Tokyo ISSP/SOFT2010 Workshop

Dynamical curvature instability
controlled by inter-monolayer
friction, causing tubule ejection inmembranes
Jean-Baptiste Fournier
Laboratory « Matière et Systèmes Complexes » (MSC),
University Paris Diderot & CNRS,France.
M. I. Angelova, A.-F. Bitbol, N. Khalifat, L. Peliti, N. Puff

Tokyo ISSP/SOFT2010 Workshop

Question :Instantaneous local modification of the lipids
of one of the two monolayers:
what happens?

e.g., local pH variation.
M. I. Angelova, N. Puff et al.

Tokyo ISSP/SOFT2010 Workshop

Outline
1. Review of the elastic and dynamical models of
membranes and monolayers.
2. Experiment by M. I. Angelova, N. Puff et al.
3. Theory of the curvature instability caused by alocal modification of the lipids of one of themonolayers
4. Comparison with the pH-micropipette experimentof M. I. Angelova, N. Puff et al.
5. Non-linear development : tubule ejection

o presented.
Tokyo ISSP/SOFT2010 Workshop
1. Review of elasticity & Dynamics
Several “pulse”experiments
were conducted on three different GUVs
κ 2
b
f
=
σ
+
c
−
κc
c
.
0
0
mplete analysis of2 these experiments is presented in Sec. 5.

Helfrichmodel
nergy density depends on the membrane curvature c, which is defi

P. Canham (1970), W. Helfrich (1973)
Theoreticalmodelcurvatures on the bilayer midsurface. The const
the two principal

b
,
κ
and
c
respectively,themembranetension,its
bending
0. Free energy
0 denote,of a bilayer membranend its spontaneous curvature. The resulting membrane free energy! per unit area f of a lipid membrane is well described on a lare free energy
F = dA fκ 2c2
f = σ0 + c − κcb0 c .
c1
2
is free-energy density depends on the membrane curvature c, which is desum of the two principal curvatures on the bilayer midsurface. The con
nstants σ0 , κ and cb0 denote, respectively, thetension, its bendin
c =membranec1 + c2✤ Bilayer
structureneglected
nstant,and its
spontaneous
curvature. The resulting membrane free energy
c1 c2 discarded (Gauss Bonnet)✤ Gaussian term !
✤ σ0 sets the area constraint (hides lipid density)
F = dA f✤ b spontaneous curvature of the bilayer (if asymmetric)c0Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics

Area Difference Elasticity (ADE) model
S. Svetina & B. Žekš (1989) –U. Seifert, I. Miao, H.-G. Döbereiner & M. Wortis (1991)✤ Lipid density is involved, but in a global manner✤ ∆A related to theintegrated curvature

Preferred (relaxed) area

A+
0
−
A0
−
∆A0 = A+
−
A
0
0

∆A
A

F =

! "

#
κ 2
k
2
b
σ0 + c − κc0 c dA +(∆A − ∆A0 )2
4A−

Fixes A = (A0 + A0 )/2+

Cost to deviate from

∆A = ∆A0

=

I (z

I

Tokyo
WorkshopdA E ISSP/SOFT2010(2H)2+ k [(p' + 2dH)2+ (p- - 2dEO2] .
2

1. Review of elasticity & Dynamics(1)

es from the bending energy of each monolayer, with the usual bilayer
We have implicitly assumed that the monolayers are symmetric and haveure Cg")<< d -'.) As written, F is a functional of the membrane shape andBilayer curvature—density elasticity.

U.small
Seifert& S. A. Langer(1993)
d only in thedisplacementsof a nearlyplanar membrane. Letting
ne lie in the (z,y)-plane, we describe its fluctuations in the Monge

density on midsurfaceρ=−1equilibrium density

ce

urfa
s
l
a
r
eut

n

e

E
RFAC
U
S
MID

F =

!

dA

"

k #$

κ 2
c +
2
2

&
%
$
%
2
2
ρ+ + ec + ρ− − ec'ometry of a bilayer membrane. The circles with squiggly tails represent the lipid
lines are the neutral surfaces of the monolayers, on which the densities #* ared line is the midsurfaceof the bilayer, on which the projected densities $' and theOn the «neutral surface» density and curvature are independent
defined.
ities p' are ✤variables (decoupled).
✤ Not if they are defined on the «mid-surface»✤ Inter-monolayer friction and membrane bending : MID-SURFACE
E. Evans & Y. Yeung (1992)hat,
and ρ̄, f andwe
ρ arethe freeenergy so that the integral of the Gaussia
byunlike
the f¯instability
wish
to study,
Tokyoper
ISSP/SOFT2010
Workshop
1. Review of elasticity & Dynamics
e massactual unit areaof the monolayer, andand thus its contribution to F , remains constant by virtue of the Gr projected unit area. We use for both monolayers
theoremnsity
and the [20].
principal curvatures defined on theurface S To
of thebilayer, so
that
thebilayer
curvaturesstructureareaccount
fortheof the membrane, its free-energ
on to the two monolayers.
bewrittenas thesumregimeof thedensities of the two monolayer
willcanconsiderphysicallyoffree-energy
curA.-F. theBitbol,L. Peliti &relevant
J.-B. Fournier(2010)
radii
thickness.
be muchnotedlarger
f + thanandthef −membrane. Since thecurvature c is defined on the bilayer mid
ll alsoto smallvariations of thetherestrict
sameourselves
for bothmonolayers.
Furthermore, we assume that the two
y around a reference density ρ0 . Note that it canhavethelipidfromcomposition
before the onset of the instability, svenient
take sameρ0 different
the equilibrium
All toquantities
definedon
the
MID-SURFACEy ρhaveplane monolayer
with fixed
mass, rigidities. They also have opposite
eq of a identicaltensions
andtotalbending2
2tance tostudyamonolayer
undertension.
Let
us
f
(r,
H,
K)
=
A
+
A
H
+
A
(r
+
H)
+
A
H
:
reference
densityρ0 != ρeq noted ±c , since the lipids in the two0
1
23
curvatures,monolayers
are oriented

Bilayer curvature—density elasticity

0

3
+A
K
+
O(!
) the exterior of
4 towards
directions:theirhydrophilic
heads
are
orientedρ − ρ0== O (&) ,(21)
while rtheirhydrophobic
tails
are
oriented towards the interior of the bilay
ρ0✤ No term ∝ r : total number of
H = (c1 + c2 ) it
e =isO (&),(22)
our instability,
necessary
to
take
into
account
theinhomogeneities
lipids
fixed
$ 2%
2
±
= c1 c2 e =ρO &defined,✤ Coupling
mass Kdensities
on the(23)bilayer
midsurface
in eachmonolayer.
term : rH-> setse±±✤ All terms ( σ ) depend on ρ0scaleddensitiesr=(ρρ0 )/ρ
whi
e is a small length in the nanometer−range
that
0 , where ρ0 is 0a reference density,
toidenticaldefine the forscaledtotalmonolayers,
curvature H and
both
wethe
may write [21]Gaussian curvature K. Since we typically expect
"2
σ−2
κc0
k! ±
≤ |r| ≤ 10−2 and 10−4 ≤ |ci±e| ≤ 10
0 , it is2$κ sensible%
f while= K is+O &2c . ±c+r ± ec .
ume that r and H are O(&)
The
2
4
2
2
ergy density f is a function of these three nonsional
In smallthis variables.
formula, k is theNB.
stretchingelastic
constantofamonolayer,
which
Minimizing with respect to the densities -> ADE

for both monolayers as they are identical, and e denotes the distancedv theand dv, whichvanishregion
everywhereA
becauseof theonto r
relates linearlyforce
dfmustthat
exertsof the shape variation, we obtain Q = m S , Q =gion B to themarbitrarinessvectorial
lengthm, yielding
ds of the1.
Review
ofm Telasticity& projected
Dynamics cu
S , t = m T and t =throughLf
Lfh(x, y) is defined and hV its border, of curvilinear abscissa sand outer normal m. The membrane elastic free-energy has thegeneral form:ð(4)
F ~ dxdy f (fLi hg,fLi Lk hg):Tokyo ISSP/SOFT2010 Workshop

,

z

j

i

zi

,

i

,i

z

i

ji

z

i zi

' Li Lk h
Sij ~f dij { & ' Li h{ &L Lj hL Lj Lk h!
Lf' Li h,z Lk &
L Lj Lk h! d"Monolayer stress tensor df! = Σ · mV(9)df = Σ · m ds .

(1

where M and M are constants. Let us introduce two
pi = ∂jconstant Lagrange multipliers λ and µ to implement
This defines the six componentsof
Lf
Lfthe projected stre
&
'
&
~
S
{L
(10)
zj
k
these two global constraints. The relation δF L+Lj hδEp L−Lj Lk h' ,tensor: Σ and Σzj ,anywherei ∈ {x, y}. where we have
δW + λδM + µδM1 = 0 must hold for
infinitesimal
Lf
¯/∂h
To
determine
theprojected
wej(11)shall
fo&' , tensor,
Tij ~eik stress
(∂
f
)h
+(∂
ij
✤
deformationofthemonolayer
patch.
The
identification
Projected
quantities(Monge)
L
L
L
h
j k
df!
low
the
methodpresented
inRef.
[29], which
is basedoto exp(11)–(13)of bulk terms in this relation yields
Lf

Calling r = (x, y), we vary the membrane shape:h(r) A h9(r) =1h(r) + dh(r), arbitrarily, while translating the membrane edges
by da(r) = dai(r)ei + daz(r)ez (recall that Latin indices standA.-F. Bitbol,only
L. forPeliti& We
J.-B.
x or y).
alsoFournier
apply to the(2010)
borderline normal n an
infinitesimal rotation dv(r) = dvi(r)ei + dvz(r)ez. Integrating
ij

L‘ h,
(12)
Tzj ~{eLetk‘ & us' consider
the principle of virtual work.
a monolay
h(x,
y)
y)
L Lj Lkρ̄(x,
h
δF
p
patch
standing
aboveadomainΩofthereference
plan
and
wk =
−h
w
,
(11)
wz =
k z
which constitutethe formal expressions of the ‘‘projected’’δhThis patch is stresssupposed
to be a closed system with fixep
and torque tensors.
∂ f¯ total mass of each lipid species. Its
Two-component
free energy reads= −(v + λ) − (u 2.1.1+ µ)φ,These relations
Stress-tensor
divergence.(12)Directly differentiatingthemembrane∂ ρ̄components= 2dF/dh andy! of the stress tensor yields h S densities
for thehS=(hh)dF/dh,
where
dF/dh
=
2h
[hf/h(h
h)]
+
h
h
[hf/
¯
∂f
2 ¯
F
=
d
r fEuler–Lagrange(ρ̄, φ,(13)hi , hterm.
.vanishes
(2
h(h
h
h)]
is
the
standard
We
recognize in at eqij )p= −(u + µ)ρ̄ .
xfact the equation
order,∂φΩ +?S = 2(dF/dh)n, since at lowest
ted tonotheextern
j

j

ij

i

i

zj

i

i j

i j

Fig. 1 Configuration and notations used in the derivation of the‘‘projected’’ stress and torque tensors. The membrane h(x, y) standing
above domain V is deformed and its edges are translated by da. The
deformed membrane, h(x, y) + dh(x, y), is drawn in transparency andits projection onto the basal plane is shown. The force density Q acting
along the border and the force dw1A2 exchanged through the cut d,9are shown (for the sake of clarity, the torques are not shown). In theij and torque tensors are defined
ij with‘‘projected’’ formulation, the stressrespect to the projected cut m d, lying on the basal plane instead of theactual cut d,9 lying within the membrane.

membrane normal is given by n . ez 2 (hih)ei. This equation
correctly states that the restoring elastic force density exerted
by the membrane is 2(dF/dh)n. This is indeed a well-known
starting point in dynamical descriptions. At equilibrium, since
dF/dh = 0, we obtain

By identifying theFor
boundaryterms
and using we(12),assume
wethe sakeof generality,
that, in additio
Ordinary isotropic
fluid
term = ofpressure(or tension)
obtain thecomponents
the membranestresstensor:totheboundary
forces(andtorques) exerted by theIII.
re
!
"
!
"
¯ monolayer,¯¯ this ∂monolayerofthepatch is submitted t
∂
f
∂
f
f
¯
Σ = f − ρ̄a distribution−external∂kδ −hihsurface
0,(13)
jSzj = hjSij = forcesofwithprojecte
∂ ρ̄
∂hj
∂hkj
A. Free-ener
density w(x, y).
each lipid is assumed to bi.e. theMoreover,stress tensor is divergence-free.
∂ f¯hsubmitted(14) energy vα (x, y) tha−
to a one-bodypotential
ki ,
1.
Other
terms
due
to
∂hkj
884 | Soft Matter, 2007, 3, 883–888
This journal is ! The Royalthe
Society of Chemistry 2007
depends on the lipid species α ∈ {1, 2}; the correspondin
curved layer structure∂ f¯ energy∂ f¯ potential
can be written (15)
as
− ∂k
,
Σzj =
Let us derive
∂hj
∂hkj
!
ai, bilayer
from
b
j
∈
{x,
y}
which generalizes the result of Ref. [29]to thed2caseEp =r [v ρ̄where+ uρ̄φ] ,monolayer. We(3

dv, and dvz, which must vanish everywhere because of theh(x, y) is defined and hV its border, of curvilinear abscissa s
the same for bothmonolayers.
Furthermore,assumetha
arbitrarinessof the shape variation,
we obtain Qjwe
= miS
ji, Qz =and outer normal m. The membrane elastic free-energy has the
miT,i and tz = mibefore
Tzi, yieldingthe onset of the ins
have the same mlipid
compositioniSzi, t, =general form:ð
have identical tensions and bending
rigidities.
They also have
Lf
Lf
&
'
&
'
L
L
~f
d
{
h{
L
h
S
ij
ij
i
i
k
(4)
F ~ dxdy f (fLi hg,fLi Lk hg):
L lipids
Lj h
L Lthej Lk h two monolayers arcurvatures,
noted
±c
,
since
the
in
0
V
!
(9)
directions:theirhydrophilic
heads
are
oriented
towards
the e
Lf
Calling r = (x, y), we vary the membrane shape: h(r) A h9(r) =' Li h,z Lk &
Lj Lk hwhile
theirhydrophobic tails areL oriented
towards the interior ofh(r) + dh(r), arbitrarily, while translating themembrane
edges
stand
by da(r) = dai(r)ei + daz(r)ez (recall that Latin
our indicesinstability,
it is necessary
toLf take into
Lf account the inhom
A.-F. Bitbol,only
L. forPeliti&J.-B.
Fournier(2010)',
Szj ~ & ' {Lk &(10)x or y). We also apply to the borderline normal n an
±
L
L
h
L
L
L
h
j
j
k
mass
densities ρ defined on the bilayer midsurface in each mo
infinitesimal rotation dv(r) = dvi(r)ei + dv
z(r)ez. Integrating
Lfscaled densities r± = (ρ± − ρ0T)/ρ
de
0 , where' , ρ0 is a reference(11)
ij ~eik &L Lj Lwritekhidentical for both monolayers, we may[21]

Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics! d"Monolayer stress tensor df! = Σ · mdf!

Lf"2(12)
σ
κ
κc
k! ±
&
' L‘ h,
~{e
T
0
0
zj
k‘
±
2
f =
+ c ± L Lj Lck h+
r ± ec .
2
422
constitute the formal expressions of the ‘‘projected’’ρ − ρ elastic constant of a monolay
In this formula, whichkisthe stretchingstress and torquetensors. 0
r=for both monolayers as they areρ0 identical, and e denotes thedivergence.
differentiating
the of th
neutral surfaces of2.1.1
theStress-tensor
monolayers[22]Directly
and themidsurface
components of the stress tensor yields hjSzj = 2dF/dh andyscaled densities hjineachmonolayer
at =a 2hdistancee from the bSij =(hih)dF/dh,
where dF/dhi[hf/h(hih)] + hihj[hf/the standard
Euler–Lagrange
term. Wein
rn± = r± ± ec ath(hfirstin thesmallvariableec, so thatif fihjh)] isorder2
2recognizec+?S=∇ρ0 )/ρh+x±±= 2(dF/dh)n,fact the equation
sinceO(!
at lowest)order, theofthecurvature
and
r
=
(ρ
−
,
these
two variables a0
Fig. 1 Configuration and notations used in the derivation of themembrane nnormal is ngiven by n . ez 2 (hih)ei. This equation‘‘projected’’ stress and torque tensors. The membraneh(x, y) standing
decoupling
betweendeformations
where
only
the
curvatureis mcorrectly
states that the restoringelastic
force
densityexertedabove domain V is deformed and its edges are translated by da. The
by the membrane is 2(dF/dh)n. This is indeed a well-known
deformations
deformed membrane,
h(x, y) +dh(x, y), is drawn
in transparency and in which only the density is affected (stretching) i
Stresstensornormalcomponents
starting point in dynamical descriptions. At equilibrium, since
its projection onto the basal plane is shown. The force
density Qsurface
actingneutral[22].= Wechoosethe sign convention for the cu
dF/dh
0,
we
obtainalong the border and the force dw1A2 exchanged through the cut d,9thata sphericalvesiclehas c <"0. Then the monolayer denoted
are shown (for the sake of clarity, the torques are
not shown).
In the!
σκ(13)
0 the stress and torque tensors+ formulation,‘‘projected’’with2 are defined
2 hjSzj = hjSij = 0,2
outermonolayer.
Σzjto the=projected cuthmjd,−∂jbasal∇planehinstead− ke
r + e∇ h + O($ )respect
lying on theof the ∂j
The expression
the isfree-energy
actual cut d,9 lying 2
within the membrane.
2
i.e. the(3)stressoftensor
divergence-free. density of a monolaye
order expansion around a reference state characterized by a fla
From Helfrich! The Royal Society of Chemistry 2007884 | Soft density
Matter, 2007, 3, 883–888
Force
uniform
density ρ± =Thisρ0journal[23].is It
is valid for small deformations±
state:r= O($) and ec =O($), where" $ is a small nond
!
σ
κ
0 2
+
+
4 the deformation.
2
2
2
characterizing
pz = ∂j Σzj =∇ h − ∇ h − ke∇ r + e∇ h +O($ )±If the2
2 densities on the neutral surfaces, ρn , are both equal
ρ0 , summing the monolayer free-energy densities f ± in (3) giv

dv, and dvz, which must vanish everywhere because of theh(x, y) is defined and hV its border, of curvilinear abscissa s
the same for bothmonolayers.
Furthermore,assumetha
arbitrarinessof the shape variation,
we obtain Qjwe
= miS
ji, Qz =and outer normal m. The membrane elastic free-energy has the
miT,i and tz = mibefore
Tzi, yieldingthe onset of the ins
have the same mlipid
compositioniSzi, t, =general form:ð
have identical tensions and bending
rigidities.
They also have
Lf
Lf
&
'
&
'
L
L
~f
d
{
h{
L
h
S
ij
ij
i
i
k
(4)
F ~ dxdy f (fLi hg,fLi Lk hg):
L lipids
Lj h
L Lthej Lk h two monolayers arcurvatures,
noted
±c
,
since
the
in
0
V
!
(9)
directions:theirhydrophilic
heads
are
oriented
towards
the e
Lf
Calling r = (x, y), we vary the membrane shape: h(r) A h9(r) =' Li h,z Lk &
Lj Lk hwhile
theirhydrophobic tails areL oriented
towards the interior ofh(r) + dh(r), arbitrarily, while translating themembrane
edges
stand
by da(r) = dai(r)ei + daz(r)ez (recall that Latin
our indicesinstability,
it is necessary
toLf take into
Lf account the inhom
A.-F. Bitbol,only
L. forPeliti&J.-B.
Fournier(2010)',
Szj ~ & ' {Lk &(10)x or y). We also apply to the borderline normal n an
±
L
L
h
L
L
L
h
j
j
k
mass
densities ρ defined on the bilayer midsurface in each mo
infinitesimal rotation dv(r) = dvi(r)ei + dv
z(r)ez. Integrating
Lfscaled densities r± = (ρ± − ρ0T)/ρ
de
0 , where' , ρ0 is a reference(11)
ij ~eik &L Lj Lwritekhidentical for both monolayers, we may[21]

Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics! d"Monolayer stress tensor df! = Σ · mdf!

Lf"2(12)
σ
κ
κc
k! ±
&
' L‘ h,
~{e
T
0
0
zj
k‘
±
2
f =
+ c ± L Lj Lck h+
r ± ec .
2
422
constitute the formal expressions of the ‘‘projected’’ρ − ρ elastic constant of a monolay
In this formula, whichkisthe stretchingstress and torquetensors. 0
r=for both monolayers as they areρ0 identical, and e denotes thedivergence.
differentiating
the of th
neutral surfaces of2.1.1
theStress-tensor
monolayers[22]Directly
and themidsurface
components of the stress tensor yields hjSzj = 2dF/dh andyscaled densities hjineachmonolayer
at =a 2hdistancee from the bSij =(hih)dF/dh,
where dF/dhi[hf/h(hih)] + hihj[hf/the standard
Euler–Lagrange
term. Wein
rn± = r± ± ec ath(hfirstin thesmallvariableec, so thatif fihjh)] isorder2
2recognizec+?S=∇ρ0 )/ρh+x±±= 2(dF/dh)n,fact the equation
sinceO(!
at lowest)order, theofthecurvature
and
r
=
(ρ
−
,
these
two variables a0
Fig. 1 Configuration and notations used in the derivation of themembrane nnormal is ngiven by n . ez 2 (hih)ei. This equation‘‘projected’’ stress and torque tensors. The membraneh(x, y) standing
decoupling
betweendeformations
where
only
the
curvatureis mcorrectly
states that the restoringelastic
force
densityexertedabove domain V is deformed and its edges are translated by da. The
by the membrane is 2(dF/dh)n. This is indeed a well-known
deformed membrane,
h(x, y) +dh(x, y), is drawndeformations
in transparency
and in which only the density is affected (stretching) i
Stresstensortangential
components
starting point in dynamical descriptions. At equilibrium, since
its projection onto the basal plane is shown. The force
density Qsurface
actingneutral[22].= Wechoosethe sign convention for the cu
dF/dh
0,
we
obtain!
$along the border and the force dw1A2 exchanged through the cut d,9#In the κcvesicle
σ0of clarity, the"torques arethat
κc0 the monolayer2 denoted
a2 spherical
0 2 has c < 0. Thenare shownnot shown).
+ (for the sake
Σij =r +tensorse∇h with−∇ h δijhjS+
hij + O($ (13))
zj = hjSij = 0,‘‘projected’’
formulation, the −stressk
and torqueare definedoutermonolayer.
respect to the projected2
cut m d, lying on the basal plane instead of the22The
expression
(3)
of
the
free-energy
actual cut d,9 lying within the membrane.
the stress tensor is divergence-free. density of a monolaye
tension of the flat membrane i.e.
with
r=0.
order expansion around a reference state characterized by a fla! The Royal Society of Chemistry 2007884 | Soft density
Matter, 2007, 3, 883–888
Force
uniform density ρ± =Thisρ0journal[23].is It
is valid for small deformations±!
" and ec2= O($), where $ is a small nond
state:r=2 O($)++
pi = ∂j Σij = −k ∂icharacterizingr + e∇ hthe +O(" )deformation.
If the densities on the neutral surfaces, ρ±
n , are both equal
ρ0 , summing the monolayer free-energy densities f ± in (3) giv

Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics

Dynamics of structureless membranes
σ0 2
κ 4
+
+
pz = ∂j Σzj =∇ h − ∇ h − ke∇222h(x, y, t)
pz (q) = −(σ0 q 2 + κq 4 )hq!

"

r + e∇ h + O($2 )2✤ Relaxation time4ητR =σ0 q + κq 3
✤ Valid if σ0 ≈ 0 at
intermediate lengthscales(inter-monolayer friction).
σ = 10−9 J/m2 → [10 µm, 3000 µm]

dhq
Tzz (q) = −4ηqdt

σ = 10−8 J/m2 → [10 µm, 300 µm]

σ = 10−7 J/m2 → never valid

τR ! 10 s at λ = 150µm

Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics

In-plane dynamics in a flat membrane(symmetric mode)
+
p+
=
∂
Σ
j ij = −k ∂ii−2ηqv q!

"r + e∇ h + O("2 )2✤ Relaxation of a SYMMETRIC
2−η2 q vq −ikqrq density modulation in a FLAT MEMBRANE

η2 + 2η/qs
τR =k✤ Crossover in the µm range

drq+ iqvq = 0
dtτRs ≈ 10 ns

Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics

In-plane dynamics in a flat membrane(anti-symmetric mode)
+
+
2
2
p+
=
∂
Σ
=
−k
∂
(r
+
e∇
h)
+
O("
)
j ij
i
i

−2ηqv q+✤ Relaxation of an ANTI-SYMMETRIC+ density modulation in a FLAT MEMBRANE−η2 q 2 vq+ −ikqrq+−−b(vq − vq )

2
η
+
2η/q
+
2b/q
2
a
τR =
k

drq±
+ iqvq± = 0dt

τRa ≈ 10 s at λ = 150 µm

Tokyo ISSP/SOFT2010 Workshop

1. Review of elasticity & Dynamics

Complete dynamics
U. Seifert & S. A. Langer (1993)−(σ0 q 2 + κq 4 )hq + keq 2 (rq+ − rq− − 2eq 2 hq )
dhq
−4ηq
dt
−ikq(rq± ∓ eq 2 hq )−2ηqv q±τR [s]
τR =4η
σ0 q + κq 3

λ[µm]
σ ! 10−8 J/m2

−η2 q 2 vq±+−∓b(vq − vq )drq±
+ iqvq± = 0
dtTokyo ISSP/SOFT2010 Workshop

Outline
1. Review of the elastic and dynamical models of
membranes and monolayers.
2. Experiment by M. I. Angelova, N. Puff et al.
3. Theory of the curvature instability caused by alocal modification of the lipids of one of themonolayers
4. Comparison with the pH-micropipette experimentof M. I. Angelova, N. Puff et al.
5. Non-linear development : tubule ejection

Tokyo ISSP/SOFT2010 Workshop

2. Experiment

Experimental setup
N. Khalifat, N. Puff, M. I. Angelova (2008)

pH 7.4pH 8-9Giant vesicle (GUV)

Produced by electroformation,
mixture of EYPC/PS 90:10,
25°C, buffer at pH 7.4

Micropipette ∅0.3 µm
NaOH Solution 1M pH 13

Tokyo ISSP/SOFT2010 Workshop

2. Experiment

Curvature instability
N. Khalifat, N. Puff, M. I. Angelova (2008)

injection (~2s)

fast approachend ofinjection

beginningof injection

slow relaxation(~10s)

Tokyo ISSP/SOFT2010 Workshop

2. Experiment

Curvature instability
N. Khalifat, N. Puff, M. I. Angelova (2008)

Giant vesicle (GUV)

Produced by electroformation,
mixture of EYPC/PS 90:10,
25°C, buffer at pH 7.4

Micropipette ∅0.3 µm
NaOH Solution 1M pH 13

Tokyo ISSP/SOFT2010 Workshop

2. Experiment

Curvature instability
N. Khalifat, N. Puff, M. I. Angelova (2008)

Giant vesicle (GUV)

Produced by electroformation,
mixture of EYPC/PS 90:10,
25°C, buffer at pH 7.4

Micropipette ∅0.3 µm
NaOH Solution 1M pH 13

Tokyo ISSP/SOFT2010 Workshop

Outline
1. Review of the elastic and dynamical models of
membranes and monolayers.
2. Experiment by M. I. Angelova, N. Puff et al.
3. Theory of the curvature instability caused by alocal modification of the lipids of one of themonolayers
4. Comparison with the pH-micropipette experimentof M. I. Angelova, N. Puff et al.
5. Non-linear development : tubule ejection

Tokyo ISSP/SOFT2010 Workshop

3. Theory of the instability

Local monolayer lipid modification?
Dynamical membrane
instability
controlledOnly
by intermonolayer✤ No curvature
molecular
insertion.
a local friction

13

PC/PS 90:10

environment.
4.7. Qualitativechange
picture ofof
thesolvent
dynamics in
two limiting cases

In Sec. 4.4, we have shown that it is physically different to change locally the planeshape equilibrium
densityand to change locallythe spontaneous
curvature. Now that
✤ Nothydrodynamic(bufferalone nothing)
we have studied✤thedynamics ofof theare going
to analyze
the difference
Specific
OHinstability,(NaCl wenot).
Effectof pH?
between -these two effects. Although both of them occur in the generic case (seeSec. 4.4), we are going to study the two limiting cases when only one of these changes✤ Amino NH3+ group of PS headoccurs.
Let us firstdeprotonates
introduce a schematicof lipids,
which takes into accountatrepresentationhigh pH (pK
a~9.8).
the preferred area
per lipid oncharged
the neutraltrimethylammonium
surface, which is (by definition)
independent
Positively
groupof the membrane curvature, and their preferred curvature- (see Fig 3). The preferred
of
PC
head
associates
withOH
at high pH
shape of a lipid is represented by a cone, superimposed on the lipid. The area pereff(pKasurface~11).
lipid on the neutral(within the monolayer) is symbolized by the upper surface
Increasedof this cone, while its preferred curvatureissymbolized
by the angle of this cone.
negativechargeOHof head group

pH 7.4pH 8-10

pH 13

φ!1

Tokyo ISSP/SOFT2010 Workshop

3. Theory of the instability

Local monolayer lipid modification

mbrane curvature instability controlled by intermonolayer friction

13

picture of the dynamics in two limiting cases

have shown that it is physically different to change locally the plane✤ Aandfractionthe! 1theofspontaneousum density
to change φlocally
curvature. Now thatd the dynamics
of theinstability,
are going to analyze the difference
lipids oftheouter wemonolayer
two effects. Although both of them occur in the generic case (seeare chemically modified.
e going to study the two limiting cases when only one of these changes✤ Depends
on the local
timeintroduce
a schematic representationof lipids,
which takes into accountea per dependent
lipid on the neutral
pH.surface, which is (by definition) independent
ne curvature, and their preferred curvature (see Fig 3). The preferredd is represented by a cone, superimposed on the lipid. The area per
tral surface (within the monolayer) is symbolized by the upper surface
hile its preferred curvature is symbolized by the angle of this cone.
???

OK

DOH − ∼ 5 × 103 µm2 /s
DOH −curvature
instability
controlled by intermonolaye
e noted f + and f − . Since the curvature c is Dynamical
definedmembraneon thebilayer
midsurface,
it isTokyo ISSP/SOFT2010 Workshop
3. Theory of the instability
he same for both monolayers. Furthermore,4.7.weassumethe two
Qualitative
picturethatof the dynamics
in two monolayers
limiting casesSec. 4.4, of
we havethat it is physicallychange loc
ave the same lipid composition before the Inonset
theshown
instability,
so different
that tothey
shape equilibrium density and to change locally the spontaneous curvat
ave identical tensions and bending rigidities.weThey
also
have opposite
spontaneous
have studied
the dynamics
of the instability,
we are going to analyze
between these two effects. Although both of them occur in the genurvatures, noted ±c0 , since the lipids in the twomonolayers are oriented in opposite
Sec. 4.4), we are going to study the two limiting cases when only one ofoccurs.
A.-F. Bitbol,
Peliti & J.-B. Fournier
rections:
their L.hydrophilic
heads (2010)
are oriented
towards the exterior of the bilayer,
Let us first introduce a schematic representation of lipids, which tak
hile their hydrophobic tails are oriented towards
the interior
thebilayer.
To isstudy
the preferredarea per lipidof
on theneutralsurface, which(by definitionof the membrane curvature, and their preferred curvature (see Fig 3).
ur instability, it is necessary to take into accountin the onlipid
shape of a thelipid isinhomogeneities
represented by a cone, superimposed
the lipid.
±
lipid on theinneutralsurface(within the monolayer)
is symbolized
by the++ass densitiesρ
definedonthebilayer
midsurfaceeachmonolayer.
Definingtheρ0 : reference
density,and r = (ρ of−thisρcone,
while
0 )/ρ
0 its preferred curvature is symbolized by the angle of±±aled densities r = (ρ − ρ0 )/ρ0 , where ρ0 is a reference density, which is chosen
entical for both monolayers, we may write [21]"2σ0κ 2 κc0
k! ±
±
f =
+ c ±
c+
r ± ec .
(3)
2
4
2
2
n this formula, k is the stretching elastic constant of a monolayer, which is the samer both monolayers as they are identical, and e denotes
theFigurethe
3. Firstdistancelipid (light grey):between
non-modified lipid.
The leng
thepreferreddiameter per lipid on the neutral surface, while the acDeterminantofthespontaneous
curvature:(obvious)0eutral surfaces of the monolayers [22] and the midsurfaceof thebilayer.
thethe preferredcurvature.
Second andIndeed,
third lipids (darkgrey or b
lipids. For the second lipid, the modification affects only theon theneutralbilayer
surface, but not
the preferred curvature.
It is thealed densities
monolayer
at a distancee ?fromthemidsurfaceareρ0 from
How farin
is eachthe equilibrium
densityLet
usmiminizes
thethird lipid.
±
±
±
=
r
±
ec
at
first
order
in
the
small
variable
ec,
so
that
if
f
is written in termsn++±free energy
for themembrane(c=0) :f/ρ
the curvatureand perrn± unit= (ρmass−ρ)/ρ,
twoflatvariables
are decoupled.
Such a00 thesen
4.7.1. Dynamics in the case where c̄0 = 0 We are first going to study thecoupling between deformations where only thecurvature
is wheremodified(bending)
andthe relaxationin the casec̄0 = 0. Theinterest of studying
this thehow modifying the plane-shape equilibrium density and not theformations in which only the density
affected
is characteristicof the0 /2to show(stretching)+ is σcurvaturecaninduceatransientshape
instability.
req =us forconsidersimpleinitial conditions
t = 0: ar̄q (0)= r̂q (0)eutral surface [22]. We choose the sign convention
thecurvature
in atsuchway
k ThisLet
corresponds to the idealized case where the injection of the NaO
hat a spherical vesicle has c < 0. Then the monolayer
denoted“+”inhaveEq.
local in time: then,following (SHφ(r, (3)t) = 0 is
for thet < 0 and1 ), we

Bilayer curvature—density elasticity

n coefficient DOH − ≈ 5×10 µm /s [25]. Thus, in the typical 5 s of relaxation
Tokyo deformation,
ISSP/SOFT2010
Workshop
3.
Theoryof the instability
Dynamical
membraneinstabilityof
controlled
by intermonolayer friction
132 µm.
mbranetheydiffusecurvature
on a length
approximatively
1.5×10
Dynamical membrane curvature instability controlled by intermonolayer frict
th is larger than theofpicture
the ofinstability
comparablewith the size of
4.7.width
Qualitative
the dynamics inand
two limitingcases! + locally−the"createdκc̃0 byIn Sec. is
4.4,probably
we have shownalso
that itaffected
is physically bydifferentchangeplane. Besides, the pH field
theto advection
σ0 c −locallyκ̃∆cthe−spontaneouske ∆ r curvature.
− r Now− that ∆φn =shape equilibrium densitypand
to change
2
oming from the micropipette,thenby ofitstheretraction
andby themovement
we have studied the
dynamics
instability, we aregoinglasttothe difference# analyze$ κc̄σthese two zone
effects. of
Althoughboth of them occur
in the1 with 0+ generic− case (seembrane. However, between
thecentraltheinstability
remains
incontact
=
σ
c
−
κ̃∆c
−
ke
∆
r
−
r
−
∆φ ,
For
aaretwo-component
monolayerSec.
4.4), wegoing to study the 0
two limiting cases when
only one of these changesφ −kof highest pH. At present,
we have no knowledge of this time-dependent
pH 2
occurs.
A.-F. Bitbol, L. PelitiLet
& us
J.-B.
± Fournier
first
introduce a (2010)
schematic representation of lipids, which takes into account
where
p
is
the
force
density inmonolayer“±”
acting inthata direction i tang
ond these rough estimates.
Hence,
tosimplify,
wewhichareis (bygoingto independentassumethe preferredi area per lipid on theneutral surface,definition)+−themembrane,
while
p
=
p
+
p
is thenormalforce density in the meofthemembranecurvature,
and
their
preferredcurvature(see total
Fig 3).More
The preferredn
es not evolve significantly during the time ofnthe ninstability.
precisely,
shape of a lipid is represented by a cone, superimposed on the lipid. The area per
2
In
these
formulas,wehavedefinedtheconstants
κ̃
=
κ
+
2ke
and c̄0 = c̃0 +the simplifyingThehypothesis
that
φ(r)
is
time-independent
after
the
end
of
the
lipid on the neutral
surface(withinthe monolayer) is symbolized
by
the
upper
surfaceφby!
1in ofmonolayer
includesapartial
fractionotherlipidsThe∂denotes
derivationthedirectioni, while ∆ is the cof this symbol
cone, while its
preferredcurvatureissymbolized
the
angle
of thiscone.
i(t = 0):

Bilayer curvature—density elasticity

Laplacian operator.
) φ(r, t) = φ(r) for t ≥Let0.us discuss the terms depending on φ in the force densities. For
composition
φ(r),equilibriumof affected
a plane monolayer
riori, the constitutive
constantsof themonolayer“+”density
are allby ±the± with fixed toobtainedby minimizing
the free energy
mass
fsmall/ρ for c = 0. Con
modification, i.e., isthey
dependon φ(r). However,since perwe unitfocus
onChange ofin ourspontaneousa fixed total mass is justified here since
the timescales
instability a+ions around the flat shape and the uniform density ρ = ρ0 , we can writetocurvature
shorter than the flip-flop characteristic time. For monolayer“+”, this give
der in ":lipid (light grey): non-modified lipid. The length ! represents
order inσ& Figure 3. First!
" on the neutral surface,σκ 2 whileκthe angle α quantifies
0
2 the2 preferred diameter per++lipid+f =+ σ1 φ + φthe preferred+ σ̃ +curvature.
1 + r Secondφ(φ)
ln third
φ−+ρlipidsc(darkσ+greyc̃0 φ) candor (cbrown):
modified
0σ1+
ρ
/2
+
φ
0 affects
eq
0
lipids.
Forthesecondlipid,
themodification
only
thepreferreddensity2
2 on the neutral
4= It is2the contrary forreqsurface,(φ) ≡but not the preferred curvature.
, the
ρ0
k
"2 third lipid.
k! ++r where+ ec we. have used the fact that σ # k. Thus, the chemical(4) modification
02
Change in equilibrium
equilibrium
densityandoftheplane monolayer“+”bypect to expression the(3),scaled
the monolayer
tensionspontaneous
curvatureof the amount4.7.1.
Dynamics
in the case wherec̄0 = 0 We
are first
going to study the dynamics
of density
the relaxation in the case wherec̄0 = 0.+The interest+of studying σ
this
1 theoretical case is+r “+” have been modified
according
to
δr
=
r
(φ)
−
r
(0)
=
. spontaneousto show how modifying the plane-shapeeq
eq equilibriumeqdensity and notφtheka transient shape instability.
σ0
σ0 curvature canσ2induce+(to O(!))
us consider
simpleinitialconditions
atφtther̄q (0) = r̂q (0) = hqequilibrium(0) = 0.
→
+We
σ1Let
φobserve+
φ2that+ σ̃(1+r)φ lnof,= 0: plane-shape(5) density
this change
appearn222This corresponds to the idealized case where the injection of the NaOH solution isTokyo ISSP/SOFT2010 Workshop

3. Theory of the instability

Complete dynamics with modified lipids−(σ0 q2σ1
κc̄0 2+−2+ κq )hq + keq (rq − rq − 2eq hq − φq ) +
q φq
4

2

dhq
−4ηq
dt

φq

−
2
−ikq(rq + eq hq )−2ηqv q−
−η2 q 2 vq−+b(vq+ − vq− )

k

2

σ1
+
2
−ikq(rq − eq hq − φq )k+−2ηqv q
−η2 q 2 vq+−b(vq+ − vq− )drq±+ iqvq± = 0dt

ed by intermonolayer friction

12
focused on theeffect of thesethe variationof equations
the plane-shapeequilibrium
density,which tha
Comparing
dynamical
with theones of of
Ref.
shows
Tokyo ISSP/SOFT2010 Workshop
3. Theory
the[16]
instability
corresponds
taking c̄0 = 0 here. Here, both the change of the equilibrium density
the ones of Ref. [16]shows
−σ1thatφtoq /k corresponds to the scalar field #(r, t) introduced in Ref. [16] to describe
troduced in Ref. [16]and tothedescribechangeof thespontaneous
are taken
account.
thechangeof theequilibriumcurvaturedensity, whichis in into
agreementwith Eq. (11). Ref. [16agreement with Eq. (11). Ref.
[16] on the effect of the variation of the plane-shape equilibrium density, which
focused-shape equilibrium4.6.
density,which of the
Resolution
equationscorresponds
to hydrodynamic
taking c̄0 = 0 here.
Here, both the change of the equilibrium densitychange of the equilibrium anddensity
the change of the spontaneous curvature are taken into account.
Letusdefinethe (symmetric) average scaled density r̄q (t) = rq− + rq+ and theaken into account.
Exponential
relaxations
ility controlled by intermonolayer friction 12+−(antisymmetric) differential scaled density r̂q (t) = rq − rq . Eliminating vq± in
4.6. Resolution of the hydrodynamic equationstowards equilibrium stateEqs.
(22)–(23)thanksthat
to Eq. (25) and adding them together gives
ations with the ones of Ref. [16] shows!
Let[16]
us todefine
average∂describer̄q the (symmetric)kq
σ1 " scaled density r̄q (t) = rq− + rq+ and theeld #(r, t) introduced
in+Ref.
−nsity r̄q (t) = rq + rq andthe=−differential r̄scaled
φq , r̂ (t) = r+ − r− . Eliminating
(26)
±
q −
(antisymmetric)
density
v
in
which
is
in
agreement
with
Eq.
(11).
Ref.
[16]
q
q
q
q
+
−
±
∂t
η
q
+
2η
k
2
= rq − rq . Eliminating vq in
Eqs.density,(22)–(23)thanksto Eq. (25) and adding them together givesof the plane-shape while
equilibrium
which
substracting
themyieldstogether gives
ere, both the change of the equilibrium density∂ r̄qγ0kq 3 !
σ21 "2κ̃qκc̃0 qr̄q keq− φq ,(26 ∂t = σ−0ηq +rvature are taken into account. (26)
−
φ
k 
q 
2 q + 2η
qhq
qh


q
4η
4η
8η
γ2
∂ substracting
 yields



while
 = −them

+ 


 ,(27)
quations
 3

32
2 
2
∂t




2
σ
q
+
κ̃q
keq
κc̃
q



0
r̂q 
r̂q
keq 0
kq −
2

 σ1 q
κc̃
q
φq 
−
+
0
−
φ8η
qhq
qhq
gescaled density r̄q (t) φ=q rq + rq and the
q


4η
4η
∂ ±
2b
2b
qhq
b




r+ −8ηr− . Eliminating
ensity
r̂
(t)
=
v
in
=
−
+



 ,(27
q

q

q
q
2

,(27) assumed
wehaveη2 q " band ηq "3 b, which2 is well
adding them+together
 where∂t verified forπ/q ≈2 40 µm

gives
r̂
r̂
keq
kq
σ
q



q
q
1
2
r̂q "
σ
q
corresponding
to
the
width
of
the
deformation
observed
intheexperiments.
−
φq
1
σ1
φ
b
2b
2b
q
− φq ,(26) that r̄q relaxes to its equilibrium value σ1 φq /k with a very
Equation
(26)
shows
2b
k
where we have assumed η2 q 2 " b and ηq " b,is well verified for π/q ≈ 40 µm
γ1whichshort
typically π/q ≈ 40 µm, andch is well verified for
π/qtimescale≈ 40 µm τ = −(η2 q + 2η)/(kq). Indeed, withcorresponding to the width of the deformation observed in the experiments.
erved
in
the
experiments.
 given above, we obtain τ ≈ 0.3 µs, which can be consideredwith the parameters
2
q3
keq 2
κc̃
q
0Equation (26) shows that r̄q relaxes to its equilibrium value σ1 φq /k with a veryσ1 φinstantaneouswith a veryilibrium/k− valueφq our
in
q
2σπ/q
qhq
short experiment.
1es −(η2 q + 2η)/(kq). Indeed, with c̃typicallytimescale
τ =
≈
40
µm,
2and
4η 
8η
=
c̄
−
,
κ̃
=
κ
+
2ke



0
0
τ
≡
τ
"
10
ns
with typically
π/q
≈
40
µm,
and
(27)
be R
solvedbyabove,diagonalizingtheτ squarematrixinvolved.
our0,(27)
 Equation
κ can be In+ with the

can
)
parameters
given
we
obtain
≈
0.3
µs,
which
considered can
 q " σ /κ̃. Indeed, the tension of a vesicle is superior to about
0.3 µs,be considered2experiments,
we
have
r̂
0 0.1 − 0.5 s
kq 2 which
σ
q
q
1
in
our
experiment.
γ0 ! γ2 −7> γ1instantaneousτ≈at qexp , σexp
)
2
φq
2
6 −1
10 J/m , 2bso
σ0 /κ̃ (27)≥ 10canm be. solved
In thisbyregime,
the eigenvaluesof matrix
the squarematrix2bEquation
diagonalizing
the squareinvolved.
In ou
a
)
he
squarematrixinvolved.
In
our
≡have
τR q≈
5 sσ0 /κ̃.atIndeed,
qexp the tension of a vesicle is superior to abou
1weηq " b, which is wellverified
forareπ/q ≈τ40µm
in Eq.
(27)
experiments,")2
on of a vesicle
is superior
to aboutormation
observedin the experiments.
−72
10 J/m , so kqσ0 /κ̃ ≥ 106 mσ−10 q. In this regime, the eigenvalues of the square matrix
e eigenvalues of the square matrix

Complete dynamics with modified lipids

Dynamical
membrane curvature instability controlled by 3.
intermonolayer
friction
12
Tokyo ISSP/SOFT2010
Workshop
Theory of theinstability

(a)

(b)

(c)

(d)

(e)

Comparing these dynamical equations with the ones of Ref. [16] shows that
membraneinstability
controlled byinintermonolayer
friction 1−σ1 φq /kDynamicalcorresponds
to the curvaturescalar field#(r, t) introduced
Ref. [16] to describe
the change of the equilibrium density, which is in agreement with Eq. (11). Ref. [16]Comparing
dynamical
with theones of Ref.
[16] shows
focused on theeffect of thesethe variationof equations
the plane-shapeequilibrium
density,which tha
−σ1 φq /k corresponds to the scalar field #(r, t) introduced in Ref. [16] to describ
corresponds
to taking c̄0 = 0 here. Here,both the change of theequilibrium density01
the change of the equilibrium density, which is in agreement with Eq. (11). Ref. [1and the change of the spontaneous curvature are taken into account.
focused on the effect of the variation of the plane-shape equilibrium density, whic
Hypotheses:
corresponds to taking c̄0 = 0 here. Here,both the change of the equilibrium densit
4.6. Resolutionof the hydrodynamic
equationsand the changeof the spontaneous
curvature
are taken into account.
(i) instantaneousmodification,(ii) density
permanent
modification,
Let us define the (symmetric) average scaledr̄q (t) =rq− + rq+ and the+−
4.6. Resolutionof the scaledhydrodynamic(antisymmetric)
differential
densityequationsr̂(iii)= rdiffusion
Eliminating vq± in
q (t) noq − rq .
Eqs. (22)–(23)to Eq.
(25) and adding
together
Let us thanks
define the(symmetric)
averagethemscaleddensitygivesr̄q (t) = rq− + rq+ and th
!
"
+
−
±
∂
r̄
kq
σ
(antisymmetric)
differential
scaleddensity
r̂
(t)
=
r
−
r
.
Eliminating
v
q
1
q
q
q
q
τ0
=−
r̄q − φq ,(26)
Eqs. (22)–(23)thanks
to
Eq.
(25)
and
addingthem
togethergives∂tη2 q + 2η
k
!
r̄q
kq
σ1 "
while substracting them∂yieldsγ=−r̄q − 2 φ
(26
q ,
0



3
2
q+
σ0ηq2+
κ̃q2η
keqk
κc̃0 q

 ∂t


γ
−
φq 
2
while substracting
themyields
qh
qh



q
4η
4η 
8η
τ2∂  q 







3
2
2 ,(27)


=
−
+




σ
q
+
κ̃q
keq
κc̃
q
0
0




∂t



φq
−
3
2
2
qhq
qh
r̂q∂
r̂




keq 4η kq
σ1 q 8η
q
4η q



−
φ
q

=−


+


 ,(27
b
2b
2b
∂t




3
2
2
r̂
r̂
2
keq
kq
σ
q
q
where we have assumed qη2 q " b and −ηq " b, which is well verified
for π/q1 ≈ φ40
µm
q
τ1
b
2b
corresponding
to the width of the deformation
observedin the experiments.2b2where weassumed" b andηq equilibrium" b,well σverified
for π/q ≈ 40 µγ1which isvalue
Equation(26)haveshowsthat r̄ηq2 qrelaxes
to its
1 φq /k with a very
corresponding
to thewidth of the deformation observed in the experiments.
short timescaleτ = −(η2 q + 2η)/(kq). Indeed, with typically π/q ≈ 40 µm, andEquation (26)thatwer̄q obtainrelaxes τto≈its0.3
equilibrium/k with a ver
1 φqconsideredwith the parameters
givenshows
above,µs, whichvalue
can σbeshort timescaleτ = −(η2 q + 2η)/(kq). Indeed, with typically π/q ≈ 40 µm, an
instantaneousin our experiment.

Case #1 — equilibrium density onlyc̄ = 0 and σ != 0) = r̂q (0) = hq (0) = 0. At t = 0, the preferred curvature per
ISSP/SOFT2010
Workshopthat are chemically
3. Theory of the instabilityor Tokyothe lipidsof the external monolayer
Dynamical membrane curvature instability controlled by intermonolayer friction 1
Here, solving Eq. (26) shows that r̄q remains equal to zero.
e membrane deformation is given byComparing these dynamical equations with the ones of Ref. [16] shows tha!
"
κc̄0−σ1 φq /k corresponds to(31)the scalar field #(r, t) introduced in Ref. [16] to describφq
1 − e−γ2 t ,2σ0
the change of the equilibrium density, which is in agreement with Eq. (11). Ref. [1. (29) in the case where σ̄1 =focused
0 and with
theeffect
aboveofinitialon thethe variation of the plane-shape equilibrium density, whic
−1
ation increases exponentially correspondswith timescale
towards2
to γtakingc̄0 = a0 here. Here, both the change of the equilibrium densit
re hq = κc̄0 φq /(2σ0 ) (see Fig. 5(c)).

Case #2 — spontaneous curvature onlyc̄0 != 0 and σ1 = 0
and the change of the spontaneous curvature are taken into account.

Hypotheses: idem(i) instantaneous modification,
4.6. Resolution of the hydrodynamic equations(ii) permanent modification,
Let us define the (symmetric) average scaled density r̄q (t) = rq− + rq+ and th(iii) no diffusion+−±(antisymmetric) differential scaled density r̂q (t) = rq − rq . Eliminating vq
Eqs. (22)–(23) thanks to Eq. (25) and adding them together gives
!
∂ r̄q
kq
σ1 "
=−
r̄q − φq ,(26∂t
η2 q + 2ηkγ2while substracting them yields



3
2 
2
σ
q
+
κ̃q
keq
κc̃
q
0
0




−
φq 
qhq
qhq



4η
4η
8η
∂ 



 = −τ2+ ,(27
∂t




r̂q
r̂q
keq 3
kq 2
σ1 q 2−φqb2b2bwhere we have assumed η2 q 2 " b and ηq " b,γ1which is well verified for π/q ≈ 40 µcorresponding to the width of the deformation observed in the experiments.
Equation (26) shows that r̄q relaxes to its equilibrium value σ1 φq /k with a ver
short timescale ττ1=essentially−(η2 q + 2η)/(kq).
Indeed, with typically π/q ≈ 40 µm, an
not involved
Qualitative description of the dynamics of the instability if σ1 = 0.
with the parameters given above, we obtain τ ≈ 0.3 µs, which can be considere
modification affects the preferred curvature, but not the preferredinstantaneous in our experiment.
the neutral surface.

Tokyo ISSP/SOFT2010 Workshop

Outline
1. Review of the elastic and dynamical models of
membranes and monolayers.
2. Experiment by M. I. Angelova, N. Puff et al.
3. Theory of the curvature instability caused by alocal modification of the lipids of one of themonolayers
4. Comparison with the pH-micropipette experimentof M. I. Angelova, N. Puff et al.
5. Non-linear development : tubule ejection

Tokyo ISSP/SOFT2010 Workshop

4. Comparison with experiment

Local dynamical shape instability
N. Khalifat, N. Puff, M. I. Angelova (2008)

Giant vesicle (GUV)

Produced by electroformation,
mixture of EYPC/PS 90:10,
25°C, buffer at pH 7.4

Micropipette ∅0.3 µm
NaOH Solution 1M pH 13

Tokyo ISSP/SOFT2010 Workshop

4. Comparison with experiment

Dynamical membrane curvature instability controlled by inte

4.7. Qualitative picture of the dynamics in two limiting cas

Adequacy? Case #1 or #2, or both?

In Sec. 4.4, we have shown that it is physically different toshape equilibrium density and to change locally the spontan
we have studied the dynamics of the instability, we are going
between these two effects. Although both of them occur✤ Instantaneous modification? Idealized. Sec.
we forare going to study the two limiting cases whenBut4.4),OK
occurs.
the relaxation stage.
Let us first introduce a schematic representation of lipids
the preferred area per lipid on the neutral surface, which is (bof the membrane curvature, and their preferred curvature (✤ Permanent modification? Strong approximation.
shape of a lipid is- represented by a cone, superimposed on
Difficult to quantify, depends on time-dependentOH
lipid on the neutral surface (within the monolayer) is symbo
concentration field and local pH vs. pKa. of this cone, while its preferred curvature is symbolized by t

#1

#2

✤ Case #2 alone? Possible, but then the diffusion of
OH- would be responsible for the relaxation towards
the flat state.

✤ Case #1 alone? Possible! Indeed even if the lipid
modification is permanent there is a relaxation: !grey):150 µmin 5 sFigure 3. FirstDlipid
non-modifiedli
OH (light
toward the flat state.
the preferred diameter per lipid on the neutral surfa−

the preferred curvature. Second and third lipids (dlipids. For the second lipid, the modification affeon the neutral surface, but not the preferred curva
third lipid.

✤ Probably both #1 and #2 involved. Increasing theeffective size of the polar head both increases theequilibrium density (#1) and the conical shape (#2).

4.7.1. Dynamics in the case where c̄0 = 0 We are first going
the relaxation in the case where c̄ = 0. The interest of study

Tokyo ISSP/SOFT2010 Workshop

∂t

r̂q



3

2



r̂q
keq
kq
4. Comparison
with experiment−b
2bynamical membrane curvature instability
by intermonolayer
where controlled
we have assumed
η2 q 2 "frictionb and ηq17" b, which is well v
corresponding to the width of the deformation observed in themembrane curvature instability controlled by intermonolayer friction 18
easured during a “pulse” experiment with Eq. (29). This equation corresponds toEquation(26)
showsr̄q relaxes
to its equilibrium va
e general solution of the dynamical equationsof ourmodel,
andthatit should
describe(!
shorttimescaleτ of=the−(η2η)/(kq).
Indeed, with typica
2 q +withe time )*+evolutionoftheheightH(t)ofthedeformation
vesiclerespectto"&with the parameters given above, we obtain τ ≈ 0.3 µs, whinitial shape.
"!
Fitsof thewith one-mode
We present
the analysis of several“pulse”
experiments
conducted
on three (q) theory:'&
instantaneous
inrelaxation
our experiment.
'!
fferent GUVs
(numbered 1, 2 and 3 in
the following).
Forbe
each
experiment,
weEquation(27)
cansolvedbydiagonalizing
the square m
−γ
t
−γ2 t
1
)
&
H(t)
=
(H
−
C
−
B)
e
+
C
ke as an initial
where
the
pipetteis present.
If +
we B
calle
GUV 1condition the last point0experiments, wehave q " σ0 /κ̃. Indeed, the tension of a vesi!!
" the# deformation
%
$
)
the
height
of
at
this
time,
which
will be referred
to as t = 0
0
−7
2
6 −1
10 withJ/m, so canconditionσbe≥ 10 H(0)m .=Inregime, the eigenvalu
0 /κ̃
initialin our theoretical analysis, this initialcondition
writtenH0this. The
Doubtful
in
Eq.
(27)
are
()*
perimental"! data is thus fitted with the formulasignification2'&
σ0 q(33)because of lateH(t) = (H0 − C − B)e−γ1 t + Be−γ2 t + C , kq
γ1 ≈and γ2 ≈.
- 20'!
Dynamical
membranecurvature
instability
controlledbyintermonolayer
frictionOH
diffusion
2b4η
hich corresponds to Eq. (29) with the above initial condition. There are thus four

Fits of the experimental data&e parametersin our fits: B, C, γ1 Thisand γ2result. Besides,
expect rapidlythat the diffusioncansince
be wecheckedby noting that, in the regi
GUV 2
3a few seconds (see Sec. 4.2), we have!!
the hydroxide
negligible after
3"ions #is no$ longer%!4"×!&
the coefficientkeq/b is much smaller than all the other
coe
5670!
−7
2
3
rried out the fits on identical intervals of duration
5 s5670"
for all the experiments, starting
σ
≈
1
−
8
×
10
J/m
!×!&
0
so thatsquarein Eq. (27)may be approximated56708matrix%eliminate
m the'()end of the injection (t = 0).
Thus,thewe
thelonger-term
evolution%×!&&%matrix.
The%model.
deformation thus evolves according tohich should no longer be well-described by our
$×!&
κc̄0
The bestfits of the experimental data to Eq.
(33) are shown−γ
in1Fig.
6 for
&!
%
t
−γ2onetFittedamount
of
#×!&
h
(t)
=
Ae
+
Be
+
φ
,
q
q
pical “pulse” experiment for each of the three vesicles
studied. Fig. 6 shows a good 2σ0%
dilation (if all):%"×!&
reement between the experimental results andtheir best fit to Eq. (33). Thus, thiswhere the constants
B!" canbe determinedfromthe initi
GUV 3& ! " # $A %and
a
few
%.
!&
!#
!$
mula describes
well theexperimental
results, which '()*+,-*./0.1-2*+
is in favor of our theoretical!!
"#$r̂q . The term κc̄0 φq /(2σ0 ) comes from the constant solution
odel.
equation(27), whichcorresponds
physicallyto the residual defo
Figure6. be
Typicalexamplesof fits 6,of theexperimental
results.
Dots:
As
can
seen
on
Fig.
the
residual
deformation
atequilibrium
is
often
close
to
Figure8. Intermonolayer
friction coefficientestimatedfrom thebest fit of theexperimental data; line: best fit to Eq. (33). The dots surrounded at t = 0

Tokyo ISSP/SOFT2010 Workshop

4. Comparison with experimentPartial conclusions✤ Good agreement between theory and experiment.
✤ Direct means of measuring intermonolayer friction coefficient b .
✤ In principle allows to discriminate between modification ofequilibrium density δρeq and modification of intrinsic curvature δc0(alas not here because of OH- diffusion).

✤ Further (theoretical) work: (i) Evolution of the width of theinstability by multimode Fourier analysis. (ii) Effect of the diffusionof modified lipids (if permanent modification).

Tokyo ISSP/SOFT2010 Workshop

Outline
1. Review of the elastic and dynamical models of
membranes and monolayers.
2. Experiment by M. I. Angelova, N. Puff et al.
3. Theory of the curvature instability caused by alocal modification of the lipids of one of themonolayers
4. Comparison with the pH-micropipette experimentof M. I. Angelova, N. Puff et al.
5. Non-linear development : tubule ejection

Tokyo ISSP/SOFT2010 Workshop

5. Non-linear: tubule ejection

Ejection of a tubule aiming at the pipette

Giant vesicle (GUV)

Produced by electroformation,
mixture of EYPC/PS 90:10,
25°C, buffer at pH 7.4

Micropipette ∅0.3 µm
NaOH Solution 1M pH 13

Tokyo ISSP/SOFT2010 Workshop

5. Non-linear: tubule ejection

Ejection of a tubule aiming at the pipette

Giant vesicle (GUV)

Produced by electroformation,
mixture of EYPC/PS 90:10,
25°C, buffer at pH 7.4

Micropipette ∅0.3 µm
NaOH Solution 1M pH 13

Tokyo ISSP/SOFT2010 Workshop

5. Non-linear: tubule ejection

Gradient of in-plane force:‘Marangoni-like’ effect?
✤ Integrated normal force√
enough to draw tubule f > 2π 2κσ0

✤ ‘Marangoni-like’ effect:
L(t)

"
σ
1
+
+
+
pi = ∂j Σij = −k ∂i r + ec − φk∇ · Σ ∼ σ1 ∇φ!

✤ Basic dynamical model:
E. Evans and A. Yeung (1994)

dL
2πrL × σ1 ∇φ = λdt
L(t) ∝ exp(γt)

lipTokyo ISSP/SOFT2010 Workshop

4.7.3. Comparison of the two types of changes of the membrane prope
the dynamical equations in the two limiting cases has confirmed that a
the plane-shape equilibrium density and a local change of the spontan
yield different dynamics. Indeed, for the length scales we consider,plane-shape equilibrium density is modified, the deformation increases
relaxing towards a non-deformed shape with a long timescale γ1−1 (typi✤ Direct means of measuring intermonolayer friction coefficient b .
seconds). When only the spontaneous curvature is affected, the deform
towards a stationary curved state with a short timescale γ2−1 (typicalls) before relaxing
slowlywithtimescale of between
about one hour.
✤ Local,1 dynamical
instability
allowstoadiscriminate
when
a globalmodificationof ofthe environment omodificationInof contrast,
equilibrium
densitymodification
δρeq andthe change of its equilibrium shape does not allow
intrinsicconsidered,curvature studying
δc0 .
between a change of the spontaneous curvature and a change of the pre
DIFFERENT
a globalmodificationof theenvironment:
lipid FROM
[15]. Asa matterof fact, theequilibrium
shape forof aa vesicle withvesicle is
with
fixed
volume, within
the equilibrium
shape, within
the ADE
fully
determined
the area-difference
elasticity(ADE) model bmodel, theis fully
determinedby the value of quantity:
combinedquantity
2 b∆a0 = ∆a0 + c0 ,α
wherenondimensionalized
preferredarea difference bet
combiningthe ∆apreferreddifference and bilayer
spontaneous
0 is theareamonolayers, α a nondimensional number involving the elastic con
curvature.
membrane and cb0 denotes the spontaneous curvature of the bilay
shape variations observed when such global modifications are perform
interpreted as coming from a change of the spontaneous curvature

Conclusions

Tokyo ISSP/SOFT2010 Workshop

Many thanks to collaborators:
Experimental
N. Khalifat (PhD), N. Puff, M. I. Angelova
Theoretical
A.-F. Bitbol (PhD), L. Peliti[ J.-B. Fournier, N. Khalifat, N. Puff and M. I. Angelova,
Phys. Rev. Lett. 102, 0181102 (2009) ]