Eur. Phys. J. E 26, 197–204 (2008)
DOI 10.1140/epje/i2007-10258-4

THE EUROPEAN
PHYSICAL JOURNAL E

The phase behavior of mixed lipid membranes in the presence ofthe rippled phase
N. Shimokawa1 , S. Komura1,a , and D. Andelman2
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397,
Japan
2
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel
Received 18 September 2007 and Received in ﬁnal form 18 December 2007
c EDP Sciences / Società Italiana di Fisica / Springer-Verlag 2008
Published online: 26 February 2008 – 
Abstract. We propose a model describing liquid-solid phase coexistence in mixed lipid membranes by
including explicitly the occurrence of a rippled phase. For a single component membrane, we employ aprevious model in which the membrane thickness is used as an order parameter. As function of temperature,
this model properly accounts for the phase behavior of the three possible membrane phases: solid, liquid
and the rippled phase. Our primary aim is to explore extensions of this model to binary lipid mixturesby considering the composition dependence of important model parameters. The obtained phase diagrams
show various liquid, solid and rippled phase coexistence regions, and are in quantitative agreement withthe experimental ones for some speciﬁc lipid mixtures.
PACS. 87.16.D- Membranes, bilayers, and vesicles – 64.75.-g Phase equilibria

1 Introduction
In recent years, domain formation in biomembranes andartiﬁcial membranes has attracted great attention in connection to the so-called “raft formation” in biological cell
membranes [1]. It is believed that the appearance of suchdomains (rafts) in membranes plays an important role forvarious cell functions [2, 3]. Beside their interest in biocellular processes, raft formation has a fundamental physical interest, because it oﬀers a special example of twodimensional phase separations coupled to internal and external membrane degrees of freedoms [4]. Using ﬂuorescence microscopy or X-ray diﬀraction techniques, the lateral phase separation between diﬀerent liquid phases, or
between the solid and liquid phases has been observed for
ternary-component vesicles consisting of saturated lipid,
unsaturated lipid, and cholesterol [5–9]. Although the biological signiﬁcance of rafts is not yet fully understood, it
serves as our primary motivation to explore, from a physical point of view, phase transitions, domains and phasecoexistence in multi-component lipid membranes.
Lateral phase separation in membranes occurs evenfor the simpler case of binary lipid mixtures without any
added cholesterol. The coexistence is between liquid-like(Lα ) and solid-like (Lβ ′ called “gel”) phases, and it canbe visualized using techniques such as two-photon ﬂuorescence microscopy [10]. For example, in a DPPC/DPPE
ae-mail: komura@tmu.ac.jp

lipid mixture, the solid domains exhibit morphologies
such as hexagonal, dumbbell or dendritic shapes [10].
For DPPC/DLPC lipid vesicle, the solid domains appear
to be stripe-like and are monodispersed in their domain
width [11]. Shapes of solid domains found on sphericalﬂuid vesicles has been also addressed theoretically [12].
By assuming the additivity of stretching and line energies, the model predicts a phase diagram including cap,ring and ribbon phase domains on a spherical surface.
Although visualization of domains embedded in ﬂuid
membranes has become possible only recently, the phase
diagram of various binary lipid mixtures in bilayer membranes has been known for some time [13, 14]. In a previous model investigated by some of us [15, 16], a coupling between composition and internal membrane structure was proposed and the resulting phase diagram wascalculated. It includes a coexistence region between Lα
and Lβ ′ phases in agreement with experiments on speciﬁc
lipid mixtures. However, for some lipids, such as DMPC
or DPPC, another distinct solid-like phase is known to occur and is called the “rippled” or Pβ ′ phase. The uniquefeature of this phase is that the membrane shape is spatially modulated while the lipid hydrocarbon tails areordered. The speciﬁc phase diagrams of DMPC/DPPS,
DPPC/DPPS [17], or DPPC/DPPE [18] mixtures havebeen explored in experiments, and show the three phases(Lα , Lβ ′ , Pβ ′ ) together with the coexistence regions
between them. However, these phase diagrams have notyet been considered theoretically.

198

The European Physical Journal E

The main objective of the present paper is to provide
a simple model that describes the liquid-solid coexistencein binary lipid mixtures while considering the possibilityof an intermediate rippled phase. Our starting point is amodel introduced by Goldstein and Leibler (GL) that accounts for the succession of phase transitions as applied
to single-component lipid membranes [19, 20]. In the GLmodel, the membrane thickness is used as a scalar orderparameter. We combined some of the ideas presented in
the GL model with our own previous model for binary(or even ternary) lipid mixtures [15, 16]. For binary-lipid
mixtures, the model parameters are taken to be dependent on the relative lipid composition, and resulting inseveral types of phase diagrams that are in quantitative
agreement with experiments on speciﬁc lipid mixtures.
In the next section, we ﬁrst review the GL model describing the phase transitions in a single-component lipid
membrane. The corresponding mean-ﬁeld phase diagram
is obtained in terms of the temperature and membraneelastic constants. In Section 3, we propose an extension of
the GL model for a two-component lipid membrane, andcalculate the binary phase diagrams. Some discussions andcomparison to other models are discussed in Section 4.

2 Single-component lipid system
In a single-component lipid bilayer, one typically observesa discontinuous ﬁrst-order phase transition from the Lα
to the Lβ ′ phase as the temperature is decreased. Thisis called the “main transition” and is associated with ordering of the hydrocarbon tails of the lipid molecules. In
the Lα phase, the hydrocarbon chains are disordered (andthe phase is liquid-like), while they order by stretching
and tilting in the Lβ ′ (solid-like) phase. Several theoretical models have been proposed in the past to describe themain transition in isolated membranes [21, 22].
In other single-component lipid systems, however, arippled (Pβ ′ ) phase appears between the Lα and Lβ ′phases for stacked bilayer membranes [23–25]. This phaseis more peculiar and many studies have been devoted tobetter understand its modulation wavelength and amplitude [26]. For a stack of lipid bilayers in water (a lamellar phase), the phase diagram as a function of relative
humidity and temperature was reported [27, 28]. Several
attempts have been made to describe the rippled phasetheoretically such as those based on a molecular level description [29, 30], or Monte Carlo simulations [31]. In continuum theories of the rippled phase, diﬀerent quantities
have been suggested for the order parameter. Examples
are the membrane thickness [19, 32] or the conﬁgurationof the hydrocarbon chains (ratio of the trans bonds in the
chains) [33]. In both cases the order parameter is scalar.
More recently, a Ginzburg-Landau theory was proposed byChen, Lubensky, and MacKintosh (CLM) who employed avectorial order parameter representing the tilt of the lipid
molecules [34, 35].
We further discuss the CLM model in Section 4 below, but our own starting point is based on a model proposed by Goldstein and Leibler (GL) for single-component

lipid membranes [19, 20]. To describe the main transition
involving chain ordering and stiﬀening, GL introduced adimensionless scalar order parameter of the membrane,m(r):δ(r) − δ0.
(1)m(r) =
δ0This parameter depends on the actual membrane thickness δ(r), and on the constant membrane thickness δ0 of
the Lα phase. The two-dimensional lateral position within
the bilayer plane is denoted by r. Notice that m encapsulates changes that may occur in several degrees of freedom,including, for example, the conformations of the hydrocarbon chains, molecular tilt and positional ordering. In
the present paper, the bilayer nature of the membrane is
not taken into account and the bilayer thickness is simply
taken as a sum of the two monolayer thicknesses. The roleof the bilayer structure in the formation of rippled phases
was explicitly considered in previous models [36, 37]. Since
the shape or height variable of the membrane is ignored in
the present treatment, we note that our model can be better applied to supported bilayer membranes. In fact manyof the recent experiments mentioned before are for vesicles, and the shape variable is an important factor evenin the limit of large vesicles. An external degree of freedom or shape deformation of phase-separated membranes
is explicitly considered such as in reference [38].
For simplicity, we assume that the rippled phase isspatially modulated only in the x-direction, and ignore
two-dimensional rippled phases such as the square latticephase [34, 35] or the hexagonal phase [39, 40]. For an isolated lipid bilayer membrane, the stretching free energy
per lipid molecule is [19]
fst =

1
1
1
a2 m2 + a3 m3 + a4 m4234
 2 2
2
dm
d m
1
1
+ C
+ D
.
2
dx
2
dx2

(2)

A very similar free energy was proposed by Marder et
al. [32]. The ﬁrst three terms are the Landau expansionin powers of the order parameter m. Only a2 , the secondorder term coeﬃcient, has an explicit temperature dependence: a2 = a′2 (T −T ∗ ) with T ∗ being a reference temperature. (It actually is the critical temperature in the absenceof the cubic term.) Because the Lβ ′ → Lα phase transition is known to be ﬁrst-order, a3 is taken to be negative,
whereas a4 is always positive to ensure stability of this freeenergy expansion. The next two terms are related to thelowest-order gradients of m. (Note that the x-coordinate
is rescaled in units of a molecular length scale ℓ so thatx → x/ℓ is dimensionless.) These gradient terms represent elastic out-of-plane undulations of the membrane andtheir coeﬃcients are the elastic constants C and D, respectively. The coeﬃcient C can be either positive or negative,
but D is always positive in order to ensure stability of the
expansion. The physical origin of a negative C value canbe related to the coupling between the conformation of
the chains and the curvature of the lipid/water interface,or to interactions between the polar head-groups of the

N. Shimokawa et al.: The phase behavior of mixed lipid membranes

70

lipids and water [19]. This point will be further discussed
later in Section 4. In the absence of spatial gradient terms
in equation (2) and if m(x) is constant in space, the maintransition temperature Tm is calculated from the conditions fst = 0 and dfst /dm = 0. It is related to T ∗ by
2a23.
9a′2 a4

Lc

50
(3)

With nonzero a3 , a ﬁrst-order melting transition at Tm
preempts the critical point at T ∗ .
When C > 0, the equilibrium phase is always homogeneous in space; either as a Lβ ′ or Lα phase. On the other
hand, for C < 0, the rippled phase with a characteristic
wave vector q ∗ = (−C/2D)1/2 (in unit of ℓ−1 ) may dominate over the two other phases. (It should be noted thatC < 0 is only a necessary condition to have the rippled
phase as we will discuss below.) To obtain the mean-ﬁeld
phase diagram, we employ the single-mode approximationin which the rippled phase is described by the most dominant wave vector q ∗ , only. This approximation is valid
near the phase transition point where the segregation tendency between the two lipids is weak enough. The orderparameter m can be written as [32]m = m0 + m1 cos(q ∗ x),

60

(4)

where m0 = m is the spatial average of m, and m1 isthe amplitude of the single q ∗ -mode of the rippled phasetaken to lie arbitrarily in the x-direction.
By substituting equation (4) into equation (2) and taking the spatial average over one period, we obtain
C2 2 11 2
2
m + a2 m0 + m1
fst  = −16D 1 22

1
3
+ a3 m30 + m0 m21
3
2


1
3
+ a4 m40 + 3m20 m21 + m41 .
(5)
4
8
The two homogeneous liquid and solid phases, Lα and Lβ ′ ,respectively, are characterized by m1 = 0; while m0 = 0 in
the disordered liquid Lα phase, and is non-zero in the solid
Lβ ′ phase. Among the three phases, only the Pβ ′ phase ischaracterized by a modulating amplitude, m1 = 0.
The mean-ﬁeld phase diagram is obtained by minimizing equation (5) with respect to both m0 and m1 , and comparing the relative stability of the three relevant phases. In
Figure 1, we present an example of such a phase diagram,as a function of the (negative) eﬀective elastic constant C
and temperature T plotted in degrees Celsius. The parameter C can be a function of relative humidity (fraction ofwater content in the lamellar phase) as discussed below.
The phase diagram is calculated for a choice of systemparameters which reproduce the main transition temperature of DMPC at Tm = 38 ◦ C: a′2 = 2.4 × 10−21 J K−1 ,a3 = −1.1 × 10−18 J, a4 = 2.2 × 10−18 J, T ∗ = −13 ◦ C(260 K), and D = 2.0 × 10−18 J. These values (except
D > 0 whose value is less important) are taken fromT [ oC]

Tm = T ∗ +

199

40

Pd’

30

Ld’

20
10

0

2

4

6

/19

/C [108

10J]

Fig. 1. Mean-ﬁeld phase diagram of a single-component lipid
membrane as a function of the (negative) elastic constant C
and temperature T (in degrees Celsius). The liquid (Lα ), solid(Lβ ′ ) and rippled (Pβ ′ ) phases are separated from one anotherby ﬁrst-order phase transition lines which meet at a triple point(marked by a full circle): Ctr = −3.33 × 10−19 J, Ttr = 38 ◦ C.
The parameters used to calculate this phase diagram are givenin the text.

reference [20]. By inserting the above Landau coeﬃcients
into equation (3), it can be readily checked that the maintransition temperature of DMPC Tm = 38 ◦ C (Fig. 1) is
obtained.
In Figure 1, three ﬁrst-order phase transition lines
separate the three phases and meet at a triple point,
Ctr = −3.33 × 10−19 J. Due to the ﬁrst-order nature of
the Lβ ′ → Lα phase transition, it can be argued, on
general grounds, that the triple point Ctr has to be located at non-zero values of C. For small magnitude of C,|C| < |Ctr |, the horizontal ﬁrst-order phase transition line
occurs at T = Tm = 38 ◦ C and is independent of C. An
increase in temperature (as long as |C| < |Ctr |) will melt
the solid phase Lβ ′ directly into the liquid phase Lα at
Tm . But for negative and large enough magnitude of C,|C| > |Ctr |, any increase in temperature will cause a sequence of phase transitions: ﬁrst the solid phase Lβ ′ melts
into the Pβ ′ phase, and only then, upon further increaseof the temperature, the Pβ ′ phase will make a phase transition into the liquid phase Lα . It is also apparent fromthe ﬁgure that the region of the Pβ ′ phase expands at theexpense of the uniform liquid and solid phases as |C| increases. This means that the phase transitions Lβ ′ → Pβ ′and Pβ ′ → Lα (at constant C) occur at larger temperaturedeviations from Tm as |C| increases.
The phase diagram of Figure 1 essentially reproduces
all the experimental facts observed for single-component
lipid membranes. It qualitatively agrees with the observed

The European Physical Journal E

phase diagram of a lamellar phase of DMPC, when thenegative elastic constant C is taken to be proportional to
the relative humidity [27, 28]. Although the exact dependence of the elastic constant C on humidity is not known,
the former can possibly be reduced by increasing hydration. For another lipid, DPPC, the Pβ ′ phase is foundexperimentally to occur at temperatures around 37 ◦ C (inRef. [18] the precise temperatures for the Lβ ′ → Pβ ′ andPβ ′ → Lα phase transitions have not been fully reported),
while the rippled phase is not observed for a stack of
DPPE lipid bilayers. Such a diﬀerence in the lipid phasebehavior can be qualitatively attributed to diﬀerent valuesof the elastic constant C.

70

Lc

60
50

T [ oC]200

Lc- Ld’

40 P

d’

30
20

3 Two-component lipid mixtures

10

In this section, we extend the above single-component
model to membranes consisting of a binary lipid mixture.
We employ a similar approach as was used in our previous study [15, 16], and consider the coupling of the melting phase transition with a lateral phase separation in themixed membrane.
A binary mixed membrane is modeled as an incompressible A/B mixture of φ mole fraction of lipid A and(1 − φ) fraction of lipid B. For simplicity, we assume thesame area per molecule for both species and ignore any
lipid exchange with the surrounding solvent. In general,
the two lipids will have diﬀerent main transition temperatures originating from diﬀerent molecular parameters suchas chain length, degree of saturation and hydrophilic headgroup.
The total free energy per lipid, ftot = fst + fmix , comprises: i) the chain stretching free energy fst given by equation (2), and ii) the free energy of mixing, fmix . The latterenergy per lipid molecule is the sum of the mixing entropy
and enthalpy. It can be written within the Bragg-Williams(mean-ﬁeld) approximation as
1
fmix = kB T [φ log φ + (1 − φ) log(1 − φ)] + Jφ(1 − φ), (6)2where kB is the Boltzmann constant, and J > 0 is an
attractive interaction parameter between the lipids thatenhances lipid-lipid demixing.
For a binary mixture the free energy fst is assumed
to have the same functional dependence on the eﬀective
elastic constant C and the reference temperature T ∗ . Although these two parameters depend on the lipid composition φ, the precise dependence cannot be calculated fromsuch a phenomenological approach. Alternatively, we proceed by further assuming the simplest linear interpolation
between the two pure lipid limiting values:
C(φ) = φCA + (1 − φ)CB ,

(7)

∗

(8)

T

(φ) = φTA∗ + (1 − φ)TB∗ ,where CA and CB are the elastic constants, and TA∗ and
TB∗ are the reference temperatures of lipid A and B,

Pd’+ Ld’

0
DPPC

(2)

(1)

Ld’

0.2

(2)
(1)
Ld’ + Ld’

0.4

h

0.6

Ld’
0.8

1
DPPE

Fig. 2. Calculated mean-ﬁeld phase diagram of a binary
lipid mixture as a function of their relative composition φand temperature T . The parameter values are chosen to ﬁt
DPPC/DPPE mixtures (φ = 0 represents pure DPPC and isdenoted by a subscript “B”): CA = −2.0 × 10−19 J, CB =−5.2 × 10−19 J, TA∗ = 9 ◦ C, TB∗ = −15 ◦ C, and J = 1.45 ×
10−20 J. All other parameter values and deﬁnitions of the different phases are the same as in Figure 1. The critical point
is indicated by a full circle and occurs at Tc = 21.3 ◦ C. The
horizontal dashed line indicates the three-phase coexistence atthe triple point, Ttr = 42.0 ◦ C.

respectively. The other elastic constant D, as well as a3
and a4 in equation (2) are assumed to be the same for thetwo lipids because the important properties of the lipids
are mostly reﬂected in the choice of their C and T ∗ parameter values. As for all other model parameters, we use
hereafter the same values listed in the previous section.
The total free energy ftot = fst + fmix as a functionof m and φ is now readily available by substituting equations (7) and (8) into equation (2). There is an explicit
coupling between the two order parameters, m and φ originating from the free energy fst = fst (m, φ). Note that theﬁrst term in equation (2) and equation (8) lead to a coupling term which scales as φm2 . Another less interestingterm is φ2 m which simply renormalizes the interaction
parameter J of fmix . After minimizing ftot with respect
to m0 and m1 , the two-phase coexistence region in the(T, φ)-plane is calculated by using the common tangent
construction to account for the constraint value of the relative concentration φ.
By properly choosing the model parameters, we attempt to reproduce the binary phase diagram of several
lipid mixtures, the ﬁrst being that of DPPC/DPPE [18]shown in Figure 2. Since a pure DPPE membrane does notshow experimentally the rippled phase, its elastic constantis chosen as CA = −2.0×10−19 J. Namely, below the triple
point Ctr (Fig. 1). The DPPE reference temperature TA∗ iscalculated from equation (3) with the same values for theLandau coeﬃcients as before, by ﬁtting its main transition

N. Shimokawa et al.: The phase behavior of mixed lipid membranes(1)

ﬁgure) as well as with the Lα phase (Pβ ′ + Lα ). There isalso a two-phase coexistence region between the two liquid(1)(2)
phases (Lα + Lα ). The three-phase coexistence between

70

50

20
10

Lc
(1)

(2)

Lc + Ld’

40 P
30

*4+

*3+

Lc

60

T [ oC]

temperature to the observed one of Tm = 60 ◦ C. On theother hand, the elastic constant of DPPC is chosen as
CB = −5.2 × 10−19 J (above Ctr ) so that the Pβ ′ phaseappears between the Lβ ′ and Lα phases. The reference
temperature of DPPC is set as TB∗ = −15 ◦ C, corresponding, from equation (3), to a Lβ ′ → Lα transition temperature Tm = 36 ◦ C. This temperature is not available
from experiments but lies in-between the experimentallyobserved Lβ ′ → Pβ ′ phase transition temperature at 32 ◦ C
and the Pβ ′ → Lα one at 40 ◦ C.
All the above parameter values are used to calculate the binary phase diagram ﬁtting the DPPC/DPPE
mixture. The relative composition φ is chosen such thatφ = 0 corresponds to pure DPPC, while φ = 1 to pure
DPPE. In addition, the interaction parameter is set as
J = 1.45 × 10−20 J. When the temperature is relatively
low, there is a coexistence region between two Lβ ′ phases(1)(2)(denoted by Lβ ′ +Lβ ′ ). This low-temperature coexistence
terminates at a critical temperature, above which there is
only one Lβ ′ phase for the entire range of composition φ.
It should be noted, however, that the appearance of thecritical point between the two “solid” phases is an artifact of the model. As the temperature is further increased,
the Pβ ′ phase appears close to the pure DPPC axis. It isbound by a coexistence region with the Lβ ′ phases as wellas with another (much smaller in its extent) coexistence
region with the Lα phase. This latter region is very small
on the scale of the ﬁgure and collapses almost into a line.
These two coexistence regions terminate at a triple pointtemperature Ttr at which all the three phases coexist. For
even higher temperature, T > Ttr , the rippled phase disappears and the only coexistence region is between the Lα
and Lβ ′ phases.
The calculated phase diagram is in quantitative agreement with the experimental one for DPPC/DPPE mixtures (e.g., Fig. 7 of Ref. [18]). Even the coexistence between the two solid phases is alluded to in reference [18]. A
similar type of phase diagram was obtained for mixtures of
DMPC/DPPS and DPPC/DPPS [17]. In these lipid mixtures, both DMPC and DPPC exhibit the rippled phase,
while pure DPPS undergoes a direct transition from theLβ ′ to Lα phases similar to DPPE.
An even stronger lipid-lipid segregation can be modeled by increasing the value of the lipid interaction parameter to J = 1.85 × 10−20 J, while keeping all otherparameter values unchanged. In the resulting phase diagram of Figure 3, the lower coexistence region between the
two Lβ ′ phases penetrates the other coexistence regions.
Such a phase diagram resembles that of DEPC/DPPE
lipid mixtures [13]. This combination of lipids have astrong segregation tendency because both head and tailmoieties are diﬀerent. In the calculated phase diagram,
there are three triple points indicated by the horizontal
dashed lines. The rippled phase can now coexist with ei(1)(2)
ther of the Lβ ′ (regions Pβ ′ + Lβ ′ or Pβ ′ + Lβ ′ in the

201

(2)

Pd’+ Ld’

d’

(1)

Ld’

0
DEPC

(1)

(2)

Ld’

(2)

Ld’ + Ld’

0.2

0.4

h

0.6

0.8

1
DPPE

Fig. 3. Calculated mean-ﬁeld phase diagram of a binary lipid
mixture as a function of their relative composition φ and temperature T . The parameters are the same as in Figure 2 except
J = 1.85 × 10−20 J. The critical point is located at 62.1 ◦ C, andthree triple points occur at 35.6, 41.3, and 56.3 ◦ C, respectively.
This choice of model parameters is in agreement with experimental phase diagram of DEPC/DPPE mixture where φ = 0
represents pure DEPC.

the two Lα phases and the Lβ ′ , and the existence of thecritical point have been indeed observed for DEPC/DPPE
lipid mixtures [13].
In Figures 4 and 5 we calculate the phase diagramsin situations where each of the two pure lipids exhibit arippled phase between their corresponding Lα and Lβ ′phases. Hence, we choose the magnitudes of the elastic constant to be larger than |Ctr | of Figure 1: CA =−5.5 × 10−19 J and CB = −6.5 × 10−19 J. The choices of
the reference temperatures TA∗ and TB∗ , as well as all otherparameters, is the same as in Figures 2 and 3. The interaction parameter is taken as J = 1.45 × 10−20 J in Figure 4,
and have a somewhat higher value, J = 1.85 × 10−20 J,
in Figure 5. In the phase diagram of Figure 4, one regionof the rippled phase extends throughout the entire rangeof the lipid composition. Above and below this region,
the rippled phase coexists with the Lβ ′ and Lα phases.
At low temperatures, another coexistence region can beseen between the two Lβ ′ phases. For a larger interaction
parameter J (Fig. 5), these coexistence regions merge toform a large two-phase region, and there are four triple
points. Here we see a new coexistence between Lα and(2)Lβ ′ phases which does not exist in Figure 4. Althoughsuch more complex phase diagrams have not been yet reported in experiments, we expect that they can be foundin the future by properly choosing the lipid mixtures. One
possible choice of lipids that may give rise to such a phasebehavior is the DPPC/DMPC mixture. Both lipids exhibit
the rippled phase between their Lβ ′ and Lα phases.

202

The European Physical Journal E

70

70

Lc

60

60
50

T [ oC]T [ oC]

50

Pd’

Lc
(1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

(2)

Pd’

Lc + Pd’

Lc+ Pd’

40

*4+

*3+

Lc

Pd’+ Ld’30

Lc + Ld’(1)

40 Pd’

Pd’ + Ld’

30
(2)

(1)

(1)

20
10

Ld’
0

0.2

(2)

(2)
(1)
Ld’ + Ld’

0.4

h

0.6

20

Ld’
0.8

Ld’

1

10

0

Ld’

Ld’ + Ld’

0.2

0.4

h

0.6

0.8

1

Fig. 4. Mean-ﬁeld phase diagram of a binary mixture of two
lipids as a function of the relative composition φ and temperature T . All parameters are the same as in Figure 2 except
CA = −5.5 × 10−19 J and CB = −6.5 × 10−19 J. The lipid interaction parameter is set to be J = 1.45 × 10−20 J. The criticalpoint is located at 21.3 ◦ C as in Figure 2.

Fig. 5. Mean-ﬁeld phase diagram of a binary mixture of two
lipids as a function of the relative composition φ and temperature T . All parameters are the same as in Figure 4 except
J = 1.85 × 10−20 J. The critical point is located at 62.1 ◦ C asin Figure 3, and four triple points occur at 30.8, 47.1, 52.2, and60.0 ◦ C, respectively.

4 Summary and discussion

curvature and the gradient in molecular tilt was considered. As a result, various types of rippled phases with
diﬀerent inplane symmetries have been predicted.
In spite of the success in predicting, e.g., the squarelattice phase that was found in the experiments [41], theCLM model suﬀers from some deﬁciencies which makes it
inappropriate to be used in our study. Within the CLM
model, the Lβ ′ → Pβ ′ phase transition cannot be induced
only by changing the temperature, while keeping all othersystem parameters ﬁxed. In other words, the phase boundaries in the CLM model is parallel to the temperatureaxis (see, for example, Fig. 9 in Ref. [35]). This result
is essentially due to the vector nature of the CLM orderparameter [42]. Unfortunately, such a phase behavior is
not in accord with the experimental phase diagrams [27,28], where the Lβ ′ –Pβ ′ phase boundary depends on temperature. This is the main reason why we have used ascalar-order parameter giving us the phase diagram as in
Figure 1.
The second problem is that the CLM model predicts
a second-order phase transition between the Lα and Lβ ′phases. However, the main transition is known to be ﬁrstorder according to various experimental investigations [19,20], as was previously mentioned in reference [41]. A possible way out would be to include the 6th-order term of
the tilting vector in the CLM Landau expansion, but thislies beyond the scope of the present work.
The two problems mentioned above do not exist in the
GL model. However, since the choice of a single scalar as
the order parameter is a major simpliﬁcation, many degrees of freedom of the lipid molecules are not properly

In this paper we proposed a model describing the liquidsolid coexistence region in membranes composed of abinary-lipid mixture. We addressed in particular the possible existence of a rippled phase between the liquid andsolid ones, and its eﬀect on the global phase diagrams.
In order to incorporate the possibility of a rippled phase,
we use a model previously proposed for single-component
membranes by Goldstein and Leibler (GL) [19, 20]. Theirmain purpose was to present a model describing the lyotropic lamellar phases of lipid bilayers as stacks of interacting membranes. Here we have shown explicitly that
the GL lipid stretching energy of an isolated membrane exhibits the sequential phase transitions Lβ ′ → Pβ ′ → Lα by
increasing the temperature. We further extend the singlecomponent model and apply it to binary lipid mixtures
assuming that the elastic parameter C as well as the reference temperature T ∗ depend linearly on the composition of one of the lipids. The calculated phase diagrams
are in quantitative agreement with the experimental ones
for speciﬁc lipid mixtures. We have also predicted othertypes of phase diagrams, which have yet to be checked
experimentally for other lipid mixtures.
Several points merit further discussion. In Section 2,
we choose the relative membrane thickness as the scalar
order parameter to describe the membrane phase transitions [19, 20]. In another model proposed by Chen, Lubensky and MacKintosh (CLM), a two-dimensional molecular tilt was used as a vector order parameter [34, 35]. In
their model, a coupling mechanism between the membrane

N. Shimokawa et al.: The phase behavior of mixed lipid membranes

taken into account. For example, one cannot distinguishbetween the similar Lβ ′ and Lβ phases because the scalarorder parameter m cannot model the molecular tilt characteristic of the Lβ phase. With this choice of order parameter, one can only distinguish between a dilute and acondensed phase. Because the membrane thickness does
not account for the diﬀerent symmetries of the Lα , Lβ ′ ,and Pβ ′ phases, m cannot represent any crystalline order,
and the solid phase is not properly described here. Hencethe appearance of the critical points between the two Lβ ′phases in Figures 2 and 4 is an artifact of our model.
Although this problem is remedied in the CLM model,
the GL model captures the essential features of the structural transitions in lipid membranes in the presence of therippled phase. Using the latter model is suﬃcient to reproduce even quantitatively some of the global features ofthe phase diagrams of binary lipid mixtures, in agreementwith experiments.
In equation (2) we assumed for simplicity that the rippled phase is modulated spatially only in the x-direction.
In general, one can consider the full (x, y)-inplane modulations as treated in the CLM model [34, 35]. This can be accomplished by including in equation (2) additional gradient terms in the y-direction. The resulting phase diagram
will be more complex because there can be more than one
rippled phase. Although a two-dimensional square latticephase has been identiﬁed for a single-component DTPC
lipid [41], it has not yet been found in binary lipid mixtures. We also note that the one-dimensional modulationof the order parameter expressed by equation (4) is sim(2)ilar to the Pβ ′ phase in CLM, with the main diﬀerence
that we include a non-zero average term m0 . The physicalinterpretation of the order parameter in the two models
diﬀers as is explained above.
Finally, we discuss why the elastic coeﬃcient C cantake negative values in equation (2) to get the Pβ ′ phaseat equilibrium. In Section 2, we have mentioned that the
coupling between the conformation of the chains and thecurvature of the membrane may be a possible reason [19].
In general, if there exists a coupling between an additionalelastic degree of freedom and the gradient of the orderparameter, the elastic constant C will be reduced. Thissituation can be expressed by including the following two
additional terms in the free energy:1dmfel = Bρ2 − γρ
.
(9)
2
dx
Here ρ represents some elastic degree of freedom, B isanother positive elastic constant, and γ is a coupling constant. For example, ρ can be taken as the curvature of thelipid/water interface, and B is the bending rigidity of themembrane. In the second term, ρ is linearly coupled to
the gradient of m which induces a spontaneous value of ρ.
Minimizing fel of equation (9) with respect to ρ, we get


γ dm
ρ=
.
(10)
B dx
This means that the equilibrium value of ρ is proportional
to the gradient of m. By substituting this equation back203

into equation (9), the eﬀective elastic constant becomes
C → C − γ 2 /B,(11)

in equation (2). Hence C is reduced by γ 2 /B and canreach negative values when the coupling constant γ is large
enough (or when the elastic constant B is small enough).
Although to suggest speciﬁc molecular mechanism to account for this added elastic degree of freedom goes beyond
the scope of the present paper, such a mechanism would, inprinciple, explain the negative value of the eﬀective elastic
constant, C.
This work is supported by KAKENHI (Grant-in-Aid for Scientiﬁc Research) on Priority Areas “Soft Matter Physics” andGrant No. 18540410 from the Ministry of Education, Culture,
Sports, Science and Technology of Japan. One of us (DA) acknowledges support from the Israel Science Foundation (ISF)
under grant No. 160/05 and the US-Israel Binational Foundation (BSF) under grant No. 2006055.

References
1. K. Simons, E. Ikonen, Nature 387, 569 (1997).
2. K. Simons, E. Ikonen, Science 290, 1721 (2000).
3. D.M. Engelman, Nature 438, 578 (2005).
4. S. Leibler, D. Andelman, J. Phys. (Paris) 48, 2013 (1987).
5. S.L. Veatch, S.L. Keller, Phys. Rev. Lett. 89, 268101(2002).
6. T. Baumgart, S.T. Hess, W.W. Webb, Nature 425, 821(2003).
7. S.L. Veatch, S.L. Keller, Phys. Rev. Lett. 94, 148101(2005).
8. S.L. Veatch, S.L. Keller, Biochim. Biophys. Acta 1746,
172 (2005).
9. E. London, Biochim. Biophys. Acta 1746, 203 (2005).
10. L.A. Bagatolli, E. Gratton, Biophys. J. 78, 290 (2000).
11. P.A. Beales, V.D. Gordon, Z. Zhao, S.U. Egelhaaf, W.C.K.
Poon, J. Phys.: Condens. Matter 17, S3341 (2005).
12. S. Schneider, G. Gompper, Europhys. Lett. 70, 136 (2005).
13. S.H. Wu, H.M. McConnell, Biochemistry 14, 847 (1975).
14. K. Arnold, A. Lösche, K. Gawrisch, Biochim. Biophys.
Acta 645, 143 (1981).
15. S. Komura, H. Shirotori, P.D. Olmsted, D. Andelman, Europhys. Lett. 67, 321 (2004).
16. S. Komura, H. Shirotori, P.D. Olmsted, J. Phys.: Condens.
Matter 17, S2951 (2005).
17. E.J. Luna, H.M. McConnell, Biochim. Biophys. Acta 470,
303 (1977).
18. A. Blume, R.J. Wittebort, S.K. Das Gupta, R.G. Griﬃn,Biochem. 21, 6243 (1982).
19. R.E. Goldstein, S. Leibler, Phys. Rev. Lett. 61, 2213
(1988).
20. R.E. Goldstein, S. Leibler, Phys. Rev. A 40, 1025 (1989).
21. S. Doniach, J. Chem. Phys. 68, 4912 (1978).
22. T. Izuyama, Y. Akutsu, J. Phys. Soc. Jpn. 51, 730 (1982).
23. J.A.N. Zasadzinski, M.J. Schneider, J. Gurley, V. Elings,
P.K. Hansma, Science 239, 1013 (1988).
24. D.C. Wack, W.W. Webb, Phys. Rev. Lett. 61, 1210 (1988).
25. M.P. Hentschel, F. Rustichelli, Phys. Rev. Lett. 66, 903(1991).

204

The European Physical Journal E

26. B.A. Cunningham, A.-D. Brown, D.H. Wolfe, W.P.
Williams, A. Brain, Phys. Rev. E 58, 3662 (1998).
27. E.B. Sirota, G.S. Smith, C.R. Saﬁnya, R.J. Plano, N.A.
Clark, Science 242, 1406 (1988).
28. G.S. Smith, E.B. Sirota, C.R. Saﬁnya, N.A. Clark, Phys.
Rev. Lett. 60, 813 (1988).
29. S. Doniach, J. Chem. Phys. 70, 4587 (1979).
30. J.M. Carlson, J.P. Sethna, Phys. Rev. A 36, 3359 (1987).
31. W.S. McCullough, H.L. Scott, Phys. Rev. Lett. 65, 931(1990).
32. M. Marder, H.L. Frisch, J.S. Langer, H.M. McConnell,
Proc. Natl. Acad. Sci. USA 81, 6559 (1984).
33. K. Honda, H. Kimura, J. Phys. Soc. Jpn. 60, 1212 (1991).
34. T.C. Lubensky, F.C. MacKintosh, Phys. Rev. Lett. 71,
1565 (1993).

35. C.-M. Chen, T.C. Lubensky, F.C. MacKintosh, Phys. Rev.
E 51, 504 (1995).
36. H. Kodama, S. Komura, J. Phys. II (France) 3, 1305
(1993).
37. U. Seifert, J. Shillcock, P. Nelson, Phys. Rev. Lett. 77,
5237 (1996).
38. J.L. Harden, F.C. MacKintosh, P.D. Olmsted, Phys. Rev.
E 72, 011903 (2005).
39. C.-M. Chen, F.C. MacKintosh, Europhys. Lett. 30, 215(1995).
40. C.-M. Chen, F.C. MacKintosh, Phys. Rev. E 53, 4933
(1996).
41. L. Yang, M. Fukuto, Phys. Rev. E 72, 010901 (2005).
42. P. Chaikin, T.C. Lubensky, Principles of condensed matterphysics (Cambridge University Press, Cambridge, 1995).