Albert-Schweitzer, 33600 Pessac, France. (~>. Institute of Biophysics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria. (Received 24. February1997,.
Phys.
J.
II
IYance
Latex
Christian
(~>
de
Bernard
PAGE
Pouligny
(~>*)
Paul-Pascal, du Docteur Albert-Schweitzer, 33600 Pessac, avenue Biophysics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
of
February1997,
24
PACS.68.10.-m
Fluid
revised
Mechanical
PACS.87.22.-q
Physics of
June
fluid-fluid
surfaces
PACS.62.20.Dc
23
properties cellular
and
1997,
accepted
15
1651
Vesicles:
recherche
Institute
(Received
Phospholipid
Giant
to
(~), Miglena Angelova (~) and
Dietrich
Centre
NOVEMBER1997,
1651-1682
Spheres Dynmnics
Adhesion of Statics and
(~
(1997>
7
France
July1997)
interfaces
of liquids
physiological
processes
of phenomena which when a solid microsphere is occur giant lipid vesicle. We used Latex beads, a few microns in which manipulated individually by means of a long-working-distance optical trap. were The evolution of the bead/vesicle system characterized in time, from 100 s. I ms to was In this time identified different expulsion and resteps, namely adhesion, ingestion, range, we In the adhesion capture. quickly in direction to the vesicle and step the sphere interior moves the surface of the becomes particle wetted by lipids. We simple model, based on propose a the counter-balance between adhesion and stretching of the lipid lamella, which explains the experimental equilibrium configuration. The bead /vesicle configuration after the adhesion step pertains to partial or complete wetting, depending on the initial vesicle state. Partial wetting be followed by a second "particle ingestion", and which leads to named step, which we can complete (or nearly complete> wetting of the particle surface. Ingestion is characterized by a further penetration of the particle the vesicle in concomitance with a decrease contour, across of the vesicle size. The phenomenon is of a dynamically stabilized attributed to the occurrence membrane, which allows part of the water initially inside the vesicle to flow the pore across (expulsion and re-capture> of the Ingestion can be followed by a back and forth out. movement particle. In the ultimate configuration, the solid surface is totally wetted by lipids, however with finite angle between the membrane and the solid surface. contact a Abstract.
brought diameter,
studied
We
in
with
contact
the
an
sequence
isolated
m~
Nomenclature
ka
elastic
~
lateral
R
vesicle
membrane
a
0
surface
sphere
solid
z
penetration
ze
equilibrium
©
Les
for
(ditions
energy
density
tension
area
excess
of vesicle
radius
angle
contact
Author
modulus
adhesion
radius
relative
e
(*>
expansion
surface
membrane-substrate
A
value
correspondence
de
(dimensionless> value (dimensionless>
penetration
Physique
(e-mail:
1997
pouligny©crpp.u-bordeaux.fr>
~
PHYSIQUE
DE
JOURNAL
1652
N°11
II
Introduction
1.
assemble in particular, they form biinto structures, aggregates of different water made of one or several membranes layers. "Liposomes" are small vesicles whose are developed for "giant lipid vesicles" methods preparing such bilayers. Recently, [1, 2j some were Such vesicles, well visible in light microscopy, ideal of sizes in the 10 to 100 ~Jm range. are mechanical and hydrodynamic tools to study basic properties of lipid bilayers. potential to offer ultiThe academic interest in liposomes is due to a large to their extent mately simplified models of biological cells. Knowledge of vesicle properties obviously helps membrane incluunderstanding such phenomena as cell deformation, shape transformation, membrane permeability, etc. sions mobility, which Membrane membrane, or membrane substrate adhesion is a great issue to solid to large work, either experimental has already motivated of theoretical [3-5] amount 7]. [6, a or Usually, a distinction is made between the "weak" and the "strong" adhesion regimes [5j. Membrane involved "weak" in the tensions regime are small (~ < 0.1 dyne/cm> and the partially driven by thermally excited undulations. This situation membrane events occurs are the (for flaccid of with substrate of vesicle having low interaction in the a energy case a instance, an other membrane> [8j. "Strong" adhesion corresponds to large energies and tensions (~ > 0.5 dyne/cm). In this regime, membrane smoothed The problem is undulations out. are by a liquid droplet. with the wetting of a substrate just mechanical and has much in common focuses on the This article interaction of lipid vesicles with the surface of solid spheres in the (a diameter> few which definitely than the vesicle (a few of particle microns in is smaller case a in diameter>. As in the above mentioned problems, the properties of the simple microns tens of sphere /vesicle system may have some relevance biophysical problems, for example, to to some
Lipids
in
molecular
understand
the
knowledge
this
microparticles
[9j.
An
other
spheres
field
can
[10],
force
as
study
membranes
to
have
transducers
membranes
important
very is
cell
the
across
be
even
of this
motivation
attached
filaments
penetration
of
way
in
[11] and
the
at
more
been
to
used
probes
as
to
size
drug
A
entities.
basic
based
vectorization
on
engineering level, since micro~ instance, for pulling tubular lipid membrane viscosities [12].
mechanical
"handles",
as
colloidal
for
optimize
for
measure
experimental part. To summarize, on a larger the particles used in our Latex microspheres, with negative surface experiments are common charges (sulfate groups). The particles are manipulated by means of an optical levitation trap with spherical giant lipid vesicles in plain water (pH m 5). Unless and brought in contact stated, the vesicles are isolated and made of SOPC or otherwise unilamellar. Membranes are DMPC, are in the fluid state and electrically neutral. Observations carried out by means are of optical microscopy, in phase or amplitude Essentially, we observe the position of contrast. the spherical particle relatively to the membrane, I.e. the degree of penetration of the sphere of the vesicle, which is equivalent to defining an inside the angle of contour apparent contact the solid surface. the membrane Data gathered at video rate (25 Hz), using standard on are This
article
image
numerization.
rate, by As
contains
we
means
will
In
of
few
a
adhesion
proceeds
of the
by
emulsion In
surface
droplet. 2, we
Section
configuration in (see for instance
set
this
the
out
the
particle
position
was
measured
at
a
1.1
kHz
device.
Latex particle on a vesicle pertains to the "strong adhesion" a through a first step which can be described as a partial wetting lipid membrane. In this step, the vesicle behaves somehow as an
a
simple
mechanical
model
Although the basic specific problem that we
situation.
[4]), the
sensitive
and
of
regime and always solid
experiments,
analog position
an
see,
modelization
section
a
to
describe
concepts address
the used
(a
sphere /vesicle equilibrium in
finite
the size
model
particle
are
not
new
interacting
ADHESION
Nall
OF
SPHERES
LATEX
TO
VESICLES
LIPID
1653
?~ lwtex
sphere
esic~
R
~
~
s
za=s+v
a)
R~~
lim
lima
co
~
co
e.itertor s
z a
vestcle
vesicle
exterior
mter>or
c
Fig.
Geometry
I.
that
vesicle
the
lipid
c):
membrane.
with
finite
a
vesicle
contact.
is
contact,
b>:
of
Adhesion
vesicle)
size
computing
at
sphere-lipid changed after
of solid
radius
to
kinetics
of
infinite
an
before
discussed
not
was
degree and
the
vesicle
a
a):
General
Adhesion
flat
a
problem (finite sphere to
solid
a an
and Ro >. Note infinite plane
substrate.
Essentially, Section 2 particle across the vesicle
details.
in
us
penetration
of
)
of the
is
aimed contour
equilibrium. experimental. The materials and methods that we used are itemized in Section 3. Our experimental reported in Section 4. We identify differobservations are interaction: adhesion, ingestion, expulsion and re-capture. ent steps in the sphere/membrane different These described in terms of particle position and vesicle configuration as a events are
and
membrane
the
following
The
function
of
tension
sections
time.
observations
These
picture
wetting
at
are
sphere/lipid
to
are
analyzed the
describe
membrane
adhesion
adhesion
of
Some for at
2.
future
work.
end
the
of the
Some of the
technical
a
above
density. We dynamically
are
left
details
in
ofthe
also
sequence
give
adequateness
the
an
stabilized
pore
and
out
and
estimate
interpretation the
of the
partial
the
value of the
of the
particle
membrane
[13]. problems. Section 6, together with prospects thd final calculations explained in two appendices are
unsolved
summarized
discuss
We
5.
step in the
energy
the creation ingestion on questions, essentially kinetics, The main points of the paper are
based
Section
in
set
across
as
open
article.
Theory
sketched in Figure 1a. We want to situation particle complex at equilibrium, I.e. well after both entities problem has some analogy and differences with the classical of the spreading of a liquid on a solid surface (wetting>. In this case, the situation which gives the angle geometry of the complex is found from the Young equation. contact 2.1.
STATIC
describe
the
EQUILIBRIUM.
We
consider
geometry of the vesicle and have got in This contact.
solid
the
(denoted
~
Fig.
the
is
~Lw
=
the
0 in
adhesion
between
tension
density
energy
interfacial
liquid
1> of the
tension
on
PHYSIQUE
DE
JOURNAL
1654
the
solid
surface.
~ji
+
o>
cos
We
N°11
Young equation
the
write
as:
A
=
liquid (L> and the liquid and
the
between
II
(1>
the
outer
the
solid
fluid
(S).
water
say
A
can
be
(W> and A is
defined
in
of
terms
as:
A
+
~LW
=
(2)
~SW.
~LS
4/3 ~R(, is conserved in the interaction. The final droplet radius The volume of liquid, V R, depends on Ro, on the contact angle 0, and on the sphere radius, a. To summarize, the equilibrium configuration of the complex is found from 2 physical constants, A and ~, and from 2 input parameters, a and Ro. The problem can be complicated by the existence of an energy related line, which is usually taken as proportional to the length term to the SLW contact of line, independently of the contact angle. Whether this line contact energy is important in practical of controversy [14-16]. situations is a matter We back the problem with the vesicle. Again we that the volto suppose come now vesicle the solid sphere, and define V inside the conserved with the is in interaction ume roughness and relate contributions membrane Ro as (3V/4~>~/~ We neglect entropic to the the lateral tension to the elastic expansion of the lipid bilayer. For the lipid species of interest, membrane for lateral The this is a very good approximation tensions > 0.5 dyne /cm. can ~ by a surface area at rest (zero tension> So then be viewed as an elastic sheet, characterized and by a modulus ka [17j. If So < 4~R(, the vesicle takes spherical expansion constant on a shape of radius R > Ro. The elastic energy of the film reads: =
4~R~. In equation (3>,
with
S
the
membrane.
=
membrane
simplicity
For is
tension
the
larger
excess
surface
excess
We
e
area,
droplet.
=
when
is
that this
initial
the
implicitly keep this
k~
(s
j
that
supposes
assumption
so'
(if so
membrane
the
4~R(,
So
1 is
does
the
the
density is uniform in paragraph. The
surface
throughout
this
small,
excess
area
on
much
larger
is
in
the
membrane
the
increases
process
j4>
to
takes
the
bilayer adheres by phospholipid
4~Rj>.
now suppose is partially
surface rule
4~R(,
than
surface
will
given by:
~
If So is
one
we
than
same
way
surface
4~a~, I,e. So membrane, with
as
in
area.
when
the
The e
>
case
accommodate If the
and
relative
that
of the
this
liquid
exception
to
this
(a/Ro~~.
In
this
completely wrapped by the elastic If cost in no energy. (a /Ro final the the spherical. shape of vesicle outside of the Equation (1) region is contact < e >~ angle directly, because ~ is not a is still valid, but it does not give the value of the contact Here the physical and the A ka, and input parameters are a, Ro, and e. constant. constants are The configuration of the system is found by minimizing the total energy case
the
particle
is
E
with
the
constraint
of
volume
=
Eel
conservation, Ead
E~d
+
V
"
=
+
El
(5)
constant.
~Asad
(6)
ADHESION
N°11
OF
SPHERES
LATEX
TO
VESICLES
LIPID
1655
membrane the solid surface. Sad is the energy gained in adhering the on line which from the fact that surface El is a contact energy term, comes area. infinite. We will skip this term the membrane along the contact line cannot be
/solid
membrane
is the
the
of
curvature
following,
in the
conditions of our We justify this point in because it is negligible small in the experiments. Appendix A. Briefly, we find that the ratio of the line energy term to the surface energy terms fi(Rola~), where kc is the elastic in equation (5) is on the order of constant curvature of the membrane. With kc m 10~~~ erg, k~ m 200 erg /cm~, Ro 30 ~Jm, a 8 ~Jm, we find only a 10~~ fraction of the total energy. that the line energy contributes =
Thus
may keep and Ro can
we
0
variables
only Est
Ead in
and
used
be
define
to
(5).
equation
As
configuration
the
=
droplet, the of 0, we solid sphere
liquid
the
with
of the
Instead
system.
of the vesicle and interpenetration (see Fig. the length defined which convenience, For interpenetration is 1). contours as za, Actually, z is the parameter which is most directly felt by number. makes z a dimensionless The following formulas will be deduced for the case the experimentalist in the real situation. when the sphere is much smaller than the vesicle cf. Fig. 1; (a/R) ~ 0 limit). In this case of the and the radius due to the sphere segment the membrane increase of the vesicle curvature features penetrating into the vesicle can be neglected. This will allow us to develop all relevant
chose
variable
the
of the
without
scenario
the
of
need
mathematical
extensive
an
work, this simplified
this
in
interest
the
represents
which
z,
adhesion, for values of a /Ro smaller than about the rigorous numerical solution is given for any Ro » a (Fig. 1b), we simply have: z
Moreover, which
the
allows
us
drop the
to
of
constraint
S
volume
which
inject into equations 0, which gives:
we =
that,
a
with of the
version
definitions
the
Young
the
adhesion
Figure energy =
slightly numerical from
shows
the
A
200
the
is
ze
found
from
~~ ~
~~~
and ~ (Eqs. (7 and 4>, respectively>, equation (9> is just z (Eq. (1)). Equation (9) expresses the balance the between
of
equation
calculated and
dyne/cm, lower ka 11
the 145
The
results.
intersection
approximation,
penetration
~~
~R~ ~ ~~
~~~~~
~~~~~
force,
densities
2
(8a> (8b>
~~
fad
ka
4~R(e
equilibrium
The
2~o
~~ and
penetration of the sphere is negligible, Simple geometry gives: conservation. the
force
elastic
When
#
~
Note
radius.
vesicle
2~a~z,
(3, 6, 5>.
~~
sphere and
of
=
Sad
dE/dz
a
(7)
~a~z~
So
of
0.
cos
to
materials
the
description of the partial demonstrated in Appendix B, where
combination
due
For
treatment.
reasonable
This is
0.3.
1+
=
volume
vesicle
in the
variation
gives
consideration
I.e.
to
surface
value
reported
fei(z)
excess
equation
areas
f~d(z)
(10>, while
SOPC
but
equilibrium and
forces
elastic
for
dyne/cm [18j>. different
of
and
adhesion
relative
(iob>
2~aA. "
this
a
does
values
Dotted
curves.
lines
represent
a
ze
lines
a/Ro
DMPC
make
not
of
sets
ratio
size
bilayers [18j.
penetration solid
different
for
e, for
are
great
the
=
exact
We
0.2.
membranes
to
took
have
difference
determined
correspond
adhesion
assumed
in
a our
graphically the
numerical
Ro
"
cc
solution
JOURNAL
1656
PHYSIQUE
DE
N°11
II
~el
C 2
~ y
d
o-o
z
enetration
Fig.
size
a
of the
Plots
2.
for
a/Ro
ratio
adhesion
f~d> and Dashed
0.2.
=
(- fei>
dilatation the
are
curves
2~aA (Eq. (lob» limit, fad(z> horizontal is just a correspond to A 2,1 and 0.5 erg/cm~, respectively. following equation (10a>, with k~ 200 dyn/cm and e =
the
penetration solutions
(numerically computed> graphs,
exact
found
is
from
shown
are
the
open
as
Appendix B). Clearly,
(cf.
conservation,
Figure
3
is
rather
shows
the
of the
intersection
circles,
and
the
small
values
error
for
this
of ze
and
the
to
dashed
0il, +1%,
f~d(z>
lines from
-fei(z>
are
left
right.
to
The input parameters. fei(z) graphs. Approximate
same
bold
as
co
three
The
-lit, the
solutions
exact
due
=
~
tilted
with
particle penetration (z>, In this approximation. lines, from top to bottom,
the
vers~s
Ro
the
The
=
are
of
line.
=
lines
forces
result
graphs
The
solid
equilibrium
(Ro
co)
~
circles.
approximation,
influence
the
I-e-
of
volume
ratio.
size
and of the
membrane
calculated A and e, with the procedure. We numerical adhesion and penetration ze decreases energy (A>, the
tension
and
for
~
that,
observe
the
different for
tension
values
of
a/Ro,
given vesicle (Ro, e)
a
when
increases
~
the
singularity along each curve from the fact that ze has to be comes smaller than 2. The left branch corresponds to complete wetting (ze 2), I.e. the particle In the limit Ro ~ cc, total demands is totally encapsulated by the membrane. encapsulation A > Ae, with: sphere
size
The
increases.
=
2ka[(a/Ro)~
Ae regime,
this
In
~
The
right
tension
l~
ka
=
encapsulation
Ro
energy,
~
cc).
(11a)
given by:
is
~
(total encapsulation,
e
Ro
of each
branch
corresponds
curve
to
partial wetting
or
Ro
~
cc).
encapsulation.
(11b) For
e
=
0,
one
equations (4 and 9):
from
finds
membrane
the
(total
El
=
2/3 ~
point,
At the
transition
direct
consequence
of
a
/Ro (cl
We tension
recall
of the
the
(partial encapsulation,
~
k(/~A~/~
=
2Ro tension
takes
on
the
value
Young equation (Eq. (1)) and
~*
Ro
A/2.
~
cc).
Note
that
to
be valid
(12) this
result
is
any
value
=
therefore
has
for
a
Appendix B>. that
which
is
this
model
driven
by
is
purely thermal
mechanical
in
undulations
[17].
that
membrane it ignores the part of the contribution Taking the entropic to
ADHESION
N°11
SPHERES
LATEX
OF
TO
A=coast
VESICLES
LIPID
1657
e=const
(
I %
I
I ~ I
-
~ i
i
~ 2
o.oi
4
6
2
o-i
4
6
2
o.oi
i
4
o-i
a/R~
i
a/R~ 2
d
d
I
~
o.oi
~
o-1
~
o.oi
i
~
b)
a)
e
size ratio (ze> verms -0.5, -0.25, 0, 0.25, 0.5,
=
panel b>, the initial excess o-1, 0.2, 0.5, 1, 2, 5 erg/cm~,
elasticity
membrane
into
lit,
from
area
is>
from
the
bottom
by Figure
indicated
Figure
3
lateral
As
tension.
which
leads
3
plots.
to
leads
to
adhesion
to
~*
m
situation
is
taken =
sphere
and
erg/cm~,
I
equilibrium while
e
is
pen-
varied:
everywhere 200 dyne/cm. graphs correspond to A
top. k~ is taken different 0; the
=
=
top.
large
deviations
step is
tension
A is
see,
the
in
than
smaller
0.5
plots of Figure 3, in the dyne /cm (data not shown>.
with
estimated
dyne/cm.
0.5
experimental
our
This is
will
we
=
la>
tension is
decreasing values of a/R is less drastic than values shown in the lower panels of of this, the penetration undulations, which keep the bilayer under due to thermal
Because
over-estimated.
are
to
chosen
is
bottom
account
after
tension
membrane
panel a>, A
In
expect, in this regime the drop in
As to
in
the
of the
(a/Ro>.
In
region where
graphs
computed
Numerically
3.
etration
i
a/Ro
a/R~
Fig.
~
o-1
on
Therefore
well
the
in
the
the
tension
mechanical
/cm~
experiments, our encapsulated sphere (or "strong" adhesion> regime, as we
order
of 1 erg
for
in
partially
a
assumed.
The urations
the a
curves
in
and
for
tension,
which
liquid droplet. exceeds
that
such
Figure testing
some
3
makes An
value
amplitudes
used
be model.
interpret
to
An
obvious
the
feature
problem qualitatively specificity of the vesicle
which is can
be
on
reached
the
order of with
energy
a
experimental of
these
membr~ne
few
sphere-membrane
curves
from
different
the
other ~c,
can
the
is
that that
the
it
may
rupture calculations
dyne /cm [18j. The
densities
on
the
order
config-
of non-constancy of wetting of a solid by
is
of
an
erg
/cm~.
when
indicate
~
PHYSIQUE
DE
JOURNAL
1658
N°11
II
DYNAMICS. In this paragraph, we want to make some prediction dynamics of particle encapsulation based on the partial wetting picture. Essentially, will just adapt the theory that the spreading of a liquid droplet on a solid surface describes we in the partial wetting regime [19, 20j. volatile. dissipation and the liquid is non when the third phase is a vapor In the simple case liquid wedge line. In situation, from the flow gradient the the just in contact our comes near therefore of have such interface made and wedges. both sides of the moving water, two we are penetrations (z « 1). In this To keep the description simple, we will consider only small which corresponds to a very large contact angle (0 Q ~) we expect large flow gradients case, essentially in the wedge between the membrane and the non wetted part of the solid surface. calculation We may then the of the dissipated P, to this region. restrict power, Figure The of sketched 1b. We denote the situation is in distance interest to the figure r (which center). particle axis through the We the membrane symmetry passes suppose approximately flat, I-e- Ro » a. We define fir) as the thickness of the wedge at distance r. In the limit of a small penetration, the radius of the line is given by rc (2za~) ~/~ Of course, contact ((rc) distance to the membrane 0. We denote x the in the wedge: 0 < z < (. that the sphere is fixed and that the We membrane particle suppose moves up to wet the For small surface. and consider that the penetrations, the wedge is very narrow we may flow velocity u is everywhere perpendicular to the symmetry axis (lubrication approximation). Following Charles and Mason [21j, we write: 2.2.
ENCAPSULATION
about
the
=
=
Ulz,r) U is
second
a
ditions:
(I)
is the
0
=
dS/dt.
=
on
the
velocity
material
2~rum
polynomial
order
u
This
solid
(z
=
membrane,
Ulz,f)~fi(r> coefficients
whose
in z,
surface
of the
=
(>; (it) and
u
determined
are
am
=
found
is
(13>
the
on
from
the
by
boundary
the
ix
membrane
=
am
condition,
conservation
mass
con~
Here
0>.
gives:
um(r)
~2
(14)
~zl.
=
r
We
find:
thus
VIZ, r) An the
=
z~filr)iflr)
xi
explicit expression for ~(r) is found from wedge. Again following reference [21j, we
Iv =
2~rdr~fi(r)
varies
in
balance
and
This
time.
leads
flux
the
variation
With
the
expression
full
is
of
where
i~
is the not
of
the
consider
incompressibility
volume
between
r
of the
and
u(x, r)
/()~, hand,
at
now
+
fluid
dr,
moves r
up,
+ dr.
in
I.e.
dv This
z).
d12
=
r
l16> calculate
we
can
~
Ill
the
power
dissipated
in
the
given by:
~
In fact, it is
condition
the
(15)
Q 2~r f)u(x,r>dx. the membrane Because compensated by the variation of Q between r and
to:
which
fi).
Umlr)(I
=
is
~blr)
wedge,
+
fluid
viscosity,
rmax
to
know
necessary
divergence of au lax
~
near
the
contact
IT is
a
~~~~~
large scale cut-off,
rmax
line.
~
~~'
on
~~~~
the
order of the sphere radius. determined essentially by the
precisely,
because
P is
Because
of this
divergence,
we
impose
a
lower
limit
ADHESION
N°11
(,
to >
r
rmin
the
on =
rc
order
of
+
SPHERES
LATEX
molecular
a
Il.
OF
This
size.
TO
VESICLES
LIPID
integration
the
restricts
1659
in
equation (17)
to
yields:
This
P
m
~~~~
4~J~a3
~~'
+
in
)~
Equation (18) is valid for za » I, I,e, for penetration The dissipated balanced by the work per power is forces acting on the membrane: P
depths unit
much done
time
(18) larger than a molecular by the adhesion and
size.
elastic
/dt
-dE
=
l~@)
(19)
with
-2~a~d(A
dE/dt The
equation of
motion
=
(for Ro
~z>
~
Equation (21) gives in
the
6i~aln(~@/l)
analytical
an
general
more
(20)
cc>.
then:
is
~
cast
~
for
solution
z~~~3
~
model
our
in
the
~~~' limit
a/Ro
of
~
cc.
It
can
be
form:
~
(3
3z +
~~~~ (@)
~~~~
ln
because the wedge penetration velocity id) starts from zero at the onset of adhesion (P (21)) diverges). of (or infinitely when Integration (22) Eq. equation is 0 narrow z leads to a characteristic S-shaped trajectory. An example of encapsulation dynamics which computed from equation (22> is displayed in Figure 7c (solid line), for comparison with we The Figure shows the real record (details are given in Sects. experimental 3 and 4>. an d(0) z(t)a of the Latex bead relatively to the vesicle, in correspondence displacement d(t> 19 ~Jm. with the experimentally recorded signal (see Sect. 4). In this exanlple, a 7.7 ~Jm, R with the We took e 0, and adjusted A (here A 0.8 erg /cm~), to make d(t ~ cc> coincide Of course, does not need to be experimental value. The cut-off chosen equal to 1 nm. was precisely defined, because it appears only in a logarithm. The computed curve starts at some initial penetration (zs > 0), whose value can be chosen rather arbitrarily. In practice, zs has molecular length, and can be taken on the order of the minimum to be much larger than a displacement that can be experimentally detected. The encapsulation time ren~ may be defined as the time elapsed between, say, 0.1 ze and makes Note that the lubrication 0.9 ze In this example, renc m 1.8 ms. approximation sense only for small values of z. Consequently the asymptotic regime of z(t) is probably poorly described by the model. As a whole, we may just retain our that A on the order of estimate The
=
=
=
an
/cm~
erg
3.
Materials
3. I.
SAMPLE
method
of
leads
to
and
renc
the
on
order
of
a
millisecond.
Methods
PREPARATION. electroformation
Giant
vesicles,
[1, 2]. All preparations beams (Fig. 4). This
manipulation by laser optical path and is equipped
for
=
=
=
with
two
parallel
a
generated by the used glass chamber are chamber (THUET optical cell) has a mm cylindrical (3 0.8 mm) platinum electrodes. few 10
made
microns
in
directly
size,
in the
are
JOURNAL
1660
PHYSIQUE
DE
II
Nall
~WV
mjectton elec~ode
needle
Latex
beads
Fig.
The
Scheme
4.
sample
of the
(axis
separation
to
chamber.
distance>
axis
pure
(MILLIPORE millio).
water
periments
DMPC
on
temperature
of
DMPC
(ITO)
covered
membranes
samples
glass slide
were was
the
between
L-a-dimyristoyl phosphatidylcholine L-o-stearoyl-oleoyl-phosphatidylcholine in
clarity, spheres
for
Note:
SOPC
membranes
carried
adjusted
electrodes
(DMPC; (SOPC; Avanti out
well
in
above
30
about
is
Avanti Polar
fluid °
Lipids>
state
C by
are
not
3
mm.
at
which
For
means
of
We
an
this
or
reason,
Indium
used
swollen
were
temperature.
room
too.
scale.
to
Lipids>
Polar
fluid
are
the
vesicles
and
Tin
Ex~
the
Ox~
conducting layer was connected (TC) which regulates the temperature by sending current pulses of different thermostat to a length through the ITO layer. The thermoresistance chamber yields the located inside the feedback signal. As usual in the method, an AC field is applied to the elec~ electroformation trodes, to generate a cluster of vesicles in contact with platinum. Giant vesicles are found at the periphery of the cluster. These ideal samples for because vesicles experimenting, not are they are interconnected, and consequently the topology of the membrane is not known exactly. Fortunately it usually happens that a few vesicles disconnect from the cluster spontaneously. takes place a few hours (up to a few days) after of the vesicle This formation cluster. process "Free" vesicles easily identified by the fact that they can be moved on arbitrary distances by are attached convection, in practice by gentle injection of water into the cell. Conversely, vesicles be moved only on short cluster distances and back to their original position to the can come after the injection has been stopped. For adhesion diameters between 2 and experiments we used Latex spheres with in the range 20 ~Jm provided by Polysciences. We do not know the charge density on the sphere surface. estimated We just the surface potentials of individual spheres in the 12 to 20 ~Jm diameter (Polybeads Polystyrene found mV solutions and -80 as range an average [22]. From the stock Microspheres, 2.5% solids in deionized water) we made very diluted suspensions in millio water (conc. 0.02% solid> which were then injected into the chamber via a small stainless steel needle, (Fig. 4>. The injection rate was chosen electrodes about 20 mm whose tip was apart from the small enough to avoid damage of the vesicles. In typical conditions, we were able to optically
ide
in
contact
with
the
chamber.
The
ADHESION
N°11
Fig.
Geometry of the along the
5.
radiation
catch
Special
contact.
trap and
the
"in
care
first
with
contact
sphere
very
towards
taken
was
a
to
trap.
laser
The
flight"
transported
then
was
before 3. 2.
injected sphere
SPHERES
LATEX
two-beam z-axis.
pressure
an
sphere
OF
the
avoid
lipid
is
near
The
trapped
the
to
electrodes, any
TO
VESICLES
LIPID
1661
values the illustrate gray between "dark" the two
injection
tip of the where
pollution of
the
sphere
The
tube.
was
selected
surface
with
vesicle
a
amplitude regions.
to
lipid
of
the
trapped produce a molecules
membrane.
Manipulation of Latex spheres is performed by means of an optical OPTICAL SET-uP. fairly similar to Buican's original design [23]. For a detailed description, see Angelova Pouligny [24]. Briefly, the glass chamber (Fig. 4> is held horizontally inside the optical
levitator.
position
Its
present
report,
are
focused
are
part of
it
inside a
horizontal
in
(Unidex 11, Aerotech).
The
suffices the
classical
to
directions
know
ix,
y> is
position (z)
vertical that
two
is
vertical
by means of motorized manually. For the purpose propagating coaxial laser
controlled tuned counter
by means of two microscope lenses. microscope for observation of the sample. cell
Compared
In
the
The
same source
time,
feeding "optical
stages of the beams
these
lenses
the
set-up
widespread tweezers" design [25], our set-up disadvantage of being more complex, but offers a much longer working distance (> 4 mm), and lower power densities in the sample (< 10~ W/cm~ in our experiments>. The beam-waists of the two beams are slightly separated longitudinally to build small region where a sphere can be stably trapped (Fig. 5>. The forces acting on the trapped a particle have been studied in details elsewhere [26]. There is no detectable laser induced heating due to of the particles in bulk radiation The horizontal force can Forces water. pressure. are For microscope be varied from about 1 to 100 pN in our conditions. observation, the lower while the objective (LD Epiplan 50 x /0.5 cc; Zeiss> is employed as a condenser one upper (LD Achroplan 40 x /0.6cc; Zeiss) is the observation objective. Images are captured by a CCD (C2400, Hamamatsu) and recorded by standard video equipment. Phase is contrast camera well-suited for observing the vesicles (the vesicle shows up as a sharp dark line> but is contour is
a
continuous
wave
argon ion has the
laser.
to
the
now
PHYSIQUE
DE
JOURNAL
1662
PSD
p
Principle
Fig.
6.
(see
Sect.
inconvenient
visualize
to
Fig. 7>. For this
better in
3.3.
DIGITAL
We
space.
a
on
reason
excursion
by
position
of the
means
written
software
quency and to
and 8 bit
contour.
PROCESSING.
IMAGE
spheres (images
sensitive
detector
interference complex patterns, amplitude contrast, which is Usually the optical trap and then the particle is held bring a vesicle in contact with the particle.
Latex
sometimes
chamber
the
move
Digital
the
we
particle
the
visualize
to
fixed
stalled
sphere
of the
measurement
3>.
rather see
of
N°11
II
to
the
switch
We
used
are
system
to
commercial
a
image
processing
program resolution
system
(DEC 3000/Axp 600 equipped with a J300 board). (Language C++) allowed us to save digitized frames at
Workstation
on
the
hard
disk.
The
same
program was Because of the
used to load
A
video
in-
home
fre~
single frames
bright image of the Latex algorithm the tracing for detecting vesicle hardly applicable. We helped is contour a ourselves by determining six points on the vesicle by hand and subsequently by fitting contour circle to these points. The procedure provides the position and radius of the vesicle (within a spheres. ~ 0.5 ~Jm> and was applied in an analogous way for determining the position of Latex determined The radius of the Latex sphere was separately, prior to the collision with the vesicle. measured Knowing the sedimentation velocity of the particle in bulk water. In each case, we allowed the Latex density (1.05 g /cm~) and the water viscosity (i~ 0.01 poise>, this procedure of the pixel resolution the particle radius Calibration within ~ 0.2 ~Jm. in x and to deduce us revealed direction that pixel corresponds to an area of 0.154 ~Jm x 0.162 ~Jm on the one y specimen. determine
vesicle
and
contours
bead
positions.
spheres,
=
As explained in the following section, particle dis~ happens much faster than the video rate. To resolve this deflection of the in time we set up a procedure based on the levitation laser beam. As shown in Figure 6, when the sphere horizontally, laser the beam deflected lateris moves ally. The method excursion of the laser horizontal amounts section in to measuring the cross a plane (P) located at some distance (20 ~Jm) above the sphere. In reality, we picked-up the small observation amount of green light (1 mW> that goes through the system and built a real image 3.4.
(P')
of (P> in
measured This
SENSITiVE
POSITION
placements in displacement
in
device
the
a
DETECTOR.
first
adhesion
separate
arm
step
near
the
video
camera
(P'> by means of an analog position yields two signals proportional to the
[22]. The
sensitive
excursion
coordinates
of the
laser
spot
was
(PSD C4674, Hamamatsu>. of the spot in (P'>, with the zero
detector
ADHESION
N°11
taken
the
at
of the
center
digitized by
were
of the
detector
points
data
AD
converter
package (Testpoint at
rate
a
was
Version
The
1663
signals ix Instruments).
output
National
checked
VESICLES
LIPID
to
be
1.1c, CEC)
clearly allowed
and y> of the
PSD
The
time
shorter for
response
than
0.1
A
ms.
recording
the
of
the
Hz.
1100
to
up
TO
area.
(PC-LPM-16,
converter
connected
software
commercial
sampled
an
SPHERES
LATEX
photosensitive
PSD
AD
and
OF
particles of diameter 2a m 15 ~Jm. We made tests by means of calibrated displacements d of the particles horizontally (this was done by moving the cell by of the that the signal motorized switched off). We found stage, while the trap was means V(d) was proportional to the particle displacement d provided that d < a. Calibration was performed for every sphere separately. 20 mV/~Jm was a typical value of V(d) Id. The signal (standard deviation of the detector signal when a sphere is trapped) was about 3 mV. noise determined the stiffness (k) of the optical trap for each particle. Using the same system, we When the laser is switched on, the particle back towards the beam axis under the action moves of the optical force fi id> m kd. The trapped sphere can be described as a highly transverse damped harmonic oscillator. The inertia effect can be neglected and the distance d(t) decreases ilk determined from the characteristic exponentially in time. k was time of this movement, r coefficient if measured The friction where ( is the Stokes 6~ i~ a) of the particle in water. k in fairly good values of k (for instance 2 pN/~Jm for a 6 ~Jm> were agreement (within 20%) with those computed using the generalized Lorenz-Mie theory [27]. In the experiments of the solid particle, as given by the PSD, was with vesicles, the excursion not exactly equal penetration length az it>, because both the particle and the vesicle move when adhesion to the that dv Id G3 a /R at equilibrium (dv is the happens. From the video recordings, we determined valid at any time, vesicle displacement at equilibrium>. We supposed that this property was We
used
method
this
with
=
=
=
=
I.e.
we
used
the
correction
=
d(t>(R
CALCULATION.
NUMERICAL
3.5.
az(t>
computer (Macintosh,
/R.
+ a>
Numerical
PowerBook
180> with
a
calculations
software
were
environment
performed on a for data graphing
analysis (Igor Version 1.2, Wavemetrics>. For numerical integration of differential algorithm of fourth width control [28]. used Runge~Kutta order with step a we
Macintosh
and
data
equations,
,
Results
4.
As
we
steps,
explain throughout this section, the solid sphere-vesicle namely adhesion, ingestion, expulsion and re-capture.
features
interaction
will
different
place when the sphere gets in contact with the by optically manipulating the particle near distance is slowly decreased, until from the outer Then this surface. the vesicle, a few microns adhesion happens. At this stage, the particle makes a horizontal jump out of the optical trap in by the video sequences of Figure 7a, This event is illustrated the direction to the vesicle interior. The positions of the sphere immediately which shows two examples for the adhesion event. before and after the frame (second frame in each row>, which adhesion means on same are seen than a video period (40 ms>. The graphs in Figure 7b show that the event is definitely shorter from the measured radius R and the distance d between the sphere and vesicle the centers Characteristic for this event, no change in the vesicle video recording in the second example. the onset indicates could be detected, while the discontinuity in d at flame 36 clearly radius (denoted f in the graph>. Figure 7c is a high temporal resolution version of the of adhesion Here the characteristic time of the position detector. recorded by means sensitive event same surface
renc
f
ADHESION.
4.1.
is
lipid
of the
about
4
ms.
is the
membrane.
Other
event
which
This
is
measurements
takes
achieved
in
similar
conditions
gave
values
on
the
order
of
a
DE
JOURNAL
1664
()flj~ ii-
PHYSIQUE
(~flfl~
/M÷i~,1 ~~~l@'
'L.
''~fill~ ~l)()k f***h~ ~
~~'~
~
~j
.#
:j)jjjjjjjj)~)
~( j
~~
~
im~
'(j_gf
~Tlflli
'[till
jjj- j
iii ..~lji1(ljt~jj~
jfrr~
".h
~"lift~°. I)i(m.
~
j~jj[j(j
i[yin.(~
ji
~ig'pj%ii~
?~
N°11
II
j.:jj'iiii, ,jjiiiji~ _j-
~..
, ~
a)
b)
~
~
-
-
fl~ '
j
~$
.
20
~N
#
t
frame
[ms]
d>
c>
Fig.
a)
7.
First
row:
The
video
to
the
Ro
frames
was
Note: recorded
of the ad
an
hoc
"adhesion" 22.4
=
40
distance
and
event
deviation line
~Jm,
sequence.
same
standard dashed
6.85
squares)
bold
whole
c)
=
subsequent
between
(R(t),
examples of
Two
a
ms.
detector
b)
The
between
frame with
step of Second
~Jm.
PSD.
signal.
The
(constant friction)
sphere
Latex
row:
a
=
onto
7.7
~Jm, to the
graphs
a
Ro
SOPC =
19
vesicle
(T
~Jm.
Time
=
24
°
C).
interval
correspond second example: vesicle radius (d(t), open circles) for the and sphere center correspond to the number within the video sequence.
vesicle
numbers the
a
center
Sampling solid model
line
(see
rate:
1000
corresponds Sect.
5).
Hz. to
The
error
paragraph
bars 2.2
represent
theory,
and
the the
ADHESION
N°11
OF
SPHERES
LATEX
TO
VESICLES
LIPID
1665
Ill I
d +
))
ii, 11
~w
§i 3
4
" 5
78
6
2
3
5
4
3
6 7 8
5
4
6
78
0.I
Fig.
Analysis giant SOPC
8.
amellar
correspond
vesicles which
(e
3
4
5
6 7 8
(T
=
24
equilibrium °
C),
for
penetrations
different
values
of
of
spherical (no visible
initially
were
0),
I
a/R~
b)
experimental
of the
vesicles
to
2
0.I
a/R~
a)
Latex
the
spheres
into
isolated
unil-
(a/Ro). Bold circles fluctuations), the triangle to
size
thermal
ratio
pre-tensioned by means of an already sphere. a) Measured depth after adhesion penetration The solid line the step. represents computed penetration depth for: ka 200 dyne/cm, A 1 erg/cm~ and e 0. b) Ratio of vesicle radii R/Ro after and before sphere The solid line adhesion. corresponds to the trace in panel a). Each configuration which led to ingestion is marked with an arrow, whose tip points to the final value of R/Ro (cf. Tab. 1). which
one
flaccid
was
>
and
squares
to
vesicles
which
were
adhered
=
few 1
milliseconds
reached
The
too.
~tm/ms (within
the
equilibrium
its
velocity d(t)
radial
resolution
=
=
of the
=
ad
detection)
(t) changes and
slows
instantaneously down
to
zero
when
from 0 to
about
sphere
the
has
position.
between the lipid important to realize that the existence of an adhesion particle is not influenced by the laser beams. We checked this several (about 20 pm) above a vesicle and then by distance times by bringing a particle at some switching off the laser trap. We observed that the particle would fall down, hit the surface of the vesicle, glide down for a few seconds, and make a characteristic brutal jump towards the vesicle interior, exactly in the way we just described. At
this
stage, it is
membrane
and
Latex
the
adhesion, the solid sphere can still be manipulated and moved by means of the optical partially encapsulated spheres (z < 2), as in the examples shown in Figure 7, the particle could be brought to any point on the surface of the vesicle, as one might expect since the membrane In this kind of manipulation, the particle was moved relatively to the is fluid. membrane. Indeed, when (or even other particles) were attached internal structures to the vesicle membrane, we could check that the manipulated particle did move relatively to these which proved that the vesicle did not whole. Movement structures, transverse to the as a move membrane is impossible with optical forces (< 100 pN)), which that the membrane-solid means surface angle, or penetration degree ze, is experimentally defined independently of the contact optical trap or of the particle weight. Pulling the particle with the optical trap in the direction opposite to the vesicle center makes the sphere-vesicle complex move as a whole. Interestingly, this is a further that the selected vesicle free. Moreover hydrodynamic drag brings test was the vesicle and sphere centers in the horizontal perpendicular to the plane automatically, same observation direction. This makes the top view of the sphere relatively to the vesicle free of parallax error, a condition which is required for a correct of the penetration. evaluation After
trap.
With
Figure following
8 the
shows
the
criteria
results
from
described
in
experiments Section
3.
performed Bold
circles
with
unilamellar
correspond
to
SOPC
vesicles
vesicles,
which
were
JOURNAL
1666
initially spherical, theory set out in well.
The
2.
figure
same
(triangle),
did
I.e.
Section
had
i.e.
Assuming
shows
tensioning of the membrane, The particle penetration is larger expected.
performed
We
same
the
adhesion
The
of
If the
that
adhered
the
to 1
or
attached
are
behavior
vesicle.
In
z)~~
to
a
the
Within conserved.
to
solid
the
to
line
solid
in
the
the
simple
vesicle
the
confirm
sphere
adhesion
Figure 8b Clearly, for
60
DMPC 30 °C
PARTICLE
4.3.
generally
is
the
to
@
EXPULSION
stable
vesicle
(e.g.
for
AND
all
However
contour.
@
RE-CAPTURE cases we
given observed
The
I).
Tab.
in a
3
The
ingested particle internally configuration in quite partly expelled out of "expulsion-recapture"
situation
with
particle
stays
of
destabilization
this
an
tangent
ingested particle can be (step @) and move back in again (step @). This the vesicle interior ends in a configuration which looks like partial penetration, as in the adhesion step sequence (step j). Note that steps @ and @ are definitely slower than steps @ and U that the that in the final between particle radius decreases monotonously steps @ and @ We observed what finiteness partially the wetted, contrary to configuration the re-captured sphere was not wetted by the lipid material. of the apparent angle might suggest, but probably totally contact two-vesicle sphere complex We went to this conclusion by repeating the procedure to build a observed that the portion of the re-captured sphere did not adhere in Figure 9. We outer as surface that this part of the solid second vesicle, which not "dry" in the was sense means a on coated by lipids, as it was re-captured fully before. probably, the sphere defined Most was we after ingestion. These experiments version of the In some experiments, we observed a "condensed" sequence. illustrated in Figure 12. small spheres (a/Ro < 0.1), as in the example carried with out were examples.
few
a
As
shown
in
Figure
11,
the
particles penetrated completely inside the vesicle in the first step fi as one might expect Expulsion @ followed nearly immediately, and the small sphere (complete adhesion). stabilized penetration value zf (zf 0.4 for the example of Figure 12, particle was at a final 0). observed a complete expulsion zf in cases we some because the giant vesicle to which The example shown in Figure 13 is perhaps very particular One of them is clearly visible contained few smaller vesicles. adhered the solid sphere was a after adhesion and nearly seconds ingested about The particle 20 in the video was sequence. Interestingly, the smaller vesicle was expelled too, at immediately expelled and re-captured. The
for
a
=
=
the
4.4.
ii)
same
FINAL
the
as
could
Latex
sphere.
REMARKS
Experimentally
distinct we
time
from never
an
the
2 was sphere-vesicle configuration after an "ingestion" with zf not 2. For lipids in the fluid state, in the complete wetting regime ze ingested (or more generally speaking a fully encapsulated sphere) an
"adhesion" detach
=
=
JOURNAL
1670
DE
PHYSIQUE
II
N°11
a)
~ ~
~ p
~ ii
b) 0 #
Fig. Flame
features to
the
of a Latex sphere (a 7.5 ~tm) into a DMPC vesicle (T 30 °C). a): Video highlight rapid sphere superimposed two successive video frames. movements we included in every panel. b): R(t) (filled squares) and d(t) (circles). The sequence f ingestion @, expulsion Sand re-capture @ which correspond adhesion process: of the video micrographs.
Penetration
11.
Note:
sequence.
numbers the
To are
4-step
different
rows
bame
=
=
ADHESION
N°11
OF
SPHERES
LATEX
TO
VESICLES
LIPID
1671
k
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