PHYSICAL REVIEW E 84, 031919 (2011)

Charged bilayer membranes in asymmetric ionic solutions: Phase diagrams and critical behavior
Naofumi Shimokawa*
Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

Shigeyuki Komura
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan, andKavli Institute for Theoretical Physics China, CAS, Beijing 100190, China

David Andelman
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel, andKavli Institute for Theoretical Physics China, CAS, Beijing 100190, China(Received 17 May 2011; published 19 September 2011)
We consider the phase separation in an asymmetrically charged lipid bilayer membrane consisting of neutral
and negatively charged lipids that are in contact with in and out ionic solutions having different ionic strengths.
The two asymmetric leaflets are coupled through electrostatic interactions. Based on a free-energy approach, thecritical point and phase diagrams are calculated for different ionic strengths of the two solutions and coupling
parameter. An increase of the coupling constant or asymmetry in the salt concentration between the in and out
solutions yields a higher phase-separation temperature because of electrostatic interactions. As a consequence,the phase-coexistence region increases for strong screening (small Debye length). Finally, possible three-phasecoexistence regions in the phase diagram are predicted.
DOI: 10.1103/PhysRevE.84.031919

PACS number(s): 87.16.D−, 64.60.−i, 64.75.−g

I. INTRODUCTION

Model membranes consisting of mixtures of saturated
lipids, unsaturated lipids, and cholesterol have received considerable attention in recent years because they can be regarded
as model systems of biological cell membranes. Below acertain temperature, the membrane undergoes a lateral phaseseparation between a liquid-ordered phase (rich in saturated
lipid and cholesterol) and a liquid-disordered phase (rich inunsaturated lipid). The resulting lipid domains are sometimes
called “rafts” and are believed to play an important role invarious biocellular processes such as signal transduction andcooperative membrane trafficking [1].
A large number of experimental studies have been carriedout to further explore the consequences of domain formationin model membranes and their relation with biomembrane
functioning. They include, among others, studies of domain
morphology [2,3], domain budding [4], growth dynamics ofdomains [5], and formation of periodic lateral structures [6]. In
particular, we note that lateral phase segregation was directly
observed for lipids at the air-water interface and for giant lipid
vesicles using fluorescence microscopy [2–6]. Furthermore, in
an attempt to understand these experimental findings, several
theoretical models have been proposed [7–9].
More recently, several experimental works have been
conducted in order to understand how electrostatic interactions
affect the phase separation of model membranes composed ofcharged and neutral lipids [10,11]. As charged lipids are everpresent in biomembranes, the role of electrostatic interactions
is important also for biological cells. The phase-coexistence
region was reported to be fully suppressed in lipid bilayers*

nshimo@iis.u-tokyo.ac.jp1539-3755/2011/84(3)/031919(10)consisting of a three-component mixture of neutral saturated
lipid DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine),
negatively charged unsaturated lipid DOPS (1,2-dioleoyl-snglycero-3-phospho-L-serine), and cholesterol, because of theelectrostatic repulsion between charged lipids [10]. Furthermore, the extent of the phase-coexistence region appears
when salt (CaCl2 ) is added, an effect that can be understoodby the screening of electrostatic interactions in presence ofadded salt. In a related experimental study, it was shown thatfor mixtures of DPPC, negatively charged DOPG (L-α-1,2dioleoyl-sn-glycero-3-phosphoglycerol) and cholesterol, thephase-separation temperature becomes higher as salt is added[11]. Such effects were also considered in several theoreticalstudies [12–14].
Another important aspect of biomembranes is the compositional asymmetry between their inner and outer leaflets.
Collins and Keller investigated the phase behavior in asymmetric lipid bilayers and showed that the domain formation ineach leaflet is coupled through an interleaflet interaction [15].
In some cases it was reported that phase separation in one of
the leaflets induces a phase separation in the second leaflet,
while in other cases the lack of phase separation in one leaflet
suppresses domain formation in the second leaflet. Theoreticalmodels based on regular solution theory [16] or Landau theory[17] attempted to take into account interleaflet interactions andexplored their consequences on the bilayer phase diagram.
In related works based on the Poisson-Boltzmann theory,a model describing the immiscibility transition in asymmetricand charged membranes was proposed by May and coworkers[18,19]. Their model took into account an interleaflet electrostatic coupling between two rigid and charged planes modelinga bilayer membrane. The spinodal line, characterizing thephase coexistence, and the critical point were derived anddepend on the ionic strength of the solution as well as on the031919-1©2011 American Physical Society

SHIMOKAWA, KOMURA, AND ANDELMANPHYSICAL REVIEW E 84, 031919 (2011)

interleaflet electrostatic coupling. One of the main conclusions
was that a stronger interleaflet coupling enhances the lateralphase separation. However, it was equally assumed by theauthors that the two aqueous solutions in contact with the twoleaflets have the same salt concentration. We note that this assumption is an oversimplification for cellular biomembranes,
where due to ionic channels and other active processes, theionic strength is different on the two sides of the membrane(the inner- and extracellular regions). This difference in ionicstrength, in turn, contributes to an additional gap between thesurface potential on the two sides of the bilayer and plays animportant role, for example, in neurotransmission processes.
In the present work, we extend the model of Refs. [18,19] inorder to mimic asymmetric biomembranes in a more completeway. In Sec. II we introduce a model based on PoissonBoltzmann theory to describe phase separation for asymmetriccharged bilayers “sandwiched” between two ionic solutions
having different ionic strength. In Sec. III the variation of thecritical point and the entire phase diagram as a function of
the ionic strength (Debye length) and interleaflet electrostatic
coupling is explored. It is shown that the phase separationis enhanced as the salt concentration is increased and is dueto the enhanced electrostatic screening. When an electrostatic
coupling between the leaflets is introduced, the lateral phaseseparation occurs at higher temperatures as compared with theno-coupling case. We also show that the phase-separation temperature increases when the concentration difference between
the two salt reservoirs becomes larger. Finally, discussion andcomparison with other works are presented in Sec. IV.

out

in

FIG. 1. Schematic representation of the electrostatic potential fora binary lipid bilayer consisting of negatively charged and neutral
lipids. The head groups of leaflets 1 and 2 are located at the plane z =d and z = 0, respectively. The (dimensionless) electrostatic potentials
in solutions 1 (z > d) and 2 (z < 0) are denoted by ψ1 (z) and ψ2 (z),respectively. The dimensionless surface potentials at the two leaflets
are denoted by 1 = ψ1 (z = d) and 2 = ψ2 (z = 0), while ψL (z)is the (dimensionless) electric potential inside the hydrocarbon coreregion (0  z  d) of the lipid membrane. The dielectric constant of
the aqueous region (water) and lipid hydrocarbon core are denotedby εW and εL , respectively.

as an incompressible two-dimensional fluid. Hence, the twoleaflets consist of the same total number of lipid molecules
N = A/.
The total free energy per lipid molecule is given by

II. MODEL

ftot (φ1 ,φ2 ) = fmix (φ1 ) + fmix (φ2 ) + fel (φ1 ,φ2 ),The system is modeled as a binary lipid-bilayer composedof a mixture of negatively charged and neutral lipids. Althoughmany experiments are done in the presence of added cholesterol as a third component, it is reasonable to stay within
the simpler case of a binary lipid mixture. As cholesterol isnoncharged, its presence will not change in any major way our
predictions on the role of electrostatics.
The bilayer consists of two undeformable and parallel
leaflets, lying in the xy plane and in contact, respectively,
with the two monovalent salt solutions, as shown in Fig. 1.
Let us label all quantities residing on the inner leaflet by thesubscript 1 and those on the outer one by 2. The inner leaflet(leaflet 1) is located at z = d and is in contact with the innerreservoir located at z > d, while the outer leaflet (leaflet 2) islocated at z = 0 and is in contact with the outer reservoir at
z < 0. In some experiments spontaneous budding of charged
domains toward the vesicular interior was reported [10]. Thiscan be explained by considering the increase in the osmoticpressure coupled with the change in spontaneous curvature of
the outer leaflet. However, in the present study we assume
that the bilayer remains flat and neglect the effect of theosmotic pressure generated across the bilayer. Our model isfurther based on three principal assumptions: (1) Because we
do not consider membrane undulations and their curvature,
the two leaflets have the same area A. (2) The neutral andnegatively charged lipids have the same cross-sectional area
per lipid . This is supported by many experiments on
fluid-like membranes. (3) Furthermore, each leaflet is taken(1)where all energies are measured in units of kB T (kB is theBoltzmann constant and T the temperature), and φ1 and φ2
are the mole fractions of the negatively charged lipid inleaflets 1 and 2, respectively. The free energy ftot consistsof two nonelectrostatic fmix terms (one for each leaflet) and anelectrostatic one fel that induces an interleaflet coupling.
The nonelectrostatic fmix is the Flory-Huggins free energyof lateral mixing in each of the leaflets separately:
fmix (φ) = φ ln φ + (1 − φ) ln(1 − φ) + χ φ(1 − φ),(2)where χ is the (nonelectrostatic) interaction parameter between the two different lipids. Quite generally, χ is taken tovary inversely with the temperature. The free energy fmix is asum of the ideal entropy and enthalpy of mixing between the
two lipid species. In the absence of the electrostatic interaction,
fel = 0, the two-phase coexistence region in the (χ ,φ) plane
is delimited by a demixing curve that terminates at the criticalpoint: χc = 2 and φc = 0.5.
The electrostatic free energy fel can be calculated throughthe charging process [20]:
 φ1
 φ2
fel (φ1 ,φ2 ) = −1 (s1 ,0) ds1 −2 (φ1 ,s2 ) ds2 , (3)00where 1 = e 1 (z = d)/kB T and 2 = e 2 (z = 0)/kB T are
the dimensionless surface potentials on leaflets 1 (z = d) and2 (z = 0), respectively, and e is the elementary charge, while
the electric potential inside the lipid hydrocarbon core region

031919-2

CHARGED BILAYER MEMBRANES IN ASYMMETRIC IONIC . . .

(0  z  d) is denoted by L (z). Within mean-field theory,
the electric potential (z) satisfies the Poisson-Boltzmann
equation
d2
2en
e
,
=
sinh
dz2
εW
kB T

(4)

where n is the salt concentration in bulk and εW the dielectric
constant of the aqueous solution (water). Using the dimensionless electrostatic potential ψ(z) ≡ e (z)/kB T , we obtain thefollowing Poisson-Boltzmann equations in regions 1 and 2 forψ1,2 (z), and Laplace equation in the core region for ψL (z):d 2 ψ1= κ12 sinh ψ1 ,dz2
d 2 ψ2= κ22 sinh ψ2 ,dz2
d 2 ψL= 0,dz2

z > d,
z < 0,(5)0  z  d,where κ1−1 and κ2−1 are the Debye screeninglength in regions
1 and 2, respectively, defined by (κ1,2 )−1 = εW kB T /2e2 n1,2 .
Using the Gauss theorem, the boundary conditions of the twoleaflets are given by
dψ1 
dψL 
e 2 φ1εW,−ε=Ldz z=ddz z=dkB T (6)dψ2 
dψL 
e 2 φ2
εW
,
−
ε
=
−
L
dz z=0dz z=0kB T 
where  is the cross-sectional area per lipid head group and εLis the dielectric constant of the lipid hydrocarbon core region.
Far from both sides of the membrane, the bulk electric potential
is taken to be zero, ψ1 (z → ∞) = 0 and ψ2 (z → −∞) = 0.
Under these boundary conditions, the above equations canbe solved analytically, yielding transcendental expressions forthe two surface potentials, 1 and 2 :
1 = −2 sinh−1 (p1 φ1 + p1 H ) ,
2 = −2 sinh−1 (p2 φ2 − p2 H ) ,

(7)

where
 = 1 − 2 is the membrane potential gap.
Throughout the paper we will make use of three dimensionlessquantities defined as
2π B∼ κ1−1 ,
p1 =
κ1 
2π B
p2 =
∼ κ2−1 ,
(8)
κ2 
ε
 εL
∼ L,
H =
4π εW B ddwhere B = e /(4π εW kB T )  7 Å is the Bjerrum length. The
ratio p2 /p1 = κ1 /κ2 is related to the ratio between the two
Debye screening lengths (κ1,2 )−1 of the two solutions, andH is the rescaled electrostatic coupling parameter between
the two leaflets. By further defining φ 1 ≡ φ1 + H  andφ 2 ≡ φ2 − H  as effective mole fractions of the negatively
charged lipid in each leaflet, we can rewrite Eq. (7) as

PHYSICAL REVIEW E 84, 031919 (2011)

Furthermore, the potential difference across the membrane = 1 − 2 satisfies a transcendental equation
 = −2 sinh−1 (p1 φ 1 ) + 2 sinh−1 (p2 φ 2 ) ,since φ 1,2 themselves depend on  as defined above.
By substituting back the surface potentials, Eq. (7), into
Eq. (3), we obtain the final expression of the electrostatic freeenergy:fel (φ1 ,φ2 ) = 12 H ( )2 + fm (φ 1 ) + fm (φ 2 ).

−1

2 = −2 sinh (p2 φ 2 ) .

(11)

The first term corresponds to the charging energy stored
in the two charged leaflets that are analogous to a twoplate capacitor. The second and third terms represent theelectrostatic energy of the two isolated charged monolayers.
In the absence of the electrostatic coupling between the twoleaflets (H = 0), fel is simply the sum of the free energies
of two isolated leaflets, i.e., fel (φ1 ,φ2 ) = fm (φ1 ) + fm (φ2 ).
Within the Poisson-Boltzmann theory, fm was derived inRef. [21], and we briefly repeat this derivation in the Appendix.
The result is φsinh−1 (ps) ds
fm (φ) = 2
0

⎧
⎫

⎨ 1 − (pφ)2 + 1⎬= 2φ+ ln pφ + (pφ)2 + 1
,
⎩
⎭
pφ
(12)
where p = p1 for φ = φ 1 , and p = p2 for φ = φ 2 . We firstsolve Eq. (10) to obtain  and then use it to calculate thetotal free energy from Eqs. (1), (2), (11), and (12).
In order to analyze the membrane stability toward a lateralphase separation, we calculate the spinodal surface given in
the (φ1 , φ2 , χ ) parameter space by the condition 2  2   2
2∂ ftot∂ ftot∂ ftot−= 0.
(13)∂φ1 ∂φ2∂φ12∂φ22
Using the total free energy of Eq. (1), we obtain
 2 
1
∂ fel
− 2χ +
φ1 (1 − φ1 )
∂φ12
 2 
 2
2
1
∂ fel
∂ fel
×
. (14)− 2χ +=
φ2 (1 − φ2 )∂φ1 ∂φ2∂φ22
The first and second derivatives of the electrostatic free energy,fel (φ1 ,φ2 ), are given by Eq. (11) and can be expressed as∂fel= 2 sinh−1 (p1 φ 1 ) = −1 ,∂φ1∂fel= 2 sinh−1 (p2 φ 2 ) = −2 ,∂φ22

1 = −2 sinh−1 (p1 φ 1 ) ,

(10)

and

(9)
031919-3∂ 2 felfm (φ 1 ) + Hfm (φ 1 )fm (φ 2 ),=∂φ12
1 + H [fm (φ 1 ) + fm (φ 2 )]∂ 2 felf  (φ ) + Hfm (φ 1 )fm (φ 2 )
,
= m 2
2
∂φ2
1 + H [fm (φ 1 ) + fm (φ 2 )](15)

SHIMOKAWA, KOMURA, AND ANDELMAN∂ 2 felHfm (φ 1 )fm (φ 2 ),=∂φ1 ∂φ2
1 + H [fm (φ 1 ) + fm (φ 2 )]2

PHYSICAL REVIEW E 84, 031919 (2011)

(16)



where fm (φ) = d 2 fm (φ)/dφ = 2p/ (pφ)2 + 1. Finally, thespinodal surface χsp (φ1 ,φ2 ) for fixed p1 ,p2 ,H can be determined according to Eqs. (10), (14), and (16).
The chemical potential is a function of the lipid concentration∂ftot
φ1= 2 sinh−1 (pφ 1 ) + ln
+ χ (1 − 2φ1 ),μ1 =∂φ11 − φ1
(17)
∂ftot
φ2
μ2 =
= 2 sinh−1 (pφ 2 ) + ln
+ χ (1 − 2φ2 ).
∂φ2
1 − φ2
Under the assumptions mentioned at the beginning of thissection, the condition for phase coexistence (the binodal line)
requires the equality of the chemical potential of the twophases, A and B, on each of the two leaflets, meaning thatboth μ1,A = μ1,B and μ2,A = μ2,B should be simultaneously
satisfied on the two leaflets. In addition, the thermodynamic
potentialg(φ1 ,φ2 ) = ftot (φ1 ,φ2 ) − μ1 φ1 − μ2 φ2(18)should be equal for the two phases, i.e, gA = gB . These
conditions are sufficient to calculate the two-phase coexistence
region as well as the corresponding tie lines inside thecoexistence region, as will be discussed below in Sec. III. We
remark that the same conditions were also used in previous
works [16,17].
A. The symmetric in and out reservoir case, p1 = p2

Let us first discuss the case of equal ionic strength (p1 = p2 )of the two solutions and review some of the results reported
already in Ref. [18]. The critical point, because of the in and out
symmetry, is located on the φ1 = φ2 line, and the expression
for the spinodal line is given by
χsp =

p
1
+
,
2
2φ(1 − φ)
1 + p φ 2 + 4pH(19)

where p = p1 = p2 and φ = φ1 = φ2 . The minimum value ofχsp corresponds to the critical point.
Let us take now explicitly the limits of high and low salt
concentrations for the symmetric p case. The high-salt limit(p  1) corresponds to the Debye-Hückel regime for which
the spinodal line is given by
χsp ≈

p
1
+
,
2φ(1 − φ) 1 + 4pH

1
1+,
2φ(1 − φ) φ + 4H

In the absence of the electrostatic coupling (H = 0), thecritical point coincides with that in the previous works[18,19,22]:√
χc = 2 + 3  3.73 ,(23)√
φc = (3 − 3)/2  0.634 .
III. GLOBAL PHASE DIAGRAMS AND CRITICALBEHAVIOR

Based on the model described in the previous section, we
present results for the critical point and phase diagrams in thecase of two coupled and charged leaflets. In particular, we
focus on the difference between a symmetric case in which thetwo solutions have the same ionic strength (p1 = p2 ) and theasymmetric case in which they differ (p1 = p2 ).
We start by giving some estimates for the physical choicesof the parameter values of p1 , p2 , and H . The cross-sectional
area of a lipid head group is set to about   65 Å2 (taken to
be the same for the two lipids), and the membrane thickness
is d  50 Å. From these values together with εW  80ε0 ,εL  3ε0 (ε0 is the vacuum permittivity) and T  300 K, we
obtain p  0.67κ −1 where κ −1 is measured in angstroms,and H = 5.6 × 10−4 . In the following, we use a range ofvalues p = 0.5,5,50 and extended the H range to cover H =0.005,0.01,0.05. In the case of monovalent salt, the values
p = 0.5,5,50 correspond, respectively, to κ −1 = 0.75, 7.5,
75 Å or to n = 16 M, 160 mM, 1.6 mM. Since the electrostatic
coupling parameter H is proportional to εL /d, the larger H
values correspond to smaller d and/or larger εL . Obviously,
the range in the p parameter of 102 gives a range of 104 inionic strength, which probably is too large to compare withexperiments. Hence it should be taken just to indicate trends
with changing the salinity.
We show first the phase separation in asymmetric charged
bilayers sandwiched between the two ionic solutions having
the same ionic strength (p1 = p2 ), repeating the results ofRef. [18]. Then we present results for asymmetric charged
leaflets that are in contact with two ionic solutions having
different ionic strengths (p1 = p2 ).

(20)

and the critical point is located at χc = 2,φc = 0.5. This meansthat the bilayer behaves as if it were a neutral membrane when
the salt concentration is sufficiently high.
In the other limit of low salt (p1), the spinodal line isgiven byχsp ≈and it is independent of p. If the electrostatic coupling is weakenough (H  1), the respective critical point is located at


√
8H
,
χc = (2 + 3) 1 −
3
(22)
√
√
3 − 3 4H ( 3 − 1)
−
.
φc =
2
3

(21)

A. Symmetric in/out reservoirs: p1 = p2

In Fig. 2 we show the critical interaction and composition(χc ,φc ) as a function of p for the symmetric reservoir case,p = p1 = p2 . Both χc and φc are obtained by numericallyminimizing Eq. (19). The solid, dashed, dot-dashed lines
correspond, respectively, to H = 0.005,0.01,√and 0.05. The
horizontal √dotted lines located at χc = 2 + 3 = 3.73 andφc = (3 − 3)/2 = 0.634 are the critical values for the H →0 limit and for high-salt conditions.

031919-4

CHARGED BILAYER MEMBRANES IN ASYMMETRIC IONIC . . .

PHYSICAL REVIEW E 84, 031919 (2011)

When χ > χc , the membrane undergoes a lateral phaseseparation, whereas for χ < χc the membrane is in a single(homogeneous) phase. For all H values, we find that χcbecomes smaller as p decreases. Namely, the phase-separation
temperature is increased, and the two-phase region has a larger
extent for smaller p. This is because the Coulombic repulsion
between the charged lipids becomes too weak to overpower
the attractive interaction that drives the phase separation. When
H is large, χc becomes much smaller, as will be discussed inSec. IV. In the high-salt limit or when p is small enough, thelimit of χc → 2 and φc → 0.5 can be seen, as was discussed
in Sec. II. In the opposite low-salt limit, χc and φc approaches
those of Eq. (22).

3.6 (a)
3.2
2.8
2.4
2
0.1

1

10

100
B. Asymmetric in and out reservoirs: p1  = p2

The calculated value of χc for the asymmetric case, in which
the two solutions have different ionic strengths, is plottedin Fig. 3. Since χsp equals to χc at the critical point, χc
is obtained by calculating numerically the minimum valueof χsp via Eq. (14). The horizontal axis is the ratio p2 /p1
in logarithmic scale, where p2 /p1 = 1 corresponds to equalionic strength of the two reservoirs. The solid, dashed, anddot-dashed lines denote fixed p1 = 0.5, 5, and 50, respectively,
while the coupling parameter H has three different values
H = 0.005,0.01, and 0.05 in the three figure parts. Similar to
the findings of Fig. 2, χc decreases as the coupling H increases.
This dependence of χc on H is similar as in the symmetriccase and is further discussed in Sec. IV. In all plotted cases
there is a peak around p2 /p1 = 1, without a break in its slope
there even for the solid line. For p2 /p1 < 1, χc decreases as
the salt concentration in solution 2 is increased, similar to thesymmetric case.
Moreover, the Coulombic repulsion between charged lipids
in the same leaflet becomes weak and χc decreases when p2 /p1
becomes smaller. For p2 /p1 > 1, χc is found to decrease andits value is more pronounced when H becomes larger, although
the electrostatic screening is weaker. The physical origin of thisbehavior will be further discussed in Sec. IV.

0.62 (b)

0.58

0.54

0.5
0.1

1

10

100

FIG. 2. (a) The critical interaction χc and (b) critical composition
φc as a function of p on a semilog plot for different H values in thesymmetric case (p1 = p2 = p). The solid, dashed, and dot-dashedlines are for H = 0.005,0.01,0.05, respectively. For H = 0, thecritical values
dotted lines located at (a)√ are represented by the √χc = 2 + 3  3.73 and (b) φc = (3 − 3)/2  0.634.

(a)

(b)
3.5

3.5

3

3

3

2.5

2.5

2.5

3.5

2
0.01

0.1

1

10

100

2
0.01

0.1

1

10

100

(c)

2
0.01

0.1

1

10

100

FIG. 3. The critical value χc as function of the asymmetry parameter, p2 /p1 , which also is equal to the ratio of the two Debye lengths,
κ1 /κ2 . The solid, dashed, and dot-dashed lines are for p1 = 0.5,5,50, respectively. The values of the H parameter are (a) H = 0.005,(b) H = 0.01, and (c) H = 0.05.
031919-5

SHIMOKAWA, KOMURA, AND ANDELMAN

(a)

(b)

1

(c)

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

(d)

PHYSICAL REVIEW E 84, 031919 (2011)

0

0.2

0.4

0.6

0.8

0

1

(e)

1

0

0.2

0.4

0.6

0.8

0

1

(f)

1
0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0.2

0.4

0.6

0.8

1

0

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

1

0.8

0

0

0

FIG. 4. Phase diagrams in the (φ1 ,φ2 ) plane for χ = 3.8 and p1 = 5. The solid lines are the tie lines and the dashed lines are the spinodal
lines. The three-phase-coexistence regions are denoted by shaded triangles. The values of the other parameters are (a) H = 0.005 and p2 = 0.5;(b) H = 0.005 and p2 = 5; (c) H = 0.005 and p2 = 50; (d) H = 0.05 and p2 = 0.5; (e) H = 0.05 and p2 = 5; and (f) H = 0.05 and p2 = 50.

The calculated phase diagrams in the (φ1 ,φ2 ) plane forχ = 3.8 and 4.0 are shown in Figs. 4 and 5, respectively.
We plot all phase diagrams for a fixed p1 = 5, while varying
p2 and H . As can be seen in Fig. 3, the system undergoesa phase separation for χ values larger than χc . The solidlines are the tie lines within the coexistence region, while
the dashed lines correspond to the spinodal lines. Each pairof points connected by the tie line satisfies the conditions
μ1,A = μ1,B , μ2,A = μ2,B [Eq. (17)] and gA = gB [Eq. (18)],
while the spinodal lines are obtained from Eqs. (10), (14), and(16). In some of the figures, inner and external spinodal lines
can be seen. However, the inner lines are preempted by theexternal ones. Since Fig. 4(b) and 4(e) and Fig. 5(b) and 5(e) arefor the symmetric case (p1 = p2 ), these four phase diagrams
are symmetric with respect to the diagonal line, φ1 = φ2 . Allother phase diagrams correspond to asymmetric situations,p1 = p2 .
The phase-coexistence region becomes relatively larger
when p2 /p1  1, corresponding to strongly screened systems.
However, χc for the asymmetric case is smaller than thatof the symmetric case. When H = 0.005, most of the tielines are nearly parallel to either φ1 or φ2 axis. This implies
that the phase separation in each leaflet takes place almost
independently and without any noticeable correlation with thesecond leaflet. On the other hand, for H = 0.05, the tie lines
are tilted, indicating a strong coupling between the two leaflets.

The three-phase-coexistence regions are indicated byshaded triangles in the phase diagrams. The tie lines in thevicinity of the three-phase-coexistence regions are almost
orthogonal to the principal diagonal φ1 = φ2 , indicating that
φ1 is small when φ2 is large and vice versa. This occurs
when the local concentration of charged lipid in one of theleaflets is increased, while the concentration in the second
leaflet is decreased conversely. This behavior is caused by theelectrostatic coupling between the leaflets, as is discussed in
the next section.
IV. DISCUSSION AND COMPARISON WITH
EXPERIMENTS

Several points merit further discussion. First, we compare
our theoretical results with experiments, where it was reported
that the phase separation is enhanced when salt is added tocharged bilayers [10]. Such a tendency is well reproducedin Figs. 4 and 5 when the salt concentration in solution 2
is increased or equivalently p2 is decreased. A similar trend
was also presented in previous works [12–14] for charged
membranes without any electrostatic interleaflet coupling. In
addition, it was reported by Vequi-Suplicy et al. [11] that thephase-separation temperature increases in the presence of salt.
In previous experimental studies [10,11], vesicles werefirst dispersed in the aqueous solution, and only then salt

031919-6

CHARGED BILAYER MEMBRANES IN ASYMMETRIC IONIC . . .

(a)

(c)

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

(d)

(b)

1

PHYSICAL REVIEW E 84, 031919 (2011)

0

0.2

0.4

0.6

0.8

0

1

(e)

1

0

0.2

0.4

0.6

0.8

0

1

(f)

1
0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

1

0.8

0

0

0

FIG. 5. Phase diagrams in the (φ1 ,φ2 ) plane for χ = 4.0 and p1 = 5. The solid lines are the tie lines, and the dashed lines are the spinodal
lines. The three-phase-coexistence regions are denoted by shaded triangles. The values of the other parameters are (a) H = 0.005 and p2 = 0.5;(b) H = 0.005 and p2 = 5; (c) H = 0.005 and p2 = 50; (d) H = 0.05 and p2 = 0.5; (e) H = 0.05 and p2 = 5; and (f) H = 0.05 and p2 = 50.

was added, affecting only the ionic strength of the solution
outside the vesicles. For this experimental procedure the valueof p2 /p1 is smaller than unity (recalling that p2 corresponds
to the outer reservoir). This situation is presented in Fig. 3
where χc ∼ 1/Tc decreases as p2 /p1 < 1 becomes smaller,in accord with the experimental observations. In addition,
we find that also for p2 /p1 > 1, χc is decreased even if
the electrostatic screening is weak. In order to confirm thisbehavior in experiments, only the outer solution should bediluted in order to yield p2 /p1 > 1. Controlled experiments
done on asymmetric charged membranes with asymmetric inand out ionic strengths are needed to confirm the predictionsof the present work. Because the lipid composition in eachleaflet prepared by gentle hydration or electroformation is notwell controlled, it is better to prepare vesicles by transferring
water-in-oil droplets coated by lipids from an oil phase to awater phase [23,24] or by using asymmetric Montal-Mueller
planar bilayers [15,25].
In the following, we discuss the physical reason why χcdecreases for p2 /p1 > 1. Such a tendency is more pronounced
for larger H ∼ εL /d, because the electrostatic coupling
between the two leaflets becomes stronger. Notice that theelectrostatic coupling depends on p1 and p2 as well as on H .
As schematically presented in Fig. 6, the electrostatic couplingis weaker when the salt concentration in solution 2 is higher(upper panel of Fig. 6), and the phase separation is suppressed.

leaflet 1
leaflet 2

leaflet 1
leaflet 2

FIG. 6. Schematic illustration of the phase separation for an
asymmetric charged lipid bilayer when p2 /p1 > 1. On the upper
panel, the salt concentration in solution 2 is high (but still obeying
p2 /p1 > 1). On the lower panel, the salinity in solution 2 is reduced
resulting in an increase of p2 /p1 . The electrostatic coupling becomesthen stronger and the phase separation is induced.

031919-7

SHIMOKAWA, KOMURA, AND ANDELMANPHYSICAL REVIEW E 84, 031919 (2011)

On the other hand, when the salt concentration in solution 2
is low (lower panel of Fig. 6), the screening is not as efficient,and phase separation is induced due to the strong electrostatic
coupling. Note that in the latter case, the strong demixing
in leaflet 1 is induced by changing the salt concentration inother reservoir (solution 2), which is coupled directly with the
opposite leaflet. This is indeed an interesting situation because
the higher salinity in solution 1 relative to 2 triggers the phaseseparation in leaflet 1 due to the strong interleaflet electrostatic
coupling. The region where the tie lines are almost parallelto the φ1 axis means that the phase separation mainly takes
place in leaflet 1. As can be seen in Fig. 4(c) and 4(f) as wellas in Fig. 5(c) and 5(f), such a region becomes much larger
than the region where the phase separation strongly occur inleaflet 2 for large p2 . On the other hand, the phase separation
also occurs in leaflet 2, although the concentration difference
between two phases is very small. As shown in Fig. 3, χc isslightly decreased when p1 is smaller and can be understood bynoting that the screening sufficiently weakens the electrostatic
coupling.
In Figs. 2 and 3 we have shown that χc is smaller forlarger H , i.e., when the electrostatic interaction across themembrane is large. Because of the electrostatic coupling,
the local charge accumulation in one of the two leaflets
suppresses the charge accumulation in the other leaflet, and it
is energetically unfavorable for the charged domains on eachleaflet to face each other. As a result, a phase separation is
induced in the other leaflet driven by the electrostatic coupling.
Next we elaborate on the physical meaning of the slope ofthe tie lines in Figs. 4 and 5. When the tie lines are located close
to the three-phase region (triangle) and/or when H is large, thetie lines tend to be almost orthogonal to the principal diagonal.
These tie lines connect two points on the binodal line thathave a large compositional asymmetry between φ1 and φ2 . It
indicates that the charged domains in one leaflet do not preferto face the charged domains in the other leaflet. The model by
Baciu and May [18] as well as the present model deal withthe compositional coupling between the two leaflets arisingexclusively from electrostatic interactions. Consequently, it isenergetically favorable for the charged domains on each leaflet
not to face each other.
In order to see the contribution of the electrostatic couplingclearly, we consider the electrostatic coupling in the symmetriccase (p1 = p2 ). The free energy of the interleaflet interaction
fcoup is obtained by subtracting the contributions of individual
leaflets from the total free energy
fcoup (φ1 ,φ2 ) = ftot (φ1 ,φ2 ) − fmix (φ1 ) − fmix (φ2 )− fm (φ1 ) − fm (φ2 ).

(24)

For small composition differences, the potential difference
across the membrane is approximately given by
≈−

2p

(φ1 − φ2 ),2
4Hp + 1 + p2 φav(25)

where φav = (φ1 + φ2 )/2 is the average composition of thecharged membrane. Substituting Eq. (25) into Eq. (24), we

obtain an interleaflet free energy that can be expanded in termsof φ1 − φ2 [26],fcoup (φ1 ,φ2 ) ≈ −2Hp2


(φ1 − φ2 )2 .
2 + 4Hp 1 + p 2 φ 2
1 + p2 φav
av(26)

From Eq. (26), it is clear that fcoup is negative and becomes
smaller when the difference between φ1 and φ2 is large,
because the coefficient of (φ1 − φ2 )2 is negative. This isconsistent with the tie lines in the phase diagrams, and a similartendency can be seen even for the more general asymmetricsystems.
In general, the physical origin of the coupling between the
two leaflets can vary and does not have to rely exclusively onelectrostatics interaction. Examples of such nonelectrostatic
effects were considered in previous works [26,27] and include
cholesterol flip-flop, dynamic chain interdigitation, van der
Waals interaction or composition-curvature coupling. The
coupling between the two leaflets can be expressed generallybyfcoup = (φ1 − φ2 )2(27)

as shown in Eq. (26). The coupling coefficient  becomes
negative in the case of the electrostatic coupling. However,
Wagner et al. [16] analyzed a model for coupled bilayers
containing such a phenomenological term as in Eq. (27)but with  > 0. It is, therefore, important to consider thesign and typical value of  for different systems containing
electrostatic and nonelectrostatic couplings and to reveal how
the interactions between the two leaflets contribute to thecoupling constant, . Further experimental and theoreticalinvestigations are required to gain more insight on thiscoupling and its origin.
The present work is concerned with macrophase separationin lipid membranes. Membranes consisting of neutral andcharged lipids can form also modulated phases [28] thatexhibit characteristic periodic structures. Hirose et al. [29]considered the coupling between two microphase separated
bilayer leaflets. Various patterns, phases and their phasetransitions were predicted by assuming a phenomenologicalcoupling between the two leaflets.
We would like to close by mentioning again the twomain approximations used in the present model. (1) We havetreated the two leaflets as undeformable flat sheets. By doing
so we neglected any possible bilayer deformation that canoccur because of osmotic pressure difference resulting froma difference in ionic concentrations (p1 = p2 ) on the two
membrane sides. In experiments, phase-separated domains
produce budding toward the interior of the vesicle, andthis is a direct consequence of the osmotic pressure on
the spontaneous curvature [10]. In future studies it may beof advantage to consider the combined effect of the phaseseparation and membrane deformation. (2) We have used
the Poisson-Boltzmann theory for symmetric monovalent salt
in order to describe the electrostatic potential. It is known
that such a mean-field theory does not treat correctly ion-ion
correlation, especially when multivalent cations are involved.
This important effect should also be taken into account infollow-up modeling.

031919-8

CHARGED BILAYER MEMBRANES IN ASYMMETRIC IONIC . . .

In conclusion, we have considered the phase separation
in a charged and asymmetric lipid bilayer, located between
two ionic solutions having different ionic strengths. Considering the electrostatic effects on the phase separation, we
have introduced the electrostatic coupling between the twoleaflets. We studied the effect of the ionic strength difference
between the two solutions, p1 = p2 , and the electrostatic
coupling H on the membrane phase behavior. When theelectrostatic coupling H is introduced, χc ∼ 1/Tc becomes
smaller than for the uncoupled case. Moreover, even if thescreening effect is small, χc decreases by increasing theasymmetry in the salt concentrations. An increase in the ionicstrength leads to an enlarged coexistence region in the phase
diagram.

PHYSICAL REVIEW E 84, 031919 (2011)

and at z = d,ψ1 (d) = −2κ1 sinh



1
2


.

(A3)

Using Eq. (6) and ψ1 (d) = ψ2 (0) = /d, it follows that
 
2π B
1
2π B  εL
=−
sinh
φ1 −
.
(A4)
2
κ1 
κ1  4π εW B dFrom this expression, the surface potential 1 is1 = −2 sinh−1 (p1 φ1 + p1 H ) ,(A5)

where p1 = 2π B /κ1 , H = εL /4π εW B d as well as otherparameters were introduced in Sec. II. The surface potential
on the second leaflet, 2 , can be obtained in the same way.

ACKNOWLEDGMENTS

2. Electrostatic free energy of an isolated leaflet

We are grateful to S. May and M. Schick for numerous
comments and suggestions. NS was supported by a JSPS
Research Fellowship for Young Scientists (23-8690) andExcellent Young Researchers Overseas Visit Program from the
Ministry of Education, Culture, Sports, Science and Technology of Japan. This work was partially supported by KAKENHI(Grant-in-Aid for Scientific Research) on Priority Area “SoftMatter Physics,” Grant No. 21540420 from the Ministry of
Education, Culture, Sports, Science and Technology of Japan,
the Israel Science Foundation (ISF) under grant no. 231/08,
and the US-Israel Binational Science Foundation (BSF) under
grant no. 2006/055.

Next we calculate the electrostatic free energy of an
isolated leaflet, Eq. (12), using the charging method. A similar
derivation can be found, e.g., in Refs. [19,21]. This energy
corresponds to the energy that a leaflet feels in solution without
the electrostatic coupling to the second leaflet. When a charged
leaflet at z = 0 is in contact with an ionic solution situated at
z > 0, the Poisson-Boltzmann equation gives
 
ψ

.
(A6)
ψ = −2κ sinh
2

APPENDIX: SURFACE POTENTIAL AND CHARGINGFREE ENERGY OF AN ISOLATED LEAFLET

This equation is the same as Eq. (A2). The nonlinear first-orderdifferential equation can then be integrated analytically:


2


ψ(z) = −2 ln 1 +.
exp(κz)coth 12 sinh−1 (pφ) − 1(A7)

1. Leaflet electrostatic potential

We present the derivation of the surface potentials 1,2
[Eqs. (7) and (9)] and the electrostatic free energy of an isolated
leaflet [Eq. (12)]. In order to find 1 , we multiply the PoissonBoltzmann equation (4) by ψ1 (z) and integrate it once, yielding


 
2
(A1)
ψ1 ψ1 dz = κ1 ψ1 sinh ψ1 dz,
where ψ1 = dψ1 /dz and ψ1 = d 2 ψ1 /dz2 . Using the appropriate boundary conditions results in the following expression: ψ1ψ1 = −2κ1 sinh
,
(A2)
2

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From this expression, the surface potential  = ψ(z = 0) is
given by
 = −2 sinh−1 (pφ).

(A8)

Through the charging process, the electrostatic free energy of
an isolated leaflet is obtained by the following integral over 
yielding an analytical expression for fm : φ
1 − (pφ)2 + 1
(s) ds = 2φ
fm (φ) = −
pφ
0




+ ln pφ + (pφ)2 + 1 .
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