Thermally induced budding of phospholipid vesicles - a
discontinuous process
J. Käs, E. Sackmann, R. Podgornik, S. Svetina, B. Žekš
To cite this version:
J. Käs, E. Sackmann, R. Podgornik, S. Svetina, B. Žekš. Thermally induced budding of
phospholipid vesicles - a discontinuous process. Journal de Physique II, EDP Sciences, 1993, 3
(5), pp.631-645. <10.1051/jp2:1993157>. <jpa-00247860>
HAL Id: jpa-00247860
https://hal.archives-ouvertes.fr/jpa-00247860
Submitted on 1 Jan 1993
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Phys.
J.
France
II
3
(1993)
631-645
1993,
MAY
PAGE
631
Classification
Physics
Abstracts
82.70
87.20
68.00
Thermally
induced
budding
discontinuous
process
a
J.
('),
Kds
E.
('),
Sackmann
R.
phospholipid
of
vesicles
-
Podgomik (2),
(') Physik Department (Biophysik E22),
Technische
D-8046 Garching,
Germany
Physics (F-I), J.
(2) Department of Theorefiical
Svetina
(2, 3) and
Universit§t
Miinchen,
S.
Strasse,
James-Franck
Jamova39,
Insfiitute,
Stefan
B. 2ek§ (2, 3)
Ljubljana,
61ii I
Slovenia
(3)
Institute
Biophysics,
of
Ljubljana,
Medical
Faculty,
University
6July
(Received
1992,
The
AbstracL-
increasing
of
process
temperature
l0February
revised
a
pear
therrnally
shaped
accepted
1993,
of the
elastic
empirically
shown
61105
energy of the
introduced.
responsible
be
to
bilayer
The
for
van
budding is studied for
changes discontinuously
of the cooling process
course
in the
widening
repeated
of
bare
the
of the
difference
mother
narrowing
and
attains
pair
a
bilayer couple model of
heating process the expansion
the
area
between
a
With
spherical
hysteresis in a
thermal
by repeated widening
the pear shape again.
into
diminishes
vesicle
vesicles.
DMPC
induced
vesicle
up to the third order
forces
der Waals
the
1993)
February
ii
vesicles
connected
by a narrow
neck. In the
observed.
The size of the daughter
vesicle
stepwise
is
manner
and narrowing of the neck until it finally remains
and
the
open
within
These
experimental findings are difficult to rationalize
phospholipid vesicles, To model the
discontinuous
behavior in
is
Lip16eva2,
Ljubljana,
of
Slovenia
and
of
monolayers
two
daughter
the
neck
vesicles
the
in
are
cooling
process.
1.
Introduction.
It is
well
of the
known
Both
energy.
cannot
vesicle
be
the
that
bending
elastic
volume
is
closed
a
of its
energy
that the
assume
changed
shape of
and
that
constant.
membrane
also
the
The
phospholipid
membrane
is
volume
[1-61.
vesicle
There
are
is
two
models
laterally incompressible and
compressibility is negligible
spontaneous
curvature
model
by
determined
[1, 21 does
for
minimization
appropriate
the
therefore
and
not
its
therefore
take
area
the
explicitly
phospholipid
which
membrane,
is
composed of two
monolayers, but implicitly
for an
asymmetric composition of the two monolayers,
allows
which
leads to a
stretching of the two monolayers, I.e.
The
relative
spontaneous
curvature.
of the two
is allowed
such changes of the
monolayers which
the
areas
conserve
average
area,
within
this
model
and
On the
hand, the bilayer couple model [3-61
other
costs
energy.
no
symmetric
membrane
therefore
but explicitly takes
and
spontaneous
curvature,
assumes
a
zero
into
into
account
account
the
the
structure
membrane
of
the
structure
by requiring
constant
areas
of the
two
monolayers
[7, 8],
DE
PHYSIQUE
area
but
JOURNAL
632
which
models
two
of
curvature
equal
to
two
well
as
minimizes
as
model
bending
the
monolayer
constant
difference.
area
spontaneous
of
the
bilayer couple
the
also
model
which
includes
cases
a general
[9-1Il.
the bilayer couple
concept
the
is symmetric,
membrane
spontaneous
limiting
vesicles
component
zero
therefore
are
and
membrane
constant
the
membrane
the
single
For
only
not
means
These
N° 5
II
convenient
to be
seems
to
is
curvature
shape.
their
describe
One
energy
~b")~cl(Cl+C2)~".
bending elastic
integration
membrane
Here k~ is the
the
membrane
and
of
is to be carried out at
difference
between
and
Minimization
surface
A
goes
values
fixed
the
the
over
vesicle
of
of
areas
ci and c~ are the principal
neutral
whole
of the
area
while
constant,
the
the
(~)
volume
V,
(AA).
invariant,
energy (Eq. (I)) is scale
appropriate
shape it does not depend on the vesicle size. It is therefore
equals unity, I.e. in units
membrane
length in such a way that the
area
A
membrane
sphere (R~) with the
area
expression
The
bending
membrane
for the
for
I.e.
of
the
neutral
given
a
unit
the
choose
to
surface.
bilayer
of the
area
monolayers
two
curvatures
radius
of
of
the
j
~
R~
It is
convenient
also
define
to
volume
4
v
~
and
the
difference
relative
the
of
areas
while
the
the
the
area
a
I.
=
(4)
arhR~
,
equation (4)
In
=
8
A~
AAs
monolayers
two
h is the
distance
between
the
surfaces
neutral
of
monolayers.
two
It is
relative
=
~~~
3
~
ha
arR(
~~
V~'
between
(2)
~
=
relative
the
2
clear
that
the
bending
membrane
relative
energy
to
of the
result
and
constant
the
minimization
bending
it
also
is
energy
of
the
procedure
appropriate
sphere
does
to
not
depend
measure
the
on
value
the
of the
bending
membrane
W~
Wb
It
be
can
seen
relative
the
The
gives
values
volume
above
all
of
that
the
v
equilibrium shapes
relative
and
minimization
possible
shapes
relative
volume
and
~
the
difference
(5)
~
corresponding
ha
energies (w~) depend only
on
[61.
procedure is usually performed for axially
bending
the
corresponding
membrane
and
relative
difference
monolayer area
ha.
introduce
classes
of
A
class
of
shapes.
to
and
v
area
"
symmetric
energies for
A
more
vesicles
any
compact
and
set
of
form
shapes contains all the
presenting the results is
shapes which can be obtained in a continuous way if v and ha are continuously changed, Each
class of shapes exists only within a certain
which is limited by
part of the v/ha phase diagram,
limiting shapes.
It has
expansion of the monolayers
been
recently shown [121 that because of the thermal
of
THERMALLY
N° 5
INDUCED
BUDDING
PHOSPHOLIPID
OF
633
VESICLES
changes lead to sequences of shapes which can be described fairly well by the
several
bilayer couple model.
Depending on the pretreatment of the initial spherical vesicle,
main
three
increasing
[131.
The
of
obtained
with
shape
be
temperature
types
sequences
can
possibilities for the vesicles of dimyristoyl phosphatidylcholine (DMPC) are :
temperature
budding
o
neck
narrow
leading
transitions
to
a
spherical
the
to
mother
formation
of
a
spherical daughter
attached
vesicle
by
a
vesicle
transitions
leading to
discocyte-stomatocyte
.
attached
spherical daughter vesicle
by a
narrow
dumbbell-pear-dumbbell
transitions.
reentrant
.
inside
an
neck
to
budded
vesicle
spherical
the
and
mother
to
intemal
an
vesicle
found [131 that in the first
the
transitions
final
the
shapes are
two
to
cases
By increasing the temperature continuously, the shape changes continuously up
without
further
the shape
becomes
unstable
and
where
temperature change
temperature
to a
transforms
relatively fast into the final mother-daughter shape. In the cooling cycle the shape
vesicle
diminishes
does not
follow
of changes but the size of the daughter
the
same
sequence
and the
vesicle
by repeated opening and closing of the neck, until it finally remains
open
initial
attains
the
shape again.
discontinuous
Namely, it
Such
transitions
be
described by the bilayer couple model.
cannot
obtains in the simplest case
changes of the
be easily
shown [12, 131 that one
temperature
can
It
has
been
discontinuous.
relative
p is the
Here
to
volume
monotonous
a
shape changes,
the
intermediate
v
thermal
relative
the
and
area
expansivity.
change of
because
pear
and 3
the
Therefore,
control
the
parameters
shape (I.e.
final
shapes.
difference
area
The
variation
ha
as
v
and
mother-daughter
of the
change of
monotonous
a
volume
ha
and
should
pair) belongs
with
temperature
temperature
lead
to
the
can
leads
continuous
to
same
be
class
as
neglected.
2
shall
analysis of the budding
transition
for DMPC
present a detailed
we
together with the shape changes in the cooling run. In section 4 we shall first show that
bilayer couple model
the
describe
discontinuous
transition
the
the
observed
and
cannot
corresponding hysteresis. We shall then extend the bilayer couple model by allowing relative
stretching of the two monolayers. Also this extended
version
of the bilayer couple model,
which
incorporates
features
of the
model,
for the
spontaneous
curvature
cannot
account
some
discontinuous
observed
behavior.
It is the
inclusion
of the third
order
in the monolayer
terms
difference
into
the
elastic
that
conform
makes
the
predictions to
with
the
area
energy
experiment for the heating run. Repeated opening and closing of the neck in the course of the
cooling run can be explained by invoking so far [141 unrecognized
contribution
to the
energy
of a budding
vesicle.
It has its origin in the van
der Waals
interactions
between
the spherical
vesicles.
mother
and
daughter
The
estimated
magnitude of these
interactions,
which
are
comparable to the
membrane
stretching energies,
shows
that they
should
be
included
into
shape equilibrium
considerations
for a budding
vesicle.
In
sections
vesicles
2.
Experimental
methods.
experiments are performed
(Birmingham, AL) in
DMPC
(dimyristoyl
phosphatidylcholine)
obtained
from
purified
by
Millipore-system.
A
detailed
description
of
water
a
the
method to obtain giant vesicles is already given in
reference
[131. The shape transitions
are
induced
by temperature changes which
in the
of the
vesicle at
constant
cause
a change
area
volume.
vesicles
The
observed
in a measuring
chamber
which is nearly free of thermal
were
convection
and
which is already
described
in
reference
[131. Each experiment consists of two
All
Avanti
with
634
JOURNAL
DE
PHYSIQUE
N° 5
II
heating and cooling
The
first
heating and cooling
performed
separate
runs.
run
was
continuously at a rate of about
0.01°C/s.
In this run the principal path of shape
transitions
occurring during the heating and recooling of the vesicle was
observed
and
with a
recorded
Sony U-matic
videorecorder.
In the second cycle the stability of the shapes was
determined
by
first heating and then cooling in steps of 0. I °C. After each step the vesicle
observed
for
was
check
15 min to
whether
the shape is stable.
In order to control
changes in surface area and volume
the area and the
volume of
constancy,
vesicle
measured
image
processing.
For
the
by
this
used
real
time
frame
were
purpose
we
a
grabber (Pixel Pipeline, Perceptics,
Knoxville, TE, USA) in a
Macintosh
IIfx. The image
processing software is based on the public domain program Image. To evaluate the volume and
the
the
area,
coordinates
of the
line
contour
6f the
vesicle
first
determined
were
and
then
the
shape. To determine the main axes one first
determines
the
center
axes
of gravity of the
vesicle
shape and then chooses the largest
diameter
the
first
axis.
The
as
second
axis is
chosen
perpendicularly to the first. The traced
line and the two
contour
axes
then
displayed on the video
During the experiment one
the focal plane
were
screen.
moves
through the shape thus deciding which of the two axes is to be the rotational axis of the vesicle.
Perpendicular to the rotational axis, the vesicle was divided into slices of one-pixel thickness.
Adding up the
volumes
of the
slices
yielded the vesicle
volume
and
summing up their
of the
main
two
circumferences
with
area
the
vesicle
with
6.5
3.
an
the
x
surface
area.
The
of 3 fG. The
accuracy
volume
is
temperature
The
temperature.
10~~
vesicle
thermal
volume
was
measurements
independent,
area
measured
show
while
expansivity
with
an
acburacy
that
within
the
membrane
coefficient
increases
area
p
of 4.5 fG and
experimental
the
~~
=
A dT
is
the
accuracy
linearly
equal
to
K~
Experimental
results.
typical budding transition of the vesicle. Heating a spherical vesicle (not
lateral
tension
24.5 °C for
time, the
vesicle
first
kept under
at
some
becomes
oblate
ellipsoid and then discontinuously
transforms
into a prolate ellipsoid
and
an
continuously into a pear. Here we analyze the last part of this heating process,
which is
then
smaller or equal to
presented in figure I, showing that stable pear shapes exist for temperatures
further
°C
°C
first
order
transition
increase
in temperature by 0. I
40. 8 °C. A
to 40.9
to
causes
a
leading
outside
budded
that
become
unstable
the
This
the pear shapes
state.
to
an
means
formation
of a daughter
vesicle
by a
The
vesicle
connected
the
mother
neck.
to
narrow
observed
of shape changes does not depend much
the heating rate and displays for
sequence
on
heating at a rate of 0.01 °C/s the same
behavior
the one
faster,
continuous
shown in figure I.
as
thermal
The
hysteresis of the vesicle of figure I during recooling is shown in figure 2.
Starting at 41.0 °C the neck between the bleb and the mother vesicle is repeatedly widening
and narrowing at 38.7 °C as well as at the
three
lower
37.2 °C, 35.5 °C and
temperatures,
32.4 °C. The shapes with the open
neck
unstable.
At each widening and narrowing the
are
radius of the daughter
vesicle gets
smaller
and the radius of the
mother
increases.
vesicle
After
each closing the vesicle pair
fluctuates
and only after an
additional
decrease
lateral
temperature
tension
develops in the
leading to a suppression of fluctuations.
membrane
At 25.3 °C the neck
finally opens and a stable pear shape is obtained again. This
is
much
lower
than
temperature
which
shape
becomes
the
the
unstable
in
the
heating
the
and
temperature,
at
pear
process,
thermal
hysteresis
19.6
°C,
but
the
is
reversible
in
that
heating
leads
amounts
to
system
a
sense
again to the sequence of shape changes as shown in figure I.
The cooling
is
different
for slow cooling
If the
vesicle is cooled
from 41 °C in
rate.
process
Figure
shown),
I
shows
which
a
was
INDUCED
THERMALLY
N° 5
OF
BUDDING
VESICLES
PHOSPHOLIPID
635
~
i
'~
,j
)
<
~
"
j
>
S
~
j
~
Ii
j
/
f
j
'~H
°C
°C
T«40.0
~
~~~~~~~~~i~
~~
_'
j.,
T.)0.9 °C
~.j~l
°C
T.4tl.9
,
A.1794~m
Vml
~~i~
la 3 t
l0Pm
I.
Typical
induced
by heating
stability
the
Fig.
prove
(A)
V
indicated
are
and
A
increase
at
in
transition
the
T
shape
shape.
pictures
The
the
in
the
°C
around
at
of
40.9
to
50
s.
40.9 °C the
All
causes
two
outside
a
in
(T)
shapes
the
determined
first
final
vesicle
heating
each
temperatures
°C
The
DMPC
a
After
above.
be
cannot
=
takes
of
°C.
of
40.8
transition
0.I
steps
temperature
picture
last
budding
in
because
budded
40.9
at
shape
pure
the
and
the
for
is
are
to
It
reached.
stable.
are
of
the
outside
an
be
The
values
of
further
shape.
that
to
areas
A
intermediate
remarked
were
min
15
and
shape.
budded
instable,
should
for
(V)
volumes
fluctuations
showing
transitions
observed
was
°C
T«40.8
transition
°C
shape
The
water.
vesicle
measured
large
of
spontaneous
pictures
step
states.
this
This
In
final
flickers.
again repeated opening and closing of the neck is
intervals,
steps of 0.I °C with 15 min
39.7 °C, 31.6 °C, 28.5 °C and 26.4 °C, which
observed, but now at temperatures
that
means
the
temperature
never
now
lowered
below
daughter
4.
intervals
transforms
vesicle
the
between
into
back
a
phase
DMPC
consecutive
two
pear shape.
transition
The
openings
neck
temperature
and
closings
remains
(I.e.
23.8
increase.
closed
till
°C)
and
the
the
The
vesicle
temperature
fission
of
is
the
occurs.
Theory.
4. I
BILAYER
vesicle
model
couPLE
choose
can
basis for the interpretation of the presented series of
systematics of the bilayer couple model.
Within
this
determined
by minimizing the
membrane
bending energy
As
MODEL.
shape changes we
vesicle
shape
the
be
the
the
636
JOURNAL
DE
t
.
.
Val
3 )
T»3o~o
~
-.
A of the
same
at
38.7
25.3
°C
stress.
a
Transitions
vesicle
and
lateral
opening
dashed
by
caused
continuously
number,
vesicle
the
at
a
rate
recooling
of
the
before the neck
e.g,
opens at
tension
exists in the
membrane.
50
s
are
needed.
=,
of
vesicles
The
for
values
noted
I'
~
,
;
budded
0.02°C/s.
volume, V, of the vesicle are
undashed, as for e.g. I and
and
_
_
~.
~jj
are
the
the
figure
micrographs
showing different
and
the
I.
daughter
above.
focus
vesicle
cooling
The
T,
temperature,
in the
shape. During the cooling the neck between the mother
°C (pictures 3, 4, 5, 6), 37.2 °C, 35.5 °C and 33. I °C.
(pictures 7, 8, 9, lo). Before the opening of the neck
For
N° 5
II
~
~
Fig. 2.
performed
PHYSIQUE
Pictures
planes
opens
the
with
of the
was
area
the
same
transiently
opening of the neck
occurred
at
is always exposed to
lateral
38.7 °C the vesicle
that
stops flickering at 39.7 °C, which
means
Each opening and closing lasts for 10-20 s, while for the final
The
the
final
vesicle
THERMALLY
N° 5
W~ (Eq. (I))
classes
shapes,
of
volume
where
which
symmetry
relative
and
v
of
class
a
in
obtained
difference
be
can
area
shapes
the
defined
is
continuous
a
of the
areas
shapes
way
to
by
membrane
two
obtained
can
contain
637
VESICLES
PHOSPHOLIPID
constant
that
showed
system [61
the
and
volume
vesicle
constant
at
analysis of
detailed
OF
BUDDING
INDUCED
be
the
continuously
ascribed
layers.
shapes of
changing
The
different
to
the
same
relative
ha.
section and in figures I and 2 belong to the class of
in the experimental
prolate ellipsoidal shapes with equatorial mirror sj,mmetry (cigar class) and to
relevant
the
the class of pear shapes. Figure 3 shows
part of the vesicle shape v/ha phase
values for some of the
of the bending energy (Eq. (I)). v/ha
diagram obtained by minimization
determined
from
the
relative
marked.
The
volume
shapes shown in figures I and 2 are
v is
estimated
by comparing the
for vesicle
volume
membrane
and ha is
measured
values
and
area,
for v
in
of shapes as the
0.85
shown
with
calculated
series
measured
the
contours
one
figure 4. Because of the small changes in v it makes perfect sense to treat v as essentially a
dependence is presented in figure 5. This
quantity. The corresponding
constant
energy
changing continuously
observation
that shapes are
presentation of data (Fig, 3) reflects the
the
demonstrates
that
limiting shape. Figure 3 also
discontinuous
transition
until
to
a
a
obtained
by heating and cooling can be
hysteresis of the phenomenon of shape changes
hysteresis in the corresponding ha and/or v values,
considered
to be the
The shapes
axisymmetrical
described
=
~~
1.0
1.2
1-1
o g
pear
l.3
class
v
.2
~~
.
.
0.8
cigar
Fig.
class
'
difference
vesicle
shape phase diagram which is
area
experimental section. The region of pear shapes is bounded on the left
side by a broken
line at which bilayer couple
predicts [lsl a continuous
from the shapes
model
transition
with the equatorial
mirror
symmetry (cigar shapes) to pear shapes. On the right side the pear shape region
is bounded by the line connecting the limiting shapes which for the class of pear shapes are compositions
of two
linked
spheres of different radii. The cigar shape region is bounded on the left side by a line of
limiting shapes which are cylinders with spherical caps. Some of the calculated limiting shapes are also
shown.
Points
denoted by 1-8 give v and ha
values
corresponding to the eight shapes in figure I, while
only three points (points 9-11) corresponding to the limiting shapes
obtained
in the cooling
run
are
shown, the one
(point11) with the largest relative
volume
corresponding to the shape before the final
opening of the neck.
3.
relevant
Thus,
to
be
A
for
as
of
in the
stated
in the
with
shape
parameters
increasing
function
volume/relative
relative
described
already
consistent
discontinuous
control
part
shapes
v
(Sect. I), the
described
observations
do not
seem
within
bilayer couple model
because
model
this
continuous
expected within one class for
changes of the
not
shown
in
figures, the bending energy is a monotonously
As
monotonously
decreasing
function
of v.
Because
the
and
a
predictions
changes are
and
of
ha.
ha
introduction
of
the
JOURNAL
638
ho
A
4.
series
ha
are
of
shapes
shown
i
o o
1,12
~
I,lfi
I,li
I-lo
1.22
/~
/$
~
0
I-I'
1.26
1,18
1.30
the
pear
relative
the
1-1'
A~
~
of
for
N° 5
I-lo
ho:
difference
II
1,ois
ho:
Fig.
PHYSIQUE
DE
2,o
class
DMPC
volume
vesicles
different
with
values
of the
relative
area
0.85.
u
=
v*o.15
w~
#~ar
1.5
~«army
~'~
Fig.
of the
areas
membrane
shown
(w~
the
this
that
2)
=
at
the
experiments
temperature
volume
relative
4.2
are
of
RELATIVE
of the
done
DMPC
and
vesicle
is then
relative
area
of
the
two
at
(T~
relative
STRETCHING
bilayer couple model
expansivity
relative
to
1.3
membrane
bending energy (in units of 8 ark~) on relative
difference
0.85. It is seen that the dependence
layers (ha ) for the relative volume v
bending energy for pear shapes on ha is a monotonously increasing function. It can be
dependence is steeper at higher relative
volumes.
Bending energy has its largest value
limiting shapes.
Dependence
5.
between
I-I
l-I
=
temperatures
23.8
=
area
AND
°C ) it
well
can
difference
SPONTANEOUS
above
the
gel
assumed
be
are
to
that
liquid crystal phase
the
temperature
transition
changes of
the
of
the
continuous.
CURVATURE.
A
possible
extension
shape transitions is to include the
of the two
membrane
layers into the analysis [9- [[i. The shape of the
determined
by minimizing the sum of the bending energy w~ and the energy due
expansivities w~, which can be equivalently expressed as the non-local bending
which
could
allow
for
discontinuous
of
energy
membrane
the
BUDDING
INDUCED
THERMALLY
N° 5
relative
expansivity
area
sphere (8 ark~)
and
the
ratio
area
obtained
in
[I II
should
It
hao
be
minimizes
0.85,
q
total
For
ha,
of
this
value
of the
extension
of the
local
bending
layers.
bending
the
to
of
energy
a
(8)
in hao
model
curvature
between
of
relative
original bilayer couple
I
the
expansion
thermal
[10, 111, and hao the
moduli
The
spontaneous
changes
exist
within
of the
of the
cigar type shape
the
pear
a
sum
interval
relative
the
of the
membrane
values
parameter
bilayer couple
to
of
different
for
=
shapes
hao)~
for q to be in
inclusion
that the
effect
the
dependence
function
10.
q(ha
the
in
Continuous
energy.
illustrate
to
the
=
a
point
=
membrane
because
respect
and
is
0.
Recent
energy
makes
q
=
10.
relative
stretching
whereas
ha
could
model
therefore
attains
lead
which
value
a
discontinuous
to
ha.
order
In
as
dependent
the
changes in
this
at
estimate
an
with
normalized
and
while
cc,
q-
gave
noted
temperature
non~local
the
(7)
w~.
+
as
non-expanded
the
limit
the
measurements
v
between
for
difference
w~
=
also
expressed
be
can
is
ternl
w~
with q,
639
VESICLES
:
w~~~
The
PHOSPHOLIPID
OF
1.13
<
of
type shape (cf. hao
a
possibility
1,325
=
~"
la
for
and
the
minimum
of w~~~ and
q the
This
example
<1.30.
ha
predicts
model
hao
expansivity term, figure 6 shows,
bending and relative expansivity
curve
of
of
the
parameter
corresponding
shows
that
discontinuous
a
in
value
Fig. 6). Such
a
for
terms
stable
pear
the
proposed
transition
from
transition
a
occurs
flow
,ai
1.135
ii
1.<15
i<
ii
ins
m
im
~.~
Fig.
i-i
ii
ii
bending energies as a function of the area
and
non-local
0.85 ). The
shapes is plotted for different
values
of hao (v
ratio
between
the
non-local
and local bending
moduli is chosen
10. A typical van der Waals
to be q
binding energy (its exact value being dependent on the size of the mother and daughter vesicle) of the two
vesicles (w~ow
0,135 ) is represented by a vertical
should be
subtracted
line and
from the elastic
energy
of a
mother-daughter pair.
6.
difference
The
ha
sum
for
the
of
the
cigar
membrane
and
pear
local
class
=
=
=
640
PHYSIQUE
DE
JOURNAL
II
dw~
values
for
of q
than
smaller
20
which
is
value
the
(Fig. 5). The
smaller
q the higher is the ha
sufficiently small values of q the
discontinuous
transition
from
this
shape. However,
starts
whereas
observations
the
shapes.
pear
that
also
on
interpret
In
that
energy
However,
for
room
such
and
such
pear
transitions
or
of the
DMPC
the
between
third
4.3
order
VAN
the
of the
in
terms
DER
of
class
in
curvature
co
elastic
FORCES.
intravesicular
term
experiment
the
should
which
continuous
are
expansivity
spontaneous
it is
to
(Fig. I).
included
be
means
possible
not
bending
in the
transitions.
not
Even
change
the
more,
it
«
behavior
of
little
leaves
the
no
or
weakly asymmetric stable vesicle shapes
within this
expression (Eqs. (7) and (9)) even the
energy
found
have
experimentally
been
to be puzzling.
appears
the bilayer couple
neither
model,
the
spontaneous
nor
through
transition
that
does
curvature
discontinuous
of
the
Therefore
data.
two
first
The
describe
can
following
the
in
energy
of
One
distinct
forces
to
consider
consideration
second
is
discontinuous
two
further
Waals
forces
der
van
based
possible
on
the
inclusion
of
be to
add
analyse
the
bilayers.
the
the
observed
the
we
into
takes
parts of the limiting shape and the
the
WAALS
contribution
vesicles.
the
relative
obtained
spontaneous
combination
experimental
vesicle
two
the
indicates
the
model
interpretations
of
pear
leads
by [1, 161
pears » [161. Thus
transition
states
analysis
above
The
curvature
the
predicted
budding
to
continuous
eggs
as
also
described
inclusion
respect
a
that
model
be
to
the
with
system
fact
has
the
transitions
from
attained
the
with
is
breaking point
symmetry
stable
For
pear shape.
actually to a nearly limiting
the
equatorial
mirror
symmetry
shape
belonging
the
class of
to
a
of
shape
a
the
at
~ha
transition
within
including
shape
value
transition
the
transitions
model
discontinuous
generalized
a
of the
basis
the
that
predicted shape
The
the
indicate
of
5
N°
line
of
elastic
approach
free
to
energy
the
problem
of the
would
system
and
minimization
satisfying the
condition
model.
This
in such a generalized
instabilities
vesicles.
Bruinsma
[171 in his study of growth
of
Also
indeed
been
done by
has
suggested that such a term could be included in the analysis of lipid bilayer
Wortis
et al. [141
different
contribution
vesiculation
shape transitions.
The
of the
vesicles
shapes including
and
is
but
since
have
intravesicular
forces to the total free
shape
dependent
these
forces
a
energy
of the
size
their
effects
would
only in some
short
the
scale
vesicle
be
seen
very
range
on
limiting geometries. Thus the free energy due to the
intravesicular
forces
would
be
small
the shape of the
vesicle is such that its
boundaries
into
proximity.
whenever
do not
close
come
Especially, for the budding
transition
this
would
has to take into
that
the
account
one
mean
shapes
of the
surface
vesicles
form in
only in the last stages of the transition
where
two
separate
blebbed
the
Limiting
ourselves
this
let
for
sake
of
the
take
to
state
argument
us
vesicles
and
(minimal)
separation
with
Since
of
radii
Ri
L,
connected
thin
neck.
R~
at a
two
a
refer
the
sign of the total free
(see Ref. [17]), W(L), is negative
to
energy
can
we
vesicle.
For plausible values of Ri, R~
10 ~Lm and
W(L as the binding energy of the blebbed
L
20 h we
convince
ourselves
that the free
compared to
can
energy of the neck is negligible
contribution
of the
der
Waals
interactions
between
the
(mother
and
daughter)
the
two
van
itself is very important,
it keeps the
mother
vesicles.
Nevertheless
the neck
because
and the
conclusions
remain
daughter vesicle in close
We
that the above
would
valid
also
contact.
note
from
general types of force laws that include forces with a
for binding energies
derived
more
intravesicular
close
range
The
forces
proximity.
smaller
force
than
curves
the
van
for
the
der
Waals
interaction
force
of
[18].
planar
membranes
can
be
tumed
via
the
Derjaguin
N° 5
approximation
[19].
al.
surface
area,
F
energies between spherical
approximation [20] simply relates
of two planar
surfaces
separation
at the
(minimal) separation L
into
E(L),
(L)
at
a
~~ ~~
Ri
free
position
of
R~
and
energy
to
are
2
ar
=
RI
of the
radii
binding
potential
relate
the
interactions
Waals
the
(L)
+
spherical
two
dominate
W(L
energy
between
we
of
the
hydrocarbon
water
over
apply
this
reasoning
now
can
of
as
Ri
For
(10)
(or for that
~~ ~~
forces
two
done by Lis et
potential per unit
acting between two
been
L
»
The
scale
membranes.
RiR2
planar
to
thus
force
the
Ri, R~
if
has
as
+
R~
spherical vesicles
the region of L
Thus
to
the
matter
we
the
where
are
in
integral
der
van
derive
RI
~
We
L to
641
VESICLES
interaction
the
vesicles.
system
a
flat
two
~vdw~~
for
E(L)
R2
W(L) that is the integral of the force)
interaction
the
vesicles
interaction
F
where
PHOSPHOLIPID
OF
Derjaguin
The
vesicles
BUDDING
INDUCED
THERMALLY
+
geometry
DMPC
vesicles.
0.2kT
R2
(11)
L
[201.
The
result
of the
measurements
of Lis
et
al.
vesicles of equal
a binding
energy of two spherical
(300A) at an equilibrium separation (giving a lower bound for the interaction
diameters
energy) of 25 A leads to a binding energy of approximately 0.3 kT. This value can
be
now
translated
via the Rj, R~ scaling ratio in the Derjaguin
approximation into a
that two
statement
spherical vesicles of radii 13.3 and 6.7 ~Lm (actually seen in the experiments) have a binding
of
approximately 90 kT at the same
(minimal)
separation.
energy
This binding
subtracted
would
thus have to be
from the
elastic
blebbed
energy
energy in the
obtain
free
of the system.
Measuring the energy in elastic units of
the total
state
to
energy
8 ark~,
where k~
I x10~'~ J, the
corresponding to the binding energy of 90 kT is
value
0.135.
Taking into account the variation of the radii of the two blebs obtained in the
w~~w
experiment one is thus led to the following upper and lower bounds for the binding energy at a
blebbed
25 A in the
(minimal) separation of L
state
[191
on
flat
DMPC
membranes
tumed
into
=
=
=
RI [iLml
explained
R2 [iLml
Wvdw(kT)
Wvdw
13.3
6.7
90
0.135
13.5
4.4
68
0.101
only able to obtain the van der Waals energy for a shape, where
by a thin neck. This is not really a major drawback
since the
separate
two
are
der
Waals
this
particular
is
by
far
the
largest
point.
On
the
phase
diagram
its
at
van
energy
contribution
total free
of
the
would
thus
show
single
point
the
end
to the
system
at
energy
as a
of the free
line leading to a very sharp
minimum
for the limiting shapes. The
energy
energy
estimated
magnitude of the van der Waals binding energy w~~w is shown as a vertical line in
figure 6. This value,
which
is
comparable to the changes in elastic energies should be
subtracted
from the
of
the
limiting shape.
energy
Though the inclusion of the van der Waals
interactions
between
the
mother
and the daughter
vesicles
does
lead to the lowering of the total
of
the
budded
unsatisfactory
state,
energy
some
As
above
blebs
we
are
connected
JOURNAL
642
features
have
dependence
maximum
vesicle
between.
in
therefore
pointed
be
to
of the
of
the
The
ha
energy
on
transition
order
first
inclusion
The
out.
PHYSIQUE
DE
should
character.
of
(Fig. 6)
to
N°
der
van
the
thus
the
As
the
II
across
height
in
minima
two
the
with
barrier
energy
an
leads
energy
of the
appearance
occur
barrier
Waals
5
and
a
be
50 kT ), a very
very large (>
expected for this
transition.
The
in the heating
in
stays
process
a
and the pear shape simply
zero
is
=
long
relaxation
metastable
the
range of
included
in
transforms
ORDER
order
in the
TERMS
mechanism.
In
w~(Aa )
elastic
first
The
three
and
almost
to
shape.
limiting
stable
in
van
their
magnitude
order
but
form
have
term
relative
the
the
elastic
introduced
is
it
hao)
co(ha
written
~
The
last
within
another
or
hao)~
[1, 161 and
term
that
is the
term
(12)
correspond to the
layers (Eq. (8)).
and
membrane
contributions
the
form
of a bilayer up to the
invoking any specific
(ha
two
curvature
all
interactions
clearly
is
one
introduced
been
of the
spontaneous
in
be
6
already
stretching
Waals
as
hao)~
q(ha
+
der
energy
without
here
be
can
expression contain
energy
considerations
up till now.
energy
behavior
the
on
general
(Eq. (5)) and
is nothing
above
the
into
account
consequences
Within
a
The
should
The
ENERGY.
difference
area
most
a
second
the
energy
order
term
terms
into
the
vesicle
the
decreases
ELASTIC
THE
w~(ha)
=
where
IN
monolayer
w~~~
bare
into
can
such
THIRD
physical
barrier
the
be
instability (Fig. 6), but
nevertheless
an
energies goveming the budding process and
realistic
of this
model
phenomenon.
any
describe
cannot
until
state
pear
and
destabilizes
third
and
instability
observed
4.4
pronounced hysteresis
a
corresponds to the case, when
time
third
thus
have
first
the
been
order
taken
and
term
its
will be investigated
below.
process
total
has a
of
values
parameters
energy
range
co, q and y the
discontinuous
transition
correspond
the
observed
from
that
would
to
a pear
limiting shape of the pear class shapes. This is shown in figure 7 where the
small
dependence on Aao
shape to the nearly
total
is plotted
energy
The phase diagram of
of the
in the
energy
of the
budding
for a
consecutive
values.
set of Aao
clearly
exhibits
the features of the
(Fig.
7)
all
process
now
leftmost
discontinuous
transition.
A single extremely
minimum
the
side of
observed
steep
at
of
(I.e.
is
being
first
of
all
replaced
for
smaller
values
low
temperatures)
the energy
Aao
at
curve
towards
larger values of the
broader
minimum,
that is slowly displaced
by a
somewhat
equilibrium ha for larger values of Aao (I.e. for higher temperature). At a certain value of
of the system is discontinuously
1.202 and 1.203) the equilibrium
Aao (lying between
state
transition
minimum
the
right
side
of
This
is
effected
other
the
moved
the
to
energy
curve.
on
barrier
closely
conforms
the
crossing
of
intervening
and
thus
without
to
energy
an
any
experiment. In the cooling run the vesicle
the potential
barrier
transform
back
to
must
cross
into a pear shape and
therefore
hysteresis appears.
the
thermal
effect
The
one
minimum
4.5
understood
the
water
decreasing
closed
Waals
6.
are
of
ha
on
the
budding
forces
They
of
the
ELASTIC
neck
of
dependences in figure
of the limiting shape
that
can
not
fluctuates
but
lateral
because
and
In
ENERGY.
would
stress
of the
and for lipid
diffusion
transport
the
mem[Srane
temperature
area
it
energy
mother-daughter
the
the shape first
suppressed by the
by assuming
neck
energy
the
decrease
STRETCHING
neck,
the
fluctuations
function
a
the
7 is the
and
same
lead
as
to
a
the
sharp
ha.
closing
be
the
figure
BILAYER
and
of
der
van
in
maximal
at
THE
opening
closing
the
of
presented
as
van
between
would
thus
after
der
develops
Waals
its
repeated
the
process
observed.
After
each
attraction
but
the
and
at
stretching.
decrease
temperature
in the
daughter
the
to
cooling
pair is
additional
some
which
decrease
lead
the
vesicle
membrane.
This
is
closed
neck
mother
constant
The
vesicle.
volume
stretching
can
for
With
and
and
with
the
BUDDING
INDUCED
THERMALLY
N° 5
OF
VESICLES
PHOSPHOLIPID
643
hog
1,io3
1.7~
1201
,jj
1,101
1,100
1-ii
1,199
1.19i
~'~~
i-i
ii
i-1
~
Fig. 7.
of shapes
The
parameters
elastic
plotted
is
are
co
=
for
energy of the bilayer
different
values of
2.8,
3.4
q
and
hao )
(ha
in
order
(v
constant
0.85 ).
for the
The
pear class
values of the
=
220.
y
=
expanded up to the third
hao, while v is kept
=
corresponding elastic energy
increase
with decreasing
until the stretching
temperature
energy
comparable to the van der Waals energy of the neck. Then the neck opens transiently
becomes
equilibrates in a new
mother~daughter pair with
unstretched
membrane.
and the
system
openings and closings
As the typical
difference
the two
consecutive
AT
between
temperature
of the
degree, AT
and the
thermal
is for the fast cooling
order of one
I K,
process
area
=6.5x10~~K~',
coefficient
fractional
dilation
expansivity
the
corresponding
p
area
=
(«
is
thus
«
This
maximal
regime,
«
and
its
tension
can
T
From
this
relation
the
increase
=
of
dominated
be
expressed
W~j
=
j~
T
da
=
is
always
fluctuations
and
the
in
the
(
gr
free
ar~ k~
~~
8
o
"kc
tension
between
(14)
+
elastic
low
relation
[221
as
k~
energy
~
A
(13)
membrane
by
1
ln
8 ark~
the
10~~
x
the
are
~~
«
6.5
=
[211 that
small
so
properties
elastic
membrane
the
is
dilation
area
where
p AT
=
(e
can
be
evaluated
as
~'
~~
~
l
«
(15)
elastic
question can be asked, what temperature
decrease
AT is needed
for an
energy
equal to the van der Waals binding energy (W~~w). Taking for this the estimate
0.135 x 8 ark~ and using the values k~
obtains
I x 10~ '~ J and A
2 800 ~Lm~, one
W~~w
from equation (15) and equation (13) AT
1.3 °C,
with
which
well
measurements.
agrees
It must be
remembered
that the AT
intervals
This
get larger, if the cooling process is slower.
slow
exists
such
time
could
that
relaxation
and
the
neck
is
long
scale
mean
very
process
on
a
a
completely closed for the exchange of water and lipids. Because of this also the elastic free
not
barrier
back to the
and not big enough to allow the crossing of the energy
energy W~, is smaller
the
Now
W~j
become
to
=
=
=
=
JOURNAL
644
pear shape
membrane
5.
This
state.
phase
Discussion
by
First
work
our
of all
which
becomes
transforms
vesicle
the
back
large,
so
into
that
the
shape.
pear
a
conclusions.
and
Attempts to understand
been presented (see e.g.
forth
before
occurs
N° 5
II
hysteresis
thermal
the
increases
transition
PHYSIQUE
DE
salient
the
that
Waals)
give a
remarkable
comparable to the
energy
are,
in
total
vesicles
already
brought
have
distinctions
novel
conclusions.
the
between
the
to
main
two
in
processes
important
however,
interactions
contribution
budding
of the
There
summarized
be
can
(van der
the
features
[161).
Ref.
mother
of
energy
the
and
daughter vesicles
magnitude is
the
whose
system
magnitude of all the other energy changes involved in the budding process.
This is something
worth bearing in mind,
since
if the van der Waals
interactions
not
even
are
driving the budding transition, their presence is an indispensable stabilizing factor acting in
parallel with other
forces is particularly
mechanisms.
The role of van der Waals
important in
the cooling
where
they
closing
the
neck
and
blocking
the
of
exchange
and
water
process,
are
lipids between the two vesicles.
Therefore
tension
develops
in
the
which
finally
membrane,
a
neck.
the
opens
The
major
transition
interposed
the
expansion
observed
itself
of the
whether
the
of the
multilamellar
be
deeper.
shown
if
not
the
der
van
directly
did
not
to
drive
consequences.
correspond
would
in
should
these
yet investigate
the
to
contribution
overcome
to
It
Aao
can
the
total
an
thus
was
budding
empirical
layers of
the
an
constitutive
two
ha
budding
the
to
experiment.
important for
model it by
the
in
term
important
is
interaction
because
are
tried
drive
have
describe
the
only
energy
state.
interactions
First
Waals
the
between
cubic
Waals
der
in
We
difference
of the
van
would
interactions
Waals
area
of the
budded
Waals
this
see
could
alone
system
mechanism.
inclusion
an
forces
the
case
not
der
van
driving
addition
der
this
does
relative
that
The
vesicles
We
of the
terrns
van
the
for its
predictable
have
vesicle
shape
should
influence
though
that
stabilizes
role
If the
One
Waals
der
van
in
that
maximum.
instabilities.
additionally
on
in
It is
membrane.
the
that
realization
account
not
can
idea
the
energy
recognize
to
they
energy
the
in
to
from
local
necessary
process
drawback
comes
cases
this
budding
favour
the
energy
conclusion.
The
to
their
outside
to
also
transition,
budding process,
importance should
in the
of all
budding
minimum
of the
Waals
der
van
least
at
inside
or
the
have
a
very
not
budded
of
forces
The
vesicles
states
thus
important
fact
depend
states.
large
budded
this
then
and
should
appear
role
in
to,
its
stabilization.
Finally
that
the
type of the shape
which
explained
by
takes
into
Aao,
on
different
number
of
phospholipids
in
monolayers.
For
example,
if
spherical
the
account
two
a
a
vesicle is exposed to lateral
before a heating cycle, it shows a budding
transition.
The
stress
equilibration
increase
in
the
lipid
flip
flop
facilitates
the
of
lateral
and
stress
rate
causes
an
initial
value of Aao
that
in
heating
lipids and the change of Aao. A different
also
the
means
Aao(T) will behave differently.
process
transition
should
we
recall
depends
the
from
statement
a
This
pretreatment.
introduction
the
could
be
References
Natu~fiorsch.
W., Z.
[11
[21
HELFRICH
DEULING
H.
[31
SVETINA
S.,
[41
[51
SVETINA
S.
SVETINA
S.,
J.
and
HELFRICH
28c
W.,
OTTOVA-LEITMA~NOVA
and
2EK% B..
BRUMEN
M.
Biomed.
(1973)
J.
693.
Phys.
A.
and
Biochim.
and 2EK% B.,
Stud.
France
37
GLASER
Acta
(1976)
R., J.
42
Biophys.
(1985)
110
1335.
Theor.
Biol.
586.
(1985)
177.
94
(1982)
13.
N° 5
[61
[71
THERMALLY
SVETINA
S.
EVANS
E.
and
A.,
[8]
SHEETz
M.
[91
SEIFERT
U.,
P,
2EK% B,,
J.
14
Biophys.
(1974) 923.
SINGER
S.
J.,
Biophys.
and
MiAo
BUDDING
INDUCED
L.,
Eur.
Proc.
DOBEREINER
Amphiphilic Membranes,
Berlin
Heidelberg 1992)
Natl.
H.-G.
Acad.
D.
645
101.
Sci.
WORTIS
and
VESICLES
PHOSPHOLIPID
(1989)
17
Lipowsky,
R.
p.
J.
OF
Richter
USA
M.,
and
71
(1974)
4457.
Structure
The
K.
Kremer
Conformation
and
Eds.
of
(Springer-Verlag,
93.
[101 Bo2k B., SVETiNA S., 2EK% B. and WAUGH R. E., Biophys. J. 61 (1992) 963.
[I11 WAUGH R. E., SONG J., SVETINA S. and 2EK% B., Biophys. J. 61 (1992) 974.
[121 BERNDL K., KAS J., LIPOWSKY R., SACKMANN E. and SEIFERT U., Euraphys. Lett. 13 (1990) 659.
[131 KAS J. and SACKMANN E., Biophys. J. 60 (1991) 825.
[141 WoRTis M., SEIFERT U., BERNDL K., FOURCADE B., MiAo L., RAO M. and ZIA R. K. P.,
Proceedings of the workshop on « Dynamical
Phenomena
Interfaces,
Surfaces
and
at
Membranes
Les
Houches,
18.2-28.2.1991,
D. Beysens, N.
Boccara
and G. Forgacs Eds.
»,
(1991).
[lsl SVETINA S. and 2EK% B., J. Theor. Biol. 146 (1990) 115.
[161 WIESE W., HARBICH W. and HELFRICH W., J. Phys..
Condens.
Mattel
4 (1992)
1647.
[171 BRUINSMA R., J. Phys. Colloq.
France
51 (1990)
C7-53.
[181 RAND R. P., Ann. Rev. Biophys. Bioeng. lo (1981) 277.
[191 LIS L. J., MCALISTER M., FULLER N., RAND R. P. and PARSEGIAN V. A., Biophys. J. 37 (1982)
657.
[20]
[211
[221
ISRAELACHVILI
EVANS
E.
HELFRICH
and
W.
J.,
Intermolecular
RAwicz
and
W.,
SERvuss
Phys.
R.-M.,
surface
and
Rev.
Leit.
Nuovo
(Academic
(1990) 2094.
Cimento
D3 (1984)
forces
Press,
64
137.
New
York,
1985).