic boundary conditions. Each run starts from a homogeneousstate with a small random noise around the average compositions φ0 and ψ0 . Time is measured in discrete time steps,
and t = 5000 corresponds to a well equilibrated system. In all
our simulations, the reduced temperature parameters are ﬁxed
to be τφ = τψ = 0.8. The characteristic
wave number in the φ√monolayer is ﬁxed as qφ∗ = 1/ 2 by setting B = A = 1, while√
the periodicity in the ψ monolayer qψ∗ = C/2D is varied andthe condition C 2 /D = 1 is used as before. In the following,
we shall consider only the two coupled stripe phases for φ0 =
ψ0 = 0. However, the combination of stripe and hexagonalphases leads to a rich variety of complex patterns [42].
When the characteristic wave numbers of the two decoupledmonolayers differ, qφ∗ = qψ∗ , the two coupled modulated
structures cannot apparently match each other. The frustration
between the two different periodicities is due to the interleaﬂet
coupling and affects their morphologies. In Fig. 8, we show
one example of the time evolution of√ a coupled micro-phase√separation for qψ∗ /qφ∗ = 3 (qφ∗ = 1/ 2 and qψ∗ = 3/ 2) and = 0.3. The spatial patterns of φ, ψ, φ + ψ, φ − ψ, andthe Fourier transformed pattern, ψ(q), are presented fortime steps of t = 25, 60, 250, and 5000. Starting from

φ−ψ

φ+ψ

ψ(q)

(a)

(b)

(c)

(d)

-1.7

0

1.7

FIG. 8. Time evolution of spatially modulated patterns in a coupled bilayer, which consists of two stripe monolayers with different
periodicities, qφ∗ = qψ∗ . Real-space patterns of φ, ψ, φ + ψ, φ − ψ, as well as the Fourier transform, ψ(q), are presented. The time steps are (a)t = 25, (b) t = 60, (c) t = 250, and (d) t = 5000. The parameters are φ0 = ψ0 = 0, τφ = τψ = 0.8, B = A = 1, D = 0.0123, C = 0.1111,
and  = 0.3, yielding qψ∗ /qφ∗ = 3. In order to emphasize the color contrast, the gray color code of the real-space patterns is chosen to vary
between −1.7 and 1.7.
021916-8

CONCENTRATION FLUCTUATIONS AND PHASE . . .

φ

PHYSICAL REVIEW E 86, 021916 (2012)

ψ

ψ(q)

φ−ψ

φ+ψ

(a)

(b)

(c)

-1.7

0

1.7

FIG. 9. Equilibrated patterns (time step t = 5000) of two coupled bilayers consisting of stripe monolayers with different periodicities. All
parameters in addition to  are the same as in Fig. 8. The  parameter values are (a)  = 0.1; (b)  = 0.3; (c)  = 0.5. In order to emphasize
the color contrast, the color code of the real-space patterns is chosen to vary between −1.7 and 1.7.

the isotropic state, the ψ monolayer forms ﬁrst stripes (a).
Then, as the φ monolayer starts to segregate at larger t, it
simultaneously chops the ψ stripes into smaller sections (b),and the ψ monolayer transforms into a ﬁngerlike patterns (c).
Reconnection of the φ stripes takes place after a long timeof annealing, and a pattern of alternating ﬁngers in the ψ
monolayer is ﬁnally obtained (d). From the time evolution of
the 2D Fourier patterns of the ψ monolayer, it is apparent thatthe intermediate structures are characterized by the two lengthscales of ratio 1:3.
In Fig. 9, we show the spatially modulated patterns at
t = 5000 of the two coupled√φ and ψ monolayers
with√different periodicities (qφ∗ = 1/ 2 and qψ∗ = 3/ 2 as before)
for different values of the coupling parameter  = 0.1, 0.3,
and 0.5. Notice that Fig. 9(b) is the same as Fig. 8(d).
In the weak-coupling case ( = 0.1), the two monolayers
exhibit two independent stripe morphologies with characteristic periodicities (called the “independent” morphology).
Here the two stripes essentially do not affect each other. As
the coupling constant is increased ( = 0.3), stripes with aﬁngerlike structure appear in the ψ monolayer, while the stripe
morphology in the φ monolayer is almost unaffected (called
the “intermediate” morphology). In the Fourier transformed
pattern of ψ, we clearly see that the structures with two
different characteristic wave numbers are coexisting. This
result is also in accord with the properties of the static
structure factors, Sij (q), having two different characteristic
lengths, as shown in Fig. 5. For a larger coupling parameter( = 0.5), very similar patterns are obtained for φ and ψ
and almost coincide with one another (called the “coincident”
morphology). It should be noted that the structure with thelarger wavelength dominates when the coupling is large

enough, and the sequence of morphological changes shownin Fig. 9, as  increases, is rather typical.
In order to quantify the three morphologies (independent,
intermediate, and coincident), we calculate the spatial averageof the product of the two compositions1(35)
d 2 r φ(r,t)ψ(r,t),φψ =A
where A is the total system area. In Fig. 10, we plot φψ at t =1000 (sufﬁcient for equilibration)
for various combinations√√of  and qψ∗ /qφ∗ = 2qψ∗ (for the case qφ∗ = 1/ 2). The
morphology of the obtained patterns is marked on the ﬁgure bycircles, crosses, and squares. The values of φψ are small for“independent” (circles) structures, while they become larger
for the “intermediate” (crosses) and “coincident” (squares)
morphologies. Although the morphological changes are gradual and do not represent a sharp transition, the intermediate
patterns appear roughly for 0.05 < φψ < 0.2.
√ The region ofintermediate structure expands as qψ∗ /qφ∗ = 2qψ∗ increases,
and the patterns coincide for  = 0.5. We note that although
the morphology cannot be solely determined by the quantityφψ , the behavior of φψ is similar to that of the peak values∗of the cross correlation, Sφψ (qφψ), presented in Fig. 6(b).
For the quantitative argument concerning the micro-phaseseparation dynamics in each monolayer, we have also calculated the two self-quantities,

1
φ2 =
(36)
d 2 r φ 2 (r,t)
A
and021916-9

ψ2 =

1
A


d 2 r ψ 2 (r,t).

(37)

YUICHI HIROSE, SHIGEYUKI KOMURA, AND DAVID ANDELMAN

(a)

3

(c)

(b)

0.6
<φψ>

0.4

(a)

0.4
φ2

2.5

0.3

2

√

2 qψ∗

PHYSICAL REVIEW E 86, 021916 (2012)

0.2

1.5

0.1

1

0

0

0.1

0.2

0.3

0.4

0.2

0 -1
0
10
10

0.5

10

Λ
0.6

FIG.
√ as a function of 
√ 10. Contour plot of φψ at t = 1000
and 2qψ∗ , for the speciﬁc case of qφ∗ = 1/ 2. (a), (b), and (c)correspond to  = 0.1, 0.3, and 0.5, respectively, as given in Fig. 9.
Circles, crosses, and squares correspond, respectively, to independent,
intermediate, and coincident structures (see text).

1

t

10

2

10

3

(b)

ψ2

0.4

In Fig. 11, we plot the square root of these quantities as afunction of t, with the same parameters as those used in Fig. 9.
In all studied cases, the modulation of the ψ monolayer having
a larger wave number (qψ∗ > qφ∗ ) grows faster than that of theφ monolayer. We also remark that the structure formation in
the φ monolayer is accelerated for larger , whereas that ofthe ψ monolayer is almost unchanged. According to the linear
stability analysis of Eq. (34), the initial growth rates of theunstable modes are essentially equivalent to the decay rates
ω± of the concentration ﬂuctuations given in Eq. (31). Thisis consistent with Fig. 7, where the characteristic growth time(1/ω) for larger q is smaller than that for smaller q. The growth
rates increases with  [see Fig. 11(a)] because the coupling
effectively reduces the temperature and enhances the phasetransition. The faster the decay of concentration ﬂuctuations,the faster the structure formation.
V. DISCUSSION AND FINAL REMARKSIn this paper we present a model for coupled modulated
lipid bilayers. We start by considering a monolayer consistingof a mixture of saturated and hybrid lipids and propose aphenomenological model that includes a coupling between thelipid composition and a 2D vectorial ﬁeld. This coupling arises
from the line-active nature of the hybrid lipid, which adjusts its
tail orientation in order to reduce the line tension. Minimizationof the monolayer free energy with respect to the vectorial ﬁeld
yields a 2D microemulsion exhibiting modulated phases. The
characteristic wavelength of modulation is determined by themonolayer coefﬁcients A and B, Eq. (8), reﬂecting molecularproperties of lipid mixture. We then construct a model for
lipid bilayers composed of two modulated monolayers thatinﬂuence each other through an interleaﬂet coupling.
Based on the model, we study concentration ﬂuctuations ofbilayers above Tc and calculate their static structure factors.
The calculated phase diagram for coupled bilayers shows

0.2

0 -1
0
10
10

10

1

t

10

2

10

3

FIG. 11. Time evolution ofthe modulation amplitudes of the twomonolayers: (a) φ 2 and (b) ψ 2 . The parameters are τφ = τψ =0.8, φ0 = ψ0 = 0, B = A = 1, D = 0.0123, C = 0.1111 yieldingqψ∗ /qφ∗ = 3. The different lines represent  = 0.1 (solid),  = 0.3(dashed), and  = 0.5 (dotted).

that the extent of ordered and structured-disordered phases
become larger as compared to the decoupled case. Whenthe two monolayers with different preferred wave numbersqφ∗ = qψ∗ are coupled (say, qφ∗ < qψ∗ ), the peak height of Sφφoccurring at smaller q numbers becomes larger as comparedto the decoupled case, whereas the peak height of Sψψoccurring at larger q numbers almost does not change. Namely
the interleaﬂet coupling strongly affects the compositional
modulation in each monolayer. Furthermore, the interleaﬂet
coupling has a clear signature on the cross structure factor,
Sφψ , as well as on the dynamics of concentration ﬂuctuations. By calculating the intermediate structure factor,
S(q,t), we show that concentration ﬂuctuations generally
exhibit a double exponential decay with two decay rates,
ω± . One of the decay times (1/ω+ ) exhibits two shoulders
at wave numbers describing the monolayer compositional
modulations.
For membranes below Tc , we studied the micro-phaseseparation of a coupled modulated bilayer. When the two

021916-10

CONCENTRATION FLUCTUATIONS AND PHASE . . .

PHYSICAL REVIEW E 86, 021916 (2012)

monolayers have different modulations, qφ∗ = qψ∗ , we obtained
numerically a variety of complex patterns. The initial growth
rates of the unstable modes are identical to the decay rates of
the concentration ﬂuctuations.
As mentioned in Sec. III, the special character ofconcentration ﬂuctuations in our model may explain theﬁnite-size domains (“rafts”) in biological membranes. For
ordinary binary mixtures above Tc , the only length scale ofthe disorder phase is determined by the correlation length,
and close to Tc , this length becomes large. But within our
2D microemulsion model, there is another length scalecharacterizing the modulations as given by Eq. (20). Thissecond length scale may also explain the ﬁnite-size domains
observed in some experiments as a result of micro-phaseseparation in the low temperature phase.
We note that our microemulsion picture will not hold if
we replace the hybrid lipids with unsaturated lipids (having
two unsaturated chains) because unsaturated lipids would notexhibit any line-activity. For mixtures containing unsaturated
lipids, we expect to observe macro-phase separation insteadof micro-phase separation, in agreement with some experiments [5]. In biological membranes, however, unsaturated
lipids are not commonly found and are present only as
minor components. The experimental situation using artiﬁcial
membranes is somewhat less clear because it was shownin some experiments [43,44] that membranes containing
a mixture of saturated lipids, hybrid lipids and cholesterol
show nanoscopic domains, whereas in other works [6], amacroscopic coarsening of domains was observed for such
mixtures.
A related model based on a microemulsion picture wasrecently proposed by Schick [45], who considered a coupling
between curvature and compositional asymmetry between
the two leaﬂets, resulting in a 2D microemulsion. Although
Schick’s model as well as ours share the microemulsion
viewpoint, the origin of the physical mechanism differs. In
Schick’s model the coupling between composition asymmetry
and curvature gives rise to domains with different spontaneous
curvature. Our model, on the other hand, assumes that for ﬂatmonolayers, the line-active nature of hybrid lipids is solely
responsible for the microemulsion formation [see Eq. (3)].
Hence, our proposed physical mechanism for the bilayer
coupling, as discussed in Sec. II B, differs. In our model, thedomains residing on the two leaﬂets are either correlated for
 > 0 or anticorrelated for  < 0, whereas they are always
anticorrelated in the model of Ref. [45].
The present work is concerned with the analysis ofequilibrium properties and relaxation dynamics toward theequilibrium state. The existence of ﬁnite-size domains may
also be explained by nonequilibrium lipid transport between
the cell interior and the membrane. Such a mechanism wasconsidered [46–49] through a coupling between the membrane
and an outer reservoir of lipid or cholesterol. Similarly to ourmodel, these works explain the appearance of the ﬁnite-size
domains as a result of micro-phase separation.
Asymmetry between lipid composition in the inner andouter leaﬂets of biomembranes has a very deep signiﬁcanceand is closely related to the cell biological functions. For
instance, the breakdown of such compositional asymmetry
is related to programmed cell death (apoptosis) [50]. For

living cells, this asymmetry is maintained by an enzyme
called “ﬂippase” that actively transports the lipids between
the two leaﬂets [51]. The half-life of lipid composition due toﬂip-ﬂop motion is measured using time resolved small angle
neutron scattering [52] and is estimated to be several hours at
physiological conditions.
Although membranal signal transduction is important forvarious biological functions, its dynamical properties are still
not so well understood. As our theory offers an explanation
for the size and dynamics of lipid domains, we hope that it
and similar models will contribute in the future toward theunderstanding of functional processes in biomembranes.

ACKNOWLEDGMENTS

We thank T. Kato, S. L. Keller, S. Ramachandran,
M. Schick, and K. Yamada for useful discussions. Y.H.
acknowledges a Research Fellowship for Young Scientists
No. 215097 from the JSPS. This work was supported by
a Grant-in-Aid for Scientiﬁc Research (21540420) from the
MEXT of Japan. S.K. acknowledges support by the JSPS
Core-to-Core Program “International research network for
non-equilibrium dynamics of soft matter.” D.A. acknowledges
support from the Israel Science Foundation (ISF) under Grant
No. 231/08 and the U.S.-Israel Binational Foundation (BSF)
under Grant No. 2006/055.

APPENDIX: DERIVATION OF G φφ (r)In this Appendix we present the derivation of Gφφ ofEq. (19). The real-space correlation function is given by the
2D inverse Fourier transform of Eq. (18):
d 2q
Sφφ (q)eiq·r
Gφφ (r) =(2π )2
 ∞1
qJ0 (qr), (A1)=dq 4
4π B 0q − (A/B)q 2 + φ /4B
where J0 (qr) is the zeroth-order Bessel function. We use therelation J0 (qr) = [H0(1) (qr) + H0(2) (qr)]/2, where H0(1) (qr)and H0(2) (qr) are the zeroth-order Hankel functions of the ﬁrst
and second kinds, respectively. The integral in Eq. (A1) thenis written as I = (I1 + I2 )/2, where
Ii =

 ∞
dq
0

qH0(i) (qr),q 4 − (A/B)q 2 + φ /4B(A2)

with i = 1,2.
In order to evaluate the above integral, we performed theintegration in the complex plane by replacing q with thecomplex variable z = x + iy. The integrand has four poles
at

zj =

021916-11

( φ /B)1/4 
2π
1
±i ,
(± 1 − γφ ± i 1 + γφ ) = ±
2
λφ
ξφ
(A3)

YUICHI HIROSE, SHIGEYUKI KOMURA, AND DAVID ANDELMAN

with j = 1, 2, 3, 4 and λφ and ξφ have been deﬁned in Eqs. (20)and (21). These poles are located in the quadrants 1, 2, 3, and4, off the real and imaginary axes. For the integral I1 , weintegrate along a quarter-circle contour in the ﬁrst quadrant inan counterclockwise direction. The contour radius is taken to
inﬁnity (Fig. 12), and, using the residue theorem, we obtain ∞xH0(1) (xr)dx 4x − (A/B)x 2 + φ /4B0
 0
iyH0(1) (iyr)= 2π i Res|z1 , (A4)dy 4+
y + (A/B)y 2 + φ /4B∞
where “Res” denotes the residue. Similarly, for the integral I2 ,we integrate along the contour of the quarter-circle of inﬁnite
radius in the fourth quadrant in the clockwise direction, ∞xH0(2) (xr)dx 4x − (A/B)x 2 + φ /4B0
 0
iyH0(2) (iyr)+dy 4= −2π i Res|z4 . (A5)y + (A/B)y 2 + φ /4B−∞
Combining Eqs. (A4) and (A5), and further using therelation H0(1) (−z) = −H0(2) (z), we obtainπ
H0(1) (z1 r) + H0(2) (z4 r) . (A6)I= 
22 φ /B 1 − γφ[1] K. Simons and E. Ikonen, Nature 387, 569 (1997).
[2] C. Eggeling, C. Ringemann, R. Medda, G. Schwarzmann,
K. Sandhoff, S. Polyakova, V. N. Belov, B. Hein, C. vonMiddendorff, A. Schönle, and S. W. Hell, Nature 457, 1159
(2009).
[3] C. Dietrich, L. A. Bagatolli, Z. N. Volovyk, N. L. Thompson,
M. Levi, K. Jacobson, and E. Gratton, Biophys. J. 80, 1417
(2001).
[4] S. L. Veatch and S. L. Keller, Phys. Rev. Lett. 89, 268101 (2002).
[5] S. L. Veatch and S. L. Keller, Biophys. J. 85, 3074 (2003).
[6] S. L. Veatch and S. L. Keller, Phys. Rev. Lett. 94, 148101 (2005).
[7] T. Baumgart, S. T. Hess, and W. W. Webb, Nature 425, 821(2003).
[8] S. Rozovsky, Y. Kaizuka, and J. T. Groves, J. Am. Chem. Soc.
127, 36 (2005).
[9] Y. Kaizuka and J. T. Groves, New J. Phys. 12, 095001 (2010).
[10] T. M. Konyakhina, S. L. Goh, J. Amazon, F. A. Heberle, J. Wu,
and G. W. Feigenson, Biophys. J. 101, L08 (2011).
[11] S. L. Veatch, O. Soubias, S. L. Keller, and K. Gawrisch, Proc.
Natl. Acad. Sci. USA 104, 17650 (2007).
[12] A. R. Honerkamp-Smith, P. Cicuta, M. D. Collins, S. L. Veatch,
M. den Nijs, M. Schick, and S. L. Keller, Biophys. J. 95, 236(2008).
[13] A. R. Honerkamp-Smith, S. L. Veatch, and S. L. Keller, Biochim.
Biophys. Acta 1788, 53 (2009).
[14] S. L. Veatch, P. Cicuta, P. Sengupta, A. R. Honerkamp-Smith,
D. Holowka, and B. Baird, ACS Chem. Bio. 3, 287 (2008).
[15] M. Haataja, FEBS Lett. 584, 1678 (2010).

PHYSICAL REVIEW E 86, 021916 (2012)

y

z2

z1
x

z3

z4

FIG. 12. Four poles in the complex plane at z = zj (j = 1, 2, 3,
and 4). The two close integral paths are indicated by the arrows in theﬁrst and fourth quadrants.

Finally, using λφ and ξφ from Eq. (A3), we obtain



2π r
ξφ λφ
r
.
Re H0(1)Gφφ (r) =
+i
32π B
λφ
ξφ

(A7)

In the above expressions, we have used the relation H0(2) (z) =H0(1) (z), where z is the complex conjugate of z.

[16] In our model we always refer to the critical point, Tc , at which
the correlation length diverges. This is related to the miscibility
transition temperature measured in experiments.
[17] A. Zachowski, Biochem. J. 294, 1 (1993).
[18] M. D. Collins and S. L. Keller, Proc. Natl. Acad. Sci. USA 105,124 (2008).
[19] J. D. Perlmutter and J. N. Sachs, J. Am. Chem. Soc. 133, 6563
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