Biophysical Journal

Volume 92

January 2007

115–125

115

Growth Dynamics of Domains in Ternary Fluid Vesicles
Miho Yanagisawa,* Masayuki Imai,* Tomomi Masui,* Shigeyuki Komura,y and Takao Ohtaz*Department of Physics, Ochanomizu University, Bunkyo, Tokyo112-8610, Japan;yDepartment of Chemistry, Tokyo Metropolitan University,
Tokyo 192-0397, Japan; andzYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

ABSTRACT We have studied the growth dynamics of domains on ternary ﬂuid vesicles composed of saturated (dipalmitoylphosphatidylcholine), unsaturated (dioleoylphosphatidylcholine) phosphatidylcholine lipids, and cholesterol using a ﬂuorescencemicroscopy. The domain coarsening processes are classiﬁed into two types: normal coarsening and trapped coarsening. For thenormal coarsening, the domains having ﬂat circular shape grow in a diffusion-and-coalescence manner and phenomenologicallythe mean size grows as a power law of ;t2/3. The observed growth law is not described by a two-dimensional diffusion-andcoalescence growth mechanism following the Saffman and Delbrück theory, which may originate from the two-body hydrodynamic interactions between domains. For trapped coarsening, on the other hand, the domain coarsening is suppressed at a
certain domain size because the repulsive interdomain interactions obstruct the coalescence of domains. The two-color imaging of
the trapped domains reveals that the repulsive interactions are induced by the budding of domains. The model free energy
consisting of the bending energy of domains, the bending energy of matrix, the line energy of domain boundary, and the translationenergy of domains can describe the observed trapped coarsening. The trapping of domains is caused by the coupling between thephase separation and the membrane elasticity under the incompressibility constraint.

INTRODUCTION
Biomembranes have characteristic lateral heterogeneities
arising from the immiscibility of the lipid components suchas sphingomyelin, unsaturated phospholipids, and cholesterol(Chol). In the so-called raft model, the heterogeneity coupledwith the protein distribution plays important roles in thefunctionality of the biomembranes (1,2). Since the raft model
has been suggested, extensive studies have been performed toreveal the composition, structure, dynamics, and physiological properties of the rafts. To realize the formation of rafts in
artificial membranes, a ternary mixture consisting of saturated
phospholipids, unsaturated phospholipids, and cholesterol has
been studied (3–11). Such artificial membranes show a lateralphase separation below the miscibility transition temperatureand form a liquid-ordered phase rich in saturated phospholipids and cholesterol, and a liquid-disordered phase rich inunsaturated phospholipids. Although the rafts in cell membranes and the domains in artificial membranes are notdirectly connected, the lateral phase separation in artificial
membranes is a milestone to understand the raft structure.
From the membrane physics point of view, the domains in
the model biomembrane vesicles show a quite unique nature.
Veatch and Keller observed the liquid domain formation in
giant vesicles consisting of DPPC (dipalmitoylphosphatidylcholine; saturated phospholipid), DOPC (dioleoylphosphatidylcholine; unsaturated phospholipid), and Chol using afluorescence microscopy with Texas Red dipalmitoylphosphatidylethanolamine (TR-DPPE) dye (7–9). When thetemperature is dropped from the homogeneous one-phaseregion to the two-phase region, the membrane is separated

Submitted May 5, 2006, and accepted for publication September 8, 2006.

into a DPPC- and Chol-rich phase and a DOPC-rich phase(TR-DPPE is localized in the DOPC-rich region). Dependingon the composition, such a phase separation exhibits two
typical kinetic pathways, i.e., a nucleation-growth type and aspinodal decomposition type, although other more complicated patterns (e.g., viscous fingering pattern) have beenreported (8). In the case of the nucleation-growth pathway,
the circular domains grow by colliding and coalescing withother domains, whereas for the spinodal decomposition, acharacteristic bicontinuous network pattern appears and thendevelops into circular domains. In both cases, the domains
develop into a small number of large domains and eventually
undergo a budding of domains (10,12). Similar domain coarsening processes have been observed in other ternary systems
such as sphingomyelin/DOPC/Chol (3,4). The above mixed
vesicles, however, sometimes show a unique long-lived
hexagonal pattern with cap-shaped domains (10). Such anordered domain structure suggests the existence of a longranged repulsive interdomain interaction, which may originate from the elastic nature of the membrane. Thus thedomain coarsening in the fluid vesicles is strongly coupledwith the membrane elasticity and the geometrical constraints.
The growth dynamics of domains in the fluid vesicles has
been investigated theoretically by means of various simulation techniques such as the generalized time-dependent
Ginzburg-Landau model (13,14), the dynamic triangulation
Monte Carlo method (15), and the dissipative particle dynamics method (16–18). Using the time-dependent GinzburgLandau model, Taniguchi succeeded to reproduce cap-shaped
domains on a vesicle (13). Due to the coupling between thevesicle shape and the phase separation, the vesicle exhibits avery slow coarsening process, where D(t) ; ta (D(t) is theaverage domain diameter and a ; 0.1). The budding dynamics

Address reprint requests to Masayuki Imai, E-mail: imai@phys.ocha.ac.jp.
Ó 2007 by the Biophysical Society
0006-3495/07/01/115/11 $2.00

doi: 10.1529/biophysj.106.087494

116

was investigated by Kumar et al. (15), using the triangulation
Monte Carlo model, and they showed that, in the buddingcoalescence regime, the number of buds decays as Nbud; t-2/3. Recently Laradji and Kumar (16,17) investigated thegrowth dynamics of the domains using the dissipative particle
dynamics and showed that the hydrodynamic interactions andthe area/volume constraint strongly affect the domain growth.
These approaches grasp the important features of the lateralphase separation in the fluid vesicles. However, it is still
far from the quantitative understanding due to the lack of experimental studies on growth dynamics in the fluid vesicles.
In this article, we present a systematic experimental investigation on the growth dynamics of the domains in thefluid vesicles using fluorescence microscopy. After explaining our experimental method, we will classify the observed
domain coarsening processes into two types, a normal
nucleation growth mechanism and a trapped mechanism inthe section ‘‘Outline of domain coarsening in fluid vesicles’’.
The normal coarsening process will be characterized experimentally and then compared with the theoretical prediction
for domain coarsening in the fluid membranes in the section‘‘Normal coarsening’’. The dynamics of the trapped coarsening will be characterized in the section ‘‘Trapped coarsening’’, and the trapping mechanism will be discussed interms of the coupling between the phase separation and themembrane elasticity under the area/volume constraint in thesection ‘‘Budding and area/volume ratio’’. The criterion for
the trapped coarsening will be examined using the freeenergy expressions for a vesicle with budding domains in thesection ‘‘Free-energy analysis on trapped coarsening’’.
Finally we will summarize our study in the ‘‘Conclusion’’.

EXPERIMENTS AND ANALYSIS
Commercial reagents
DPPC (.99% purity) and DOPC (.99% purity) wereobtained in a powder form from Avanti Polar Lipid(Alabaster, AL). Chol (.99% purity) was purchased from
Sigma-Aldrich (St. Louis, Mo). All lipids were used without
further purification. Mother solutions of lipids were stored inchloroform at 20°C until use. Texas Red dihexanoylphosphatidylethanolamine (TR-DHPE) and perylen were obtained from Molecular Probes (Eugene, OR), and used as adye for contrast between liquid phases. TR-DHPE is localized in a liquid disordered (DOPC-rich) phase and showsa red color, whereas perylene partitions preferentially into aliquid ordered (DPPC- and Chol-rich) phase and shows a blue
color (10).
Formation of giant vesicles
In this work, we fixed the ratio between the components
comprising the vesicles at DPPC/DOPC/Chol ¼ 4:4:2. Giantvesicles were prepared by the gentle hydration method
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Yanagisawa et al.

(19,20). At first we dissolved the lipids, DPPC, DOPC, andChol in 10 ml of chloroform (10 mM). To dye the DOPC richphase, TR-DHPE was added at the ratio of 0.8:10 (dye/lipid).
When we observe the both phases, we added TR-DHPE andperylene at the ratio of 0.8:10 and 1:500, respectively. The
solvent was evaporated in a stream of nitrogen gas and theobtained lipid film was then kept under vacuum for 1 night toremove the remaining solvent completely. The dried lipid
film was prewarmed at 60°C, and then we added 30 ml ofpure water at 60°C. After 1 min of prehydration, the sample
was hydrated with 970 ml of pure water at 60°C. During thehydration process, the lipid films spontaneously form giant
vesicles with diameters of 10–150 mm.
Observation of giant vesicles using
ﬂuorescence microscope
The vesicle suspension was put on a glass plate with a siliconrubber spacer having the thickness of 0.5 mm, and thensealed with a cover glass immediately. This sample cell wasset on the temperature control stage (Carl Zeiss, Oberkochen,
Germany) with a precision of 60.2°C. We observed thedomain coarsening process using an inverted conformal
microscope (Carl Zeiss, LSM 5) with either the laserscanning mode or the Hg lamp mode. To avoid the domain
formation before the observation, we paid special attention tokeep the sample temperature above the melting temperatureof DPPC (Tm, 41°C). For the observation of the vesicles, wedecreased the temperature from the one-phase region (42–43°C) to the miscibility transition temperature with the rateof ;0.5°C/min or less. When the phase separation took
place, we fixed the temperature immediately and the time
evolution of domains was recorded with a charge-coupleddevice camera (Carl Zeiss, Axio Cam). For the observationof TR-DHPE dyeing systems, we used the laser scanningmode (l ¼ 543 nm) with 41002-B probe filter (CHROMA,
Rockingham, VT; excitation/emission maxima ;575:640
nm). For the TR-DHPE and perylene dyeing systems, weused the Hg lamp mode with a modified XF-006 filter, which
has the transmission of 80% in blue color region and 20% in
red color region. To avoid photo-oxidation, we minimized theexposure to light.
Image analysis
The fluorescence microscope observation gives images ofgrowing liquid-ordered domains (black regions) on a vesicle
as shown in Fig. 1 a, where the arrows indicate the velocityof the domains. From the images, we made two histograms:
distribution of the moving direction of domains and distribution of the interdomain distances. In the case of theimage analysis, we focused on vesicles having a nearly
spherical shape with a diameter of ;20 mm, because the field
depth of the optical microscope was ;15 mm. To avoid
image distortion effects owing to the curvature, we restricted

Domain Growth Dynamics

117(Narishige (Tokyo, Japan), MMO-202ND and MN-4) with amicroinjector (Narishige, IM-9B) system. The floating target
vesicle having a diameter of ;100 mm is aspirated by the
micropipette with an internal diameter of 20 mm. We increased the external suction pressure mildly until the vesicle
deformed to a tight shape and then recorded its shape using
the charge-coupled device camera.

RESULTS AND DISCUSSION
Outline of domain coarsening in ﬂuid vesicles

FIGURE 1 (a) Fluorescence microscope image of a vesicle with growing
domains. Arrow indicates the velocity of the domain, and the largest domain
is marked by a circle. Scale bar is 5 mm. (b) Parameters used to describe
domain growth. The domain migration angle ui is that between the ith
domain velocity direction (indicated by an arrow) and the line connecting
the centers of the 0th largest domain and the ith domain. The interdomain
distance dij is defined by the edge-to-edge distance between the centers of ith
domain and jth domain.

the sampling area to the top hemisphere region with a radiusof 5 mm, which gives maximum error of ;10%. From theseries of images, we estimated the distribution of the moving
direction of domains with respect to the largest domain on
the vesicle (marked by a circle in Fig. 1 a). In Fig. 1 b, wegive the definition of the moving direction of a domain. The
migration angle ui is that between the ith domain velocity
direction (indicated by an arrow) and the line connecting thecenters of the largest (0th) domain and the ith domain. For
one vesicle, we extracted a total of 50 domains from thesequential images with a 1.5 s interval and obtained the
distribution of the migration angle. The distributions from 10
vesicles were accumulated to obtain the distribution of themoving direction of domains. For the interdomain distance
distribution analysis, to avoid the effect of domain size
polydispersity, we measured the edge-to-edge distance dij
between the centers of the ith domain and the jth domain asshown in Fig. 1 b. The histogram of the interdomain distances h(d) was obtained from ;100 pairs and the counts
were divided by 2pd.

First we show the short coarsening processes of the domains
on the fluid giant vesicles having the composition of DPPC/
DOPC/Chol ¼ 4:4:2. When we decrease the temperaturefrom the homogeneous one phase, the lateral phase separation between the liquid-ordered (Lo) and the liquid-disordered (Ld) phases takes place at 34°C (miscibility transition
temperature, Tmix) with a nucleation-growth mechanism. In
this study, we adopted the gentle hydration method to form
the giant vesicles. The obtained vesicles have large polydispersities both in size and shape. This brings the diversity ofthe domain coarsening processes between the different
vesicles (22). To improve statistical accuracy, we observed
vesicles having nearly a spherical shape with a diameter of;20 mm. As a result of a large number of examinations, wefound that the domain coarsening processes are classified
into two types.
Fig. 2 shows the two types of the time evolutions of thedomain growth at 30°C observed by the fluorescencemicroscope with TR-DHPE. For the both cases, we set
t ¼ 0 when the domain size reached to an optical resolutionof the microscope (;0.8 mm). In Fig. 2 a, the circulardomains (Lo phase) grew in a diffusion-and-coalescencemanner while keeping the circular shape and reached to thesize of ;10 mm within several minutes. A similar coarsening

Micropipette aspiration
To estimate the area/volume ratio of a vesicle, we adopted a
micropipette aspiration technique (21). By micropipetteaspiration, a flaccid spherical vesicle deforms to a tightvesicle. In this case, we are able to calculate the geometricalparameters. In this study, we used a micromanipulator

FIGURE 2 Time evolution of domain growth on a vesicle at 30°C observedby the fluorescence microscope; (a) normal coarsening and (b) trapped coarsening mechanisms. Scale bar is 5 mm.
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process was reported in by Veatch and Keller (8). We callthis coarsening process normal coarsening. In Fig. 2 b, on theother hand, the domain coarsening was trapped at a certain
size for several 10 minutes. During the trapping period, thedomains having almost the same domain size migrate on thevesicle without coalescence. We call this coarsening process
as trapped coarsening. It should be noted that both coarsening processes are observed at the same time in the same
microscope field.
To clarify the difference between normal coarsening andtrapped coarsening, we plot the time evolution of the mean
domain diameter D(t) for the both cases in Fig. 3. For normalcoarsening, phenomenologically the mean domain diameter
D(t) grew with a power law of D(t) ; ta until the domains
develop into a single large domain. The final monodomainstate was stable for at least 1 day. For trapped coarsening, themean domain size grew to ;2 mm with the similar powerlaw, and then the coarsening ceased for ;100 min. After thetrapping period, the domains restarted to grow toward a monodomain structure. We confirmed that the trapped coarsening can be reproduced by redoing the sample thermal history.
When the temperature of the vesicle showing the trappedcoarsening was increased above the transition temperature(corresponding to the homogeneous one-phase region) andthen decreased below the phase separation temperature, we
observed a similar trapped coarsening again. Analogous
trapping phenomena (or slowing down of domain coarsening) were observed for vesicles having a hexagonal pattern(10) and vesicles having a large Ld phase fraction at low
temperature (8). In our case, however, the domain (Ld phase)
fraction was not so high, and each domain diffused freely on
the vesicle surface as shown in Fig. 2 b. Thus we consider
that trapped coarsening is a frequently observed coarsening
process in fluid vesicles. In the following, we will discuss themechanisms of the two different coarsening processes.

FIGURE 3 Time dependence of mean domain diameter D(t) on a vesicle
shown in Fig. 2, a and b. The open and closed circles denote normal andtrapped coarsening, respectively.
Biophysical Journal 92(1) 115–125

Yanagisawa et al.

Normal coarsening
To show the generality of the power law scaling of thenormal coarsening, we plot in Fig. 4 a the time evolution of
the domain diameter obtained from various normal coarsening vesicles with the composition of DPPC/DOPC/Chol ¼
4:4:2. Here we introduce the normalized diameter defined by
Dnf ðt f Þ ¼ Dðt f Þ=Df ; where D(t f) is the mean diameter of theobserved domains at time t f, and Df is the diameter of the
final monodomain. The normalized time is defined by t f ¼
t=tf , where tf is the time when the vesicle attains the monodomain structure. Roughly speaking, all the data collapse

FIGURE 4 (a) Time dependence of normalized mean domain diameter ofvarious vesicles with the composition of DPPC/DOPC/Chol ¼ 4:4:2. The
normalized diameter is expressed by Dnf ðtf Þ ¼ Dðtf Þ=Df , where D(tf) is themean diameter of the observed domains at time tf and Df is the final
monodomain diameter. The normalized time is expressed by t f ¼ t=tf ,where tf is the time when the domains develop into the monodomain. A solid
1=4line and a triangle in the figure indicate the growth law of Dnf ;tf and theslope of tf 2/3. (b) Time dependence of normalized mean domain diameter
Dnf ðtf Þ in a vesicle with the various compositions. The open circles,
triangles, and closed circles denote the composition of DPPC/DOPC/Chol ¼
4.5:4.5:1, 4:4:2, and 3:3:4, respectively. DPPC/DOPC/Chol ¼ 3:3:4 sample
showing spinodal decomposition mechanism, the data are taken from thelate stage where domains have circular shape. A solid line and a triangle in
1=4
the figure indicate the growth law of Dnf ;tf and the slope of tf 2/3.

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119

onto a single straight line, and the coarsening obeys thephenomenological power law growth DðtÞ;t2=3 . Furthermore, we examined the normal coarsening vesicles madewith other compositions. Namely, we varied the fraction ofcholesterol from 10% to 40% while keeping the ratio between DPPC and DOPC as 1:1. The obtained growth data areshown in Fig. 4 b. Here the vesicles containing Chol 35–40%
showed the spinodal decomposition mechanism with bicontinuous textures. In these cases, we used the data fromthe late stage period in which the Lo phase forms circulardomains. These results confirm the generality of the normalcoarsening process characterized by the phenomenological
growth law behavior D(t) ; t2/3.
To our knowledge, little is known about the growth lawof the microdomains on the vesicles (16,17,23). Here we
discuss the obtained growth law from a simple dynamicalscaling point of view. The domain travels on the vesicle until
it meets another domain. Once these two domains meet, they
coalesce into one large domain. If we assume that eachdomain is regarded as a free Brownian particle and theaverage distance between neighboring domains is proportional to the average droplet diameter D(t), the collision time
Dt is expressed by
1=2DðtÞ;ðKDtÞ ;(1)

where K is the diffusion constant of the domain. Since
the droplet diameter increases by an amount roughly D foreach coalescence, the increment of the diameter obeys theequationDDðtÞ DðtÞ
;
:
Dt
Dt

rate of a large domain is not small compared with that of asmall particle like a protein or lipid molecule with the size ofseveral nanometers. When the momentum dissipates rapidly
from the membrane to the surrounding medium, i.e.,h9D=hd  1, the diffusion constant was derived by Komura
and Seki (25) and is given by

2
1
hd
:
(5)
K=kB T 
phd h9D
In this case, we obtain the growth law of D ; t1/4. This
growth law is shown in Fig. 4, a and b, by a solid line. Unfortunately, this scaling law does not describe our experimental data. One candidate to explain this disagreement isthat the assumption of the free Brownian particle (Eq. 1) does
not hold for the migrating domains on a vesicle.
We consider that the key concept to understand ourexperimental result is the hydrodynamic interactions between the domains. In Fig. 5 a, we show the movement of thecenter of gravity of each domain by an arrow during the timeinterval of Dt ¼ 1.5 s. Instead of moving randomly, eachdomain seems to be attracted toward the largest target
domain following the flow around the domain. We estimatedthe distribution of the moving direction of domains withrespect to the largest domain in the vesicle (marked by acircle). Details of the image analysis were described above.
The migration angle ui is that between the domain velocity
direction (indicated by an arrow) and the line connecting thecenters of the largest domain and the ith domain as shown in(2)

Then we obtained the expression
1=2DðtÞ;ðKtÞ :(3)

For the diffusion constant of a particle in a membrane,
Saffman and Delbrück derived the following expression
taking a two-dimensional hydrodynamic interaction coupled
to an ambient fluid through the momentum dissipation mechanism (24,25):


1
hd
log
g ;
(4)
K=kB T ¼4phd
h9ðD=2Þ
where kB is the Boltzmann constant, T is the temperature, hand h9 are the viscosities of the membrane and the exteriorliquid, respectively, d is the thickness of membrane, and g is
Euler’s constant (¼ 0.5772). Here we should note that theabove expression is an asymptotic form when the rate of
momentum dissipation to the surrounding medium is small,i.e., h9D=hd  1. By putting typical values for the membrane system, h ¼ 1 poise, h9¼ 0.01 poise, d ¼ 5 nm andD ¼ 5mm, we obtain h9D=hd ¼ 10, which does not satisfy
the necessary condition for the Saffman and Delbrück
equation (Eq. 4). This is due to the fact that the dissipation

FIGURE 5 (a) Time sequence of migrating domains on a vesicle withnormal coarsening. Arrow indicates the movement of the center of gravity ofthe domain during the time interval Dt ¼ 1.5 s. The largest domain is marked
by a circle and the scale bar denotes 5 mm. (b) Distribution of domain
migration angle u defined in Fig. 1 b. The vertical axis indicates the numberof domains, nd.
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Fig. 1 b. In Fig. 5 b, we plotted the distribution of u for thenormal coarsening domains in the early stage, where nd is
the number of domains. When the domains behave as freeBrownian particles, the distribution of u should be uniform.
However, the experimentally obtained distribution has aGaussian profile with a mean value of ;0° and a variationof ;76°, indicating that the domains tend to move toward
the target domain. We consider that this motion is due to thetwo-body hydrodynamic interaction, which results in thedeviation from the theoretical prediction described above.
The coupling between the phase separation and the diffusion
in membranes with hydrodynamic interaction may play animportant role to understand the observed normal coarsening
behavior (26–28) and will be addressed in our forthcoming
article.

Yanagisawa et al.

FIGURE 7 Loci of several trapped domains during 60 s projected on avesicle image.

In contrast to normal coarsening, trapped coarsening ischaracterized by a significant slowing down of the domain
growth as shown in Fig. 3. In Fig. 6, we plot the timedependence of the domain diameter obtained from various
vesicles with the composition of DPPC/DOPC/Chol ¼ 4:4:2.
The mean size of the trapped domains and the onset times of
the trapping differ among the vesicles. Hence we introduce
the normalized diameter defined by Dni ðt i Þ ¼ Dðt i Þ=Di ;where Di is the mean domain diameter at the onset of thetrapping. Similarly, the normalized time is defined byt i ¼ t=ti ; where ti is the onset time of the trapping. To
elucidate the mechanism of the trapped coarsening, the tracesof several trapped domains during 60 s are shown in Fig. 7.
The traces show that each domain is localized in a certain

cage, and the domains tend to repel each other when they
approach. Such a repulsive interaction can be effectively extracted from the radial distribution function of the domains.
Unfortunately, owing to the limitation of the microscope
visibility and the finite number of the domains on thevesicles, we cannot obtain the reliable radial distributionfunction from the microscope images. Instead, we measured
the distance distribution h(d), which represents the numberof domains nd with the interdomain distance d ( to avoid theeffect of domain size polydispersity, we measured the edgeto-edge distances between the centers of the domains).
Details of the image analysis were described above. The
normalized distance distribution histogram has a peak at;1.6 mm as shown in Fig. 8. This indicates the existence of arepulsive interdomain interaction at short length scale. If weassume that the trapped domains are in a temporal equilibrium state and the area fraction of domains is low, the

FIGURE 6 Time dependence of normalized mean domain diameterobtained from various trap coarsening vesicles with composition of DPPC/
DOPC/Chol ¼ 4:4:2. Normalized diameter is expressed by Dni ðt i Þ ¼
Dðt i Þ=Di . Di is the mean domain diameter at the onset time of the trapping
and t i is the normalized time expressed by ti ¼ t=ti where ti is the onset timeof the trapping.

FIGURE 8 Histogram of the interdomain distance h(d) for trappeddomains. To avoid the effect of domain size polydispersity, we measurededge-to-edge distances between centers of two circular domains (see Fig.
1 b). A solid curve indicates h(d)/2pd ; exp(u(d)/kBT) with uðdÞ ¼
 ¼1.8 mm, ur ; 12 kBT,
ur ðDd Þ2  ua ðDd Þ. The values used in this fitting are D
and ua ; 11.9 kBT.

Trapped coarsening

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121

normalized histogram is approximately expressed by the interaction potential u(d) through the Boltzmann distribution
hðdÞ=2pd;expðuðdÞ=kB TÞ:(6)

The histogram can be reproduced by assuming a phenomenological interdomain potential such that
 2
 


D
D
;
(7)
ua
uðdÞ ¼ urdd is the mean diameter of the domains of the ensemwhere D
ble. The first and the second terms represent the repulsive
and the attractive interdomain potentials, respectively. As
shown by the solid curve in Fig. 8, the observed histogram iswell described by Eq. 7 with the parameter values of ur ; 12
kBT and ua ; 11.9 kBT. The obtained repulsive potential is
responsible to the loci of the trapped domains in Fig. 7.
Moreover, the value of ur ; 12 kBT is comparable to thebending modulus of the DOPC matrix membrane, which is;20 kBT. The phenomenological attractive interaction in thelarge interdomain distance may arise from two contributions:
one is the limitation of the Boltzmann distribution approximation (low area fraction approximation) and the other isthat the domains tend to move toward the target domain asshown in Fig. 5 b.
To elucidate the difference between normal and thetrapped coarsening, we dyed not only the matrix (Ld phase)
but also the domains (Lo phase). TR-DHPE localized in the
Ld phase emits red fluorescence, whereas the perylene
prefers the Lo phase and emits blue fluorescence. The tworegions dyed with different colors can be observed simultaneously using a modified filter that has high transmission
for blue color and low transmission for red color. Fig. 9, aand b, are the time evolution of the two color cross-section
images for the normal coarsening and the trapped coarsening
domains, respectively. For normal coarsening, the domains
do not bud and have the average curvature that is equal tothat of the surrounding matrix. In this case, the coarsening
proceeds straightway and the vesicle keeps its spherical shapeduring coarsening. On the other hand, in the case of trappedcoarsening, the domains have the same curvatures as thesurrounding matrix in the initial coarsening stage, although
the cross section of the vesicle is flaccid. With the elapse oftime, the domain grows to a certain size and then buds to a
cap-like shape. This observation indicates that the line
energy of domains starts to play a dominant role on the domain morphology (12). It is important to note that the onset
time of the domain budding coincides with that of the domaintrapping.
The simplest model to explain the observed repulsive
interaction is the bending penalty of the matrix membrane
between the two approaching cap-shaped domains. Whenthe domains slightly bud toward the outside of the vesicle(cap-shaped domains), the matrix membrane surrounding thecap-shaped domain is deformed to skirt-like shape, accord-

FIGURE 9 Time evolution of two color (TR-DHPE and perylene) crosssection images for normal coarsening vesicle (a) and trapped coarsening
vesicle (b). To show the budding domains clearly, the focusing points are at
the top of vesicles for 3.0 and 10 min images in b. (c) Two color crosssection images for budding domains in the trapped coarsening vesicle. Scalebar in figures is 50 mm.

ingly. A cross-section image of the vesicle with the trappeddomains (Fig. 9 c) supports the fact that the geometry of thesurrounding matrix is a skirt-like shape. When the two capshaped domains approach, the curvature of the skirt-likeregion becomes large (;1/d), which results in a repulsive
potential between the two domains (;1/d2). In contrast, if
the domains are not bud and have the same curvature withthe matrix membrane, the approaching two domains are notblocked by the matrix membrane, which gives rise to thecoalescence of the domains. Thus we conclude that thebending of the matrix membrane between the two approaching cap-shaped domains induces the repulsive interaction,
which in turn results in the trapped coarsening.
Budding and area/volume ratio
According to the theory of budding (12), the budding isgoverned by the balance between the membrane bending
energy and the line energy at the domain boundary. The
important aspect of the trapped coarsening is the coupling
between the phase separation and the membrane elasticity.
When the budding occurs, however, there is another important constraint, i.e., the incompressibility of the membrane.
Because of the equality of the osmotic pressure between
inside and outside of the vesicle membranes, we consider
that the area/volume ratio of the vesicle is constant during
the coarsening process. This assumption was confirmed
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Yanagisawa et al.

experimentally by a micropipette aspiration technique asshown later. Here we discuss the geometrical constraints forthe budding based on the following assumptions: 1), alldomains are cap-like shape of the same size with a curvatureof 1/a; 2), the matrix is represented as a perforated sphericalfilm with a curvature of 1/b (there is a discontinuity of thecurvature at the boundary of the domain); and 3), the total
surface area and the total volume of a vesicle are conserved.
The geometrical parameters used in the following calculation
are defined in Fig. 10.
We consider a vesicle with nd cup-shaped domains. The
total surface area of the vesicle S is a sum of those of thedomains Sd and the matrix Sm,

where RV is the volume-based radius. The area/volume ratio
plays an important role in the budding behavior of thevesicle. In this study, we introduce a parameter ‘‘dimensionless excess radius j’’ to measure the area/volume ratio,
which is defined byj[

RS  RV:
RV(14)

where the angle b is defined in Fig. 10. On the other hand,
the total volume of a vesicle with nd cap-shaped domains V isexpressed as a sum of the domain part Vd and the matrix partVm, thus

In case of j ¼ 0, the vesicle takes a perfect spherical shapeand there is no room for the budding, even if the gain in theline energy overcomes the cost in the membrane bending
energy. Increase of j means increase of the area/volume
ratio, or the excess area, which allows the domains to bud.
The dimensionless excess radius j can be estimated using
a micropipette aspiration technique. By the aspiration, theexcess area of the vesicle forms a subsystem in the micropipette as shown in Fig. 11. The deformed vesicle consists ofa spherical cap part with the radius R1 (subsystem 1), and acylindrical part (length l) with a hemispherical cap with theradius R2 (subsystem 2). Assuming that the total volume andthe total surface area are conserved during the aspiration, we
can calculate the total volume and the total surface area fromthe following geometrical relations:   R2 2
2 u2
1 2 1 2pR2 l ¼ 4pRS : (15)S ¼ 4p R1 1  sin22   R3 
4
3
2 u
1 2
V ¼ p R1 1  sin
3
2
2
1 2
4 3
2
(16)
1 pR2 R1 cosu 1 pR2 l ¼ pRV :33

4 3
V ¼ Vd ðnd Þ 1 Vm ðnd Þ ¼ pRV
(11)
3


4 3 2 a 1
2
 pðasinaÞ ðacosaÞ
(12)
Vd ðnd Þ ¼ nd pa sin
3
2
3
 


4 3
4 3 2 b12
 pðbsinbÞ ðbcosbÞ ;Vm ðnd Þ ¼ pb  nd pb sin
3
3
2
3
(13)

The excess radius j can be calculated from the experimentally obtained R1, R2, l, and f. First, to examine theconservation law of the total area and the total volume of avesicle, we measured them before and after the phaseseparation and confirmed the conservation law. Then we
obtained the distribution of j for the vesicles showing thenormal coarsening and the trapped coarsening as shown in

FIGURE 10 Geometrical model describing a cap-shaped domain on a vesicle. The domain has a curvature of 1/a and the matrix has a curvature of 1/b.

FIGURE 11 Schematic representation of aspirated vesicle with micropipette. By the aspiration, the vesicle is divided into two subsystems.
Subsystem 1 has spherical cap geometry with radius R1 and subsystem 2 is acylinder with a hemispherical end cap, characterized by radius R2 and length l.

S ¼ Sd ðnd Þ 1 Sm ðnd Þ ¼ 4pR2S ;(8)

where we introduce a surface area-based radius RS. The total
domain area Sd is given by
n
 o
2
2 a
2
¼ pL ;
Sd ðnd Þ ¼ nd 4pa sin(9)2
where L is the radius of a flat circular disk with the area Sd
and the angle a is defined in Fig. 10. The total area of thematrix Sm is given by

 
2
2
2 b
;
(10)
Sm ðnd Þ ¼ 4pb  nd 4pb sin
2

Biophysical Journal 92(1) 115–125

Domain Growth Dynamics

Fig. 12, where nv is the number of vesicles. For the vesicles
with the normal coarsening domains (white bar), the valuesof j are distributed in the range of 0–0.01. The small j for thenormal coarsening indicates that the vesicle has a nearly
spherical shape. On the other hand, for the trapped coarsening (gray bar), the values of j are distributed in the rangeof 0.01–0.06. Thus the trapped coarsening vesicle has aflaccid spherical shape. These results are consistent with thefluorescence microscope observation shown in Fig. 9, a andb. These results confirm that the condition whether thevesicle shows the trapped coarsening or the normal coarsening is determined by the value of j and the experimentallyobtained threshold is j* ; 0.01. It is interesting to note thatthe excess area of the spherical subsystem 1 of the vesicle isextremely decreased by the aspiration. Then the trapped domains start to coalesce and finally form a large monodomain
at the neck of the aspirated vesicle.
Free-energy analysis on trapped coarsening
Here we perform a free-energy analysis of the phaseseparating vesicle with the excess area. The free energy per
vesicle having nd cap-shaped domains consists of i), thebending energy of the domains Ecapd ; ii), the bending energyof the matrix Ecap
;
iii),
the
line
energyat the domain boundm
aries Ecap
;
and
iv),
the
energyassociatedwith the translal
tional entropy of the domains Ecap.
Under
the geometricaltrconstraints expressed by Eqs. 8–13, each contribution is
given by
(  
)
2
1
2 Sd ðnÞ
2kd pL2cap¼
Ed ¼ nd kd
(17)
2
2
a
nd
a
 2

 
1
2
cap
2 b
(18)
Em ¼ kmSm ðnd Þ ¼ 8pkm 1  nd sin
2
b
2

123

cap
l

E

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
LL¼ 2pspﬃﬃﬃﬃﬃ 1 
pﬃﬃﬃﬃﬃ
nd
2a nd
cap

Etr ¼ nd kB Tlogf;(20)

where kd and km are the bending moduli of the domain andthe matrix, respectively, s is the line tension at the domainboundary, and f is the area fraction of the domains. Here thespontaneous curvature of the membrane is assumed to be
zero. The excess radius j is taken into account by the relation222

pL ¼ fS ¼ f4pRs ¼ f4p½RV ð1 1 jÞ :(21)

Then the total free energy of a vesicle with nd cap-shaped
domains is expressed by
cap
cap
cap
cap
Ecap
total ¼ Ed 1 Em 1 El 1 Etr :(22)

Here we assume that all the domains inevitably bud in thepresence of the excess area as shown in Fig. 10.
To obtain the condition whether the domains bud or not,we have to estimate the free energy of a vesicle having thesame j but without any budding domains. In this case, themean curvature of the domain is equal to that of the matrix,
i.e., hereafter we call it a flat (nonbudding) domain. When we
calculate the total free energy of such a vesicle, we approximated the vesicle as a sphere with radius Rs, i.e., a ¼ b ¼ RS ¼
RV(1 1 j). The free energy per vesicle with nd flat domains
is expressed by
flat

flat

flat

flat

flat

Etotal ¼ Ed 1 Em 1 El 1 Etr(  
)
2
2
1
2
4pRV
flat
f ¼ 8pfkd
Ed ¼ nd kd 2
2
nd
RV
 2
1
2
flat
2
Em ¼ kd 2 4pRV ð1  fÞ ¼ 8pkm ð1  fÞ
2
RV
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ﬃ
f
f
flat
1
El ¼ 4psRV ð1 1 jÞ
nd
nd
Eflat
tr ¼ nd kB T logf:

FIGURE 12 Distribution of excess area j for normal coarsening vesicles(white bar) and trap coarsening vesicles (gray bar) obtained by micropipetteaspiration technique. The vertical axis denotes the number of vesicles nv.

(19)

(23)
(24)
(25)

(26)
(27)

In the free-energy analysis, the lengths a and L are scaledby the radius RV and we introduce the dimensionless total
free energy Eitotal ¼ Eitotal =2pkd (i ¼ cap or flat).
The obtained free energies of a vesicle with cap-shapedflatdomains Ecap
total and with flat domains Etotal are plotted as afunction of the number of domains nd on a vesicle in Fig. 13,
where we used the values of f ¼ 0.4, kd/km ¼ 1.25, and kd/s ¼
0.2 mm (10,29). For the vesicle with a large excess radius(j ¼ 0.116; black lines), the free energies for Ecap
total (solidline) and Eflat(dashedline)
decreasewithdecreasingnd. Thetotal
flat domains have lower energy for nd . 110, whereas thecap-shaped domains are more favorable for nd , 110. Thus
the budding transition is expected to take place at nd ;110.
This means that when the domain grows to the threshold
Biophysical Journal 92(1) 115–125

124

FIGURE 13 Free-energy curves of a vesicle with cap-shaped domains
flat
Ecap
total (solid lines) and with flat domains Etotal (dashed lines) as a functionof number of domains nd. Black and gray lines correspond to a vesicle
with large excess area (j ¼ 0.116) and small excess area (j ¼ 0.025),
respectively.

size determined by ðpL2 =nd Þ1=2 , the domain starts to bud.
Recalling that the onset time of the budding transition
coincides with that of the trapped coarsening, nd corresponds
to the number of the trapped domains on a vesicle andðpL2 =nd Þ1=2 is the mean size of the trapped domains.
This threshold strongly depends on the value of j. In thecase of the smaller excess radius (j ¼ 0.025; gray lines inFig. 13), the budding transition takes place at nd ;70. Further
decrease of j results in nd ¼ 1, where budding transition
does not take place on the vesicle, and hence the normalcoarsening occurs. This critical excess radius j* determines
whether the vesicle exhibits trapped coarsening or normalcoarsening. To estimate the value of j*, we measured thenumber of domains in the trapped state, nd, as a function of jas shown by open circles in Fig. 14. The normal coarsening
vesicles (corresponding to nd ¼ 1) have very small values of

FIGURE 14 Number of trapped domains nd as a function of excess area j.
Open circles denote experimentally obtained nd and solid curve denotes thetheoretical prediction based on the free-energy analysis.
Biophysical Journal 92(1) 115–125

Yanagisawa et al.

j , 0.01, whereas the vesicles with j . 0.01 exhibit thetrapped coarsening and the number of the trapped domains
increases dramatically with increasing j. Thus the criticalexcess area j* is ;0.01.
The transition domain number nd at the transition obtained
from our free-energy analysis is plotted against the excess
radius j by a solid line in Fig. 14. Using the values of f ¼
0.4, kd/km ¼1.25, and kd/s ¼ 0.58 mm, nd is unity up to j ;0.01 and then starts to increase steeply with increasing j.
This calculation describes the experimental data qualitatively. Thus the domain growth kinetics on a vesicle isgoverned by the three parameters, kd/km, kd/s, and j. Here it
should be noted that the theoretical critical excess area j*strongly depends on the value of kd/s. Typical values of thebending modulus k and the line tension s are ;1019 J and;1011–12 N, respectively. These values give kd/s ;0.01–
0.1 mm, which is an order of magnitude smaller than the usedvalue in this calculation. This deviation may arise from theconventional assumptions used in the model and/or ambiguity in the estimation of k and s for the ternary system.
Very recently, a similar slowing down of the domaincoarsening was reported for the fluid membranes with asymmetric bilayer using the dissipative particle dynamics simulation(30). In this case, surface tension originated from thespontaneous curvature of the domain brings the long-range
repulsive interaction. Thus the coupling of phase separationand elasticity brings anomalous dynamics of the domains on
the fluid vesicle.
Experimental conditions for the normalcoarsening and the trapped coarsening
Finally we argue about the experimental condition to observenormal coarsening or trapped coarsening. As described before, we observed both coarsening processes in the same
microscope field of the same sample. Thus whether it forms
the tense vesicle with normal coarsening domains or theflaccid vesicle with trapped coarsening domains seems todepend on the uncontrollable swelling process. Quite recently, however, to form large unilamellar vesicles, a gentle
hydration technique using NaI was reported by Tsumoto et al.
(31). In this technique, they added NaI methanol solution to
the ternary lipid chloroform solution for preparing the driedlipid film, which may control the osmotic pressure difference
between the inside and the outside of the vesicle. We foundthat using this technique, ;90% vesicles show normal coarsening domains, whereas using the standard gentle swelling
technique described in the experiment section, ;70% vesicles show trapped coarsening domains. In addition, we examined the excess area of the vesicles produced by the NaI
swelling technique and found that most of the vesicles have asmall excess radius j ,0.01. Thus we can control the area/
volume ratio of vesicles, and distinguish between normalcoarsening and trapped coarsening, by adjusting the ionicstrength of NaI solution.

Domain Growth Dynamics

CONCLUSION
We have investigated the growth dynamics of circulardomains in the vesicle consisting of DOPC, DPPC, andcholesterol ternary mixture. The coarsening process can be
categorized into two types: normal coarsening and trappedcoarsening. For normal coarsening, the domains having flat
circular shape grow according to the phenomenologicalpower law of t2/3. The observed growth law is not describedby a two-dimensional diffusion-and-coalescence growthmechanism following the Saffman and Delbrück theory,
which may originate from the two-body hydrodynamic interactions between domains. For trapped coarsening, domaincoarsening is suppressed by the repulsive interdomain interaction. If the vesicle has enough excess area, the growing
domains bud due to the line energy of the domains. The
bending of the matrix membrane between the two approaching cap-shaped domains induces the repulsive interdomaininteraction. The theoretical calculation based on the budding
model reveals that the condition determining whether thevesicle shows trapped coarsening or normal coarsening depends on the excess area of the vesicle, the line tension
between phase separating domains, and the bending modulusof the membrane. Thus trapped coarsening is a result of astrong coupling between the phase separation and the elastic
nature of the membrane under the area/volume constraint.
We thank Dr. Youhei Fujitani (Keio University) for valuable discussions,and Dr. Norifumi L. Yamada and Mr. Mafumi Hishida for gratefulsuggestions about NaI.
This work was supported by a Grant-in-Aid for Scientific Research (B)from the Ministry of Education, Science, Sports and Culture of Japan (No.
15340138). A part of this work was carried out when one of the authors(M.Y.) stayed at Yukawa Institute for Theoretical Physics, Kyoto University. She is grateful for the support by the Grant-in-Aid for the 21stCentury COE ‘‘Center for Diversity and Universality in Physics’’ from the
Ministry of Education, Culture, Sports, Science and Technology of Japan.

125
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