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Frustration-induced ripple phase in bilayer membranes
Hiroya Kodama, Shigeyuki Komura
To cite this version:
Hiroya Kodama, Shigeyuki Komura. Frustration-induced ripple phase in bilayer membranes. Journal
de Physique II, EDP Sciences, 1993, 3 (9), pp.1305-1311. �10.1051/jp2:1993104�. �jpa-00247907�
Classification Physics Abstracts
64.60 68.15 87.20
Short Communication
fhustration-induced ripple phase in bilayer membranes Hiroya
Kodama (~) andShigeyuki
Komura(~)
(~)
Department of Applied Physics, Faculty of Engineering, Nagoya University, Nagoya 464-01, Japan(~) Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan
(Received
29 Marcli 1993, accepted in final form 5 July1993)
Abstract A model describing the phase transition from the L~ or
Lp
phase to the modulatedPp
"rippled" phase in abilayer
membrane is considered within mean-field approximation. Thecoupling
between the curvature and the asymmetry in the area per molecule can destabilize the flat phase by releasing its frustration energy. We obtain the pressure-temperature phase diagram which qualitatively accounts for part of the pressure experiment of DPPC in excess water. The existence of the modulated structure within our model is also confirmed by a cell dynamicalsimulation.
Properties
of membranes withhigh flexibility
are of current interest in connection with the statisticalphysics
offluctuating
surfaces andbiophysics
oforganic
membranes [1].Lipid bilayers
constitute a rich set ofphysical
andbiological
systems,exhibiting
a widevariety
of behavior undervarying
circumstances. One well-known fact about such systems is the presence of a first-orderphase
transition associated with themelting
of thehydrocarbon chains,
separat-ing
ahigh-temperature
disordered fluidephase
called theLa
and alow-temperature
orderedgel phase
called theLp phase.
Between these two flatphases,
an intermediatestructurally
modulated
("rippled") phase
has been detected in a fewphospholipids.
Thisphase
is termed thePp phase
and has stimulated considerable theoretical interest [2-5] as well as several relatedexperimetal investigations
[6].In this
article,
we shall present asimple
butphysically
clearexplanation
of theripple
for- mation mechanismby considering
the combines effect of a curvatureinstability
[7-9] and a frustrated flatbilayer [10-13].
The former accounts for thepossible
destabilization of the membraneshape by introducing coupling
between the curvature and the internaldegrees
offreedom,
such as, the localcomposition.
On the otherhand,
a flatbilayer composed
of twoidentical
amphiphilic monolayers
with finite spontaneous curvature isalways energetically
frus- trated and can be also destabilized. In the case that thebilayer
iscomposed
ofa mixture of two
surfactants,
forinstance,
the allowedasymmetries
in the localcomposition
can lead to a1306 JOURNAL DE PHYSIQUE II N°9
spontaneous vesicle formation [14].
As far as the
Pp phase
isconcerned,
both of these aspects areexpected
toplay
animportant
role. In thephysical picture
weconsider,
the local curvature and the asymmetry in the area permolecule are
coupled.
The area per unit molecule represents the order parameterdescribing
the main transition. We obtain a pressure-temperature
phase diagram
within the mean-fieldtheory.
Our resultsqualitatively reproduce
part of thephase diagram
obtained in recent pressurestudy
on water-richcompositions
of DPPC [15]. In order to confirm the existence of the modulated structure in theexpected region
of thephase diagram,
we have alsoperformed
a cell
dynamical
simulation of the present model [16].We shall first discuss the main transition between the La and
Lp phases,
in which the discontinuousjump
in the area per molecule S can betypically
observed.According
to the pressure-areaphase diagram
ofmonolayers
ofphospholipids
determinedby
the film balancetechnique
[17],typical
values of the critical temperature, lateral pressure, and area per moleculeare T~ = 42
+~
44
°C, fl~
= 47
+~ 50
dyn/cm,
and S~ = 40 +~ 50l~, respectively. Although
this critical
point
has been identified as a tricriticalpoint forming
ajunction
between a first order and a second order transition curve [17], we assumehere,
forsimplicity,
that the main transition can berepresented
within the framework of the van der Waalstheory
ofordinary
gas-liquid phase
transitions near the criticalpoint.
As a convenient set ofindependent variables,
we use the temperature T and n, where n = I
IS
is the number of molecules per unit area.The dimensionless molecular
density, #
=(n n~)/n~,
is taken here as the order parameter.By using
the notation r =(fl fl~)/fl~
and t =(T T~)/T~,
theequation
of state near the criticalpoint
can be written asr = bt +
at#
+B#~, (l)
with dimensionless constants b > 0, a > 0 and B > 0 [18]. When t < 0 and r
=
bt,
theL~
(#
=-(-at/B)~/~)
andLp (#
=
(-at/B)~/~) phases
coexist.Equation (I)
can be obtainedby minimizing
thethermodynamic potential
per molecule 4l= e
-(r bt)#
+)at#~
+(B#~
with respect to
#,
where e is some molecular energy of orderS~H~.
Theinhomogeneity
in themonolayer
can beconveniently
taken into accountby
anotherpotential
which relates to some chosen andspecified
area A in the monolayer,consisting
of a variable number of molecules.Upon keeping
in mind that r is then a function of temperature and chemicalpotential,
we consider thefollowing Ginzburg-Landau
type Hamiltonian in the unit of energy scalee/S;
HGL
=/
dA
I-(r bt)#
+at#~
+B#~
+(i7#)2)
,
(2)
2 4 2
with g > 0.
Up
to thispoint,
we have described theproperties
of amonolayer
membrane. We now turn our attention tobilayer. By introducing
the suffices 1 and 2 associated with eachmonolayer,
we assume that the main transition of the
bilayer
issimply represented by
Hi
=/
dA~j I-(x
bt)#~
+at#(
+Bit
+ g(V#I)~ (3)
~=i,2
~ ~ ~
In
general,
whenconsidering
aparticular
suchbilayer,
one must take into account its interaction withneighboring bilayers
[3]. In this paper,however,
we consider water-rich systems for which such interactions arenegligible
[4,5].The total curvature
bending
energy per unit area of the twomonolayers comprising
thebilayer
is[11,
12]Hb = Jc
~j (c(° cil)~, (4)
2
~
where Jc is the
bending rigidity,
and c(~) and (~~ denote the mean curvaturemultiplied by
2 and the spontaneous curvature of the I-thmonolayer, respectively.
If twomonolayers
are setapart, the curvature energy
(4)
takes its minimum value 0by adopting
their curvatures to c(~) =c(° independently.
In the case ofbilayer, however,
the curvatures of eachmonolayer
have the same
magnitude
butopposite signs,
c(~)=
-cl~)(e c),
due to the fact thatthey
arestuck
together [12, 13]. Suppose
bothmonolayers
are characterizedby
the same spontaneous curvature co. Then the curvature energy(4)
can be minimized at c= 0, but this flat
bilayer
still has an energy Jcc(. This frustration energy can be decreased if an asymmetry in the spontaneouscurvature exists, and turns out to vanish when c
= c(~~ =
-c(~~.
The reasonwhy
mixed vesiclescan be
energetically
stabilized follows from the aboveargument [11-13].
As was consideredby
MacKintosh and Safran [13], we assume here the
following simple
lineardependence
of the spontaneous curvature on the order parameter:ci~
= $ ~/#~, where ~/ is a constant, and j is the spontaneous curvature when #~ = 0.The
shape
energy of thebilayer integrated
over the total area A is thenH2=
/dAl2a+jJc((c-ci~~)~+ (c+c/~)~j), (5)
in the energy scale
e/S
as before. In theabove,
a non-zero constant surface tension a has been includedby assuming
that the Shulman condition is not satified for the area per molecule under consideration [19].While
allowing
thebilayer
to curve, we consider the case in which itsshape
is close toplaner
without anyoverhangs
or discontinuities. If we use asingle
valued function£(z,g)
whichmeasures the
displacement
of thebilayer
from a flat referencexv-plane (Monge representation),
the total Hamiltonian H is
H
=Hi
+ H2=
/ dzdg (a(Vi)~
+ Jc(V~i)~
+2Jc~/#- (V~£)
+g
(v#+)~
+(vi-
)~ +f (#+, #- )j
,(6)
with
f (#+, #-
=
-2(x bt)#+
+ at(#(
+ #~ + B#(
+3#(#~
+#~
,
(7)
2 2
where
#~
=(#2
~ii /2.
Here we have eliminated the terms whichmerely
shift the criti- calpoint.
Thedegree
of freedom(£)
can be traced outby performing
a Gaussianintegral
J D(£)
exp(-H/kBT)
[8]. If weexpand
up to the forth order in the wave number, the effective Hamiltonian becomesH~w =
/ dzd» ljz (vi-)~
+
jK iv21-)~
+ g(vi+)~
+i (i+, i-)j
,
(8)
where Z = 2
(g
Jc~~/~la)
,
and K = 2 (~~~/~
la2)
The necessary condition for the presence of the modulatedphase
is Z < 0 [3].Hereafter we shall use the reduced notations; x'
=
(1/B) (2BK/22)~~~
x, T =(2K/22)
at,m~ =
(2BK/22)~~~#~,
b'=
(lla) (2BK/22)~~~b, g'
=(-2/Z)g
and V'=
(-K/Z)~/2V.
The effective Hamiltonian
(8)
is minimized with respect to the reduced order parameters at~ ~*
~ ~,
0 =
(x' b'T)
+(g'V'~
T3mf~) mj mj~, (9a)
1308 JOURNAL DE PHYSIQUE II N°9
0 =
(-i7'~ i7'~
T3m)~)
mlmf~ (9b)
The constant solution m[ = 0 is
lineally
stable whenlx'
b'T( >~)
~ ~~(10)
12
~~~
12
In
particular,
if T <-1/8,
the main transition between the L~ andLp phases
occursby changing
x'. At x'= b'T, the reduced mean order parameter m+ takes the uniform values
m[
=~(-T)~/~, being
ofequal
energy, and the L~ andLp phases
coexist.Let us consider the parameter
region
where m[= 0 is
unstable, namely,
when(10)
is not satisfied. If we assume that(m((
issufficiently small,
or moreprecisely,
3(m((~
<1/4
T,(9b)
now has a stable modulated solution for which orientational symmetry is broken. Close to the criticalpoint
of thehomogeneous
system T = 0, the above condition is well satisfied. To first order in(1- 4T)~/~,
this solution isgiven by
[20]ml m (1
4T)~/~
cos(z'll)
,
(11)
where z' is chosen
along
the axisparallel
to the wave vector of the modulation. The modulation of mL will induce a modulation of the local curvaturecorresponding
to therippled Pp phase.
When g is
sufficiently small, m[
is alsoperiodic along
z' withapproximately
half the wavelength
of m[Accordingly,
aftertaking
thespacial
average, we obtain the isotherm as a function of(m[)
for a
given
~ as~i
=
~'~ ~ ~
~~~~
~~~~~~
~~' ~'~~ ~~~~
~~~~~~~
'(i~)
b'T + ~ ~~
(m[)
+(m[)~
otherwise.8
Here we have defined
(m[)
as the average of absolute maximum and absolute minimum values ofm[. Figure
shows thephase diagram
in the (T,(m[))-plane
obtained from(12).
It is remarkable that the
Pp phase
appears as an intermediatephase
in the temperature range-1/8
< T <1/4.
Infigure
2, the samephase diagram
is shown in the(x', T)-plane
with b'= 1.
Three first order transition lines
join together
at thetriple point
x'= T =
-1/8.
Under the condition 3(m[(~
«1/4
T, the isotherms(12)
are discontinous and cannot beprecisely
determined in the coexistence
region.
In the presentphase diagrams,
we have assumed that the coexistence of thePp phase
and the two flatphases
occurs when theequality
in(lo)
holds.Although
we are confident that the transition should be continuous near the criticalpoint (T
=1/4),
and thus that there exists a tricriticalpoint connecting
the first and second order transitionlines,
it is difficult to determine theprecise position
of thispoint
within the presentapproximation
and it is hence omitted infigure
2.In order to confirm the existence of the modulated structure as an
equilibrium
pattern, weperformed
a celldynamical
simulation [16]. We start from a random initialconfiguration
andimpose periodic boundary
conditions.Figure
3 exhibits atypical fully developed profile
ofm-
ix, g)
for which the parameters are chosen from thePp phase region
in thephase diagram.
If the parameters
corresponding
to either theL~
orLp phase
areused,
m+simply
reaches a finite uniformvalue,
and no asymmetry appears(m-
m0). Although
we used the celldynam-
ical method
only
to obtain thestationary profiles
of m~, their time evolutionmight
be alsointeresting
toinvestigate.
Detailed results of the simulation will bereported
elsewhere.oi
L~ P~ L~
oLj
~
ir'
P~+L~ P~+L~ L~
oi
L~+Lj
0 4 0.2 00 0.2 0 4 0.2 0.1 0.2
<
ml
> ~Fig.
IFig.
2Fig. 1. Phase diagram in the (T, <
m[ >)-plane
whereT is the reduced temperature and <
m[
>is the spatial average of the reduced mean molecular density. The
Pp
phase appears as an intermediatephase in the temperature range
-1/8
< T <1/4.
The direct transition between the La andLp
phasesis also seen for T <
-1/8.
Two-phase coexistence regions between the three single phases are indicated by La +Pp
etc.Fig. 2. The same phase diagram as in figure I, in the
(x', T)-plane
where x' is the reduced lateralpressure. Here we have set b'
= I. Three first order transition fines join together at the triple point x'= T =
-1/8.
As
regards
the relevance to thepreviously existing theories,
the mechanicalorigin
of theperiodic
structure in the modelby
Honda and Kimura [4, 5] differs from that in ours since we have assumed that the membrane has a small butnon-vanishing
surface tension. Meanwhile our workcorresponds
to a natural extension of those of Leibler and Andelman [8], and MacKintosh and Safran [13] to thebilayer
and non-local cases,respectively.
Infact,
Leibler and Andelmanargued
thePp phase
in terms of one scalar order parametercoupled
to the local curvature [8], whereas ourtheory
includes two order parameters,namely, #+
controlledby
the lateralpressure and #-
coupled
to the curvature. Theresulting
main difference turns out that theregular
2-dimensional modulated structurehaving
the symmetry of ahexagonal phase
[8] does not show up in the whole parameter space, and henceonly Pp phase
appearsjust
between the two flatphases.
This can beinterpreted
as follows. The total Hamiltonian should beinvariant under the
exchange
of the twomonolayers
or, in other words,#-
- -#-. Therefore the linear term in#-
is absent in(6), corresponding
to thevanishing
chemicalpotential
casein reference [8] so that
only
thestripe phase
is relevant as a modulatedphase. Although
these aspects of thebilayer
have been also discussed in the references[I1-13], they only
considered the transition between thehomogeneous
lamellar and vesiclephases.
As a
generalization
of ourmodel,
we could also introduce an external fieldconjugate
to#-,
forexample,
a uniformbending
force that breaks thebilayer
symmetry. Then it would beinteresting
to consider aphase diagram
in ahigher
dimensional parameter space, such as the1310 JOURNAL DE PHYSIQUE II N°9
m
x Y
Fig. 3. Typical fully developed ripple pattern of the reduced molecular density asymmetry m-
(~,
y) obtained by the cell dynamical simulation with periodic boundary conditions. The simulation has been started from a random initial configuration. The used parameters are chosen from thePp
phase regionin the phase diagram.
temperature and two external fields
coupled
to#~.
Thephase diagram presented
infigure
2can be viewed as one
particular
cross section of such ageneralized phase diagram.
Another extension is to consider thecoupling
effect between themonolayers.
This can be doneby including
additional terms in the Hamiltonian, such as#+#2
,
in accordance with the allowed symmetry. The addition of such terms may introduce an asymmetry, but is not
expected
todramatically change
thephase diagrams
[13].Figure
2reproduces
part of thephase diagram
obtained in recent pressurestudy
on water- richcompositions
of DPPCby
Krishna Prasad et al. [15]. From theexperimental point
ofview,
the lateral pressure in abilayer
system is not aseasily
controlled as that in a(Langmuir) monolayer
system sincebilayers
are often formed in asolvent,
whilemonolayers
are formed at aliquid-air
interface [17]. In reference[15],
thehydrostatic
pressure rather than the lateralpressure was
changed.
Onepossibility
to control the lateral pressure in abilayer
may be found in themicropipet technique
of Needham and Evans [21].Finally,
it should be mentioned that our model does not account for the orientationaldegree
of freedom of
lipid
molecules with respect to the membrane surface in therippled phase.
Thisdegree
of freedomgives
rise twopossible phases.
In addition to thePp phase
we have assumedthroughout
this paper in which thelipids
arealigned orthogonally
to themembrane,
there may be aPp, phase
in which thelipids
are not soaligned.
Infact,
there are severalexperiments
and models which suggest that thePp, phase
does indeed exist[22-25].
From ourpoint
ofview, however,
the real molecular structure of therippled phase
is an openquestion [26, 27].
In thispaper we have tried to suggest the minimal
ripple
formation mechanism inbilayers. Extending
the model
by considering
a vector field order parameter, it should be possible to describe thetilting
effect or defect structure.In summary, we have discussed the
phase
transition from the La orLp phase
to the mod- ulatedPp phase
in abilayer
membrane within mean-fieldapproximation.
We obtained a pressure-temperaturephase diagram
whichqualitatively reproduces
part of thephase diagram
obtained in recent pressure studies of
bilayers.
The celldynamical
simulation of our modelalso demonstrates the existence of the modulated structure as an
equilibrium
pattern under certain conditions.Acknowledgements.
We would like to express our
appreciations
to Prof. A. Onuki and Prof. D. Andelman for their interests and critical comments. Valuable discussions with Prof. I.Hatta,
Prof. M.Doi,
Prof.K.
Honda,
Prof. H.Kimura,
Dr. M. Cates and Dr. J. Harden are alsoacknowledged.
We thank Dr. G.Paquette
for his criticalreading
of the manuscript. One of us(H.K.) acknowledges
the support
by
Grant-in-Aid for Scientific Researchby
theMinistry
ofEducation,
Science andCulture, Japan,
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