Polymorphism of Lipid-Water Systems: Epitaxial
Relationships, Area-per-Volume Ratios, Polar-Apolar
Partition
Vittorio Luzzati
To cite this version:
Vittorio Luzzati. Polymorphism of Lipid-Water Systems: Epitaxial Relationships, Area-perVolume Ratios, Polar-Apolar Partition. Journal de Physique II, EDP Sciences, 1995, 5 (11),
pp.1649-1669. <10.1051/jp2:1995205>. <jpa-00248261>
HAL Id: jpa-00248261
https://hal.archives-ouvertes.fr/jpa-00248261
Submitted on 1 Jan 1995
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J
Phys
II
France
(1995)
5
1649-1669
1995,
NOVEMBER
1649
PAGE
Classification
Physics
64 60
Abstracts
My
Md
64 70
Md
68 35
Polymorphism of Lipid-Water Systems:
Area-per-Volume Ratios, Polar-Apolar
Vittorio
Epitaxial
Relationships,
Partition
Luzzati
de Gdndtique Moldculaire,
Gif-sur-Yvette,
France
Centre
91198
(Received
1995,
March
31
R4sum4.
reiised
nombreuses
De
lors
coexistent
d'une
Centre
National
respectent
phdrtomArte
des
souvertt
July 1995)
31
exp6rimentales
observations
transition
Scientifique,
Recherche
accepted
1995,
June
28
de la
montr6
ont
phases
deux
les
que
dpitaxiales
relations
le
:
initial
but
qui
de
d'exphquer ce
dtrartge
Puisque les
irttdractiorts
irtterfaciales
semprddomirtant darts l'dquilibre drtergdtique il y a lieu de supposer
que les deux
phases I l'dquihbre
d'41dments
de
dortt le volume et l'aire
superficielle sortt
consistent
structure
Ces deux
irtvariants
paramktres peuvertt s'expnmer en fortction de la terteur en eau et des dides mailles
416mentaires
mensions
des deux phases
valeurs
leurs
ddtermindes
peuvertt dortc Atre
I partir
de
donndes
exp6rimentales
On note,
le
volume
des
outre,
rapport
erttre
en
que le
414mertts de
lipidique du syst6me 4quivaut k urt coefficient
structure
et celui de la
composante
de
Lorsqu'on applique ces iddes aux
donrtdes
de la
littdrature
observe
partage.
on
que le coefficient de partage
varie
fortement
valeurs
manifestent
des
corr41ations
remarquables
et que
ses
d'autres
parambtres chimiques et physiques du systbme Ceci suggkre que la sdparatiort des
avec
chairtes
paraffiniques et des t6tes polaires des lipides n'est pas aussi nette qu'ort a l'habitude de le
Cette
d'un portage po1alrelapo1alre
variable
dans ce domaine;
nouveaut4
notion
est
pertser
une
le
coefficient
En
de partage,
thermodynamique
intdressant
pourrait jouer un r61e
par allleurs,
relations
de
qui
les
4pitaxiales
analyse
des
donn4es
publ14es
telles
montre
cc
conceme
une
que
relations
Pour expliquer ces
fr4querttes, mais avec beaucoup d'exceptiorts.
observations
sont
on
4met
l'hypothbse que le r61e des
Cette
hypothbse
relations
6pitaxiales est
cirt6tique.
surtout
travail
cc
blent
dtait
jouer
conduit
r61e
un
I
associer
exemples
Abstract.
nant
the
role
phases
two
ratio
determined
be
thus
equivalent
able
in
to
with
segregation
Les
Editions
particular
Volume
the
coefficient
partition
hydrocarbon
1995
and
sens
un
phase
and
area
The
volume
ideas
coefficient
the
chains
transition
These
was
physical
away
from
often
interactions
there
of the
parameters
explanation
display
an
equilibrium
phases,
the
cell
seek
to
was
interfacial
stabilize
experimentally
chemical
Physique
polarlapolar
and
content
The
the
of the
de
any
m
invariant
work
thermodynamic
that
forces
partition
a
literature
the
correlated
@
is
water
the
that
the
this
of
purpose
in
(dans
sdlectif
avarttage
un
large)
Plusieurs
hypothAse
cette
phases
of
involved
of the
functions
cart
all
among
area/volume
is
pairs
Considering
tionship
de
original
The
that
observation
l'appui
I
dpitaxiales
relations
aux
citds
sort
appear
for
ground
is
of
consist
ratio
two
coexisting
(structure
applied to
found to display
were
parameters
the
of
the
a
can
whose
elements
structure
elements
structure
of the
to
the
empirical
the
epitaxial relaplay a predomiproposition that
of
an
expressed
be
phases.
as
values
their
elemertts)/(lipid
molecules)
large variety
data
of
avail-
remarkably
variations,
that
the
system,
suggesting
wide
polar headgroups
is
not
as
sharp
as
it is
1650
commonly
The
assumed
this
moreover,
partition
PHYSIQUE
JOURNAL
DE
of
variable
notion
coefficient
a
well
may
II
polarlapolar
turn
into
N°il
partition
interesting
an
is
a
novelty
m
thermodynamic
the
field,
parameter
relationships, a search through the
literature
shows
that its
observations
order to explain these
the
conjecture is put
effect on the phase
forward
that
the epitaxial
coincidences
have a kinetic
In partictransitions
ular, it is suggested that any
involving epitaxially related phases is unlikely to display
transition
metastable
The possibility is also evoked
selective
advantage (be it technological,
that a
states
biological or experimental) may be associated
with the
of epitaxial
relationships
This
existence
illustrated
by several examples drawn from the
literature
conjecture is
significance
As to
the
very
existence
has
of the
many
epitaxial
In
exceptions
Introduction
1.
Many of the authors ,vho have reported X-ray scattering
lipid-water systems
experiments
on
noted, especially in phases of type I. that if the spacing of the first
reflection
of the
have
lamellar
and of the hexagonal phase and the spacing of the [21ii reflection of the cubic phases
then the points often follow a smooth
Q~~° and Q~~~ are plotted as a function of concentration,
when
boundaries
(see,
phase
crossed
for
example,
curve,
even
are
11, 2])(~ ). Besides, the study
of
quasi-monocrystalline
samples has shown that the spacing
coincidences
often
correspond
relationships
genuine
3-D
epitaxial
This
has
of
observation
led
elegant
analyses
to
to
[3, 4]
the
relationships
between
the two phases [3,4].
Similar
geometric
observations
have
been
extended
recently to phases of type II (see [5] and Sect. 3 2)
Apparently, all the authors
take this puzzling phenomenon as a
of
fact,
,vithout
,vondering
why pairs of phases in
matter
thermodynamic equilibrium should be epitaxially related to each othf<r (the explanation put
forward by Manani,
Amaral
and
coworkers 11,6], that I find utterly unsatisfactory, is discussed
The present work
prompted by this question- it later developed into a
in the Appendix)
was
elaborate
touching
endeavour
other
facets of lipid polymorphism
more
upon
Regarding the question of the epitaxial relationships a review of the literature
confirmed
that
phases in thermodynamic
equilibrium are often epitaxially i-elated to each other, but it also
showed
that the rule has
In an
exceptions.
rationalize
this puzzling pheattempt
to
many
I
forward
kinetic
explanation,
whereby
the
oi
epitaxial
relationships
put
nomenon
a
presence
has the effect of lowering the kinetic
barrier of the
I also held out the conjecture
that
transition.
the very
of epitaxial
existence
relationships may impart a selective advantage on lipid-water
systems
Confronted
with
such
a
collection
of
likely
thermodynamic
phase
transitions
I
was
led
to
wonder
1n.hether
any
par-
directly related to chemical
potential as to be
so
phases at
equilibrium
invariant
At first sight the best
candidate
in
appears
mterfaoal
indeed
the
predominant forces at ,vork in lipid-water systems
to be the
as
area,
polarlapolar
originate from the interfacial
interactions.
Naturally, the possibility of
to
seem
testing this hypothesis hinges upon the position of the interface along the lipid molecules
This
problem has often been
discussed
in the past.
For example, it has early been
shown iii and
amply confirmed since, that if the polarlapolar
interface
between
the
is set at the
separation
hydrocarbon chains and the polar headgroups, then the area-per-chain is found to increase or
ticular
to
structural
remain
constant,
parameter
but
never
is
to
to
decrease
be
as
the
amount
early days of lipid polymorphism this
observation
hexagonal
of what
the
then
called
structure
~vas
prevalent at that time~ that all
of the
proposition.
recollection)
(~
In
against
support
the
the
of
water
increases,
sometimes
was
produced
as
an
phase
argument
(now phase Hi) and m
(personal
lamellar
lipid phasis are
"iuiddle
the
when
even
phase"
PHASES,
LIPID
N°11
RELATIONS,
EPITAXIAL
SPECIFIC
AREAS
1651
In this
it must be stressed, the
area-per-chain at the interface
case,
phase
with
different
the
and
take
values in pairs of
water
content
vary
may
one
may
authors
focused the
coexisting phases. More recently several
have
the
virtual
attention
on
surfaces (called "neutral"
"pivotal",
whose
within
phase,
supposed
is
to be
[8, 9]
or
area,
one
with respect to the water
of the system. In this case also the
invariant
area-per-chain
content
different
take
values in pairs of coexisting phases.
may
The analysis presented m this work
from the proposition
that
wherever in a lipid-water
stems
thermodynamic
equilibrium, these phases can be
in
visualized
system two phases
coexist
to
of
having the same
elements
chemical
and the
consist
structure
composition
area-persame
volume
The
elements
ideal objects, whose shape is assumed to be
ratio.
structure
consistent
are
with the
symmetry of the phase (planar slabs for phase L, cylindrical or prismatic rods for phase
HI- etc ), that contain
either the hydrocarbon
chains (sometimes accompanied by a fraction of
hydrocarbons
In short, and
somewhat
arbitrarily
the polar moiety) or only a fraction of the
of the
and "polar"
the
elements
I shall call respectively "apolar"
moieties
contents
structure
boundaries
crossed.
are
within
and
of the
interstices.
be
phase of known
the surface/volume
ratio of the
elements
structure
structure
a
can
fraction
dimensions
expressed as a function of their volume
and of the cell
For a pair of phases
equilibrium simple arguments sho~v that the volume fraction of the
elements
structure
can
in
of the lipid component, the
and the cell
determined
if
the,,olume
be
concentration
structure
and if the
made
that
the
of the two phases are
known.
dimensions
structure
assumption
is
For
elements
have
volume
ratio
chemical
the
the
applied
I have
procedure
this
chemical
composition
correlated
with
Finally,
surface
of the
equal
D
~
volume
P,
~
the
and
the
m
G,
of the
the
surface
same
elements
structure
and
surface
excellent
most
and I
of the
area.
with
to
two phases (Sect
specify the volume
the
out
of the
highly
be
The
thus
Most
literature
to
2 3).
(and
two phases
interestingly, the
variable
and strongly
elements
structure
turns
bicontinuous
the
three
result
the
in
system
common
compared
The
agreement
of the
area
suffices
from
data
elements
structure
three
of
sets
properties
P)
and
D
several
to
chemical
considered
I
of type
iPMS
volume
same
(structure elements) /(lipid component)
composition) and the surface area of the
is
average
phases
that
the
assumed,
two
the
frequency
and
partial specific
with
and
in
phases
three
the
of the
concentration,
increase
curvatures
the
with
curvature
mean
each
at
1n.hich
(associated
phases
cubic
Gaussian
to
the
be of
G
order
observed
are
experimentally
Methods
2.
2
NOTATION
1.
MW,
vpa,.
~m~i,
c
@:
molecular1n.eight
,>olume (in l~
cwT/[cw ii
=
ABBREVIATIONS
AND
i)
+
ii
(m Dalton)
of
one
volume
lipid
molecule
concentration
and of its
volume
(in cm~ g~~
(lipid /hpid+water),
lipid,
of the
moiety (see Tab.
hydrocarbon
cw is
the
weight
I);
concentra-
tion,
lattice
a
L
and
iPMS
H
parameter
I-D
infinite
yin
lamellar
periodic
lj,
and
?-D
minimal
hexagonal phases,
surfaces;
cubic phases of space
bicontinuous
Q~~~(D), Q~~°(G), Q~~9(P). 3-D
gyroid
and primitive iPMS),
im3m (D, G and P stand for diamond,
Q~~~(M) 3-D micellar cubic phase of space group Pm3n (M stands
elements
phases whose
non-lamellar
contain
structure
type I (and it)
(oil-in-water) (and nice uersa);
group
for
Pn3m,
ia3d
micellar),
hydrocarbon
the
and
chains
JOURNAL
1652
DE
PHYSIQUE
N°li
II
the partial specific volume of the lipid. ~m~i is the volume
MW/0.602,
weight). vt~ar is the
where
MW is the
molecular
(~m~j
determined
assuming that the volume
volume of the hydrocarbon moiety of the lipid molecule,
occupied by each of the CH3-> -CH2- and -CH= groups is (in i~ ): ~cH~
53.5, ~cH~
27.3,
°C.
The
temperature-dependence
of
the
volumes
All
volumes
assumed
20
is
20.3.
at
are
~cH
~20[1+ 4.6 x 10~~(T 20)]. Some of the values of
taken from
to take the form: ~T
were
assessed
discussed
the original papers (see references in Tabs. II, IV, V, VI), others
in
were
as
ether-linked
f$oj. (j in this lipid the hydrocarbon chains are
to the glycerol and only the
sugar
moiety is ascrtbed to the polar headgroup).
Table
I.
Chemical
of
lipid
molecule
the
data.
is
=
"
"
"
"
CI2TAC
PLPC
OLPC
C12K
CI6TAB
C12E06
C12EOio
(ml/g)
1.10
0.94
0.93
0.90
0.95
0.98
0.94
(l~
(l~
460
773
801
356
555
732
975
436
476
326
463
354
354
lipid
~m~j
~t~ar
354
lipid
OOG
OSG
DDGG
MO14
MO15
MO18
(ml/g)
0.84
0.85
0.98
1.03
1 04
1.06
1.00
408
436
961
513
542
628
1229
245
245
783#
367
394
476
953
(l~)
(l~)
~m~j
~t~ar
S/V:
DOPE
area-per-volume ratio of the structure elements;
(structure
element)/(lipid molecules);
xm~j
(structure
volume
elements) /(hydrocarbon chains);
ratio
Xmoi~m~i/~t~ar:
Xpar
(S/V)Xmoi~m~i: area-per-molecule at the surface of the structu<re element;
Sm~j
CnTAB (or CnTAC): n-alkyl trimethyl
bromide (or chloride);
ammonium
PLPC (or OLPC): palmitoyl (or oleoyl) lyso phosphatidylcholine;
qJ,
volume
qJ/c:
=
fraction
volume
and
ratio
"
=
C12K: potassium
CnEO~n: n alkyl
laurate;
oxyethylene glycol monoether;
m
n-octyl-I-O-fl-D-glucopyranoside;
n-octyl-I-S-fl-D-glucopyranoside;
monomyristolein;
monopentadecenoin;
OOG:
OSG:
MO14:
MO15
monoolein;
MO18:
DOPE:
di-dodecyl-fl-D-glucopyranosyl-rac-glycerol;
di-oleoyl-phosphatidylethanolamine;
DDAB:
didodecylmethylammonium
DDGG:
2.2.
THE
FUNCTIONS
Hyde [10]
1introduce
ax
(v?).
bromide.
Using
dimensionless
a
a
procedure
similar
to
advocated
that
by Engblom
a=aS/V
relevant
the
to
the
structure
functions
of the
structure
elements
is
phase
elements.
For
each
specified,
the
volume
V
Ii)
of
and
known
the
shape of
expressed as
Replacing these expressions in (I
of qJ (see Eqs. (2, 3, 9, 11)). The
structure,
surface
area
and
S
once
can
be
cell parameter a and of the volume ratio qJ.
expressions of the parameter (aqJ) as a function
mathematical
operations are trivial in the case of phases L and H (Eqs. (2, 3)).
yields the
&
parameter:
the
N°il
PHASES,
LIPID
Lamellar
2.2. I
for
Phase
of both
structures
RELATIONS,
EPITAXIAL
The
elements
structure
types I and II
(~5aH )1
1§5aII)II
elements
expression
The
of
(qJa),
12)
Hexagonal Phases.
The choice of the shape
arbitrary
In keeping with
polyhedral shape of
the
that
the
hexagonal
phases
the
in
structure
assume
the sides of the cell
A simple
calculation
yields
structure
lamellae
1653
2
=
2.2.2.
the
planar
are
AREAS
is:
v~aL
Were
SPECIFIC
"
cylinders
circular
elements
structure
micelles
the
elements
hexagonal
are
is
to
extent
some
phase Q~~~ (see below)
of
parallel
prisms
I
to
(3)
4~J~/~
411
=
of the
13b)
~J)~/~
equation (3) should
multiplied by
be
the
factor
0 952
BIco~1tinuous
2 2 3.
included
between
standard
Cubic
Phases
two
of the
expression
Q~~~, Q~~° and Q~~9.
parallel to, and
surfaces
S
area
and
distant
,>olume
the
The
d and
-d
concentration
element
structure
from
~J
the
as
a
is the
volume
surface
minimal
function
of d
are
The
(see,
example, [9, io]
for
x
dla
(4)
2Sox+(4~/3)jz~
2So+4~xz~
(5i
=
~J
=
S(z)la~
(6)
=
where
these
The
a~so
the
is
parameters
real
are
of
root
equation
(5i
6
a
function
z
in
(6) by
cell
and
y
is
the
its
is
=
=
phase
x
So
D
Q~~~
-2
1 919
P
Q~~~
-4
2.345
~
~230
_~
~ o~i
of
the
IP&IS.
The
values
of
[10]
[-So/(2~~)]~/~(cos 6 Vssin Hi
(1/3) arcos ([-27~~/(245()]~/~~J)
expression
as
a
function
of ~
ii) yields
the
(7a)
(7b)
expression
of
S(.r) la~
as
of ~J.
F~j~oi
=
IPMS
constant
surface
sjxila2
The
Euler
ill]:
z
Replacing
unit
per
area
suffix X specifies
yields the
the
expressions
nature
of
of the
IP&IS
j81
(X=D,G,P).
Applying
equation
(I)
to
the
three
(~Ja).
(vJax II
(vJax III
Fx(i
=
=
Fx(vJl
v~)
(9a)
(9b)
JOURNAL
1654
J/Iicellar
2.2 4
bedded
in
Q~~~(M)
Phase
Cubic
is
v
the
lipid
reduced
in
N°11
contains
and
volume
micelles
the
same
the
s
identical
is
to
3G~v~/6~
=
average
that
proportion
Since
S/V
(8s)/(a~~s)
with
unit
cell of
the
symmetry
ratio
I
em-
The
3 [12]
to
be
can
(10)
polyhedron. We
one
space-filling polyhedra,
of
this
0 764
=
the
area
of
lipid micelles,
of
kinds
two
cell
of type I. A
q
the
II
unit
matrix; the
structure
is
types of polyhedra of respectively 12 and 14 faces, in the
quotient q of the average space-filling polyhedron is [13]:
by two
isoperimetric
where
phase
This
water
a
filled
of
PHYSIQUE
DE
parameter
contains
a
shape
that the
suppose
and
that
their
size
polyhedra
8
is
one
has
=
(~SaM )1
The
2 2 5.
v7aS/17
=
ax/y(v7x,
Functions
(lla)
2(36n/q)~/~~J~/~
=
Let X
c, /y
and
thermodynamic equilibrium, cx land cy
in
respectively the lipid moiety and the
structure
system
of
ax/y
Assuming
two phases
that
volume
the
ax(vJx)lay(vJy)
=
and
surface
the
ratio
two
phases of the
(and
the
~Jy
The
ratio
lipid-iv.ater
same
volume
lay
ax
concentrations
the
takes
form.
of the
(12a)
elements
structure
the
are
same
in
the
gets
one
v7x/v7y
the
(llbi
580~S~/~
(s/~~)xax/(s/vjyay
=
area
(S/V)x
and
Y be
and ~Jx
element.
10
=
la;
ax
takes
(12b)
=
(12C)
form.
the
ax(vJx)lay(vJyl
Cx/Cy
(S/~')y
=
ax(vJxllay(vJx/cx/yl
=
=
ax/y(vJx,cx/y)
axlay
=
j12dj
where
cx/y
The
mathematical
When
the
concentrations
centration
y~~~
~Jx
and Smoi
cx
and
of the
can
BICONTINUOUS
2.3
of
expressions
Y are
phases X and
be
cy
are
determined
CUBIC
(see
PHASES.
ax(v7x)
(12)
determined.
Sect.
SURFACE
~t
((RiR2)~~)
=
"
(1/2)1(1/Ri
and
ax
and
ay
volume
the
can
solved
and
the
~Jx
is
known
the
parameters
volume
con-
ymoi,
2 11-
CURVATURE
Gaussian
expressions of the average
and
mean
surfaces parallel to, and
elements (that are, by definition,
IJIPS) are (see [9]).
"
(2, 3, 9, 11).
equations
in
be
Once
The
~~
given
are
parameters
cell
the
equation
then
element
structure
and
(12ei
cx/cy
functions
the
identified
known,
"
+
OF
curvatures
at
a
STRUCTURE
THE
of the
distance
d from
=
the
ELEMENTS.
of the
structure
corresponding
(13)
-2~y/S(~S)
i/R~ii
surface
-2~xaz1v7i/S(v7i
(141
N°11
PHASES,
LIPID
RELATIONS,
EPITAXIAL
1655
~
log
jog
AREAS
SPECIFIC
Pe
a
type
a
,
,5
~
1
"f~
~l'-
L'SD
o,5
L
~
o
o
o
o,2
Fig
and
Iamellar,
of
a
(3))
surface.
~
notation
p
given
is
1/R2
and
and
are
be
can
Note
2 2
=
~J for
I
phases considered m this work, both
the phase is specified by the capital
structures
(Pn3m),
D for Q~~~
function
M for Q~~~
corresponds
aH
to
(Pm3n)
prismatic
a
of type I
letter
The
structure
expressions
element
and
and
~J
(iV) means the average of iV over
(~~/S) (see Eqs (1, 8, 13, 14))
v7l-2~i)~/~(S/V)F~~/~(T)
1-2v7~~)(S/V)F~~(T)x(T)
=
=
(left
L for
2 1
the principal
curvatures
expressed as functions of
/1lT)
T
the
Section
m
of
nature
(Ia3d),
that
~lT)
where
various
the
The
G for Q~~°
Section
m
~J for
us
(right frame)
II
hexagonal,
given
The
I/Ri
iv-here
type
H for
are
(Eq
of
logo
function
The
I
frame)
of type I and
T
~S
=
for
lls)
(16)
of type
structures
the
II
Results
3.
The
functions
ax
(i7)
relevant
to
the
phases
different
mentioned
/y(~Jx,cx/y) (Sect
this
in
2.2)
(Sect
work
are
3) are plotted m Figure 2, for
m
ax
strongly
dependent
with the
Note
functions
all
that
the
conspicuous
on ~J,
are
cx
cy
ax/;
(G/D)II (more on this later).
transition
exception of the
lattice
which the
for those
transitions
searched
the
examples of phase
I have
literature
in
often
the
These
and
determined
by
X-ray
dimensions
scattering
experiments.
are
scarce,
are
presented in Tables II, IV, V,
from figures.
Some of the examples are
data
be inferred
must
lacking that the two phases are
evidence is
though, direct
VI. Even in these
sometimes
cases,
the
values of c and a
equilibrium, in these
precisely at thermodynamic
circumstances
were
(see the legend of
Besides, in some
adopted that seemed closest to the phase transition
cases
of the other
II) the cell parameter of one phase was extrapolated to the
concentration
Tab
Section 4.
considered
of these
phase The
approximations
in
are
consequences
involving the phases X and Y (a point in Tabs. II, IV, V, VI the data are
transition
For every
Figure
plotted
The
I.
functions
2
"
the
lattice
cx/cy
The
can
values
assumption
structure
parameters
thus
that
elements
and
computed,
be
of all
ax
these
the
of
avy
and
equation
parameters
are
the
concentrations
(12)
solved
reported
shape of the micelles
phase H, both of type
of
in
and
the
cx
~Jx,
Tables.
phase Q~~~
I and
is
of type II,
and
xmoi,
Note
cy
xpar
The
ratios
and
Smoi
that,
assumed
to
Sect.
be of
lay
and
determined
keeping
in
polyhedral (see
were
ax
with
2.2.4)
the
the
prismatic
JOURNAL
1656
PHYSIQUE
DE
N°11
II
type
typ~
II
3,5
O~/y
o~ ~~
~~~
~
i
~ ~
/
~'~
M/H
_
-
'
'
GIL
~
~
lG/D
~/~
~
'wj
,
_
,
i,5
''
'
G/H
'
1,5
'
~
H/f,
i
o,5
'
o,s
o
o,2
o
o,6
o,4
o,8
o,2
o
o,4
o,8
0,6
Qx
Qx
Fig
The
2.
the
nature
(aG/D)Ii
is
functions
almost
shape (3),
ax/y
m
the
qJ
(12))
the
over
for
cx
(see
transition
cylindrical,
shape
the
were
(Eq
~Jx
us
phases involved
independent of
the
of
The
of
capital
of
pairs
Ii
Fig
Note
letters
the
that
specify
function
0 < ~J < 0 75
range
should
aH(~2)
function
the
cy
legend
#
the
multiplied by
be
the
factor
0.952.
chemical
of hydrocarbon
the
discriminate
of clarity it is
notion
convenient
to
polar headgroups (that applies to the lipid molecules) from the operational notion
2), that applies to the lipid-~vater system and refers to
of polar and apolar
moieties (see Sect
sake
the
For
and
chains
the
of the
content
SySTE&is
3.I.
Table
II.
of the
results
lipids (CI2TAC
of the
Tit
The
I
TYPE
OF
and
elements
structure
and
interstitial
relevant
the
to
CI6TAB)
space.
systems
cationic,
are
of type
one
I
are
(C12K)
presented
anionic,
is
in
the
(PLPC, OLPC, C12E06, C12EOio, DOG and OSG) do not carry net electrical charges
statistical
analysis of the experimental sets ax lay is presented m Table III. In all the phase
A
(G/H)j and (M/H)I (points f to n m Tab. Iii the values of the ratio axlay are
transitions
sG(211)
narrowly clustered; the mean,
close to the epitaxial
relations
is very
moreover,
(G/LiI.
sH(10i (li As for the
aGlaL are
sH(10) and sM(211)
the
ratios
transition
more
sL(I) suggested by
sG(21ii
relation
widely scattered
and fairly
distant
from the epitaxial
others
=
=
=
previous
authors
and x~~~ vary
parameters
~moi
too
scanty to bring out the
The
data
for
with
chemical
the
nature
the
of
the
lipids, whose headgroups have
liquid paraffins (points a, b, f, g, h,
ionizable
fraction
temperature
with
general
are
correlations
the
[3]
that
may
be
as
small
as
0.42
trend
and
of the
~vith
concentration,
phenomenon.
Much
although
clearer
are
the
the
headgroups of the lipid molecules
In all
high affinity for water and a very low affinity
elements
but a
of Tab. II) the
contain
structure
polar
a
of the
hydrocarbon
moiety
This
fraction
increases
(compare points a-f, b-h, g-I), but ~~~~ is always smaller than
the water
increases
content
as
electrical
charges, and its value is strongly
I
net
~p~~ is larger when the lipids do not
carry
paraffins
(compare c,d with e; I,j
correlated
with the affinity of the polar headgroup for the
n).
M/H
of C12EOio (point n), in which the
~vith k; iii with
The
is the
extreme
transition
case
I); this
observation
whole of the lipid
Ii-e- ~moi
molecule
belongs to the
elements
structure
paraffins.
the
relatively
polyoxyethylene
glycol
for
with
high
affinity
of
is
consistent
"
N°11
PHASES,
LIPID
RELATIONS,
EPITAXIAL
SPECIFIC
AREAS
1657
Systems of type I. The lipid (volume)
and cy and the lattice paconcentrations
cx
and ay of two phases X and Yin, or at least close to, thermodynamic equilibrium
taken from the references
In
of the examples was the phase separation properly
none
were
(points g,i,I,m, m italics) the two phases were supposed to have the
explored; m some
cases
and the lattice
concentration,
parameter of the hexagonal phase was extrapolated to the
same
of the cubic phase f3$, ij. Equation (12) was used to determine ~Jxi the paramconcentration
fij
determined
described m Section 2.i.
The references are
eters
as
xmoi,
xpar and Smoi were
for g,1; /2j for c,d,j; /29j for n; /3$j for I,m; Ill) for a, f; /$2j for b,h,,. /$3j for e,k.
Table
II.
rameters
ax
Transition
b
c
d
e
C12K
CI6TAB
OOG
OSG
C12E06
25
22
cG
0 64
0 80
0 75
0 85
0 64
cL
0 70
0 85
0 78
o.93
0 67
78 6
98 7
76 4
74 0
l10 2
43 5
point
a
(1)
(1)
aG
29.0
35 6
29 6
27 8
2 71
2 77
2 58
2.66
2.53
~JG
0 37
o.42
0.36
0 34
o 34
xrr~j
0 58
0 52
0 48
0 40
0 53
Xpar
o.63
0.62
0 80
0 71
1.09
36.7
37,1
35 3
33 6
50.2
aL
aG IL
Srroj
Transition
point
i
j
k
OPLC
OOG
C12E06
g
C12K
CI2TAB
CI6TAB
100
20
70
20
15
22
cG
0.64
0 91
0 80
o 76
0 75
o.64
cH
0 58
0 §1
0 73
0
0 63
0 61
78.6
79 6
98.7
l13
38.1
37.2
48 3
53.$
aG/H
2 06
2 14
2 04
2 12
1.98
2.06
~JG
0.32
0.29
0 29
0 30
0 27
0.32
xrr~i
0 49
0 32
0 36
0 39
0 36
0 49
x~~~
0.53
0 42
0 43
o.66
0.60
1 04
35.0
29.4
32 9
43 1
31.9
48 8
aG
aH
(1)
(1)
76
1
Transition
point
lipid
C)
C12K
m
n
PLPC
C12EOio
20
20
22
CM
o 52
o 48
o.42
cH
0. 52
0
85 4
136 7
]25
39
62
59 0
T
(1)
Ii)
aM
aH
7
$8
7
o 49
0
2 15
2.18
2 12
~JM
0 29
o 32
o 42
xrr~j
0 56
o 67
1.00
x~a~
o 73
19
2 75
47 7
57 6
llo.2
aM /H
Srr~i
76.4
l10.2
38 6
53 6
JOURNAL
1658
Table
III.
Statistical
dard
demation
of the
All
transitions.
the
pooled together.
of the epitamal
are
some
sG(211)/sH(10)
5H(210)/6L(1)
N°11
II
analysis of the lattice parameter
The
ratios.
relevant
ezperimentnl values of the ratios axla;
points of Table II (type I) and those of Tables
n
is
coincidences
2/vi
relevant
this
to
@/2
=
each
G /H
M/H
G/D
2 650
6
2
3
2
0.086
~
~
17
067~0.052
II.
6
247~0
2
relevant
data
101
024
150~0
The
~ 0 373
2 931
18
TYPE
OF
1.572~0.042
the
to
195~0.109
1
of type
systeius
lipids belong to different classes: monoglycerides
(DOPE).
glycolipids (DDGG) and
dioleyl-phosphatidyl-ethanohmine
charges.
II
presented
are
(liOl4,
The
IV to VI
=
type
n
4
SySTE&is
2.450;
1.528,
=
=
=
~
n
5
Tables
stan-
(type II)
of
ratios
VI
spacings
are-
type
transition
3.2.
V,
@
sG(211)/sL(1)
sG(321)/sD(211)
(14/6)
work
2121,
=
IV,
the
different phase
the
The
group.
and
average
to
155.
=1
=
of
of points
number
the
sM(211)/sH(10)
=
PHYSIQUE
DE
bears
None
in
MO18),
MO15,
electrical
net
(G ID )II are
note~vorthy that all the experimental points axlay relevant to the
transition
clustered
result is
~vith the
narrowly
around
157 (Tab. III)
This
mathematical
consistent
properties of the function aG ID us ~JG aG ID is indeed
almost
independent of ~JG over the
relevant
then from ~JG
assumed
0
to be 1 025,
range of ~J (see Fig 2) If,
moreover,
cD /G is
function aG ID barely departs from the
0.9 the
observed
value
57
An
obvious
1
to ~JG
mean
of aG ID being independent of ~J is that in this phase
the cell parameter
transition
consequence
the
ratio
be used to
determine
cannot
parameters ~JG, xmoi, xpar and Smoj.
It is
"
"
The phase diagrams of three
monoglyceride-water systems
MO18): these are plotted in Figure 3 The data relating to
reported
are
and
MO18:
j~~~ is
m). In
to
The
G
close
~~~~
Section 4
m
The
IV
contrast,
discussed
is
Table
in
I
to
phase diagraiu of DDGG
/H, GIL,
the
G
transition
ID
have
G/H land
been
also
close to I and
decreases
with
the
domain of phase G.
into
(with respect
branch
branch,
and
m
the
increasing
with
MO15
transition
at
the
transition
increasing
the
transition
temperaturei
GIL
teiuperature.
GIL
of
In
(see Fig 4)
G/H observed
this
system
an
j~m is close
at lon~er
water
e)
a
phase
reported
are
see
phase L
GIL
and
MO14
in
to
and
d
i
observation
This
three
MO18,
and
ID
similar
(points
and
a
MO14
displays
thus
very
are
temperature
(compare points
G
transitions
GIL
displays several phases (L, H, G, D).
properly explored (Fig. 4) The data
also
Along
h-I) and
happens
larger
to
explored (MO14, MO15,
been
phase
transitions
Table
m
IV)
Tab
V
At
~~~~ is
deeply
descending
ascending
penetrates
ascending and
to I all along
a
the
along the
(compare points b
transition
content
descending branch ~~m is smaller (compare the points of the pairs g-j and
decreases
~vith increasing
with
what
temperature (see the points
to I), in
contrast
and
hi
to
decreases
and
much
is
relevant
results
have
the
at
the
the
transitions
G/H
of
DDGG
and
GIL
of
MO14
and
MO15
(see above).
systems provides another example of phase L penetrating into the domain
phase, H in this case (see Fig. 5). The tr~nsition H/L thus displays an ascending
and a descending
branch.
In this system, and in
with
DDGG
contrast
~~a~ is very close to I
in both the ascending and the descending
branch (compare points g to j of Tab
V and points
The
of
a
DOPE-iv.ater
another
to
d of
Tab.
VI
).
N°11
Table
Figure
MO18,
LIPID
IV
The
3.
PHASES,
RELATIONS,
EPITAXIAL
SPECIFIC
AREAS
1659
Systems of type II, monoglycemdes.
Notation
and symbols as m
Table II. See
references are /39j for MCI$; /38) for MO15; /32j for MO18, 1,1,m; Ill) for
v,w.
GIL
Transition
point
a
d
c
lipid
point
T
f
e
lipid
M014
T
h
g
M015
C
50
60
70
80
60
70
80
cG
O 55
0 59
0 67
0 71
cG
0 59
0 65
0 70
o 73
CL
0 69
0 69
0 74
0 74
CL
0 71
0 74
0 75
0 76
]53
133
]20
]]3
aG
]44
]23
]]]
]07
42
40
38
39
aL
44
4]
40
37
aG/L
3 66
3 36
3 ]3
2 92
aG/L
3 27
2 98
2 76
2 93
~JG
0 40
0 40
0 42
0 46
~JG
0 53
0 58
0 60
0 43
~moj
0 11
0 68
0 63
0 64
xm~j
0 9]
0 89
0 86
0 59
0 98
0 94
0 87
0 88
42
1 39
1 34
0 92
34 8
37 7
37 7
36 1
35 2
35 9
36 9
40
Ii)
Ii)
aG
aL
xpar
Srr~i
~
point
T
~par
Srr~j
i
lipid
C)
m
MO]8
0
20
40
cG
0 60
0 75
0 86
CL
0 76
0 80
0 87
]7]
]30
99
49
47
39
3 47
2 78
2 50
(1)
Ii)
aG
aL
aG/L
~c
o 50
0 58
0 63
'mot
0 83
0 77
0 73
09
1 01
0 96
33 3
33.4
37 3
xpar
Sm~j
Transition
point
n
G
ID
point
q
p
o
aD
T
50
60
70
80
cc
0 54
0 58
0 60
0 65
cc
cD
0 52
0 55
0 59
0 63
CD
]59
145
129
121
97
90
81
75
161
1 59
Ii)
Ii)
64
r
Ii)
Ii)
ac
aD
62
point
w
v
M018
T
ac
aD
40
cc
0 7]
0 76
cD
0 68
0 74
Ii)
134
]]8
Ii)
87
74
1 55
1 59
t
s
lipid
M014
ac
Ii)
(I)
u
MO15
50
60
70
80
0 55
0 59
0 62
0 64
0 54
0 57
0 61
0 63
]57
140
132
126
99
90
83
159
] 56
58
78
] 62
PHYSIQUE
DE
JOURNAL
1660
N°11
II
~
100
°~
9
pi
°
80
j
60
2
)
~°
40
g
20
~ ~
~
o
100
~
80
g
Pn3m
Pi
6o
(
I
40
[
L~
~
20
o
100
Hi
~
80
~
g
Pi
S
Pn3m
+
HID
6o
z
(
I
Wd
40
E
~
~
20
1<
o
lo
0
20
3o
Composifion,
(w/w)
%
60
50
40
70
Water
The phase diagrams of the
monoglycende-water
and
the
3.
systems
monoglycendes
A
monoolem, B
monomyristolein,
C monopentadec~nom
from Figures I and 2 of [38] and Figure 4 of [39]
permission,
Fig
the
4.
Discussion
It is
expedient
4.I.
formula
Reproduced;
of
with
Conclusions
and
to
chemical
focus
POLAR/APOLAR
the
discussion
PARTITION
on
following points.
the
WITHIN
THE
LIPID
MOLECULES
This
paper
derives
thermodynamic equilibrium can be described m
any
elements
that have the same
chemical
composition and the same area/volume
terms of
structure
One
apply
this
phase
proposition
particular
and
the
ratio
determine
to
transi<tion
can
any
(structure elementsi/(lipid molecules) ymoi (or the equivalent
volume
ratio
parameter ~~~~
the
hydrocarbon chains).
analyzed in
relevant
The
data
from
the
literature
have
been
to
from
the
proposition
that
pair of phases in
N°11
PHASES,
LIPID
RELATIONS,
EPITAXIAL
SPECIFIC
AREAS
1661
loo
90
Hn
+
Hio
Hn
Bo
w
___
j
~
f
~
40
LO~
3o
L~
--
2o
.
-
o
H~O
Fig
The
4
phase diagram
of the
Reproduced,
DDGG-water
system
with
permission,
from
Figure
2 of [9]
waters
no
20
16
12
per
DOPE
8
6
lo
2
4
so
-
OOOO
-
.
6O3
.
O
O
-O~
376
510
coo.-
-.
.
o
o
--
30
~~
T°C
O
6~ ~ O
O
.
O
-.
-
O
~
-
OO
lIl
OO
O
O O
-
-'
-
~~
O
O
~~
6680
~
06
O
o
O
o
o
o
O
O
O
O
~j
~i~j
O
*
°
O
O
~
~
o
o
~J~
Q
Q
al
fraction
40 O
.
'
*
*
Z
O
O
~
~
d
~~
d
d
09
08
wt
.
iO
DOPE
numbers
The
and
by DOPE as a function of water
content
temperature
single
lamellar
phase
of
the
The
shaded
block
indicates
hexagonal
region
spacings
give
Diamonds
cubic phase (Q)
coexisting
phase L Crossed
Full circles
phase H Open squares
squares
from Figure 3
Reproduced, with
Triangles
Q and H phases
coexisting Q and H phases
permission,
Fig
5
Phases
representative
of [33]
formed
JOURNAL
1662
Table
Systems of type
V.
DDGG.
II-
PHYSIQUE
DE
Notation
N°11
II
symbols
and
as
Table
m
f9).
From
II
See
Figure
G/H
Transition
b
a
45
50
55
60
65
70
cG
0.90
0 90
0.90
0.89
0 87
0.86
cH
0 94
0.93
0.91
0 91
0.90
0.90
97 0
95.0
89 1
86.6
T
(ii
Ii)
aG
aH
91.8
95 6
40 4
40 6
413
41 7
41 2
41 8
2.27
2.35
2 35
2.28
2.16
2.07
aG /~
~JG
O 66
0.74
0.78
0.72
0.61
0 50
ymoj
0 73
0 82
0 87
0 81
0.70
0 58
y~~~
0.89
1 00
1 07
0.99
0.86
0.71
57 2
49.5
46 6
53 3
64.5
72.4
Smoj
GIL
Transition
h
I
45
50
50
cG
0.83
0 83
0 75
0 74
cL
0.81
0.81
0 77
98.6
98 1
125 0
43 8
43.0
45 2
Ii)
Ii)
aG
aL
k
35
scanty
and
to
xmoi
0
1~45
.3
07
2.28
2 77
2 76
2.94
~JG
0.70
0 52
0 53
0 39
2
~moj
0 86
0.85
0.69
0 72
0 54
32
1.05
1 04
0 85
0 88
0.66
54 8
56 2
56
55 7
55.9
(i~
4
G/D
o
p
q
r
s
t
45
50
55
60
65
70
cG
0.66
0.66
0 68
0.70
0.70
0.72
0 72
cD
0.65
0.65
0 66
0.69
0 70
0 71
0 71
146.8
150 2
143 0
128 0
125.3
124.0
123.0
97 9
93 0
90 0
83 5
83 0
82 7
79 2
1.50
1.61
1 60
1 53
1 51
1.50
1.55
Iii
Ii
Actually,
the
that
bring
with
46.8
n
some
errors
reported
clear
46.0
40
aD
deviation
0.70
137.5
m
aG
It is
0.73
127.0
35
T
3.
0 76
2.25
Transition
Section
0.70
0.71
x~~~
equilibrium;
0 71
aG /~
Smoj
point
f
e
c
in
due
Table
of
these
data
to
this
approximation
III
(approximately
and ~~~~ vary with
trend of the
out> the
~m~j
the
relative
affinity
of
refer
to
pairs of phases
can
0 04 for
concentration
to,
be
assessed
the
ratio
and
~luch
phenomenon.
the polar headgroups
by
the
quite
not
at
standard
axlay).
temperature,
clearer
for
but
smallest
water
is
yet,
the
and
the
data
correlation
paraffins.
are
too
of x~~~
The most
N°11
LIPID
Table
PHASES,
Systems of type II:
Rand.
See Figure
VI.
RELATIONS,
EPITAXIAL
DOPE.
promded by R-P-
Notation
b
c
22.5
22.5
15
cH
0.76
0.80
0.86
0 88
cL
0 81
0.82
0 84
0 86
67.8
60.8
53 1
51.9
50.5
48.4
48 4
48.2
cH/L
0.94
0 98
1.02
03
9JH
0.60
0 62
0.69
0 69
~moj
0 78
0 77
0.80
0 79
l 00
0 99
1.03
02
46 5
48 4
47.0
47 0
(I)
Ii)
aL
~~~~
Sn~oj (i~
and
least
drocarbon
in
(that
chains
headgroups
paraffin-soluble
lipids the
these
may
be
1663
Table
m
II.
kindly
Data
H/L
a
C)
aH
CI6TAB):
as
15
point
T
water-soluble
symbols
and
AREAS
5.
Transition
CI2TAC,
SPECIFIC
small
as
as
0.42,
of
those
are
elements
structure
point
g
but
Tab
II),
m
lipids (C12K,
of the hy-
ionizable
the
contain
fraction
a
the
of
rest
the
chain
belonging to the polar moiety. At the other extreme, the CnEom's display bulky headgroups
highly miscible with water but also somewhat
miscible
with paraffins- accordingly, y~~~ is alincreases
with the size of the
polyoxyethylene moiety (from x~~~ m I
ways larger than I and
fraction of the polyalcohol
at point e to ymoi m I at point n of Tab. II). Apparently, a sizable
behaves in this case as if it belonged to the apolar moiety., along with the hydrocarbon
chains.
The headgroups of the other lipids
phosphatidylethanolamine,
phosphatidyl choline
sugars,
display
intermediate
miscibilities; accordingly, the
elements
structure
contain
most of the
hydrocarbon chains (points a to d in Tab. VI), sometimes
accompanied by a small fraction of
These
the headgroups.
results
clearly illustrated by the sequences of points a to e, f to k
are
and
to
m
of Table
II
puzzling phenomenon
display similar properties
A
increases
with
local
in
increasing
polarity
the
case
is
the
in
is close
-~~~~
of MO15
temperature
within
observed
y~~~
(,~~m
hydrocarbon
=
monoglycerides (Tab. IV).
to I at
room
temperature
larger than
0 92 at 90 °C). The
moiety; this polarity,
is
much
Whereas
and
(142)
I
50
at
of ~~~~
other
many
value
like
MO14
decreases
and
temperature
as
°C,
MO18
and
decreases
to
seems
mirror
a
properties of fatty
precisely, the
More
atoms
[14] depends upon the parity of the number of carbon
of the phenomenon
temperature-dependence
suggests that in MO15 the double bond is more
strongly oriented with respect to the interface than in the two other monoglycerides (see Fig
3), and that the preferential
decreases
increases.
orientation
temperature
as
In two of the systems the
deeply into the domain of another
domain
of phase L penetrates
phase. G in DDGG (Fig. 4), H in DOPE (Fig 5). In the case of DOPE, Xpar is very close to
(points a to d of Tab
branches of the
transition
I in both the ascending and the descending
VI). In contrast, m the case of DDGG ~~~~ is close to I m the ascending branch (as well as at
molecules
the
transition
G/H,
at
lower
water
content,
see
points
a
descending branch (compare points g, h and i to k
in
correlation
of this
phenomenon
the possible
to bring out
lipids
the
to
d in
of Tab.
with
Tab.
V)
the
V)
The
chemical
and
smaller
data
are
than
I
too
scanty
properties
of the
JOURNAL
1664
All
these
results
and
that
indicate
related
parameter,
tant
polarlapolar
the
PHYSIQUE
N°11
II
within
partition
the
lipid
properties of the lipid and
chemical
the
to
DE
molecules
variable
is
with
an
impor-
temperature
concentration.
ARE
4 2
PAIRS
SHOULD
WHY
strated
t~vo
PHASES
OF
THEY
BE
cases,
involve
Clerc
epitaxially
phases
have
drawn
clear
RELATED?
EPITAXIALLY
Straightforward Crystal grOWtl1
phase
transitions
some
Rangon and
moreover,
that
EQUII7BRIUM
THERMODYNAMIC
IN
SO?
experiments
related
each
to
correlations
demon-
l1aVe
[2-5].
other
between
In
epitaxial
pathways
the
and the
of the phases, and have proposed elegant
geometric
structure
other [3,4]. It is also
noteu<rthy that in all the cuwhereby the phases tTansform
into the
one
bic phases
mentioned
coincidences
involve
reciprocal lattice
in the Tables the epitaxial
to
seem
points of the plane normal to the [I I I] direction and containing the origin ii-e the indices fulfill
0i This coincidence, like those
of
the rule h ~ k ~ l
discussed
in [I], suggests the
presence
structural
relationships that allow the two phases to interlock and to grow one at the expenses
coincidences
=
of
the
other
For
the
of the
presence
(G ID
transition
correspond
)iI
the
the
(M/H)I
transitions
epitaxial
observed
epitaxial
ratios
and
(G/H)I the
sH(10)
sM(211)
coincidences
=
la; are narroiv.ly
ax
sG(321)
sD(211
IL )iI
for
(G
more
so
data
and
(see Tabs. II to VI)
sG(211i
6H(10) [1]
=
clustered
around
a
value
that
confirm
For
the
could
phase
transitions
experimental
ratios
even
more
are
scattered
and in
sharply at
~vith all the simplest
epitaxial
relations
some
cases
are
variance
(Tab III). The rule that two phases in thermodynamic equilibrium are epitaxially related to
each other thus
exceptions
to have
seems
many
futile
It
seek
the
of
the
epitaxial
coincidences
thermodynamic equilibrium.
to
seems
causes
in
difficult
It is indeed
comprehend how two phases stable in different regions of the phase
to
diagram may influence one another just because the two are doomed to be epitaxially related
still
another
conclusion is thus difficult to escape
that either
region of the phase diagram. The
in
epitaxial relationships are fortuitous
interestingly,
that
their
kinetic
events
or,
more
causes
are
Presumably, the presence of epitaxial
coincidences
has the effect of lowering the kinetic
barrier
of phase
and thus of avoiding
metastable
If this is the case, then the
transitions
states
existence
of epitaxial
relations
advantage (or disadvantage) on a particular
may well confer a selective
lipid-water system.
The
selection
could be based
technological
detergent, for
criteriaon
a
example, is expected to undergo fast phase transitions in water (washing one's hands requires
the
several phase transitions, all within seconds). As for biological
soap-water
system to cross
lipids the kinetic aspects of some phase
advantage: phase
transitions
may also entail a selective
taking place in some lipid-rich organelles could be critically
transitions
in living
important
(more on this below). Finally, lipids displaying sluggish phase transitions
organisms
are
prone
problems and are thus doomed to be thrust aside in ordinary laboratory practice, at
to raise
the
to
coincidence
(G IL )I, (G /H)II, (H IL )Ii,
least
I
can
in
use
those
the
a
absence
few
of special
examples
that
transitions
to
arid
As
to
the
other
the
motivations.
illustrate
involve
=
phases
these
suggestions
containing
It is
different
advisable
amounts
to
of
avoid, for this
water,
since
the
purpose,
kinetic
phenomena may then
the long-range diffusion of water
of the
rather
than the
time-course
mirror
phase
transitions
[15-17] One example is the extremely sluggish
temperature-induced
DII IL
(the relaxation
phase
several weeks)
observed
from
transition
in lipid
times
extracts
may be
(nd the low-temperature
S
solfataricus [18j. The fact that the lattice
parameters of the highphases vary with the water
of the system (see Fig
that at all
llc of [19]) suggests
content
of the system
the
phases
the
of ~vater.
water
content
As we have
two
contain
amount
same
conjectured [18], metastability
confer
this
the ability to
at low
may
on
organism
preserve
temperatures
the
cells
from
the
native
thermal
high-temperature
structure
(S. sulfataricus
fluctuations
of
the
grows
membranes
naturally
and
at
85
mav
°C
but
thus
protect
its
natural
N°11
PHASES,
LIPID
exposed
RELATIONS,
EPITAXIAL
SPECIFIC
AREAS
1665
fluctuations).
It is noteworthy
that the value of the
approximately
1.5
1.8
is quite different
[18],
to
aD
(Tab. III).
from any of the values corresponding to the epitaxial
coincidences
Other
examples can be found in gangliosides, a widespread class of biological glycolipids
with unusually bulky polar headgroups [20]. The phase diagrams of ganglioside-water systems
One of the
features
complex and in some
unusual.
is the
respect
uncommon
presence
are
embedded
micellar
cubic phases formed by disjointed
micelles of type I
of several
water
m
a
of
Another
of these
features is the widespread
metastable
continuum.
states, to
occurrence
identification
that the
of the phases is a serious
problem m some of these systems
the
extent
the phases of space
Im3m and
For the sake of the argument, let me
consider
Fm3m of
groups
observed
the system GMI(ac )-water (experiments E and F in [20] ). These phases have been
micelles,
and
almost
the
The
that
the
concentration.
contain
temperature
at
at
same
same
m
number of lipid molecules,
packed in the body-centred mode m
the two phases the
same
are
mode m phase Q~~~. The
lattice
ratio is
phase Q~~~, in the face-centred
27, quite
parameter
low-index
coincidences
of Table III. In this
different
from any of the
epitaxial
also the
case
extraordinary metastability coincides with the absence of an epitaxial relationship. Less clear
transitions
of the selective advantage, biological or otherwise, that sluggish phase
is the
nature
habitat
is
ratio
laL,
that
confer
could
on
temperature
with
from
concentration
gangliosides.
CUBIC
BICONTINUOUS
4 3
wide
to
varies
SURFACE
PHASES.
CURVATURE
THE
OF
STRUCTURE
ELEMENTS.
generally evoked in lipid-water systems. Two
Three types
are
work:
associated
with
the gyroid (G) IPMS, and Q~~~, associated
considered
this
in
Q~~°,
are
been firmly
established
of
these
diamond (D) IPMS.
The
with the
structure
two phases has
freeze-fracture
electron
by a variety of techniques: X-ray scattering [21],
microscopy [22, 23j,
self-diffusion
fluorescence
[25, 26]. Also, examples of phase Q~~° have been
[24] and NMR
reported of both types I and II, whereas all the known examples of phase Q~~~ belong to type
Im3m (Q~~~) and is
bicontinuous
cubic phase belongs to space
II [12]. The third type of
group
Im3m
with the primitive (P) IPMS.
Several examples of cubic phases of space
associated
group
bicontinuous
have been reported in the
literature; these have generally been assumed to be
although the hypothesis lacks so far firm experimental support [12]. Furthermore, two recent
of
should
observations
a
bicontinuous
be put
system;
ganglioside-water
phases
cubic
record.
on
this
phase
The
first is
has
recently
bears
of type
observation
[20j. The second
the system monoolem-(cytochrome C)-water,
originally
interpreted
in
of
terms
a
P-type
relevant
been
the
on
whose
cubic
phase of
the
to
shown
to
phase
consists
reported
freeze-fracture
bicontinuous
electron
[27]
structure
group
space
disjointed
of
a
few
years
of
Im3m
micelles
in
ago
micrographs were
micrographs,
Those
light of the recent developments of image-filtering techniques m freezenow
electron
microscopy [20,22,23,28] show that the structure is most likely micellar (Gulik
fracture
Therefore, the very
in lipid-water
systems of
existence
& Delacroix, personal communication).
questionable
of the P-type is at present
bicontmuous
structures
of
surface
forward in this work the
curvatures
of the proposition put
the
constraints
Under
of the
with the frequency of
correlations
display remarkable
elements
the
occurrence
structure
Gaussian
and
the
Compare for this
different
mean
types of cubic phases.
average
purpose
re-evaluated
of
curvature
phase
most
that
same
surface
frequently
Gaussian
average
are
the
the
phases (G, D, P)
cubic
~
at
given
in
and
equations
any point ~J the
S/V ratio, increase
at
of
the
mean
(15,
in the
to
the
values
same
to
convenient
use
to
three
the
virtual
bicontinuous
Since G is
of ~J and of S/V
the
ratios ~x/~G and ~1xlllG of
the
the
expressions of ~ and
plotted in Figure 6. Note, on the one hand,
ratios
are
elements, supposed to display the
of the
structure
curvatures
order G ~ D ~ P. On the other hand phase G (both of types
curvatures,
16). The
surface
relevant
elements
structure
corresponding
observed, it is
where
X
stands
for D
or
P
The
JOURNAL
1666
II
X=G
_T
Yx/YG
PHYSIQUE
DE
~
o,9
N°11
X=G
i
Px/~G
~
-
~X=D
'
,
~
o,9
,
'
X"?
'
~ ~
I
,
'X=D
'
~ ~
I
0,7
'
0,7
I
o 6
o 6
'
'
'
o,s
Surface
Fig 6
corresponding to
Gaussian
~x/~c
mean
of
type I,
II)
and
existence
surface
the
phase
in
P
The
OF
and
molecules
the
the
paraffins,
must
be
closer
In
a
the
It would, of
tion
are
than
I
(see
element
is
over
an
same
The
increase
approach
polar and an
a
is
of the
Sect.
for
G,
the
are
D
P
or
the
=
I
the
G
order
average
T
represent
curves
phases
cubic
~x
D
~
~J
ratios
P
~
apolar
elements)
relax
is to
to
the
contain
a
that
constraint
dimensions
apparent
impossible
be
course,
excellent
agreement
to
the
prove
II, IV, V, VI), so that the
eluded.
Second, the experimental
two
range
(see
of the
bicontinuous
elements
with
problem
of the
that
the
same
validity
3
ax
lay
phases
fairly
in
values
in
observa-
of xmoj
is
apolar
the
the
chains.
several
(G ID )Ii
transition
the
case,
larger
structure
observed
in
different
thickness
curvature
are
of
relevant
Third,
partial
surface
of
none
presence
2).
hydrocarbon
discrepancies
of the
Yet,
of the
First,
of
the
and
elements
structure
of the
disturbing
to
elements
points
partition of the
operational
an
of allowing the apolar
supposedly miscible
provide
extended
suffice
validity
Sect.
cubic
a
virtue
points of the
fully
the
awkward
[30j. Fourth, the
of
notion
of the
of two
lamellar
structures
of the phase diagram, the
structures
some
1.57
end
all the
proposition.
Tabs.
around
the
would
structure
the
the
to
proposition
the
fraction
than
of the
with
left
moiety, its main
An interesting
coefficient.
partition
based
value
introduce
to
work is
the
lipid
of the
lamellae
elements
frequently
analysis, and in
more
4.3). Finally, the whole of the results of the
partition coefficient jmoj are in excellent
agreement with the
properties of the systems. These points are discussed in this and in
(see
and
m
apolar
of
I have
ratio.
polar lapolar interface
good many cases [20, 21, 29] this
explain why
~(
stands
curvatures
This
the
to
extended
in the
two
consist
heuristic
of its
of the
narrowly clustered
system DDAB-water,
are
the
X
(15, 16))
o,8
bicontinuous
virtual
S/V
of
elements
structure
PROPOSITION.
area/volume
all
the
to
in
the
type I (see Eqs
that
three
value
of
equilibrium
least
at
structure
with
between
the
of
same
o,6
elements.
BASIC
of this
moiety (I.e. the
chains.
the
to
of
Note
T
structure
same
or
into
determination
and
o,4
is
THE
novelty
main
lipid
phases
thermodynamic
in
proposition,
of the
curvatures
of the
VALIDITY
phases
composition
elements
structure
of ~J
of
o,2
most
curvature
4.4.
o,s
frequent, phase D (but only of type II) is fairly
and the very
common
framework
of the
problematic.
Therefore, and within the
proposition
with the
this work, the free energy of the lipid-water
increase
to
systems
seems
is
two
T
I
o
the
=
~J for
functions
as
of
forward
put
of
i
I
O,8
value
same
surface
~x/~lc
and
O,6
curvature
the
and
phases
for
O,4
I
~"~
i
O,2
j
'
,
observed
particular
chemical
previous
and
than
the
the
Sections.
is
the
seems
others
values
physical
N°11
PHASES,
LIPID
RELATIONS,
EPITAXIAL
SPECIFIC
AREAS
1667
proposition are closely related to the neutral (or pivotal) surfaces
refer to
elements
with
two types of surfaces
structure
constant
sense
S/V ratio. The difference is that our proposition applies to pairs of phases in thermodynamic
equilibrium, and thus to singular points of the phase diagrams, whereas the notion of neutral
The
surfaces
[31, 32],
involved
that
applies
surface
in
the
in
to
one
the
the
phase throughout
its
of
range
The
existence.
of
existence
very
neutral
a
form on the c-dependence of the lattice
mathematical
In
parameter.
a precise
(phases
HII
of
DOPE
of
few
and
MO18
the
(Q~~°)II
experimental
points
[32])
[33]
a
cases
a
vs.
have been
shown to be in fair agreement
with the law that follows the
of a neutral
existence
c
surface; in others (phases HI of CI2TAC and PLPC [34] the data are grossly at variance with
al should be constant) (result not shown).
this
assumption (in this case
In more
explicit thermodynamic terms our proposition is equivalent to reducing the free energy
imposes
surface
of the
system
to
an
interfacial
located
energy
at
the
surface
of the
Again,
elements
structure
radically different from, nor more arbitrary than reducing
is
deformation
of a virtual
elastic
plate, infinitely thin
to an
energy of bending
surface [8,9]
Our proposition,
the
neutral
is
meant
to apply to all
moreover,
phases with the hydrocarbon chains m the disordered (a) conformation, and is
surface approach, to the lamellar, hexagonal and
like the neutral
bicontinuous
approach
the
not
the
free
energy
centred
on
and
lipid-water
restricted,
cubic phases of
the
not
type II
Acknowledgments
thank
Drs
grateful
Caffrey
Hyde
M
Drs
to
S.
and
P.
and
P.
kindly
for
Rand
Rand
for
critical
a
unpublished
providing
reading
of the
material.
manuscript
I
also
am
helpful
for
and
suggestions.
Appendix
Against
Arguments
Disjointed
Micelles
Some
The
structure
rods
(length
of the
the
hexagonal
Proposition
phase of type
diameter) packed in the
partial specific volumes
»
2-D
that
Hexagonal
the
I is usually
described
m
hexagonal mode. Knowing
Consists
Phase
terms
of
long
of
stiff
and
concentration,
lattice
determine
Experiments
the radius of the rods.
one
can
performed on a variety of lipids show that the radius R~~~ of the circular cylinders ideally
occupied by the hydrocarbon chains is always shorter than the fully extended length of the
chains, and that the law R~~~ vs. c varies from lipid to lipid [35]
and
coworkers
Amaral
has recently been challenged by Mariani,
This
[1, 6]. These
structure
of rod-like
micelles of finite
forward
that phase HI
have put
authors
the proposition
consists
length whose radius is equal to the fully extended length of the chains, separated along their
fraction of the
the
volume
length by gaps filled by water.
In order to fit the (a vs. c) data
parameter
gaps
was
proposition
and
allowed
to
the
to
vary
with
immediate
c
It
should
be
stressed
that
those
transition
vicinity of a phase
that they explicitly applied it
authors
(where
the
did
not
their
restrict
fluctuations
are
large
whole
concentration
eventually diverge) but
to the
range of
I find this proposal unsatisfactory.
phase For several
reasons
electron
in a
observed by
freeze-fracture
First, hexagonal phases have been often
microscopy
variety of lipid-water systems (see, for example [36]). The micrographs always display very
structure.
long and fairly stiff rods and none tells in favour of a micellar
originate from the postulate that the radius of the rod-like
the proposal
Second,
to
seems
length of the chains [1,6]. This postulate is
micelles is precisely equal to the fully extended
and
the
IOURNAL DE
FBYSIQW
D
T. 5, N° it,
NOVEMBER
lW5
©
JOURNAL
1668
astonishing
phases (and
lipid-water
all
In
PHYSIQUE
DE
(in
order
in
II
N°11
voids)
of
presence
is indeed
the
avert
to
of the
one
radius)
expected not to
structure
case
unambiguously
exceed the fulllength ofthe
chains.
When,
this
dimension
measured
moreover,
is
(for example, the
of the
half-thickness
lamellae in phases L, (Q~~°)II and (Q~~~)II) the length
invariably turns out to be substantially shorter than the chain length. This important result,
and also the fact that
the
dimension
question expands with decreasing
temperature,
is
a
m
Why
of
the
disordered
conformation
of
the
chains
should
phase
HI behave
[35j.
consequence
differently?
Third, from the crystallographic viewpoint the intensity
scattered
by such a micellar
structure
happens
when
infinitely
restricted
the
plane
normal
the
rods,
the
rods
not
to
to
it
as
are
is
long. To the best of my knowledge, no such out-of-plane scattering has ever been reported, at
considered
least in 2-component
lipid-water systems like those
[1,6]
m
Finally, the reflections
contained in the reciprocal plane normal to the hexagonal axis
correWhen
infinitely long,
spond to the projection of the
that plane.
the rods
structure
on
are
and
coincide (in real as well as in reciprocal space)
If on the
sections
projections
contrary
the
micellar,
is
structure
histograms
the
elements
of the
dimensions
of the
3-D
Amaral
projections and sections differ from each other. In this case the
density map and of its 2-D projection no longer coincide and
approach advocated by Luzzati and coworkers [34,37], and amply used
then
electron
recognition
pattern
by Mariani,
rod
the
this
[lj
coworkers
and
loses
its
justification.
main
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