Langmuir 2006, 22, 6771-6774

6771

Tension-Induced Morphological Transition in Mixed Lipid Bilayers
S. Komura* and N. Shimokawa
Department of Chemistry, Faculty of Science, Tokyo Metropolitan UniVersity, Tokyo 192-0397, Japan

D. Andelman
School of Physics and Astronomy, Raymond and BeVerly Sackler Faculty of Exact Sciences,
Tel AViV UniVersity, Ramat AViV 69978, Tel AViV, Israel
ReceiVed NoVember 19, 2005. In Final Form: March 5, 2006
Recently, Rozovsky et al. reported on the morphology and dynamics of superstructures in three-component lipid
bilayers containing saturated lipid, unsaturated lipid, and cholesterol (Rozovsky, S.; Kaizuka, Y.; Groves, J. T. J. Am.
Chem. Soc. 2005, 127, 36). We suggest that the observed sequence of the striped-to-hexagonal morphological transitionin mixed bilayers can be attributed to an enhanced membrane surface tension that is induced by the vesicle adhesion
onto the solid surface.

I. Introduction
Raft domains in biological cell membranes are associated withmembrane signaling pathways and have attracted a great deal ofinterest in recent years.1,2 Because of the complexity of biologicalmembranes, a minimal model to investigate rafts (or more
precisely, domain formation) consists of three-component lipid
bilayers containing saturated and unsaturated lipids as well as
cholesterol. These three-component systems have been investigated both at the air-water interface and in artificial vesicles.3-5
Complex phase separation and the appearance of domains havebeen observed using fluorescence techniques. Such a phaseseparation occurs between a liquid-disordered phase rich inunsaturated lipid (e.g., DOPC) and a liquid-ordered phase richin saturated lipid (e.g., sphingomyelin) and cholesterol.3-5
Recently, Rozovsky et al.6 reported on the morphology anddynamics of superstructures in a ternary mixture of sphingomyelin, DOPC, and cholesterol. A giant 10 µm unilamellar vesicle
is placed in contact with a lipid bilayer that is supported on topof a silica surface. The experimental setup is shown schematicallyin Figure 1. Fluorescence microscopy is used to monitor thecreation and temporal evolution of domains on the adhering
vesicle. The bound vesicle exhibits a gradual pattern transformation under the adhesion constraints. In a number of cases, thenonadhering top region of the vesicle, which can be directly
imaged by the microscope, first exhibits a striped pattern ofwell-characterized stripe width. As time evolves, the striped
pattern transforms quite abruptly into another pattern consistingof an ordered hexagonal lattice of circular domains, as isschematically represented in Figure 1. The transformation of the
stripes into circular domains is seen first as a “pinch-off” instabilityof the stripe tip. In a matter of a few seconds, the entire striped
pattern is replaced by circular domains having roughly the same
diameter as the original stripe width. As time progresses evenfurther, the hexagonal pattern of circular domains becomes less
* Corresponding author. E-mail: komura@comp.metro-u.ac.jp.
(1) Simons, K.; Ikonen, E. Nature 1997, 387, 569.
(2) Brown, D. A.; London, E. J. Biol. Chem. 2000, 275, 17221.
(3) Veatch, S. L.; Keller, S. L. Phys. ReV. Lett. 2002, 89, 268101.
(4) Baumgart, T.; Hess, S. T.; Webb, W. W. Nature 2003, 425, 821.
(5) Veatch, S. L.; Keller, S. L. Phys. ReV. Lett. 2005, 94, 148101.
(6) Rozovsky, S.; Kaizuka, Y.; Groves, J. T. J. Am. Chem. Soc. 2005, 127,
36.

Figure 1. Schematic representation of a mixed vesicle adhering
onto a supported lipid bilayer. The nonadhering upper region of thevesicle exhibits a morphological transition from a striped phase toa hexagonal phase with the elapse of time. The size of the vesicle
is on the order of 10 µm, and the period of the modulation is roughly
1 µm. Details of the experiment are described in ref 6.

ordered. Because of the large size of the vesicle, its upper partshown in Figure 1 of ref 6 is a few micrometers above thesupported lipid bilayer and is not in contact with the silica surface.
The shape of the vesicle and its contact area change as the pattern
evolves. Note that the field of view is adjusted to capture theshape evolution.
It is known that surface tension can induce the adhesion ofa vesicle onto a substrate or to another vesicle,7 the reason beingthat the membrane surface tension suppresses thermal fluctuations
and promotes its adhesive properties. A self-consistent treatmentof such a problem predicted the conditions for tension-induced
adhesion.7 On the other hand, it was reported that the spreadingof red blood cells on a substrate produces finite surface tensionthat can even cause them to rupture.8 More recent experimentson the pulling of nanotubes from adhered vesicles revealed that
membrane tension strongly increases during the pulling andelongation of nanotubes.9(7) Seifert, U. Phys. ReV. Lett. 1995, 74, 5060.
(8) Hategan, A.; Law, R.; Kahn, S.; Discher, D. E. Biophys. J. 2003, 85, 2746.

10.1021/la053135x CCC: $33.50 © 2006 American Chemical Society
Published on Web 06/30/2006

6772 Langmuir, Vol. 22, No. 16, 2006

Komura et al.

In view of these works, we explore the relation between an
increase in surface tension and lateral morphological transitions
that can occur on the membrane plane. The aim of this paper is
to point out that the observed sequence of the striped-to-hexagonal
morphological transition in mixed bilayers may be attributed toan enhanced membrane surface tension that is induced by thevesicle adhesion onto the solid surface. In the next section, we
adopt such a model describing the modulation instability in mixed
membranes.10 We then analyze the consequence of this model
as surface tension is increased. Some discussion and extensionsof our model are provided in section III.

II. Model and Results
The main component of the model is to introduce a coupling
between the local composition on the membrane plane and the
local curvature. Rather than considering a closed-form vesicleof finite area and enclosed volume, we deal here with a simpler
planar and extended membrane. Its shape is then described by
the displacement h(x, y) relative to the reference x-y plane. The
local relative composition (assuming a simple binary mixture oftwo components) is denoted by φ(x, y). The total free energy is
written in terms of these two local fields. For simplicity, we
consider only small undulations above the x-y plane in the formof a phenomenological expansion proposed in ref 10

H{h, φ} )[a∫ d2x σ2(∇h)2 + 2κ(∇2h)2 + 2b(∇φ)2 + 22φ2 +]a4 4
φ - µφ + Λ(∇2h)φ (1)4where σ > 0 and κ > 0 are the surface tension and bending
rigidity of the membrane, respectively. The parameter b > 0 is
related to the line tension between different domains, and a2 ∝T - Tc is the reduced temperature (Tc is the critical temperature).
Below the phase transition, a2 is negative, and we need to include
in the expansion a positive fourth-order term, a4 > 0, for stability
purposes. The coefficient of the linear term, µ, is the chemical
potential. Finally, Λ is the composition-curvature coupling
constant. This coupling term represents the situation where thespontaneous curvature of the membrane depends on the local
composition. For simplicity, other coupling terms such as (∇4h)φ(that has a higher gradient order) as well as higher-order couplingterms are omitted.
Starting from the above free energy, we first integrate out theh-field in Fourier space. Then the effective free energy is expandedin powers of the gradient of φ10,11

Heff{φ} ≈

[

a

a

]

∫ d2x B2 (∇φ)2 + C2 (∇2φ)2 + 22φ2 + 44φ4 - µφ (2)
where

B≡b-

Λ2
σ

C≡

Λ2κ
σ2

(3)

In deriving the above expression, we performed an expansion
up to fourth order in the wavevector q (or equivalently up to
fourth order in the differential operator in real space). WhenB < 0 (or Λ2 > bσ), the coefficient of (∇φ)2 becomes negative,
leading to an instability of the homogeneous system with respect(9) Cuvelier, D.; Chiaruttini, N.; Bassereau, P.; Nassoy, P. Europhys. Lett.
2005, 71, 1015.
(10) Leibler, S.; Andelman, D. J. Phys. (Paris) 1987, 48, 2013.
(11) Brazovskii, S. A. SoV. Phys. JETP 1975, 41, 85.

Figure 2. Mean-field phase diagram in the (m, ε) plane. m represents
the reduced composition, and ε represents the reduced temperature
as defined in eq 6. There are four different phases: the disorderedphase (denoted D1 and D2 on the two sides of the transition lines),
the striped phase (S), the hexagonal phase (H), and the inverted
hexagonal phase (IH). These phases are separated by first-ordertransition lines. For simplicity, we show here by solid lines only thecrossover in the free energies while avoiding plotting two-phaseregions. The filled circle located at (m, ε) ) (0, 1) indicates thecritical point. Reproduced from ref 10.

to long-wavelength fluctuations. In this case, the magnitude of
the most unstable wavevector is given by q* ) (-B/2C)1/2. Noticethat the sign of Λ can also be negative because B and C depend
only on Λ2.
The phase diagram can be constructed by comparing the meanfield free energies of the striped and the hexagonal phases. The
stripe order parameter is

φS ) φ0 + φq* cos(q*x)(4)

where φ0 ) 〈φ〉 is the spatially averaged composition (imposedby the chemical potential µ) and φq* is the amplitude of the q*mode in the x direction. Similarly, the local composition of thehexagonal phase φH(rb) is given by a superposition of three modes:
3

φ H ) φ0 +φq cos(q
bi‚b)
r
∑
i)1

(5)

*

Here bq1, bq2, and b
q3 are arranged symmetrically on a 2D circle
3such that |qbi| ) q* and ∑i)1bqi ) 0. Only the most unstable
wavevector q* (single-mode approximation) is used in thecalculation. This can be justified for the weak segregation region
close to the critical point, T j Tc. In Figure 2, we reproduce theoriginal phase diagram of ref 10. It is presented in terms of the
reduced temperature and the reduced average composition
defined, respectively, by

ε≡

4Ca2
B2

m≡

( )

4Ca4 1/2
B2

φ0

(6)

whereas the surface tension σ and other parameters included in
B and C are fixed. (We have corrected the numerical prefactorsof “2” in eq 10 of ref 10 in the above equations.)
Four distinct phasessdisordered (D), striped (S), hexagonal(H), and inverted hexagonal (IH)scan be seen in Figure 2.
Regions of coexistence are omitted here for clarity. Thus, thetransition lines indicate the locus of points at which the freeenergies of different phases cross each other. The calculated
phase diagram is in qualitative agreement with experiments

Transition in Mixed Lipid Bilayers

Langmuir, Vol. 22, No. 16, 2006 6773

performed on mixtures of lipids in Langmuir monolayers at the
air-water interface.12
To see the role of surface tension clearly, we cannot directly
use the above phase diagram of Figure 2 because σ enters into
the two reduced variables, ε and m, via prefactors B and C.
Instead, we construct a new phase diagram in terms of two
different reduced variables defined by

Ω≡

a2
a4φ0

2

Σ≡

bσ
Λ2

(7)

These two variables are related to ε and m by

Ω)

ε
m2

Σ)1-

(4κa4)1/2φ0
Λm(8)

We note that Ω is independent of the surface tension parameterσ and that the only variation of σ enters into the second variable,Σ ∼ σ. With the use of these new variables, the phase diagram
can be drawn for a given value of M ≡ (4κa4)1/2φ0/Λ. Different
values of M correspond to different values of the averagecomposition φ0 when the coupling constant Λ, a4, and the bending
modulus κ are kept fixed. Physically, this is quite reasonablebecause we expect the latter parameters to have only a weak
dependence on temperature and composition.
Figure 3 is the calculated phase diagram in the (Ω, Σ) plane
for two choices of the parameter M: M ) 0.1 and 0.2. The reason
behind choosing these M values will be discussed later. BecauseΩ is a function of φ02, this phase diagram does not distinguishbetween the hexagonal (H) and inverted hexagonal (IH) phases
or between the disordered condensed (D1) and dilute (D2) phases
in Figure 2. (The inverted hexagonal phase and the disordered
condensed phase appear when φ0 < 0.) Generally, the stripedphase (S) exists when Σ ∼ σ is small. The dashed arrow in Figure
3 indicates a horizontal path along which Σ (or, equivalently, thesurface tension σ) increases for a fixed value of Ω. Clearly,
along this path the striped phase (S) transforms into the hexagonalphase (H) via a first-order phase transition. As Σ increases evenfurther, there will be another first-order phase transition into thehomogeneous disordered phase (D).
It should be noted that although the calculated phase diagram
depends on the value of M the sequence of the modulated
structures remains unchanged. Such a sequence of patterntransformation, S f H, upon increasing the surface tension isconsistent with the sequence observed in the mixed lipid bilayers.6
Hence, we consider that the enhanced surface tension during theadhesion process is responsible for the morphological evolutionof the domain shapes in the adhering vesicle. This is the main
claim of the present paper. In the experiment of ref 6, theyobserved that the area of the vesicle increased in time. This may
be due to the enhanced surface tension which acts to pull andexpand the vesicle. Moreover, the experiment shows that theordered hexagonal lattice further evolved to disordered circulardomains with elapsed time. Such a situation may correspond toa transient state when the membrane undergoes a transition fromthe hexagonal phase (H) to the disordered phase (D) as the surface
tension increases.
To make a comparison with ref 6, we give here the estimatedvalues for the model parameters in eq 1. The surface tension ofan adhering vesicle can be in the range of 5 × 10-6 to 10-4J/m2,
and we have chosen here σ = 10-5 J/m2.13 The other parameters
can be found in refs 7 and 14 as κ = 25kBT = 10-19 J and b =(12) Keller, S. L.; McConnell, H. M. Phys. ReV. Lett. 1999, 82, 1602.
(13) Hu, J.-G.; Granek, R. J. Phys. II 1996, 6, 999.

Figure 3. Mean-field phase diagram in the (Σ, Ω) plane for (a) M) 0.1 and (b) M ) 0.2. Σ represents the reduced surface tension,and Ω represents the temperature-composition coupling parameter,
as defined in eq 7. The definitions of the different phases are thesame as in Figure 2. The effect of increasing the surface tension isrepresented by the dashed right-hand arrow.

5κ = 5 × 10-19 J. Although there is no direct measure of thecoupling constant value, Λ can be estimated by fitting theexperimentally observed modulation period (roughly 1 × 10-6
m for the striped phase) with 2π/q*. Then we obtain the coupling
constant as Λ = 4.9 × 10-12 J/m. This estimation is consistent
with that mentioned in ref 14. Using these values, the surface
tension required to induce the striped-to-hexagonal transition
can be evaluated as σ = Σ × (4.7 × 10-5) J/m2. For instance,σ ) 1.4 × 10-5 J/m2 when Σ ) 0.3. Along the dashed right-handarrows in Figure 3, the most unstable mode changes as q* ∼[Σ(1 - Σ)]1/2.
Furthermore, if we assume a4 to be of the same order as σ andset a4 = 10-5 J/m2, then the parameter values M ) 0.1 and 0.2
are equivalent to setting φ0 ) 0.24 and 0.48, respectively. These
values are physically relevant and represent asymmetric binary
mixtures. Naturally, the modulation instability does not occur if
the model parameters do not satisfy the necessary condition B) b - Λ2/σ < 0, but with our above choice of parameters, weare in the instability regime B < 0.

III. Discussion
We have studied the effect of surface tension on a curvature
instability that is coupled to a lateral phase separation. Although(14) Rädler, J.; Feder, T.; Strey, H.; Sackmann, E. Phys. ReV. E 1995, 51,4526.

6774 Langmuir, Vol. 22, No. 16, 2006

eq 1 provides a minimum and sufficient model to see such aneffect, several points merit further discussion.
(i) We have considered only an average flat and extensive
membrane rather than closed-shaped vesicles of finite area andvolume. In the case of closed-shaped vesicles, the principle
mechanism of the tension-induced transition should prevail, but
a pressure difference across the vesicle should be taken into
account because it plays an important role in the selection of themost stable modes on the vesicle.15-17(ii) The present model is valid for membranes composed ofa binary mixture, although a ternary mixture is used in ref 6.
Nevertheless, the local composition φ can be taken as thecomposition of one of the lipids, say sphingomyelin or DOPC,
with respect to its critical value. This is because the liquidordered phase may be only moderately enriched in cholesterol.
An extension of the present model to a ternary mixture is also
possible following ideas similar to those in ref 18.
(iii) As stated before, we considered the weak segregation
limit of the lateral phase separation, which is valid when the twophases are only weakly immiscible. Such an approximation was
taken in refs 10 and 17, whereas the opposite strong-segregation
limit was investigated in refs 15, 16, 19, and 20. At present, it
is not clear which approximation is more appropriate to accountfor the experimentally observed patterns because the experimental
system probably lies in an intermediate regime. In a recent paper
dealing with the strong segregation limit,20 however, the stripedphase exists between the hexagonal and the disordered phases,
and the increased surface tension causes a reVerse transition
from the hexagonal to the striped phase. Although the experimentally observed sequence of the transition is in qualitative
agreement with the trend found in our weak segregation theory,
more work needs to be done before further conclusions can be
drawn. In particular, there is a need to do experiments where thesurface tension is either controlled or measured directly. That
will allow a more quantitative agreement between theory andexperiment.
(iv) In other generalizations of the model, the bilayer natureof the membrane can be taken into account explicitly. In the caseof mixed vesicles, the spontaneous curvature arises from theasymmetry in the composition between the inner and outer leafletscomprising the bilayer membrane.21,22 Asymmetry in thecomposition occurs in situations where the two leaflets experiencea different environment as well as in biological cells because ofactive (in vivo) processes.23 Although the evidence of theasymmetric composition between the inner and outer leaflets
has not been reported in ref 6, it might exist for an adhering
mixed vesicle because only the outer leaflet is in contact with(15) Andelman, D.; Kawakatsu, T.; Kawasaki, K. Europhys. Lett. 1992, 19,57.
(16) Kawakatsu, T.; Andelman, D.; Kawasaki, K.; Taniguchi, T. J. Phys. II
1993, 3, 971.
(17) Taniguchi, T.; Kawasaki, K.; Andelman, D.; Kawakatsu, T. J. Phys. II
1994, 4, 1333.
(18) Kumar, P. B. S.; Gompper, G.; Lipowsky, R. Phys. ReV. E 1999, 60,
4610.
(19) Harden, J. L.; MacKintosh, F. C. Europhys. Lett. 1994, 28, 495.
(20) Harden, J. L.; MacKintosh, F. C.; Olmsted, P. D. Phys. ReV. E 2005, 72,
011903.
(21) Safran, S. A.; Pincus, P. A.; Andelman, D.; MacKintosh, F. C. Phys. ReV.
A 1991, 43, 1071.
(22) MacKintosh, F. C.; Safran, S. A. Phys. ReV. E 1993, 47, 1180.
(23) Keller, S. L.; Pitcher, W. H.; Huestis, W. H., III; McConnell, H. M. Phys.
ReV. Lett. 1998, 81, 5019.
(24) Kodama, H.; Komura, S. J. Phys. II 1993, 3, 1305.
(25) Qi, S.; Wang, Z.-G. Phys. ReV. Lett. 1996, 76, 1679. Qi, S.; Wang, Z.-G.
Phys. ReV. E 1997, 55, 1682.
(26) Nonomura, M.; Ohta, T. J. Phys. Soc. Jpn. 2001, 70, 927.

Komura et al.

the surface. More detailed theories dealing with the bilayer natureof the membranes are given in refs 18 and 24 and can begeneralized to the present case of mixed vesicles.
We next discuss how the present work is related to previous
models beside refs 19-20. Hu and Granek14 investigated thebuckling of mixed amphiphilic monolayers by considering similar
coupling between the composition and the curvature. In theirwork, the surface tension is introduced as a Lagrange multiplierin order to ensure the fixed total area of the monolayer (fixedarea ensemble). In our case, however, the surface tension isregarded as a control parameter of the membrane shape (fixedsurface tension ensemble). Despite the difference in the working
ensembles, they also predicted that a transition from the stripedto the hexagonal phases can be induced either by decreasing thecoupling constant (Λ in our paper) and/or increasing the line
tension b (Figure 3 in ref 14). Notice that both parameter changes
are equivalent to increasing Σ, as seen in eq 7, and our result is
consistent with theirs.
In previous experiments of ref 4, similar striped and hexagonal
patterns have been shown, but no similar temporal evolution wasreported. This may be because the vesicles in ref 4 were suspended
in an aqueous solution (rather than adhering onto a surface) andtheir surface tension is not increasing as function of time. It isalso not entirely clear how much the patterns and domains seenon the vesicular surface are truly equilibrium ones or just longlived metastable ones.
Finally, we comment on the dynamical aspect of the observed
morphological transition. In the past decade, kinetics of phasetransitions between different ordered mesophases have beenstudied intensively, especially for block copolymer systems.25
Among various works, Nonomura and Ohta investigated themorphological transition between the lamellar and the hexagonalphases in two dimensions.26 Their dynamical equation is basedon the time-dependent Ginzburg-Landau equation using a freeenergy that includes a long-range repulsive interaction term.
This free energy is known to be similar to the one used in thepresent work.
In ref 26, the transition is induced by changing the temperature.
Note that in our Figure 3 a change in temperature is simply avertical scan where Ω ∼ a2 ∝ T - Tc changes, whereas Σ ∼ σ,M, and φ0 are kept fixed. The vertical scan can also lead to acrossover between the striped and hexagonal phases. In theexperiments,6 however, the temperature is kept fixed, and we
consider that the origin of the transition relies on the temporal
increase in the surface tension as detailed above. Despite thedifference in the control parameters (either temperature or surface
tension), the time evolution of the structure in the simulation26
is very similar to that observed for the lipid mixtures. For example,
the stripes undergo a pearling phenomenon, and the hexagonal
domains appear at the grain boundaries of the lamellar structure.
We hope that more quantitative measurements of the surface
tension during the adhesion process will shed light on thisintriguing phenomena and will lead to further theoreticalinvestigations.
Acknowledgment. We thank P. Bassereau, R. Granek, J. T.
Groves, J. L. Harden, P. Nassoy, S. Rozovsky, and T. Ohta for
useful discussions. This research was partially supported by aGrant-in-Aid for Scientific Research (C), the Japan Society for
the Promotion of Science under grant no. 15540395, the Israel
Science Foundation (ISF) under grant no. 160/05, and the USIsrael Binational Foundation (BSF) under grant no. 287/02.
LA053135X