INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 17 (2005) S2951–S2956

doi:10.1088/0953-8984/17/31/023

Phase behaviour of three-component lipid mixtures
S Komura1,3 , H Shirotori1 and P D Olmsted2
1 Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397,
Japan
2 School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK

E-mail: komura@comp.metro-u.ac.jp

Received 30 December 2004, in final form 25 January 2005
Published 22 July 2005
Online at stacks.iop.org/JPhysCM/17/S2951
Abstract
To describe the lateral phase separation in biological membranes, we propose
phenomenological models for binary and ternary lipid mixtures. We consider acoupling between the local lipid composition and internal membrane structure,and its influence on transitions between liquid and gel membrane phases.
Assuming that the gel transition temperature of a pure lipid is shifted by thepresence of other lipids, we obtained a variety of binary phase diagrams which
are in semi-quantitative agreement with the experiments. With the use of themodel parameters deduced from the binary cases, we predict the quantitative
phase behaviour of the ternary lipid systems.

1. Introduction
Biological membranes are lipid bilayers in water, and typically contain various components
such as lipid mixtures, sterols, and proteins that are indispensable to cell functions [1].
Rather than being uniformly distributed in the membrane, there is growing evidence that somemembrane components are incorporated in domains arising from lateral lipid segregation inmembranes. This phenomenon has attracted great interest in the context of ‘rafts’ [2], i.e.,liquid domains rich in cholesterol, saturated lipids (typically sphingomyelin lipids), and insome cases particular proteins [3]. Moreover, cholesterol-rich domains have been directly
observed in model membranes composed of lipid mixtures and cholesterol, using fluorescencemicroscopy [4–7].
From a physical viewpoint, a strategy for understanding the basic structure of ‘rafts’ inbiological membranes is as follows [8]. First, it is necessary to have a simple model for binary
mixtures of saturated and unsaturated lipids. Second, a minimal model describing binary lipid–cholesterol systems is required in order to understand the effects of cholesterol on membrane
3 Author to whom any correspondence should be addressed.

0953-8984/05/312951+06$30.00© 2005 IOP Publishing Ltd

Printed in the UK

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S2952

S Komura et alphase behaviour. Finally, these two viewpoints may be combined to fully investigate threecomponent systems. In the previous paper, we have focused on the first two steps and proposed
simple phenomenological models for lipid–lipid and lipid–cholesterol binary mixtures [9].
Assuming that the gel transition temperature of the saturated lipid is shifted by the presenceof the unsaturated lipid, and that cholesterol acts as an external field on the chain melting
transition, we obtained a variety of phase diagrams which are in semi-quantitative agreementwith the experiments. Such an approach proved to be quite useful since we can predict, at least
qualitatively, the complex phase behaviour of the ternary lipid–lipid–cholesterol system fromthe three binary sub-systems. However, a complete model which quantitatively describes theternary phase behaviour is still lacking, although there are several experimental attempts todetermine it [10–13].
In this paper, we propose a model for three-component phospholipid mixtures (withoutcholesterol). This model is a natural extension of our previous model for binary lipid
mixtures [9], and provides us with the first step to construct a general ternary model including
sterols. Our model can be useful, since red blood cell membranes, for example, contain over
250 lipid species [14]. In our approach, we first determine the phenomenological parametersfrom the three experimental phase diagrams of binary systems. These values are used to
predict the full ternary phase behaviour based on the extended model for three-component
lipid systems. Although there have been many experimental works on binary lipid mixtures,
we are not aware of any corresponding systematic study for ternary lipid systems. Hence ourtheoretical predictions should be checked in the future. In the next section, we first describe
the model for binary phospholipid systems. Our previous model is slightly modified in orderto take into account the asymmetry between the two lipids. The calculated phase diagrams
reproduce experimental ones, without specifying the detailed microscopic state of the lipids.
In section 3, the model for three-component phospholipid mixtures is presented together withsome typical ternary phase diagrams.
2. Binary lipid systems
Lipid membranes undergo a freezing or gel transition, in which the hydrocarbon tails order. It
is generally believed that the gel phase transition is driven by the significant reduction in lateral
diffusion as well as conformational ordering of lipids. However, because these two degrees of
freedom are strongly coupled, the gel phase transition occurs at a single temperature for purelipid systems. As explained below, one of our major assumptions is that the membrane state,even for lipid mixtures, can be described by one internal degree of freedom, coupled to thelateral phase separation.
Let us first focus on a binary lipid system for which many experiments have been
performed [15]. If different lipids exhibit only a small difference in their gel transition
temperature, their composition–temperature phase diagram will include a ‘cigar-like’ shapeof liquid–gel coexistence (figure 1(a)). Below this liquid–gel coexistence, there is a gel–gel coexistence region (figure 1(b)) unless the water freezes around this temperature. Whenthe gel transition temperature difference between the two lipids becomes larger, the phase
diagram becomes more complex because the two coexistence regions partially overlap(figure 1(c)).
We consider a single bilayer membrane composed of x mole fraction of lipid A andy = 1 − x of different lipid B. The two lipids are taken to have different gel transition
temperatures originating from different chain length, degree of saturation, or hydrophilic headgroup. The total free energy f = f 1 + f 2 comprises (i) the free energy f 1 of mixing and binaryinteractions, and (ii) the ‘stretching’ energy f 2 , which controls changes in bilayer thickness.

Phase behaviour of three-component lipid mixtures

S2953

(a)

(b)

340

340

D

D

300

280

D+O

320

T[K]

T[K]

320

D+O

300

280

O

O1
260
0

1

0.5

xDPPC

O1+O2

260
0

0.5

xDPPE

O2
1

(c)
340

D1
Tr

D1+D2

D2

T[K]

320
D1+O2
300
Tr

280

O2
O1

260
0

O1+O2
0.5

xDPPE

1

Figure 1. Calculated phase diagrams of binary lipid mixtures corresponding to (a) DEPC–DPPC,(b) DPPC–DPPE and (c) DEPC–DPPE. The interaction parameters are (a) J = 1.0 × 10−20 J,(b) J = 1.0 × 10−20 J, (c) J = 1.37 × 10−20 J. The parameters used for each lipid are∗∗∗= 234 K, TDPPC= 264 K, TDPPE= 284 K. The
n DEPC = 1, n DPPC = 1, n DPPE = 2, TDEPC

−21
−1
−18
other common parameters are a2 = 2.4 × 10
J K , a3 = −1.1 × 10
J, a4 = 2.2 × 10−18 J,
δ0 = 3.5 × 10−9 m. The ordered (large ψ) and disordered (ψ = 0) phases are respectively denotedby O and D. The critical point is indicated by a filled circle, and the triple point by Tr.

The free energy per site f 1 is the sum of the entropy of mixing and enthalpy. It can be
written within a Flory–Huggins (mean-field) theory as


x
1−x
1
(1)
f1 (x) = kB T
log x +
log(1 − x) + J x(1 − x),nAnB2
where kB is the Boltzmann constant, T is the temperature, and J > 0 is an attractive interaction
parameter that enhances demixing. The overall asymmetry between the two lipids that
contributes to the entropy is taken into account through n A and n B . These phenomenologicalparameters will be adjusted so as to reproduce the location of the experimentally determined
critical point.
To describe the gel transition, which involves chain ordering and stiffening, we introduce
a rescaled membrane thickness ψ ≡ (δ − δ0 )/δ0 as an order parameter, where δ is theactual membrane thickness, and δ0 is the constant membrane thickness in the disorderedphase corresponding to the liquid phase [16]. Note that ψ summarizes changes in various
degrees of freedom, including the conformations of the hydrocarbon chains and their interchain correlations. Since the gel transition is first order, an appropriate phenomenologicalS2954

S Komura et alLandau expansion of the ‘stretching’ free energy in powers of ψ is [16]f2 (x, ψ) = 12 a2 [T − T ∗ (x)]ψ 2 + 13 a3 ψ 3 + 14 a4 ψ 4 ,(2)

where a2

> 0, a3 < 0, and a4 > 0 are the phenomenological coefficients which arequantitatively determined according to the experiments. For a single component membrane(i.e., x = 0 or 1), T ∗ is a reference temperature, and the first-order gel transition temperatureis given by Tg = T ∗ + 2a32 /(9a2 a4 ). For a binary mixture, on the other hand, we describe thereference temperature T ∗ (x) as a linear interpolation between the two pure limits:T ∗ (x) = x TA∗ + (1 − x)TB∗ ,(3)

where TA∗ and TB∗ are the reference temperatures of the pure A and B lipids, respectively. Note
2that equation (3) leads to a coupling term xψ . We have neglected a lower order bilinear termxψ, which induces a small temperature and composition dependence in δ0 . Another possible
coupling term x 2 ψ simply renormalizes the interaction parameter J .
Combining equations (1)–(3), we obtain the total free energy f . After minimizing withrespect to ψ, the two-phase region is obtained by the Maxwell construction. As regards
the choice of the parameter values, we tried to reproduce the phase diagrams of the binary
lipid mixtures reported in [17, 18]. In figure 1, we show three phase diagrams representing(a) DEPC–DPPC, (b) DPPC–DPPE and (c) DEPC–DPPE mixtures. The mixture of DEPC–
DPPC, as shown in figure 1(a), corresponds to that with relatively weak segregation tendency
characterized by small J . In this case, a cigar-like coexistence region is obtained between adisordered phase (D) where ψ = 0 (liquid) and an ordered phase (O) where ψ > 0 (gel).
With the same value of J but with a different combination of transition temperatures, as infigure 1(b), there appears another coexistence region between two ordered phases, O1 + O2,
below the cigar-shape region. These two ordered phases are characterized by the different
order parameter values, ψ1 < ψ2 , which coincide at the critical point.
The phase diagram for DEPC–DPPE mixtures can be reproduced by choosing larger J as
shown in figure 1(c), in which the O1 + O2 coexistence region extends into and beyond the
D + O coexistence region. We chose here n DEPC = 1 and n DPPE = 2. It should be noted thatthe DEPC–DPPE lipid mixtures have a stronger segregation tendency because both head andtail moieties are different. In this more complex phase diagram, there are two triple points(denoted as Tr) at which the two disordered phases and ordered phase (D1 + D2 + O2), or thedisordered phase and two ordered phases (D1 + O1 + O2) coexist.
3. Ternary lipid systems
Based on the model for binary lipid systems, we now discuss the phase behaviour of threecomponent lipid membranes. Suppose a bilayer membrane consists of three types of lipids, A,
B and C, whose mole fractions are x, y, and z, respectively. Note that there is a conservation
law: x + y + z = 1. The simplest and most natural extension of the binary case is to construct
a free energy f = f 1 + f 2 such that


JBC
JCA
x
y
z
JAB
xy +
yz +
zx,
(4)
f 1 (x, y, z) = kB T
log x +
log y +
log z +
nA
nB
nC
2
2
2
f 2 (x, y, z, ψ) = 12 a2 [T − T ∗ (x, y, z)]ψ 2 + 13 a3 ψ 3 + 14 a4 ψ 4 ,

(5)

and
T ∗ (x, y, z) = x TA∗ + yTB∗ + zTC∗ .
In addition to TA∗ and TB∗ , the reference temperature of the lipid C is denoted by TC∗ .

(6)

The
asymmetry of the lipid C with respect to the other lipids is taken into account by n C in thePhase behaviour of three-component lipid mixtures

S2955

DPPC

(a)

T=290K

O1
O2
D

D+O1+O2

0

0.5

DEPC

DPPE
1

DPPC

(b)

T=320K

D
O
DEPC

DPPE
0

0.5

1

Figure 2. Ternary phase diagrams of the three-component lipid mixture (DEPC–DPPC–DPPE) at(a) T = 290 K and (b) 320 K. The ordered (large ψ) and disordered (ψ = 0) phases are respectively
denoted by O and D. The solid lines in the two-phase regions represent tie lines. The three-phasecoexistence region is denoted by D + O1 + O2. The parameters are the same as those listed in thecaption of figure 1.

entropy terms. Equation (6) brings the couplings terms xψ 2 and yψ 2 . Since the above model
naturally reduces to that of the binary systems when one of the species is absent, all theinteraction parameters JAB , JBC , and JCA can be determined from the corresponding threebinary phase diagrams.
To illustrate this procedure, we have investigated the ternary mixture of DEPC–DPPC–
DPPE. In the previous section, we have obtained three binary phase diagrams which reproduce
experimental ones. We use here the same values for the interaction parameters and otherphenomenological parameters. For a given composition, the total free energy f is minimized
first with respect to ψ to determine the internal state of the membrane. Then the free energiesof the two-phase and the three-phase coexisting states are calculated to obtain the whole phasebehaviour.
Figure 2 shows two typical phase triangles at (a) T = 290 K and (b) 320 K. In figure 2(a),the phase separation occurs on the two binary sides (DEPC–DPPE and DEPC–DPPC) of
the triangle, and there is a three-phase coexisting region (D + O1 + O2) within the triangle.

S2956

S Komura et al

The two-phase coexisting region between two ordered phases (O1 + O2) extends toward theDPPC–DPPE binary side, and ends at a critical point. Note that the O1- and O2-phases havethe same symmetry and are continuously connected. At a higher temperature, as in figure 2(b),the phase separation takes place between the DPPC–DPPE binary system, and the extended
two-phase region (D + O) occupies most of the phase triangle. Such a phase diagram resultsfrom the strong segregation tendency of DPPE with respect to both DEPC and DPPC at thistemperature.
In summary, we have calculated the phase triangles of the three-component lipid mixture
based on the proposed phenomenological model. The model parameters were determined fromthe experimentally obtained binary phase diagrams.
Acknowledgments
We thank D Andelman and T Kato for useful discussions. This work is supported by the Japan
Society for the Promotion of Science, the Royal Society, the Ministry of Education, Culture,
Sports, Science and Technology, Japan under grant No. 15540395.
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