Membranes
18.354 - L11
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Cell membranes (D=2)
Illustration by J.P. Cartailler. Copyright 2007, Symmation LLC.
transport:
stochastic
escape problems
shape: differential
geometry
red blood cells affected by
sickle-cell disease
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Cell membranes
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AFM
http://www.sbmp-itn.eu/sbmps/research_method/
source: wiki
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Bio-membrane
Selective barrier
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Morphological changes
e.g., fusion through stalk-formation
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Endo- & Exocytosis
material exchange with environment
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Active transport
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Intercellular gap junctions
exchange of molecules and ions between animal cells
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Virus envelop
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envelop fuses with host membrane
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Capsid
Baker et al (1999) MMBR
cryo-EM
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Diatoms (algae)
H4SiO4
source: wiki
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More planktonic diatoms
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Plants
unlike animal cells, every plant cell is surrounded by a polysaccharide cell wall
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source: wiki
typical plant cell has between 103 and 105 plasmodesmata !
connecting it with adjacent cells !
equating to between 1 and 10 per µm
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Chara fragilis
http://www.youtube.com/watch?
feature=player_detailpage&v=kud4qUhsCxg
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1
2
Characean algae
Fig. 4. Cyclosis developing in our model for αp = 1 and αu = 0:5, with other parameters fixed as in the text. Color coding co
cytoskeleton
(37) by afor
process
growth
once thefor
main lower magnitude. The white lines represent IZs sep
B purple
the orderAvector P, with
green
Pofz twisting
<to0,
darker
~1mm for Pz > 0 and
2
3
cables are established and bound
the cortical chloroplasts 1
(38). The twist appears to be advantageous primarily for large,
regions. Superimposed
arefor
streamlines
ofαthe
flow
u induced
the filament
wherecoding
the flow
is directed tf
closis developing
in our model
αp = 1 and
0:5,where
with
other
fixed
as in thefield
text.P,Color
corresponds
mature
cells,
it enhances
vacuolarparameters
mixing
and nutrient by
u = cytoplasmic
uptake (25, 26).
cytoplasm
the individual
(Upper)
Time
sequence
offormalize
six frames,
showing
progression
from lines
random
disorder
localupo
will
the
hydrodynamics.
Endow
the shell
ector of
P, with
purple forlines.
Pz > 0
and green
for
Pwith
<cylindrical
0, now
darker
for
lower
magnitude.
The white
represent
IZsthrough
separating
z We
coordinates ðθ; zÞ and corresponding unit vectors
(Lower)are
“Unwrapped”
streaming
patternse ;flow
three
indicated
frames.
eof
. Letthe
uðxÞu
beinduced
the velocity field
ofby
the suspending
cytoplasmic
perimposed
streamlines of
the cytoplasmic
the filament
field P, where the flow is directed from the th
fluid. The small velocity of streaming allows us to work in the
vacuole
zero Reynolds number limit. We then take u to obey the forced
vidual lines. (Upper) Time sequence of six frames,
showing
Stokes equations
with friction, progression from random disorder through local order to com
nwrapped” streaming patterns of the three indicated frames.
−μ∇ u + νu + ∇Π = F;
θ
z
2
Cyclosis Parameter Scan.
p
u
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
APPLIED PHYSICAL
SCIENCES
BIOPHYSICS AND
COMPUTATIONAL BIOLOGY
Fig. 1. Cytoplasmic streaming in Chara corallina. (A) Internodes and
branchlets of Chara. Individual internodes can grow up to 10 cm long. (B)
Rotational streaming in a single internodal cell of Chara. The stripes indicate
the polarity of actin cables at the periphery driving flow in the cytoplasm and
vacuole. (C) Microscopic mechanism driving plant cell streaming. Cargo-carrying myosin motors bind to actin filaments and entrain flow as they walk.
APPLIED PHYSICAL
SCIENCES
ðsÞ
ðrÞ
d
=
d
= 0:025, κ = 0:5, and μ = 0:
indifferent
∂P
ðrÞ
zoned
(IZ) P + ðI − PPÞ · ½eð∇uÞ · P
+ eu · ∇P = dðsÞ ∇2 P −
portant coupling constants αp and α
∂t
ðsÞ
ðrÞ appropriate radius-to-length ratio f
d
=
d
= 0:025, κ = 0:5, and μ = 0:05 to focu
!
P
· dÞd· P;
ably organized
cables
u u −·κðP
+ eu · ∇P = dðsÞ ∇2 P − dðrÞ+Pα+p PðI+−αPPÞ
½eð∇uÞ
αu .(36).
We als
portant coupling
constantsactin
αp and
t
Fig. 4. Cyclosis developing
in our radius-to-length
model for α = 1 andratio
α = 0:5,
with
othe
appropriate
for
a
youn
!
C
Time
Fig.
displays
a tyl
order vector P, with
for
P >Progression.
0 and green for
P 4
< 0,
darker for
where +
all
constants
are
now the
nondimensional.
Thepurple
paαpcoupling
P
+
α
u
−
κðP
·
dÞd
;
ably
organized
actin
cables
(36).
u
cargo
flow
regions.diffusion
Superimposed
are streamlines
of the
cytoplasmic
u induced
b
merical
integration
of flow
the model
with
rameters dðsÞ and dðrÞ are
spatial and rotational
constants,
motor
binds & walks
unbinds
of the individual lines. (Upper) Time
sequenceαof
six
showing pr
The
experimen
parameters
respectively.
The
final
hydrodynamics
read
p; α
u . frames,
filament
Time Progression.
Fig.
4
displays
a
typical
tim
coupling constants
are now nondimensional.
The
pa(Lower)
“Unwrapped”
streaming
patterns
of
the
three
indicated
frames.
progression
(28)model
is clearly
reproduce
merical
integration
of
the
with
an illust
dðsÞ and dðrÞ are spatial and2 rotational diffusion constants,
2
through
small
patches
of
locally
ord
−μ∇ u +read
u + Π0 ez + ∇Π′ = jPj P;
parameters αp ; αu . The experimentally obse
ely. The final hydrodynamics
spontaneous polarization, and sett
progression
(28) is clearly reproduced as it mo
streaming by way of an example and a parameter space scan to
subject2 to
closis
as the passive and active reor
∂P
identify regimes of robust cyclosis development. Finally, we 2
will
ðsÞ
2
ðrÞ
through
patches
ordered
discuss
and disruptions
−μ∇ u
+theumodel
+ inΠthe0context
ez +of pathologies
∇Π′ =
jPj P;
+ eu · ∇P
= d ∇small
Pfull
−d
P + ðIof
− locally
PPÞ · ½eð∇uÞ
· P “stre
to understand the importance of each of the components of the
potency.
Although
the
polar
flow
∂t
filament dynamics.
spontaneous
polarization,
and
settles
into
f
Z2π Z ℓ
establishing the!global streaming pa
Model
A young Chara cell is modeled as a cylinder of radius R with
o
closis
and
reorienting
ef
+ αas
+that
αpassive
− κðP
· dÞdactive
; nematic
p Pthe
u u the
periodic boundaries at a distance L apart. The fixed chloroplasts
traditional
shear
al
u
·
e
dz dθ
=
0:
∇
·
u
=
0
and
z
lie at the edge, with the subcortical cytoplasm a thin cylindrical
full potency.
Although
thefactor
polare,flow
alignme
layer between the chloroplasts and the vacuolar membrane; the
the
restriction
still
plays a
vacuole then comprises the bulk of the cell. The cytoplasm is
0 actin0filtaken to consist ofZ
a2π
layerZ
of ℓshort, initially disordered
establishing
the
streamingbetween
pattern,
it pi
where all coupling
constantscurved
are global
now
nondimensional.
The
aments beneath a layer of myosin-coated endoplasmic reticulum
IZ
boundaries
adjac
vesicles, which drive flow by forcing the cytoplasm as they proðsÞ
ðrÞ
cess
on
the actin
filaments (Fig.u
1C).
Because
the subcortical
rameters
d
and
d
are
and rotational
diffusion
that
thespatial
traditional
nematic
shear
·
e
dz dθ
=
0:
∇
·
u
=
0
and
regions,
where
flow
shearalignment,
is constan
high, in
zboundary
layer ishave
very thin compared
with the vacuole,
we approximate the
All fields
periodic
conditions
on
θ
=
0;
2π
and
cytoplasmic layer as a purely 2D cylindrical shell. We also asrespectively.
Thethe
finalrestriction
hydrodynamics
read
factor
e,stages
still of
plays
role:
it a
sume that the effective viscosity contrast between cytoplasm and
in
the
later
the atime
evolu
z = 0; ℓ, where
ℓ
=
L=R
is
the
nondimensional
cell
length.
0 0large to take vacuolar flow as purely
vacuolar fluid is sufficiently
&butGoldstein
(2013)Results
PNAS
dunkel@math.mit.edu
passive, Woodhouse
induced by cytoplasmic flow (25)
not affecting it.
To gauge the model’s
robustness, we up- a
curved IZ boundaries
between
adjacent
This yields a truly 2D problem.
executed a numerical parameter sweep of nonzero α and α
To test the model and understand more about the regimes of
where ΠðxÞ is the 2D pressure and FðxÞ is the filament-induced
forcing, discussed shortly. The frictional term νu captures the
effect of the no-slip boundary proximity in our thin-shell approximation, in a manner similar to Hele–Shaw flow (SI Text).
There are two crucial constraints on the flow field that must be
incorporated. First, because the tonoplast membrane, which separates the cytoplasm and vacuole, behaves as a 2D incompressible fluid (39), we
1 must have incompressibility ∇ · u = 0. 2
3
Second, the presence of end caps on the cell must be acknowledged, either through explicit modeling of the end geometry or,
as we do here, qualitatively
through the simple extra constraint
p
u
R 2π R L
of zero net flux 0 0 u · ez dzdθ = 0, essentially due to the
incompressibility of a finite domain in the ez direction. This is
z
z
balanced by allowing a longitudinal pressure gradient Π0 and
writing ΠðxÞ = Π0 z + Π′ðxÞ, with Π′ðxÞ the remaining fully periodic pressure field.
We now turn to the filament suspension, whose dynamics will
incorporate several crucial effects. The filaments are taken to be
restricted: passive advection and shear alignment in the flow u is
inhibited by a factor e due to frictional or binding effects of the
filaments on the chloroplasts or with cortical polymer networks.
Fig. 4. Cyclosis developing in our model for αp = 1 and αu = 0:5, with other parameters fixed as in the text. Color coding corresponds to the z-component of
As a consequence oftherestriction,
the filaments are non–selforder vector P, with purple for Pz > 0 and green for Pz < 0, darker for lower magnitude. The white lines represent IZs separating up- and down-streaming
advective: a vesicle walking
forward onare
a filament
induce
regions. Superimposed
streamlineswill
of the
cytoplasmic flow u induced by the filament field P, where the flow is directed from the thin end to the thick end
only a negligible backward
of (Upper)
the filament
itself. The
of the propulsion
individual lines.
Time sequence
of six frames, showing progression from random disorder through local order to complete steady cyclosis.
filaments are also taken
to spontaneously
bundle:patterns
filaments
willthree indicated frames.
(Lower)
“Unwrapped” streaming
of the
locally align with each other, controlled by a coupling constant αp
(a rate constant for the exponential growth of small local podðsÞ = dðrÞ = 0:025, κ = 0:5, and μ = 0:05 to focus on the most imlarization), mimicking the ∂Ppresence of ðsÞbundling
proteins.
2
ðrÞ
+
eu
·
∇P
=
d
∇
P
−
d
P
+
ðI
−
PPÞ
·
½eð∇uÞ
·
P
portant coupling constants αp and αu . We also chose ℓ = 5 as an
(Inhibiting the action of bundling
proteins thus corresponds to
∂t
appropriate
radius-to-length ratio for a young cell with observsetting αp = 0.)
!
+ αp P or
+ αsubstrate
dÞd ;
ably organized actin cables (36).
u u − κðP · patTo represent nonspherical cellular geometry
terning we include a repulsive direction d, which filaments will
Time Progression. Fig. 4 displays a typical time sequence for nuwhere
all coupling
constants
The papreferentially avoid, with
coupling
constant
κ (Fig. are
3A);now
this nondimensional.
will
ðsÞ
ðrÞ
merical
integration of the model with an illustrative choice of the
rameters
d
and
d
are
spatial
and
rotational
diffusion
constants,
be set here to d = eθ , the circumferential direction. Although
parameters αp ; αu . The experimentally observed regeneration
finalis hydrodynamics
read
apparently a strong respectively.
assumption, The
there
remarkable experiprogression (28) is clearly reproduced as it moves from disorder,
mental precedent for such an effect whereby
filaments can 2
2
through small patches of locally ordered “streamlets” caused by
−μ∇
u +inhibition
u + Π0 ez +of
∇Π′
= jPj P;
reorient circumferentially around cells
upon
bindspontaneous polarization, and settles into fully developed cying or director components (40–42).
subject to
closis as the passive and active reorienting effects of flow reach
full potency. Although the polar flow alignment is important for
Z2π Z ℓ
establishing the global streaming pattern, it is worth remarking
old
new
that the traditional nematic shear alignment, though damped by
u
·
e
dz dθ
=
0:
∇
·
u
=
0
and
z
IZ
IZ
IZ
the restriction factor e, still plays a role: it acts to smooth out
0 0
curved IZ boundaries between adjacent up- and down-streaming
regions, where flow shear is high, into straight lines. This is seen
All fields have periodic boundary conditions on θ = 0; 2π and
in the later stages of the time evolution in Fig. 4.
z = 0; ℓ, where ℓ = L=R is the nondimensional cell length.
Blood cells: shape & function
source: wiki
red blood cells affected by sicklecell disease
http://learn.genetics.utah.edu/
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Optical tweezer
source: wiki
http://www.nature.com/ncomms/journal/v4/n4/extref/ncomms2786-s1.swf
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Red blood cell in tweezer
Basu et al (2011) Biophys J
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The transition between rigid-body–like and fluidized cell occurs
enables a material element to deform locally to form a dimple,
−
+
in a hysteretic domain of shear rates, fγ_ c − γ_ c g. In this domain
because hydrodynamic constraints tend to profile global ellipboth flipping/rolling and
tankdynamics
treading are of
stable,
each dynamic
Full
a red
blood cell
in shapes,
shearasflow
soidal
observed for lipid vesicles or elastic capsules in
state depends on how Jules
it isDupire,
reached,
either
by increasing
or deflow. We propose an interpretation involving a buckling pheMarius Socol,
and Annie
Viallat
creasing γ_ . The mechanisms
transition
toward
F Scientifique,
and toward
nomenon
forMixte
which
the7333,
biconcave
shape is of minimal energy,
Aix Marseilleof
Université,
Centre National
de la Recherche
Laboratoire Adhésion
et Inflammation Unité
de Recherche
Inserm
UMR1067, 13009 Marseille, France
TT are different. In the TT-to-F transition, TT stabilizes the cell
and a weak shape memory. The shape memory results from an
Edited by T. C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved October 30, 2012 (received for review June 20, 2012)
orbit. In the R-to-TT transition, the cell axis of symmetry has to
anisotropic stress-free shape of the membrane. The anisotropy
At the cellular scale, blood fluidity and mass transport depend on1 which the axis of symmetry of the cell lies in the shear plane.
Jules
Dupire,
and
Annie
Viallat
rotate through a π/2 angle.
The
streamlines
on the
membrane
may come
from
of the spherical shell—for
the dynamics
of
redMarius
blood cellsSocol,
in blood
flow, cell
specifically
on their
Some observations,
however,
suggestthe
that inhomogeneity
other cellular orideformation and orientation. These dynamics are governed by
entations may be more stable (17, 18). Furthermore, numerical
have to change abruptly
with
aUniversité,
sharp
increase
of
shear
strain
instance,
strengthened
region
or7333,
anisotropic
elastic
Aix
Marseille
Centre
National
de la
Recherche
Laboratoire
Adhésion
et Inflammation
Unité Mixte de
Recherche
Inserm
cellular
rheological
properties,
such
as internal
viscosity
andScientifique,
predictions
of RBC deformations
show
significantequatorial
discrepancies
cytoskeleton
elasticity.
in which
cell rheology
altered
UMR1067,
13009
Marseille,
France
with experimental
observations.
Indeed, experiments
report
only
energy. This can be achieved
only
at Inadiseases
higher
shear
rateisvalue.
properties
(27)—or
from
athenonspherical
shell. The latter hygenetically or by parasitic invasion or by changes in the microenthe
stationary
stretched
shape
of
cells
steadily
aligned
in
flow
−
vironment,
blood
flow may
be severely
impaired.
The nonlinear
Finally, the values found
the
critical
shear
stresses
η0 γ_ atc highPA,pothesis,
supported
by(received
the for
work
Lim,
Wortis, and Mukhoshear
whereas
recent30,numerical
studies
predict
Edited
byfor
T.
C. Lubensky,
University
of
Pennsylvania,
Philadelphia,
and rates,
approved
October
2012
reviewof
June
20, 2012)
+ interplay between cell rheology and flow may generate complex “breathing” dynamic states with strong shape deformations at
(0.023 Pa) and η0 γ_ c (0.086
Pa)remain
are
inunexplored
agreement
with
that
(28),
allows
one to
find
observed
shapes
by using a
dynamics,
which
largely
Under
low
shearpadhyay
rates
RBCs
elastic
(9–16).
which
theand
axis
of capsules
symmetry
of the
cell lies inRBC
the shear
plane.
At
the cellular
scale,
blood
fluidityexperimentally.
and mass
transport
depend
on for both
simple shear flow, only two motions, “tumbling” and “tank-treadHere,
we
couple
two
videomicroscopy
approaches
providing
the dynamics of red blood cells in blood flow, specifically onmechanical
their
Some model
observations,
however,
suggest that
otherand
cellular
ori-elasticity.
reported in ref. 7.
including
bending,
stretch,
shear
ing,” have been described experimentally and relate to cell memultidirectional pictures of RBCs to elucidate the full dynamics
1
Full dynamics of a red blood cell in shear flow
deformation
orientation.
These
dynamics
entations
may be
stable
(17,
chanics. Here, weand
elucidate
the full dynamics
of red
blood cellsare
in governed
of an RBC inby
shear flow,
and we analyze
themore
mechanical
origin
of 18). Furthermore, numerical
shear flowrheological
by coupling two
videomicroscopy
approaches
provid- viscosity
cellular
properties,
such
as internal
and
the observed
dynamics
(shapes andof
regimes
motion).
predictions
RBCofdeformations
show significant discrepancies
ing
multidirectional
pictures
of
cells,
and
we
analyze
the
mechanUnder
physiological
conditions,
a
mature
cell
is
a
biconcave
cytoskeleton elasticity. In diseases in which cell rheology is altered
with experimental observations. Indeed, experiments report only
ical origin of the observed dynamics. We show that contrary to
disk about 6–8 μm in diameter and 2 μm thick. Its membrane
genetically
or
by
parasitic
invasion
or
by
changes
in
the
microenthe bilayer
stationary
shape
of cells steadily aligned in the flow
common belief,
when red blood cells flip into the flow, their oriconsists of a fluid lipid
and anstretched
elastic spectrin
network
Fig. 5. Rolling-to-tank-treading
transition
vironment,
blood by
flow
may rate.
be severely
impaired.
nonlinear
atbilayer
high and
shear
rates,
whereas
numerical studies predict
entation is determined
the shear
We discuss the
“rolling” The
lying
just beneath the
attached
to the
membranerecent
inobserved on RBCs bearing interplay
a bead;
dextran
motion,
similar
to
a
rolling
wheel.
This
motion,
which
permits
the
tegral
proteins.
The
inner
cell
volume
is
filled
with
a
solution
of
between cell rheology and flow may generate complex
“breathing” dynamic states with strong shape deformations at
cells to avoid energetically costly deformations, is a true signature
hemoglobin. Although the structure of the RBC is one of the
2 106 g/mol, c = 9% (wt/wt);
scale
bar,
8
μm;
dynamics,
which
remain
largely
unexplored
experimentally.
Under
low shear rates for both RBCs and elastic capsules (9–16).
of the cytoskeleton elasticity. We highlight a hysteresis cycle and
simplest among cells, it nevertheless involves several mechanical
−1
simple
shear
flow,
only
two
motions,
“tumbling”
and
“tank-treadHere,
we couple
twoandvideomicroscopy
approaches providing
.
top-view observation. (A) Shear
rate
=
3
s
two transient dynamics driven by the shear rate: an intermittent
parameters: viscosities of
the hemoglobin
solution
of the lipid
regime
during
the
“tank-treading-to-flipping”
transition
and
a
Frisand bending elasticity
of the
lipid bi- to elucidate the full dynamics
ing,” have
been described experimentally and relatebilayer,
to cellincompressibility
memultidirectional
pictures
of RBCs
The symmetry axis of the rolling
cell (images
bee-like “spinning” regime during the “rolling-to-tank-treading”
layer, and
compressibility
and
shear
elasticity
of
the
spectrin
chanics.
Here, we elucidate the full dynamics of red blood
cells
in
of an RBC in shear flow, and we cyanalyze the mechanical origin of
transition.
Finally, we
reveal that the biconcave red cell shape is
1–7) rotates gradually (images
8–10).
The
toskeleton. The nonspherical biconcave shape of RBCs enables
shear
flow
by
coupling
two
videomicroscopy
approaches
providthe
observed
dynamics
(shapes
and
highly stable under moderate shear stresses, and we interpret this
shape changes at constant volume and area. Moreover, the mem- regimes of motion).
spinning about the symmetry
axis
is
detecresultmultidirectional
in terms of stress-free
shape and
ing
pictures
of elastic
cells, buckling.
and we analyze the
branemechanhas a memory ofUnder
its shapephysiological
(19): after a shapeconditions,
deformation a mature cell is a biconcave
ted by the bead motion (images
10–19).
Fiinduced
by
an
external
force,
the
membrane
returns
to its initial
ical origin of the observed dynamics. We show that contrary to
disk about 6–8 μm in diameter
and 2 μm thick. Its membrane
elastic capsule | low Reynolds number | shape memory | erythrocyte
biconcave shape and the membrane elements return to their initial
common
when red blood cells flip into the flow, their orinally, the streamlines change
andbelief,
the cell
consists
of
a
fluid
lipid
bilayer
and
position after removal of the force. The rim, for instance, is always an elastic spectrin network
entation
determined
by the shear
We
discuss
“rolling”
justelements.
beneath
the bilayer
and
formed
by the same lying
membrane
However,
the actual
de-attached to the membrane intank-treads (images 20–30).
A
bar suspension
loodvertical
is is
a concentrated
of cells:rate.
45% in
volume
is the
formation
state
of
the
membrane,
even
in
the
biconcave
state,
is
motion,
similar
toblood
a rolling
wheel.
This
motion,
permits
the
occupied
by red
cells
(RBCs).
Its fluidity
tegral proteins.
The inner
celltransition
volume
is filled
with a solution
of an overall
separates the different movements.
Sequence
of
46.6
s; scale
bar,strongly
7 which
μm. (B)
The
tank-treading
movement
at
the
sometimes
presents
not
known
because
the
stress-free
shape
of
the
membrane
for
depends
on its behavior
in flow, which
is deformations,
a key factor of proper
cells
to avoid
energetically
costly
is a true signature
hemoglobin.
Although
the structure of the RBC is one of the
−1
which the strain
vanishes,
determined.
rotation of part of the membrane,
which
behaves
locally
like
solid
by
as a energy
whole.
γ_ = has
6 snot
,been
time
sequence of 1.98 s.
tissue perfusion.
At the
cellular scale,
blood
flowa behavior
is rotating
of
the
cytoskeleton
elasticity.
We
highlight
a
hysteresis
cycle
and
simplest
among
cells,
it
For many years, the viscosity ratio λ, where λ isnevertheless
the viscosity involves several mechanical
affected primarily by the RBC response to the hydrodynamic stress
two
transient
dynamics
driven
shear
inside the cell relative
to the viscosity
of the suspending
parameters:
viscosities
of thesolution,
hemoglobin solution and of the lipid
in terms
of cell orientation
relative
to theby
flowthe
direction
andrate:
of cellan intermittent
hasand
been
the only mechanical
parameter
used
to
describe
the
deformation.
For example,
on one hand, at low shear rates, transition
similar
regime
during
the “tank-treading-to-flipping”
a Frisbilayer, incompressibility and bending
elasticity of the lipid bibehavior
of
RBCs
in
shear
flow
observed
in
pioneering
studies
cell
orientations
may
favor
the
formation
of
stacks
(rouleaux)
(1)
|
20812
www.pnas.org/cgi/doi/10.1073/pnas.1210236109
Dupire et al.
bee-like “spinning” regime during the “rolling-to-tank-treading”
layer,
and
compressibility
and
shear
elasticity
of the spectrin cy(2, 20): at high λ, RBCs have been reported to tumble (T), reof RBCs, like rolls of coins, which increases blood viscosity. On the
transition.
werates,
reveal
that the biconcave
red cell
Theflipping
nonspherical
shape of RBCs enables
ferredshape
to hereisas thetoskeleton.
particular unsteady
motion (F) biconcave
when
other hand, atFinally,
high shear
the individualization
of RBCs,
the cell axisthis
of symmetry
rotates
in
the
shear
plane.
The
nature
highly
stable and
under
and
we interpret
their alignment,
theirmoderate
stretching in shear
the flowstresses,
(2) decrease
blood
shape changes at constant volume and area. Moreover, the memof this motion (rigid-body–like or with membrane movement)
viscosityin
(3).terms
The orientation
and the deformation
flow of RBCs
result
of stress-free
shape andinelastic
buckling.
brane has a memory of its shape (19): after a shape deformation
B
are governed by their rheological properties. They result from the and its stability are not experimentally known. At low λ values
induced
by an
external force,
the membrane returns to its initial
and high shear rates,
RBCs have
a “fluid-like”
tank-treading
viscoelastic contributions of all components of the cell composite
elastic
capsule
low
Reynolds
number
shape
memory
erythrocyte
movement in whichbiconcave
the membrane
rotates
around
the center of elements return to their initial
structure. Moreover, RBC rheological properties also depend on
shape
and
the membrane
mass of the cell and
has
a
quasi-stable
inclination
(TT).
The The rim, for instance, is always
the microenvironment and on metabolic functionality (4). Both
position
after
removal
of the
force.
Keller
and
Skalak
(KS)
analytical
model
(21),
which
describes
an
local and systemic disturbances of homeostasis (in diabetes melformed
by the
same
membrane
elements. However, the actual deis a concentrated
suspension
of cells:
45% in
volume
is ellipsoid
RBC
as a viscous
of fixed
shape,
qualitatively
recovers
litus, lood
hypertension)
have the potential
to induce RBC
rheological
(T), strongly
(TT) as a function
of λ. state
However,
recent
experiments even in the biconcave state, is
formation
of the
membrane,
alterations
and consequently
to impaircells
blood (RBCs).
circulation. ItIts
thereoccupied
by red blood
fluidity
revealed new dynamic
states
specifically
due
to
the
shear elas- shape of the membrane for
fore
is
crucial
to
understand
the
relationships
between
the
rhenot known because the stress-free
depends
on its behavior in flow, which is a key factor of proper
ological properties of RBCs and their orientation and deformation ticity and the shape memory of the red cell membrane (7, 8): (i)
which the strain energy vanishes, has not been determined.
tissue
perfusion.
scale,
flow behavior is
in flow. This
question isAt
far the
from cellular
trivial because
evenblood
in a simple
For many years, the viscosity ratio λ, where λ is the viscosity
shear flow, primarily
RBCs present
variety
of dynamic
states,to
such
steady
affected
bya the
RBC
response
theashydrodynamic
stress
tank-treading,
swinging,
unsteady
tumbling,
and
chaotic
motion.
Author contributions: A.V.inside
designed research;
J.D. and
M.S. performed
research;
J.D.,
the cell
relative
to the
viscosity
of the suspending solution,
in terms of cell orientation relative to the flow directionM.S.,and
of cell
and A.V. analyzed data; and J.D. and A.V. wrote the paper.
To date, there has been little experimental work on the conhas been the only mechanical parameter used to describe the
deformation.
example, properties
on one hand,
at low
similar
The authors
declare no conflict of interest.
nection between For
the mechanical
of RBCs
and shear
their rates,
of RBCs in shear flow observed in pioneering studies
cell
orientations
may
favor
the formation
stacks (rouleaux)
(1) Directbehavior
dynamics
in shear flow
(5–8),
compared
with the manyof
numerical
This article is a PNAS
Submission.
|
|
|
BPNAS 2012
dunkel@math.mit.edu
Full dynamics of a red blood cell in shear flow
Jules Dupire, Marius Socol, and Annie Viallat1
Aix Marseille Université, Centre National de la Recherche Scientifique, Laboratoire Adhésion et Inflammation Unité Mixte de Recherche 7333, Inserm
UMR1067, 13009 Marseille, France
Full dynamics of a red blood cell in shear flow
Edited by T. C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved October 30, 2012 (received for review June 20, 2012)
which the axis of symmetry of the cell lies in the shear plane.
Some observations, however, suggest that other cellular orideformation and orientation. These dynamics are governed by
entations may be more stable (17, 18). Furthermore, numerical
Aix
Marseille
Université,
Centre
National
de laviscosity
Recherche
Adhésion show
et Inflammation
Unité Mixte de Recherche 7333, Inserm
cellular
rheological
properties,
such
as internal
andScientifique,
predictionsLaboratoire
of RBC deformations
significant discrepancies
cytoskeleton13009
elasticity.
In diseases
in which cell rheology is altered
UMR1067,
Marseille,
France
with experimental observations. Indeed, experiments report only
genetically or by parasitic invasion or by changes in the microenthe stationary stretched shape of cells steadily aligned in the flow
vironment,
blood
flow may University
be severely of
impaired.
The nonlinear
at high
whereas
recent30,numerical
studies for
predict
Edited
by T.
C. Lubensky,
Pennsylvania,
Philadelphia,
PA,shear
and rates,
approved
October
2012 (received
review June 20, 2012)
interplay between cell rheology and flow may generate complex
“breathing” dynamic states with strong shape deformations at
dynamics,
which remain
largely
unexplored
Under
low
shear rates
RBCs
elastic
(9–16).
which
theandaxis
of capsules
symmetry
of the cell lies in the shear plane.
At
the cellular
scale,
blood
fluidityexperimentally.
and mass transport
depend
on for both
simple shear flow, only two motions, “tumbling” and “tank-treadHere,
we
couple
two
videomicroscopy
approaches
providing
the dynamics of red blood cells in blood flow, specifically on their
Some observations, however, suggest that other cellular oriing,” have been described experimentally and relate to cell memultidirectional pictures of RBCs to elucidate the full dynamics
deformation
and
orientation.
These
dynamics
are
governed
by
entations
may be
stable
(17,
chanics. Here, we elucidate the full dynamics of red blood cells in
of an RBC in shear flow,
and we analyze
themore
mechanical
origin
of 18). Furthermore, numerical
shear flowrheological
by coupling two
videomicroscopy
approaches
provid- viscosity
cellular
properties,
such
as internal
and
the observed
dynamics
(shapes andof
regimes
motion).
predictions
RBCofdeformations
show significant discrepancies
ing
multidirectional
pictures
of
cells,
and
we
analyze
the
mechanUnder
physiological
conditions,
a
mature
cell
is
a
biconcave
cytoskeleton elasticity. In diseases in which cell rheology is altered
with experimental observations. Indeed, experiments report only
ical origin of the observed dynamics. We show that contrary to
disk about 6–8 μm in diameter and 2 μm thick. Its membrane
genetically
or
by
parasitic
invasion
or
by
changes
in
the
microenthe bilayer
stationary
shape
of cells steadily aligned in the flow
common belief, when red blood cells flip into the flow, their oriconsists of a fluid lipid
and anstretched
elastic spectrin
network
vironment,
blood by
flow
may rate.
be severely
impaired.
nonlinear
atbilayer
high and
shear
rates,
whereas
numerical studies predict
entation is determined
the shear
We discuss the
“rolling” The
lying
just beneath the
attached
to the
membranerecent
inmotion,
similar
to
a
rolling
wheel.
This
motion,
which
permits
the
tegral
proteins.
The
inner
cell
volume
is
filled
with
a
solution
of
interplay between cell rheology and flow may generate complex
“breathing” dynamic states with strong shape deformations at
cells to avoid energetically costly deformations, is a true signature
hemoglobin. Although the structure of the RBC is one of the
dynamics,
which
remain
largely
unexplored
experimentally.
Under
low shear rates for both RBCs and elastic capsules (9–16).
of the cytoskeleton elasticity. We highlight a hysteresis cycle and
simplest among cells, it nevertheless involves several mechanical
simple
shear
flow,
only
two
motions,
“tumbling”
and
“tank-treadHere,
we couple
twoandvideomicroscopy
approaches providing
two transient dynamics driven by the shear rate: an intermittent
parameters: viscosities of
the hemoglobin
solution
of the lipid
regime
during
the
“tank-treading-to-flipping”
transition
and
a
Frisand bending elasticity
of the
lipid bi- to elucidate the full dynamics
ing,” have been described experimentally and relatebilayer,
to cellincompressibility
memultidirectional
pictures
of RBCs
bee-like “spinning” regime during the “rolling-to-tank-treading”
layer, and
compressibility
and
shear
elasticity
of
the
spectrin
chanics.
Here, we elucidate the full dynamics of red blood
cells
in
of an RBC in shear flow, and we cyanalyze the mechanical origin of
transition. Finally, we reveal that the biconcave red cell shape is
toskeleton. The nonspherical biconcave shape of RBCs enables
shear
flow
by
coupling
two
videomicroscopy
approaches
providthe
observed
dynamics
(shapes
and
highly stable under moderate shear stresses, and we interpret this
shape changes at constant volume and area. Moreover, the mem- regimes of motion).
resultmultidirectional
in terms of stress-free
shape and
ing
pictures
of elastic
cells, buckling.
and we analyze the
branemechanhas a memory ofUnder
its shapephysiological
(19): after a shapeconditions,
deformation a mature cell is a biconcave
induced
by
an
external
force,
the
membrane
returns
to its initial
ical origin of the observed dynamics. We show that contrary to
disk about 6–8 μm in diameter
and 2 μm thick. Its membrane
elastic capsule | low Reynolds number | shape memory | erythrocyte
biconcave shape and the membrane elements return to their initial
common belief, when red blood cells flip into the flow, their oriconsists
of
a
fluid
lipid
bilayer
and
position after removal of the force. The rim, for instance, is always an elastic spectrin network
entation
determined
by the shear
We
discuss
“rolling”
justelements.
beneath
the bilayer
and
formed
by the same lying
membrane
However,
the actual
de-attached to the membrane inlood is is
a concentrated
suspension
of cells:rate.
45% in
volume
is the
formation
state
of
the
membrane,
even
in
the
biconcave
state,
is
motion,
similar
toblood
a rolling
motion,
which permits the
occupied
by red
cells wheel.
(RBCs). This
Its fluidity
strongly
tegral proteins. The inner cell volume
is filled with a solution of
not
known
because
the
stress-free
shape
of
the
membrane
for
depends
on its behavior
in flow, which
is deformations,
a key factor of proper
cells
to avoid
energetically
costly
is a true signature
hemoglobin. Although the structure of the RBC is one of the
which the strain energy vanishes, has not been determined.
tissue perfusion. At the cellular scale, blood flow behavior is
of
the
cytoskeleton
elasticity.
We
highlight
a
hysteresis
cycle
and
simplest
cells, λitisnevertheless
For many years, the
viscosityamong
ratio λ, where
the viscosity involves several mechanical
affected primarily by the RBC response to the hydrodynamic stress
two
transient
dynamics
driven
shear
inside the cell relative
to the viscosity
of the suspending
parameters:
viscosities
of thesolution,
hemoglobin solution and of the lipid
in terms
of cell orientation
relative
to theby
flowthe
direction
andrate:
of cellan intermittent
hasand
been
the only mechanical
parameter
used
to
describe
the
deformation.
For example,
on one hand, at low shear rates, transition
similar
regime
during
the “tank-treading-to-flipping”
a Frisbilayer, incompressibility and bending
elasticity of the lipid bibehavior of RBCs in shear flow observed in pioneering studies
cell
orientations
may
favor
the
formation
of
stacks
(rouleaux)
(1)
bee-like “spinning” regime during the “rolling-to-tank-treading”
layer,
and
compressibility
and
shear
elasticity
of the spectrin cy(2, 20): at high λ, RBCs have been reported to tumble (T), reof RBCs, like rolls of coins, which increases blood viscosity. On the
transition.
werates,
reveal
that the biconcave
red cell
Theflipping
nonspherical
shape of RBCs enables
ferredshape
to hereisas thetoskeleton.
particular unsteady
motion (F) biconcave
when
other hand, atFinally,
high shear
the individualization
of RBCs,
the cell axisthis
of symmetry
rotates
in
the
shear
plane.
The
nature
highly
stable and
under
and
we interpret
their alignment,
theirmoderate
stretching in shear
the flowstresses,
(2) decrease
blood
shape changes at constant volume and area. Moreover, the memof this motion (rigid-body–like or with membrane movement)
viscosityin
(3).terms
The orientation
and the deformation
flow of RBCs
result
of stress-free
shape andinelastic
buckling.
brane has a memory of its shape (19): after a shape deformation
are governed by their rheological properties. They result from the and its stability are not experimentally known. At low λ values
induced
by an
external force,
the membrane returns to its initial
and high shear rates,
RBCs have
a “fluid-like”
tank-treading
viscoelastic contributions of all components of the cell composite
elastic
capsule
low
Reynolds
number
shape
memory
erythrocyte
movement in whichbiconcave
the membrane
rotates
around
the center of elements return to their initial
structure. Moreover, RBC rheological properties also depend on
shape
and
the membrane
mass of the cell and
has
a
quasi-stable
inclination
(TT).
The The rim, for instance, is always
the microenvironment and on metabolic functionality (4). Both
position
after
removal
of the
force.
Keller
and
Skalak
(KS)
analytical
model
(21),
which
describes
an
local and systemic disturbances of homeostasis (in diabetes melformed
by the
same
membrane
elements. However, the actual deis a concentrated
suspension
of cells:
45% in
volume
is ellipsoid
RBC
as a viscous
of fixed
shape,
qualitatively
recovers
litus, lood
hypertension)
have the potential
to induce RBC
rheological
(T), strongly
(TT) as a function
of λ. state
However,
recent
experiments even in the biconcave state, is
formation
of the
membrane,
alterations
and consequently
to impaircells
blood (RBCs).
circulation. ItIts
thereoccupied
by red blood
fluidity
revealed new dynamic
states
specifically
due
to
the
shear elas- shape of the membrane for
fore
is
crucial
to
understand
the
relationships
between
the
rhenot known because the stress-free
depends
on its behavior in flow, which is a key factor of proper
ological properties of RBCs and their orientation and deformation ticity and the shape memory of the red cell membrane (7, 8): (i)
which the strain energy vanishes, has not been determined.
tissue
perfusion.
scale,
flow behavior is
in flow. This
question isAt
far the
from cellular
trivial because
evenblood
in a simple
For many years, the viscosity ratio λ, where λ is the viscosity
shear flow, primarily
RBCs present
variety
of dynamic
states,to
such
steady
affected
bya the
RBC
response
theashydrodynamic
stress
tank-treading,
swinging,
unsteady
tumbling,
and
chaotic
motion.
Author contributions: A.V.inside
designed research;
J.D. and
M.S. performed
research;
J.D.,
the cell
relative
to the
viscosity
of the suspending solution,
in terms of cell orientation relative to the flow directionM.S.,and
of cell
and A.V. analyzed data; and J.D. and A.V. wrote the paper.
To date, there has been little experimental work on the conhas been the only mechanical parameter used to describe the
deformation.
example, properties
on one hand,
at low
similar
The authors
declare no conflict of interest.
nection between For
the mechanical
of RBCs
and shear
their rates,
of RBCs in shear flow observed in pioneering studies
cell
orientations
may
favor
the formation
stacks (rouleaux)
(1) Directbehavior
dynamics
in shear flow
(5–8),
compared
with the manyof
numerical
This article is a PNAS
Submission.
At the cellular scale, blood fluidity and mass transport depend on
1
Jules
Dupire,
Annie Viallat
the dynamics
of redMarius
blood cellsSocol,
in bloodand
flow, specifically
on their
B
|
|
|
B
dunkel@math.mit.edu
Blood cell - simulations
McWhirter et al (2012) New J Phys
dunkel@math.mit.edu
Vesicles (“artificial” cells)
Mai & Eisenberg
Chem Soc Rev 2012
dunkel@math.mit.edu
between
force 87.16.dm,
per unit 47.55.N!,
area on 83.85.Cg
a surface and the adjacent
PACS numbers:
quantified
measuring
far upstream from
vesicles
thewas
flow. The flow induced in the membrane took the form of stack
of suchbyslices
a three-dimensional
velocity
field
shear rate, it is natural to ask whether the experimental
speed of microspheres as a function of week
heightending
above the
two vortices, rather than the simple overturning flow that
setup
of
Vézy
et
al.
[21]
suggests
a
means
to
study
memP H Y S I Cdroplet
A L of
R Eone
V Ifluid
E W in L Ecoverslip.
T T E R SShear rates were typically in the
range 2013
1 $ "_ $
19 JULY
occur in a hemispherical
PRL 111, 038103would
(2013)
%1
brane
fluid
mechanics
in
detail.
end,
we
describe
6 s . PIV was done with Matlab by adapting standard
the in
background
ofcells
an immiscible
second
fluid
without
vesicles
known:
internodal
of To
thethis
aquatic
plant
elbrück on the
a method
that
thedifference
setdiameter
by shear toofthe
code [27] to track small dilute tracers by finding the
the membrane
[23,24].
This
isup
attributable
Charahere
corallina,
these
canquantifies
be cylinders
1flows
mm in
mbranes [1] it
time-averaged
fieldVesicles
[28]. For three-dimensional
adherent
vesicles,
and, This
through
a recent
calculation
of thetonoplast
membrane,
restricts
the
flow
and up
toMembrane
10incompressibility
cm
long [16].
iswhich
subject
to [25],
n be viewed as
Viscosity
Determined
from
Shear-Driven
Flow velocity
in Giant
reconstruction, movies were recorded at !30 frames per
field
to oneof
that
is two-dimensionally
provides
a means
determining
Thei.e.,
continuous
hydrodynamic
shear
throughmembrane
the actiondivergence
ofviscosity.
cyto- free,
rface viscosity.
at intervals of 2–3 !m throughout and above
area
thevelocimetry
vesicle
surface
method
uses
particle
(PIV)
to measureVasily second
plasmic
streaming,
motion image
ofonthe
cytoplasm
surrounding
, this viscosity
Aurelia
R. conserving,
Honerkamp-Smith,
Francis
G.[25].
Woodhouse,
Kantsler,
and Raymond E. Goldstein
vesicles containing microspheres [Fig. 2(b)], giving
Since
viscosity
is
the
coefficient
of
proportionality
the vacuole
[17].
Because
of flows
its potential
role
transport
he translational
the three-dimensional
inside
andinoutside
vesicles,
Department
of
Applied
Mathematics
and
Theoretical
Physics,
Centre for Mathematical
Sciences,
University
of Cambridge,
two-dimensional
velocity
field slices
[Fig. 2(a)]. From a
between
force
per
unit
area
on
a
surface
and
the
adjacent
[18] and
thereparticle
is great
interest
in the the
three-dimensional
ions within the
tracking
to monitor
shear-induced
Wilberforce
Road,
CambridgemoveCB3 0WA, United
Kingdom
stack of such slices a three-dimensional velocity field was
shear
rate, shear-induced
it is natural to flows
ask whether
the the
experimental
characteristics
of such
k [3,4] suggests
(Received
14[19]
January
2013;
published
17 July 2013)
ment of phase-separated
domains
within
theand
membrane,
in
of Vézy et al.tonoplast
[21] suggests
role played
bysetup
the intervening
[20]. a means to study memn external flows
a microfluidic
environment.
The
viscosity
of lipid bilayer
membranes
brane
fluid mechanics
in detail.
To thisplays
end, an
weimportant
describe role in determining the diffusion constant
is viscosity [6].
Figure
1
shows
the
experimental
setup:
a
vesicle
of
radius
here a method
thatand
quantifies
the flows
up by shear
of
of embedded
proteins
the dynamics
of set
membrane
deformations,
yet it has historically proven very
essed as d " !,
R, typically
in
the
range
of
10–40
!m,
adheres
to
the
surface
adherent
vesicles,
and,
a recent
calculation
[25],
difficult
to measure.
Here
wethrough
introduce
a new method
based
on quantification of the large-scale circulation
! is the bilayer
of a patterns
microfluidic
chamber
in
the
presence
of
a
flow
with
shear
provides
a
means
of
determining
membrane
viscosity.
The
induced inside vesicles adhered to a solid surface and subjected to simple shear flow in a
enders !m very
_ The
ratemicrofluidic
".
chamber,
typically
2 mm
wide
and 200
!m
method
uses
particle
image
velocimetry
(PIV)
to deep,
measure
device.
Particle
image
velocimetry
based
on spinning disk confocal imaging of tracer
cult to measure
the
three-dimensional
flows inside
and
outside vesicles,
is made
from
polydimethylsiloxane
by
soft
lithography
and
particles inside and outside of the vesicle and tracking of phase-separated membrane domains are used
veloped include
and aparticle
tracking that
to monitor
thetreated
shear-induced
movesealed
with
glassthe
coverslip
had been
to promote
to reconstruct
full three-dimensional
flow pattern
induced by the shear. These measurements show
e-bound microment of phase-separated
domains
within
the membrane,
vesicle
adhesion.
Vesicles
were
produced
by
standard
meth- in analysis, and allow direct determination of
excellent
agreement
with
the
predictions
of
a
recent
theoretical
s in multicoma microfluidic environment.
ods the
of membrane
electroformation
[26] in 100 mM sucrose with or
of fluctuation
Figure 1viscosity.
shows the experimental setup: a vesicle of radius
without R,
0:5typically
!m microspheres
(Invitrogen, Carlsbad, CA).
point [10,11].
in the range of 10–40 !m, adheres to the surface
10.1103/PhysRevLett.111.038103
We DOI:
chose
compositions
tothe
obtain
twoof substantially
al experimental
of
alipid
microfluidic
chamber in
presence
a flow with shear PACS numbers: 87.16.dm, 47.55.N!, 83.85.Cg
differentrate
membrane
viscosities.
One2 mm
composition
gives
_ The chamber,
".
typically
wide and 200
!mprideep,
urface rheology
marily liquid-ordered
(Lo ) vesicles with by
a small
fraction ofand
is made from polydimethylsiloxane
soft lithography
observing dy" C):
liquid-disordered
at room
aSaffman
glass
coverslip
thattemperature
had been
to promote
] or membrane- Ever since
d ) phase
vesicles known: in internodal cells of the aquatic plant
thesealed
workwith
of(L
and
Delbrück
ontreated
the(!23
vesicle
adhesion.
Vesicles
were
produced
by
standard
meth- corallina, these can be cylinders 1 mm in diameter
cholesterol
(Sigma-Aldrich,
St.
Louis,
MO),
55%
periments havedynamics40ofmol%
Chara
inclusions in biological membranes [1] it
ods of electroformation [26] in 100and
mM5%
sucrose
with or
non-Newtonianhas been DPPC
(dipalmitoylphosphatidylcholine),
and
up to 10 cm long [16]. This tonoplast is subject to
recognized
that lipid bilayers can be viewed
asDiPhyPC
without
0:5
!m
microspheres
(Invitrogen,
Carlsbad,
CA).
DPPC,
DOPC, continuous
and
hydrodynamic shear through the action of cytoultrathin (diphytanoylphosphatidylcholine).
fluid layers
endowed
with a surface
viscosity.
We
chose
lipid
compositions
to
obtain
two
substantially
xtends to flowsAlong with
DiPhyPC
were
purchased
from
Avanti
Polar
Lipids
plasmic
that different
of the membrane
surrounding
fluid, this
viscosities.
One viscosity
composition gives
pri- streaming, motion of the cytoplasm surrounding
e, as the plantplays an (Alabaster,
AL) inand
used without
further purification.
the vacuole
[17]. Because of its potential role in transport
important
role
determining
the translational
marily
liquid-ordered
(L
)
vesicles
with a small fraction
of
o
membrane (or
Vesicles
containing
primarily
Ld phase with
a small
" there is great interest in the three-dimensional
[18]
and rotational
diffusion
constants
inclusions
within
the fraction
liquid-disordered
(Lof
d ) phase at room temperature (!23 C):
he largest lipid
FIG. 1of [2].
(color
online).
Microfluidic
gel
were
made
fromshear
85% experiment.
DOPC
and
15%
online).
Flow fieldsflows
inside[19]
an adhering
characteristics
of such
shear-induced
and thevesicle in
membrane
Adomains
ofcholesterol
theoretical
work
[3,4]
suggests
40body
mol%
(Sigma-Aldrich,
St. Louis,
MO),
55% FIG. 2 (color
(a) Schematic
of
the
chamber
(not
to
scale)
and
flows.
shear.
DPPC. The
Ld(dipalmitoylphosphatidylcholine),
phasesofwere
labeled
with 0.5%
DPPC
andTexasRed5% DiPhyPC
role played
by(a)
theExperimental
intervening two-dimensional
tonoplast [20]. PIV velocity fields at
that (b)
nonequilibrium
dynamics
vesicles
in external
flows
Confocal imaging reconstruction of an adhering hemispheriDPPE
(Invitrogen).
Coverslips
cleaned
(diphytanoylphosphatidylcholine).
DPPC,aggressively
[5] can
also
be vesicle
sensitive
to
the
value
of were
this
viscosity
[6].DOPC, and heights z=R ¼ 0:26, 0.47, 0.71 above coverslip. (b) Confocal
phase
with
small
L
domains
visible
on
its
surface.
cal
L
o
d
nder the terms of
slices at same fractional heights as (a) show vesicle (red)
were
Polar Lipids
inTracking
NaOHDiPhyPC
and
soaked
apurchased
solution
offrom
0.001%
for
be
asAvanti
d polylysine
"
membrane
viscosity
!mincan
background
(c)
and!,Ld
(c)–(d)
of gel
domains
in
Ldexpressed
e. Further distri-As the
(Alabaster,
AL)
and
used
purification. containing fluorescent microspheres. (c) Theoretical twominutes
for usethickness
with
Ld phase
vesicles,
orapex
in at
0.0005%
d 30
is in
the
and
!without
is vesicle
the further
bilayer
domains
Lomembrane
background
(d),
flowing
across
the
the author(s) andwhere
Vesicles
containing
primarily
L
a small
fraction dimensional velocity fields [25] for a sheared hemispherical
d phase
polyethylenimine
for scale
5inminutes
for
uses).
with
L
phase
nd DOI.
(tracks
color-coded
time
over
$2:6
¼ 2:6
s!1 the
o
very
fluid"_ viscosity,
nanometric
of d
renders
!with
m
z=R online).
¼ 0:3, 0.5,
0.7.fields
Interior
exterior vesicle
PIV vectors
ofVesicles
gel domains
were made
from 85%
DOPC
15% vesicle
FIG. 2at(color
Flow
insideand
an adhering
in
vesicles.
were
gently
osmotically
deflated
byand
dilutsmall. Not surprisingly,
it
has
proven
difficult
to
measure
in
each
panel
of
(a)
and
(c)
have
been
rescaled
for
visual
clarity.
shear. (a) Experimental two-dimensional PIV velocity fields at
DPPC. The Ld phases were labeled with 0.5% TexasReding
into
130
mM
glucose
and
10
mM
HEPES
shortly
before
;
ingenious
techniques
that
have
been
developed
include
!
(d)
Experimental
streamlines
the three-dimensional
velocity
m
heights
z=R ¼ 0:26,
0.47, 0.71ofabove
coverslip. (b) Confocal
038103-1
Published
by the American
Society
DPPE (Invitrogen).
CoverslipsPhysical
were cleaned
aggressively
loading
into
the
chamber.
measurements
ofin
the
motion
of membrane-bound
microobtained
by fractional
integrating
two-dimensional
fields,
slices
at same
heights
asdunkel@math.mit.edu
(a) show flow
vesicle
(red)comNaOH
and soaked
in a solution of 0.001%
polylysine for field
were made
on a Zeiss
Cell Observer spincontaining
fluorescent
microspheres.
pared
with theory
(e). Large
arrows in (c)
(a), Theoretical
(d), and (e) twoindicate
spheres [7],Measurements
diffusion constants
of domains
in multicom-
03
PRL 111, 038103 (2013)
PHYSICAL REVIEW LETTERS
week ending
19 JULY 2013
Membrane Viscosity Determined from Shear-Driven Flow in Giant Vesicles
Aurelia R. Honerkamp-Smith, Francis G. Woodhouse, Vasily Kantsler, and Raymond E. Goldstein
PHYSIC
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, United Kingdom week ending
(Received
AL REVIEW LE
T T E14RJanuary
S 2013; published 17 July 2013) 19 JULY 2013
The viscosity of lipid bilayer membranes plays an important role in determining the diffusion constant
of embedded proteins and the dynamics of membrane deformations, yet it has historically proven very
y relation. Figure 2(d)
difficult to measure. Here we introduce a new method based on quantification of the large-scale circulation
uch streamlines. patterns induced inside vesicles adhered to a solid surface and subjected to simple shear flow in a
microfluidic device. Particle image velocimetry based on spinning disk confocal imaging of tracer
nd around the vesicle,
particles inside and outside of the vesicle and tracking of phase-separated membrane domains are used
cent calculation [25].
to reconstruct the full three-dimensional flow pattern induced by the shear. These measurements show
erical cap of radiusexcellent
R agreement with the predictions of a recent theoretical analysis, and allow direct determination of
the membrane viscosity.
to the plane z ¼ 0,
DOI: 10.1103/PhysRevLett.111.038103
PACS numbers: 87.16.dm, 47.55.N!, 83.85.Cg
centered at the origin.
he vesicle (r < R), the
vesicles known: in internodal cells of the aquatic plant
Ever since
rnal fluid viscosity
#the
þ work of Saffman and Delbrück on the
Chara corallina, these can be cylinders 1 mm in diameter
of inclusions in biological membranes [1] it
y fields: u"dynamics
inside
the
and up to 10 cm long [16]. This tonoplast is subject to
has been recognized that lipid bilayers can be viewed as
m
u of the ultrathin
membrane,
continuous hydrodynamic shear through the action of cytofluid layers endowed with a surface viscosity.
plasmic streaming, motion of the cytoplasm surrounding
Alongexternal
with that of the surrounding fluid, this viscosity
a)]. The two
the vacuole [17]. Because of its potential role in transport
plays an important role in determining the translational
compressibility
equa[18] there is great interest in the three-dimensional
and rotational
diffusion constants of inclusions within the
characteristics and
of suchexternal
shear-induced
flows [19] and the
membrane
[2]. A bodyFIG.
of theoretical
work [3,4]
suggests Membrane
u$ ¼ 0, with
far-field
3 (color
online).
flows.
role played by the intervening tonoplast [20].
that nonequilibrium dynamics of vesicles in external flows
the no-slip[5]condition
(a)toSelected
streamlines
along one side of an Lo vesicle
can also be sensitive
the value ofexternal
this viscosity
[6].
be expressed
as dclosed
" !, orbits above the surface. (b) TimeAs the membrane viscosity
!m canflow,
in shear
showing
no radial penetration,
d is the membrane
thickness
and ! stack
is the bilayer
lapse
confocal
of an Lo vesicle, viewed from above,
es must bewhere
continuous
fluid viscosity, the nanometric scale of d renders !m very
"
of Ld domains.
at r ¼ R,
and
small.
Notthus
surprisingly,illustrating
it has proven circulation
difficult to measure
; ingenious
that have been developed include
m¼
um ¼ 0 at !!measurements
%=2. techniques
of the motion of membrane-bound microdunkel@math.mit.edu
uids’ normal
stresses
spheres [7], diffusion constants of domains in multicom-
PRL 111, 038103 (2013)
week ending
19 JULY 2013
PHYSICAL REVIEW LETTERS
Membrane Viscosity Determined from Shear-Driven Flow in Giant Vesicles
Aurelia R. Honerkamp-Smith, Francis G. Woodhouse, Vasily Kantsler, and Raymond E. Goldstein
Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, United Kingdom
(Received 14 January 2013; published 17 July 2013)
The viscosity of lipid bilayer membranes plays an important role in determining the diffusion constant
of embedded proteins and the dynamics of membrane deformations, yet it has historically proven very
difficult to measure. Here we introduce a new method based on quantification of the large-scale circulation
patterns induced inside vesicles adhered to a solid surface and subjected to simple shear flow in a
microfluidic device. Particle image velocimetry based on spinning disk confocal imaging of tracer
particles inside and outside of the vesicle and tracking of phase-separated membrane domains are used
to reconstruct the full three-dimensional flow pattern induced by the shear. These measurements show
excellent agreement with the predictions of a recent theoretical analysis, and allow direct determination of
the membrane viscosity.
DOI: 10.1103/PhysRevLett.111.038103
Ever since the work of Saffman and Delbrück on the
dynamics of inclusions in biological membranes [1] it
has been recognized that lipid bilayers can be viewed as
ultrathin fluid layers endowed with a surface viscosity.
Along with that of the surrounding fluid, this viscosity
plays an important role in determining the translational
and rotational diffusion constants of inclusions within the
membrane [2]. A body of theoretical work [3,4] suggests
that nonequilibrium dynamics of vesicles in external flows
[5] can also be sensitive to the value of this viscosity [6].
As the membrane viscosity !m can be expressed as d " !,
where d is the membrane thickness and ! is the bilayer
PACS numbers: 87.16.dm, 47.55.N!, 83.85.Cg
vesicles known: in internodal cells of the aquatic plant
Chara corallina, these can be cylinders 1 mm in diameter
and up to 10 cm long [16]. This tonoplast is subject to
continuous hydrodynamic shear through the action of cytoplasmic streaming, motion of the cytoplasm surrounding
the vacuole [17]. Because of its potential role in transport
[18] there is great interest in the three-dimensional
characteristics of such shear-induced flows [19] and the
role played by the intervening tonoplast [20].
dunkel@math.mit.edu
Dynamics of a vesicle in general flow
viscosities quantitative
of the inner discrepancy
and outer fluids.
Another analytical
microfluidic 4-roll mill device (23,24) manufactured in silicone
observable
with simulations.
approach based on a quasi-spherical vesicle approximated by a
elastomer by soft lithography (Fig. 1). Particle tracking veloci1
J.spherical
Deschamps,
V. Kantsler,
Segre,
and V. Steinberg
harmonics
expansion E.
used
a perturbation
scheme
metry (PTV) measurements show fair agreement with numerical
around the Lamb solution of the Stokes flow near a spherical
Department
of and
Physics
of
Complex
Systems,
Weizmann
Institute
of Science,
Rehovot, 76100
Israel
body in external
shear
flow
(1). Further
refinement
of the
modelSI
simulations
high
flow
uniformity
(Fig.
2 and
Appendix).
The
experiments were performed in the following way. A
!1
resulted
in
dynamic
equations
for
vesicle
shape
and
inclination
Author
contributions:
V.K.
and
V.S.
designed
research;
J.D. performed
research;
J.D., V.K.,
that of
!
/sV.S.
was
The
accessible
range ofUniversity
s was [0.05–0.32]
s and
vesicle
with
given
and #,
measured
Edited
Harry
Swinney,
of Texas
at motion
Austin,
Austin,
and
approved
May
2009
(received
forRreview
March
11, 2009) initially in situ in the same
and
analyzed
data;6,
and
V.S. wrote
the paper.
angle by
(3,4)
and L.
described
rather well both
the TT
and
the TX, E.S.,
[1.18–9.23].
reduce
error by
bars
S and
", the geometrical
device,
was followed at a prescribed value of !/s and s in the
The authors declare no conflict
of interest.
"c(!),
as verified
the on
recent
experiments
transition lineTo
An
approach
toand
quantitatively
study vesicle
dynamics
asinwell
as
stronger
vesicle
shape
deformations
a new
1
observation
window.
A feedback(3,17).
in the Moreover,
flow velocity
was used to
parameters
R
#
of
each
vesicle
were
measured
situ
from
(16–18).
However,
the
recent
experimental
key
finding
of
a
new
This article is a PNAS Direct Submission.
J. Deschamps, micro-objects
V. Kantsler,
E.fluid
Segre,
and V.
Steinberg
biologically-related
in
a
flow,
which
is
based
on
aspect
is
the
dependence
on
$̇
of
the
separate
regions
1To whom
type
of
unsteady
motion,
which
we
dubbed
trembling
(TR),
led
correspondence
shouldabevesicle
addressed. E-mail:
victor.steinberg@weizmann.ac.il.
hold
in the
field of view for up to 10 TU/TRofperiods.
a 3D reconstruction of its shape (20) in TT motion at low S, with
to reconsider
bothof
theoretical
models
(17).
TRadiffers
from
TU
the
combination
a dynamical
trap
and
control
parameter,
the
existence
of
TR
and
TU
(17).
Precisely these features changed
This article contains supporting information online at www.pnas.org/cgi/content/full/
Department
Physics
Systems,
of Science,
76100
Israelspace of parameters (S, "), vesicles with
To Rehovot,
explore the
whole
mean
errors ofin1.7%
onofthan
RComplex
and
6.1%
on #Weizmann
within
theInstitute
range
[0.3,
by oscillations
0902657106/DCSupplemental.
% of to
less
!/2of(rather
2!
) and by
ratio
of the vorticity
the strain
rate, isthan
suggested.
The flow
is
the idea of vesicle dynamics as smooth and shape-preserved
Dynamics of a vesicle in general flow
APPLIED PHYSICAL
SCIENCES
" $ 1 wereshearing,
used in this
work, theremotion
various
offor
R and
#adequate
were loaded
and individually
observed,
2.5].
Since only
vesicles
with rotational,
continuously
varied
betweenUniversity
and elongaand values
called
an (received
theoretical
description.
Edited
by
Harry
L.
Swinney,
of
Texas
at
Austin,
Austin,
TX,
and
approved
May
6,
2009
for
review
March
11,
2009)
!www.pnas.org"cgi"doi"10.1073"pnas.0902657106
/s was varied
in were
steps suggested
during thetoexperiment
changing
is
no –11447
error
from
this
(20). trap,
Errors
fromSeveral
and theoretical
11444
! contribution
PNAS ! July 14,
2009
! vol.
106 parameter
! no.
28 dynamical
tional
in a microfluidic
4-roll
mill
device,
the
that
models
describe thebydyis theinpressure
difference
between
2 inlets(3,17).
(see
1).
non-uniformity
of
fieldstudy
were
below
2%;
thati.e.,
cumuall 3stronger
states
shear
flow,
their regions
of existence,
and Fig.
allows
scanning to
of the
the velocity
entire phase
diagram
of motions,
An approach
quantitatively
vesicle
dynamics
as namics
well#P,
asofwhich
vesicle
shape
deformations
Moreover,
a new
In
this
way
the
space
of
parameters
(S,
")
was
populated
with
lates
to
overall
mean
errors
of
8.7%
on
S
and
of
2%
on
".
transitions
(5,7,13).
As
we
have
verified
recently,
only
the
1
of
tank-treading
(TT),
tumbling
(TU),
and
trembling
(TR),
using
a
biologically-related micro-objects in a fluid flow, which is based on
aspect is the dependence on $̇ of the separate regions of
them presented in ref. 5 describes adequately the experimental
single vesicle even at ! " #in/#out " 1, where #in and #out are the
the combination of a dynamical trap and a control parameter,
existence
of ofTR
TUwhich
(17).isPrecisely
these features changed
data the
(20). The
main result
thisand
model,
based on the
viscosities of the inner and outer fluids. This cannot be achieved in
ratioshear
of the
the strain
rate,TTisand
suggested.
flow is
theof !#
idea
vesicle
as smooth
#1, of
second
orderdynamics
spherical harmonics,
andand shape-preserved
pure
flow,vorticity
where theto
transition
between
either TU orThe approximation
continuously
varied
rotational,
shearing,
and elongamotion
and
called forsolution,
an adequate
theoretical description.
neglecting thermal
noise,
is a self-similar
which reduces
As a result,
it is found that
the vesicle
TR
is attained only
at !>1.between
the
number
of
the
dimensionless
control
parameters
to
just
2:
S to describe the dydynamical
states
in
a
general
are
presented
by
the
phase
diagram
tional in a microfluidic 4-roll mill device, the dynamical trap, that
Several theoretical models were suggested
3
$
$
$
/ 3&! and
%'
4(1 & 23
30!
, where
&
' 7!$̇#
in
a space scanning
of only 2 dimensionless
control
parameters.
The findings
of all
3 states
in"/32)
shear!/
flow,
their
regions
of existence, and
allows
of the entire
phase
diagram
of motions,
i.e.,outR namics
is the bending elasticity [taken further as & " 25 kBT'10(12 erg
are in semiquantitative accord with the recent theory made for a
transitions (5,7,13). As we have verified recently, only the 1 of
tank-treading (TT), tumbling (TU), and trembling (TR), using
a
(21)]. The phase diagram of the vesicle dynamical states is
quasi-spherical vesicle, although vesicles with large deviations
them
presentedbyinthe
ref.
5 describes
adequately
the experimental
" #in/experimentally.
#out " 1, where
#out 2-dimensional,
are the
single
vesicleshape
evenwere
at !studied
in andof
parameterized
variables
(S,%), and
indefrom
spherical
The#
physics
data
(20).
The
main
result
of
this
model,
which
is based on the
viscosities
of the inner and outer fluids. This cannot be achieved
in of other geometrical parameters. To scan all 3 regimes
pendent
TR
is also uncovered.
of TU
motion
to trace transitions
among
in aorder
shear spherical
flow,
approximation
of !#
#1, them
second
harmonics, and
pure shear flow, where the transition between TT and either
or and
$̇
and
"
,
that
is
change
the
viscosities
of
one
should
vary
both
neglecting thermal noise, is a self-similar solution, which reduces
!>1. As
result, itremains
is found
TR isnderstanding
attained only
the at
rheology
of abiofluids
a that
great the vesicle
inner
and
outer
which is an impossible task to realize on
challenge,
whose
relies,
a large part,by
onthe
detailed
thefluids,
number
of the dimensionless control parameters to just 2: S
dynamical
states
inprogress
a general
areinpresented
phase diagram
an
individual
vesicle.
Because
of topology of the phase diagram
3/ $
studies
of
the
dynamics
of
a
single
cell.
Vesicles
are
a
model
$!/$30!, where &
!$̇#outRpossibility
3&! and
4(1vesicle
& 23"is/32)
' 7remaining
in a space of only 2 dimensionless control parameters. The findings
(5,20),
the
only
with%a '
single
to
system used to study the dynamic behavior of biological cells,
(12
is the
bending
elasticity
[taken
erg
are in semiquantitative accord with the recent theory made
fortransitions
a
scan
from
TU to TR
by varying
$̇. further as & " 25 kBT'10
similar in some respects to red blood cells, and their dynamics in
This limitation
overcome
a general of
flow,
the dynamical states is
(21)].is The
phasein diagram
thewhere
vesicle
quasi-spherical
vesicle,
although
vesicles
with large
shear
flow have been
the subject
of intensive
theoretical
(1–8), deviations
velocity
gradient
can
be
written
as
'
V
"
s
&
(
)
,
where
sik
i k
ik
ikj the
j
2-dimensional, parameterized
by
variables
(S,%), and indefrom spherical
were studied
experimentally.
numerical
(9–13),shape
and experimental
(14–18)
investigations.The physics of
is
the
symmetric
strain
tensor,
)
the
vorticity
vector,
and
s
"
j
pendent of other geometrical parameters. To scan all 3 regimes
is a droplet of viscous fluid encapsulated by a
TRA isvesicle
also uncovered.
$ 2
A
B
U
tr(sik)/2 the strain rate. The corresponding control parameters
phospholipid bilayer membrane suspended in a fluid of either
motion
and
transitions
among
for
vesicles inofgeneral
flow
(5)to
aretrace
S '14
!s#outR3/3$
3&! andthem in a shear flow,
the same or different viscosity as the inner one. Both the volume
$̇ and
", we
that
is change
the viscosities of
one
vary
$!()/s)/
$30both
% " 4 (1 & 23
"/32)should
!. In this
paper,
report
the
nderstanding
of biofluids
and the
surface area ofthe
the rheology
vesicle are conserved.
The remains
former a great
inner
and
outer
fluids,
which
is
an
impossible
task
to realize on
phase diagram in such general flow. This approach uses an
whose
progress
relies, in atolarge
part, on detailed
meanschallenge,
that the vesicle
membrane
is considered
be impermeadditional control
parameter vesicle.
)/s, whichBecause
is fixed toof
unity
in shearof the phase diagram
an individual
topology
able,
at least
time scale of
of the
experiment,
the latter
studies
of on
thethedynamics
a single
cell. and
Vesicles
are a flow
model
(s " ) "
$̇
/2),
to
study
vesicle
dynamics
[it
was
suggested
(5,20), the only remaining possibility with a single vesicle is to
means
the membrane
dilatation
is neglected
since
is 2D
systemthat
used
to study the
dynamic
behavior
ofitbiological
cells,
first
by G. I. Taylor to study emulsions in a 4-roll mill (22)]. The
scan
transitions
from TU in
to the
TR experiment,
by varying $̇.
fluid
(1,2).
theoretical,
and computational
efforts
similar
inExperimental,
some
respects
to redwhose
blood
cells,defines
and their
dynamics
in
ratio
can
be
easily
variedbycontinuously
tic of the microfluidic 4-roll mill device; Q1during
and Qthe
the
flow
discharges,
ratio
the
flow
type.
The
flow
is
driven
gravity,
2 are
last decade led to the observation and characterizaThis limitation
is overcome
in from
a general
flow, where the
evidencing
TT
to either
TU or TR and
TU
shear
the
subject
of Q
intensive
theoretical
(1–8),
n the pressure drop P0 and the pressure difference
between
2been
inlets
#P determines
(B) fexistence
$ (1 ! !/s)/(1%
!/s) as transitions
a functionfrom
of the
reduced
1/Q2.The
tion
of 3flow
stateshave
in vesicle
dynamics
in shear
flow.
velocity
cangiven
be written
as 'i"V. k The
" sik & (ikj)j, where sik
to TR on the
same gradient
vesicle with
R, !, and
numerical
of
the
firstsquares,
2,(9–13),
tank-treading
(TT) squares,
and tumbling
and the
P0). Inset: s as a function of (1 ! #P/P0). Large
filled
P0and
$ P; experimental
open
P(14–18)
4/3P;investigations.
small
filled circles, P0 $ 5/3P, open circles, P0 $
0 $(TU),
experimentalispath
the phase
diagram
depends
the
the across
symmetric
strain
tensor,
)j theonvorticity
vector, and s "
A3Dvesicle
is athem
droplet
offlow.
viscous
fluid
by
a to soft
transition
between
were
already
predicted
by aencapsulated
phenomonfiguration P&750 Pa). The solid line is the
FEM
simulation
of the
Experimental
imperfections
due
lithography
lead
to varied. The possibility to
2way )/s and
initial
state
and
the
s
are
$
tr(sik)/2 the strain rate. The corresponding control parameters
enological
modelbilayer
of Keller
and Skalak
(19) and its
phospholipid
membrane
suspended
in afurther
fluid ofobserve
either all dynamical
tive discrepancy with simulations.
statesinwith
the same
vesicle,
evenSfor
" "!s# R3/3$3&! and
for vesicles
general
flow
(5) are
'14
out
extensions
(2,11,12).
Twoviscosity
control parameters,
theone.
excess
areathe volume
the same
or
different
as
the
inner
Both
1 used in the current experiment, $
complements
the previous
2-4! and the viscosity contrast " " # /#
$
% the
" 4shear
(1 &
23dynamics
"/32) !(
)/s)/ 30
!.the
Inother
this paper, we report the
! " A/R
out, determine
based on
flow
(2–4,7–20).
On
and the surface area of the vesicle arein conserved.
The views
former
the transition line "c(!) between TT and TU, which is indepenphase diagram
such
flow. This approach uses an
hand, the experimental
approachin
used
heregeneral
will be advantageous
means
that
therate
vesicle
membrane
is considered
to be imperme$̇ The
in the
approximation
of were
a fixed performed
vesicle
dent
of the
shear
high flow uniformity (Fig. 2 and SI
Appendix).
experiments
in
the
following
way.
A
additional
control
parameter
)
/s,
which
is fixed to unity in shear
to study the dynamics of other flexible microobjects, including
able,
at
least
on
the
time
scale
of
the
experiment,
and
the
latter
%
with
respect
to
the
flow
shape,
with
the
vesicle
inclination
angle
!1 and that
Fig. 2.
(A)
streamlines
images
of theR
velocity
for pure rotational
(first
column,
$
43),
(second
column,
!/s $ 2.6)
and pure shear
biological
membranes
and
red
cells,
in flow.
of !
/sasExperimental
wasonly dynamical
nge of s was [0.05–0.32] s
vesicle
with
given
andfields
#, measured
initially
in situ
in)
the
same
flow
(s!/s"
"mixed
$̇blood
/2),
to
study
vesicle
dynamics
[it was suggested
direction
variable
(2,9,10,16–19).
R it
means
that
the (B)
membrane
dilatation
is neglected
since
is 2D
(third,
!/s $
1)the
flows;
Zoom of the
same experimental
flows;Here,
(C) velocity
vector
field representation of the same flows (PTV).
first
I. Taylor
to study emulsions in a 4-roll mill (22)]. The
/s G.
and
s in the
educe error bars on S and ", theisfluid
geometrical
was followed
at aviaprescribed
value
of !by
the effective
vesicledevice,
radius, theoretical,
related
to theand
volume
V "
Results
and Discussion
(1,2).
Experimental,
computational
efforts
ratio
canvesicle
be used
easily
continuously in the experiment,
R3, A from
is the vesicle
surface area, window.
and #in andA#feedback
out are the inMeasurements
observation
the flow velocity
to varied
d # of each vesicle were measured4/3
in!situ
of the was
dynamics
were conducted in a
during
the
last
decade
led
to
the
observation
and
characterizaDeschamps
al. the inner and outer fluids. Another analytical
PNAS
! July 14,from
2009 !TT
vol.to
106
! no.TU
28 !or11445
viscositiesetof
evidencing
transitions
either
TR and from TU
microfluidic
4-roll
mill device
(23,24) manufactured
in silicone
vesicle
ininthe
field
of The
view
for
up to 10
TU/TR
periods.
ion of its shape (20) in TT motion at
lowofS,3based
with
tion
states
vesiclea dynamics
shear
flow.
approach
onin
a hold
quasi-spherical
vesicle
approximated
by
a existence
on the(Fig.
same
vesicletracking
with given
elastomer byto
softTR
lithography
1). Particle
veloci- R, !, and ". The
ofrange
the first
2, tank-treading
(TT)
tumbling
(TU),
and
the
To
explore
the
whole
space
of parameters
(S,
"),
vesicles
with
7% on R and of 6.1% on # within the
[0.3,
spherical
harmonics
expansion
used
a and
perturbation
scheme
metry (PTV)experimental
measurements show
agreement
numerical
pathfairacross
the with
phase
diagram depends on the
transition
between
them
were
already
predicted
byloaded
a phenomLamb
solution
of the
Stokes
near #
a spherical
work,thethere
various
values
offlow
R and
were
and individually observed,
esicles with " $ 1 were used in thisaround
U
dunkel@math.mit.edu
Dynamics of a vesicle in general flow
viscosities of the inner and outer fluids. Another analytical
microfluidic 4-roll mill device (23,24) manufactured in silicone
approach based on a quasi-spherical vesicle approximated by a
elastomer by soft lithography (Fig. 1). Particle tracking veloci1
J.spherical
Deschamps,
V. Kantsler,
Segre,
and V. Steinberg
harmonics
expansion E.
used
a perturbation
scheme
metry (PTV) measurements show fair agreement with numerical
around the Lamb solution of the Stokes flow near a spherical
Department
of Physics
Complex
Systems,
Weizmann
Institute
body in external
shearof
flow
(1). Further
refinement
of the
modelof Science, Rehovot, 76100 Israel
resulted in dynamic equations for vesicle shape and inclination
Author contributions: V.K. and V.S. designed research; J.D. performed research; J.D., V.K.,
Edited
Harry
Swinney,rather
University
of Texas
at motion
Austin, and
Austin,
andand
approved
May
2009
(received
for review March 11, 2009)
V.S. analyzed
data;6,
and
V.S. wrote
the paper.
angle by
(3,4)
and L.
described
well both
the TT
the TX, E.S.,
The
authors
declare
no
conflict
of
interest.
transition line "c(!), as verified by the recent experiments
An
approach
to quantitatively
study vesicle
dynamics
as well
as
stronger
vesicle shape deformations (3,17). Moreover, a new
1 Direct Submission.
(16–18).
However,
theV.
recent
experimental
key
finding
of a new
This article is a PNAS
J.
Deschamps,
Kantsler,
E.
Segre,
and
V.
Steinberg
biologically-related
micro-objects
in a fluid
flow,(TR),
whichled
is based
on correspondence
aspectshould
is be
the
dependence
on $̇ of the separate regions of
1To whom
type of unsteady motion,
which we dubbed
trembling
addressed.
E-mail: victor.steinberg@weizmann.ac.il.
to reconsider
bothof
theoretical
models
(17).
TRadiffers
from
TU
the
combination
a dynamical
trap
and
control
parameter,
the
existence
of
TR
and
TU
(17).
Precisely these features changed
This article contains supporting information online at www.pnas.org/cgi/content/full/
Department
of
Complex
Systems, Weizmann
Institute
of Science, Rehovot, 76100 Israel
by oscillations
in Physics
0902657106/DCSupplemental.
% of to
lessof
than
!/2 (rather
2!) and by
ratio
of the vorticity
the
strain
rate, isthan
suggested.
The flow
is
the idea of vesicle dynamics as smooth and shape-preserved
Dynamics of a vesicle in general flow
A
continuously
varied
betweenUniversity
rotational,
shearing,
and elongaand called
for2009
an (received
adequatefor
theoretical
description.
Edited by Harry
L. Swinney,
of Texas
at Austin,
Austin, TX,motion
and approved
May 6,
review March
11, 2009)
11444 –11447
! PNAS ! July 14,
2009 mill
! vol.device,
106 ! no.
28 dynamical trap, that
www.pnas.org"cgi"doi"10.1073"pnas.0902657106
tional
in a microfluidic
4-roll
the
Several theoretical
models were suggested to describe the dystates in shear
flow,shape
their regions
of existence,
and Moreover, a new
allows
scanning to
of quantitatively
the entire phasestudy
diagram
of motions,
i.e., as namics
An approach
vesicle
dynamics
well asof all 3stronger
vesicle
deformations
(3,17).
transitions
have
verified recently,
1 of separate regions of
tank-treading
(TT), tumbling
(TU), andin
trembling
(TR), which
using ais based
biologically-related
micro-objects
a fluid flow,
on (5,7,13).
aspectAsiswethe
dependence
on only
$̇ ofthethe
them presented in ref. 5 describes adequately the experimental
single vesicle even at ! " #in/#out " 1, where #in and #out are the
B combination of a dynamical trap and a control parameter,
the
existence
of ofTR
TUwhich
(17).isPrecisely
these features changed
data the
(20). The
main result
thisand
model,
based on the
viscosities of the inner and outer fluids. This cannot be achieved in
ratioshear
of the
the strain
rate,TTisand
suggested.
flow is
theof !#
idea
vesicle
as smooth
#1, of
second
orderdynamics
spherical harmonics,
andand shape-preserved
pure
flow,vorticity
where theto
transition
between
either TU orThe approximation
continuously
varied
rotational,
shearing,
and elongamotion
and
called forsolution,
an adequate
theoretical description.
neglecting thermal
noise,
is a self-similar
which reduces
As a result,
it is found that
the vesicle
TR
is attained only
at !>1.between
the
number
of
the
dimensionless
control
parameters
to
just
2:
S to describe the dydynamical
states
in
a
general
are
presented
by
the
phase
diagram
tional in a microfluidic 4-roll mill device, the dynamical trap, that
Several theoretical models were suggested
3
$
$
$
/ 3&! and
%'
4(1 & 23
30!
, where
&
' 7!$̇#
in
a space scanning
of only 2 dimensionless
control
parameters.
The findings
of all
3 states
in"/32)
shear!/
flow,
their
regions
of existence, and
allows
of the entire
phase
diagram
of motions,
i.e.,outR namics
is the bending elasticity [taken further as & " 25 kBT'10(12 erg
are in semiquantitative accord with the recent theory made for a
transitions (5,7,13). As we have verified recently, only the 1 of
tank-treading (TT), tumbling (TU), and trembling (TR), using
a
(21)]. The phase diagram of the vesicle dynamical states is
quasi-spherical vesicle, although vesicles with large deviations
them
presentedbyinthe
ref.
5 describes
adequately
the experimental
" #in/experimentally.
#out " 1, where
#out 2-dimensional,
are the
single
vesicleshape
evenwere
at !studied
in andof
parameterized
variables
(S,%), and
indefrom
spherical
The#
physics
data
(20).
The
main
result
of
this
model,
which
is based on the
viscosities
of the inner and outer fluids. This cannot be achieved
in of other geometrical parameters. To scan all 3 regimes
pendent
TR
is also uncovered.
of TU
motion
to trace transitions
among
in aorder
shear spherical
flow,
approximation
of !#
#1, them
second
harmonics, and
pure shear flow, where the transition between TT and either
or and
$̇
and
"
,
that
is
change
the
viscosities
of
one
should
vary
both
neglecting thermal noise, is a self-similar solution, which reduces
!>1. As
result, itremains
is found
TR isnderstanding
attained only
the at
rheology
of abiofluids
a that
great the vesicle
Fig. 1. (A) Schematic of the microfluidic 4-roll mill device; Q1 and Q2 are the flow discharges, whose ratio defines the flow type. The flow is driven by gravity,
inner
and
outer
which is an impossible task to realize on
and the ratio between the pressure drop P0 and the pressure difference between
2challenge,
inlets #P determines
Q1/Q
(B)
(1 ! !/s)/(1%
!/s) as
function
of the
reduced
whose
relies,
inapresented
a large
part,
onthe
detailed
thefluids,
number
of the dimensionless control parameters to just 2: S
dynamical
states
in2.progress
af $general
are
by
phase diagram
an
individual
vesicle.
Because
of topology of the phase diagram
pressure drop (1- #P/P0). Inset: s as a function of (1 ! #P/P0). Large filled squares, P0 $ P; open squares, P0 $ 4/3P; small filled circles, P0 $ 5/3P, open circles, P0 $
3/ $
studies
of
the
dynamics
of
a
single
cell.
Vesicles
are
a
model
$!/$30!, where &
!$̇#outRpossibility
3&! and
4(1vesicle
& 23"is/32)
' 7remaining
2P (in our specific configuration P&750 Pa). The solid line is the 3D FEMin
simulation
of theof
flow.
Experimental
imperfections due to soft
lithography
lead to
a space
only
2 dimensionless
control
parameters.
The findings
(5,20),
the
only
with%a '
single
to
observable quantitative discrepancy with simulations.
system used to study the dynamic behavior of biological cells,
(12
is the
bending
elasticity
[taken
erg
are in semiquantitative accord with the recent theory made
fortransitions
a
scan
from
TU to TR
by varying
$̇. further as & " 25 kBT'10
similar in some respects to red blood cells, and their dynamics in
This limitation
overcome
a general of
flow,
the dynamical states is
(21)].is The
phasein diagram
thewhere
vesicle
quasi-spherical
vesicle,
although
vesicles
with large
shear
flow
been
the subject
theoretical
(1–8), deviations
simulations and high flow uniformity (Fig. 2 and SI Appendix).
The have
experiments
were
performedofinintensive
the following
way. A
velocity
gradient
can
be
written
as
'
V
"
s
&
(
)
,
where
sik
i k
ik
ikj the
j
2-dimensional, parameterized
by
variables
(S,%), and indefromvesicle
spherical
shape
studied
experimentally.
The physics of
was
The accessible range of s was [0.05–0.32] s!1 and that of !/s numerical
with given
R andexperimental
#,were
measured
initially
in
situ in
the same
(9–13),
and
(14–18)
investigations.
is
the
symmetric
strain
tensor,
)
the
vorticity
vector,
and
s
"
j
!
/s
and
s
in
the
1.18–9.23]. To reduce error bars on S and ", the geometricalA vesicle
device, was
followed
at
a
prescribed
value
of
pendent of other geometrical parameters. To scan all 3 regimes
is a droplet of viscous fluid encapsulated by a
TR isobservation
also uncovered.
2
$tr(sik
window. A feedback in the flow velocity was used to
parameters R and # of each vesicle were measured in situ from
)/2 the strain rate. The corresponding control parameters
phospholipid
bilayer
membrane
suspended
in
a
fluid
of
either
motion
and
transitions
among
hold a vesicle in the field of view for up to 10 TU/TR periods.
a 3D reconstruction of its shape (20) in TT motion at low S, with
for vesicles inofgeneral
flow
(5)to
aretrace
S '14
!s#outR3/3$
3&! andthem in a shear flow,
the same
or different
viscosity
as the inner
one.
Both
the volume
To explore
the whole
space of parameters
(S, "),
vesicles
with
mean errors of 1.7% on R and of 6.1% on # within the range [0.3,
$̇ and
", we
that
is change
the viscosities of
one
vary
$!()/s)/
$30both
% " 4 (1 & 23
"/32)should
!. In this
paper,
report
the
nderstanding
of biofluids
various
values of
R and
# the
wererheology
loaded and
individually
observed,
2.5]. Since only vesicles with " $ 1 were used in this work, there
and the
surface
area
ofthe
vesicle
are
conserved.
The remains
former a great
inner
and
outer
fluids,
which
is
an
impossible
task
to realize on
phase diagram in such general flow. This approach uses an
!/s was varied
in steps progress
during the experiment
by changing
s no error contribution from this parameter (20). Errors from
and
challenge,
whose
relies, in
atolarge
part, on detailed
means
that
the
vesicle
membrane
is
considered
be
imperme#P, which is the pressure difference between 2 inlets (see Fig. 1).
non-uniformity of the velocity field were below 2%; that cumuadditional control
parameter vesicle.
)/s, whichBecause
is fixed toof
unity
in shearof the phase diagram
an individual
topology
at
time
scale of
of the
experiment,
studies
ofwayon
the
dynamics
a(S, single
cell. and
Vesicles
are a flow
model
In least
this
thethe
space
of parameters
")
was populated
with the latter
ates to overall mean errors of 8.7% on S and of 2% on ".able,
(s " ) "
$̇
/2),
to
study
vesicle
dynamics
[it
was
suggested
(5,20), the only remaining possibility with a single vesicle is to
means
the membrane
dilatation
is neglected
since
is 2D
systemthat
used
to study the
dynamic
behavior
ofitbiological
cells,
first
by G. I. Taylor to study emulsions in a 4-roll mill (22)]. The
scan transitions from TU to TR by varying $̇.
fluid
(1,2).
theoretical,
and computational
efforts
similar
inExperimental,
some respects
to red blood
cells, and their
dynamics
ratio in
can be easily varied continuously in the experiment,
during the last decade led to the observation and characterizaThis from
limitation
is overcome
in from
a general
flow, where the
TT to either
TU or TR and
TU
shear
beendynamics
the subject
of intensive
theoreticalevidencing
(1–8), transitions
tion
of 3flow
stateshave
in vesicle
in shear
flow. The existence
velocity
cangiven
be written
as 'i"V. k The
" sik & (ikj)j, where sik
to TR on the
same gradient
vesicle with
R, !, and
numerical
and experimental
(14–18)
of
the first 2,(9–13),
tank-treading
(TT) and tumbling
(TU),investigations.
and the
experimentalispath
the phase
diagram
depends
the
the across
symmetric
strain
tensor,
)j theonvorticity
vector, and s "
A vesicle
is athem
droplet
of viscous
fluid
by a
transition
between
were already
predicted
by aencapsulated
phenom2way )/s and s are varied. The possibility to
initial state and
the
$
tr(sik)/2 the strain rate. The corresponding control parameters
enological
modelbilayer
of Keller
and Skalak
(19) and its
phospholipid
membrane
suspended
in afurther
fluid ofobserve
either all dynamical
statesinwith
the same
vesicle,
evenSfor
" "!s# R3/3$3&! and
for vesicles
general
flow
(5) are
'14
out
extensions
(2,11,12).
Twoviscosity
control parameters,
theone.
excess
areathe volume
the same
or
different
as
the
inner
Both
1 used in the current experiment, $
complements
the previous
2-4! and the viscosity contrast " " # /#
$
% the
" 4shear
(1 &
23dynamics
"/32) !(
)/s)/ 30
!.the
Inother
this paper, we report the
! " A/R
out, determine
based on
flow
(2–4,7–20).
On
and the surface area of the vesicle arein conserved.
The views
former
the transition line "c(!) between TT and TU, which is indepenphase diagram
such
flow. This approach uses an
hand, the experimental
approachin
used
heregeneral
will be advantageous
means
that
therate
vesicle
membrane
is considered
to be imperme$̇ in the
approximation
of a fixed vesicle
dent
of the
shear
additional
control
parameter
)
/s,
which
is fixed to unity in shear
study the dynamics of other flexible microobjects, including
able, with
at least
on the
time scale
of% the
and theto
latter
withexperiment,
respect to the flow
shape,
the vesicle
inclination
angle
biological membranes
and
cells,
in flow.
flow (s "
) red
" $̇blood
/2), to
study
vesicle dynamics [it was suggested
direction
as thethe
onlymembrane
dynamical variable
(2,9,10,16–19).
Here,
R it is 2D
means that
dilatation
is neglected
since
first by G. I. Taylor to study emulsions in a 4-roll mill (22)]. The
isfluid
the effective
vesicle radius, theoretical,
related to theand
volume
via V "
Results
and Discussion
(1,2).
Experimental,
computational
efforts
ratio can be easily varied continuously in the experiment,
4/3!R3, A is the vesicle surface area, and #in and #out are the
Measurements of the vesicle dynamics were conducted in a
during the last decade led to the observation and characterizaviscosities of the inner and outer fluids. Another analytical
transitions
from TT to either
TU or TR and from TU
microfluidic evidencing
4-roll mill device
(23,24) manufactured
in silicone
tion of 3based
states
vesicle dynamics
in approximated
shear flow. The
approach
onin
a quasi-spherical
vesicle
by a existence
on the(Fig.
same
vesicletracking
with given
elastomer byto
softTR
lithography
1). Particle
veloci- R, !, and ". The
of the first
2, tank-treading
(TT)a and
tumbling
(TU), and
the
spherical
harmonics
expansion used
perturbation
scheme
metry (PTV)experimental
measurements show
agreement
numerical
pathfairacross
the with
phase
diagram depends on the
transition
between
them
were
already
by a phenomaround
the Lamb
solution
of the
Stokes
flow predicted
near a spherical
U
APPLIED PHYSICAL
SCIENCES
U
dunkel@math.mit.edu
NATURE | VOL 423 | 8 MAY 2003 | www.nature.com/nature
© 2003
letters to nature
been identified as possible associates of Přı́bram11,12. In summer
2000, 4486 Mithra was well placed for observations by radar from
Arecibo and Goldstone. The object was found to be in unusual spin
state, double-lobed, and more severely bifurcated than any other
near-Earth asteroid targeted previously17. Such objects are typically
believed to be ‘rubble piles’, loosely bound by gravitation. During
close encounters with planets, these objects would be exposed to
tidal forces that could lead to their disruption or to the detachment
of significant amounts of near-surface rubble that would then be
found in orbits similar to that of the parent. This model could
possibly resolve the paradox of young streams containing different
classes of objects of long cosmic-ray exposure.
Further discussions of the origin of the ‘Přı́bram stream’ must
await the final results of the laboratory analyses of Neuschwanstein;
more data could be gained from clear identification of other
members of this stream (either meteorites or asteroids) in the
future. The Neuschwanstein and Přı́bram meteorites bring new
information to the discussions of the mechanisms of meteorite
delivery to the Earth, and demonstrate the importance of long-term
fireball observing programmes.
A
Received 23 January; accepted 28 March 2003; doi:10.1038/nature01592.
1. Ceplecha, Z. Multiple fall of Přı́bram meteorites photographed. Bull. Astron. Inst. Czech. 12, 21–47
(1961).
2. Bischoff, A. & Zipfel, J. Mineralogy of the Neuschwanstein (EL6) chondrite—first results. Lunar
Planet. Sci. Conf. XXXIV, abstr. 1212 (2003).
3. Ceplecha, Z. Geometric, dynamic, orbital and photometric data on meteoroids from photographic
fireball networks. Bull. Astron. Inst. Czech. 38, 222–234 (1987).
4. Borovička, J., Spurný, P. & Keclı́ková, J. A new positional astrometric method for all-sky cameras.
Astron. Astrophys. Suppl. Ser. 112, 173–178 (1995).
5. Ceplecha, Z., Spurný, P., Borovička, J. & Keclı́ková, J. Atmospheric fragmentation of meteoroids.
Astron. Astrophys. 279, 615–626 (1993).
6. Borovička, J., Spurný, P. & Ceplecha, Z. The Morávka meteorite fall: Fireball trajectory, orbit and
fragmentation from video records. Meteorit. Planet. Sci. Suppl. 36, A25–A26 (2001).
7. Drummond, J. D. A test of comet and meteor associations. Icarus 45, 545–553 (1981).
8. Halliday, I., Griffin, A. A. & Blackwell, A. T. in Highlights of Astronomy Vol. 6 (ed. West, R. M.) 399–404
(D. Reidel, Dordrecht, 1983).
9. Halliday, I. Detection of a meteorite “stream”: Observations of a second meteorite fall from the orbit
of the Innisfree chondrite. Icarus 69, 550–556 (1987).
10. Halliday, I., Blackwell, A. T. & Griffin, A. A. Evidence for the existence of groups of meteoriteproducing asteroidal fragments. Meteorit. Planet. Sci. 25, 93–99 (1990).
11. Drummond, J. D. Earth-approaching asteroid streams. Icarus 89, 14–25 (1991).
12. Drummond, J. D. The D discriminant and near-Earth asteroid streams. Icarus 146, 453–475 (2000).
13. Dodd, R. T., Wolf, S. F. & Lipschutz, M. E. A H chondrite stream: Identification and confirmation.
J. Geophys. Res. 98, 15105–15118 (1993).
14. Treiman, A. H. An improbable concentration of basaltic meteorite falls (HED and mesosiderite) in the
mid-20th century. Meteorit. Planet. Sci. 28, 246–252 (1993).
15. Stauffer, H. & Urey, H. C. Multiple fall of Přı́bram meteorites photographed III. Rare gas isotopes in
the Velka stone meteorite. Bull. Astron Inst. Czech. 13, 106–109 (1962).
16. Jenniskens, P. et al. Orbits of meteorite producing fireballs. The Glanerbrug—a case study. Astron.
..............................................................
Controlled vesicle deformation and
lysis by single oscillating bubbles
Philippe Marmottant & Sascha Hilgenfeldt
Faculty of Applied Physics, University of Twente, PO Box 217, 7500AE Enschede,
The Netherlands
.............................................................................................................................................................................
The ability of collapsing (cavitating) bubbles to focus and concentrate energy, forces and stresses is at the root of phenomena
such as cavitation damage, sonochemistry or sonoluminescence1,2. In a biomedical context, ultrasound-driven microbubbles have been used to enhance contrast in ultrasonic
images3. The observation of bubble-enhanced sonoporation4–6—
acoustically induced rupture of membranes—has also opened up
intriguing possibilities for the therapeutic application of sonoporation as an alternative to cell-wall permeation techniques
such as electroporation7 and particle guns8. However, these
pioneering experiments have not been able to pinpoint the
mechanism by which the violently collapsing bubble opens
pores or larger holes in membranes. Here we present an experiment in which gentle (linear) bubble oscillations are sufficient to
achieve rupture of lipid membranes. In this regime, the bubble
dynamics and the ensuing sonoporation can be accurately controlled. The use of microbubbles as focusing agents makes
acoustics on the micrometre scale (microacoustics) a viable
tool, with possible applications in cell manipulation and cellwall permeation as well as in microfluidic devices.
Most experiments assessing sonoporation have used strong
ultrasonic fields giving rise to a multitude of simultaneous phenomena. The micrographs of ref. 4 suggest that the formation of
microjets may be involved, a process that is very difficult to control
and model9–11. Other candidates for damaging agents during bubble
collapse involve shockwaves emitted from the bubble12 and/or
subsequent heating of the cell wall13. All of these processes are
significant only when the bubble undergoes a fast, inertial collapse.
By constrast, some experiments14 demonstrated cell transfection
with moderate, off-resonance driving which should have resulted in
much weaker bubble oscillations. Our experiments use single
microbubbles fixed on a substrate, and use far smaller ultrasound
driving amplitudes, replacing the nonlinear dynamics of the inertial
collapse with gentle, linear oscillations. Linear oscillations can be
dunkel@math.mit.edu
Competing interests statement The authors declare that they have no com
interests.
Correspondence and requests for materials should be addressed to P.S. (sp
NATURE | VOL 423 | 8 MAY 2003 | www.nature.com/nature
letters to nature
been identified as possible associates of Přı́bram11,12. In summer
2000, 4486 Mithra was well placed for observations by radar from
Arecibo and Goldstone. The object was found to be in unusual spin
state, double-lobed, and more severely bifurcated than any other
near-Earth asteroid targeted previously17. Such objects are typically
believed to be ‘rubble piles’, loosely bound by gravitation. During
close encounters with planets, these objects would be exposed to
tidal forces that could lead to their disruption or to the detachment
of significant amounts of near-surface rubble that would then be
found in orbits similar to that of the parent. This model could
possibly resolve the paradox of young streams containing different
classes of objects of long cosmic-ray exposure.
Further discussions of the origin of the ‘Přı́bram stream’ must
await the final results of the laboratory analyses of Neuschwanstein;
more data could be gained from clear identification of other
members of this stream (either meteorites or asteroids) in the
future. The Neuschwanstein and Přı́bram meteorites bring new
information to the discussions of the mechanisms of meteorite
delivery to the Earth, and demonstrate the importance of long-term
fireball observing programmes.
A
Received 23 January; accepted 28 March 2003; doi:10.1038/nature01592.
1. Ceplecha, Z. Multiple fall of Přı́bram meteorites photographed. Bull. Astron. Inst. Czech. 12, 21–47
(1961).
2. Bischoff, A. & Zipfel, J. Mineralogy of the Neuschwanstein (EL6) chondrite—first results. Lunar
Planet. Sci. Conf. XXXIV, abstr. 1212 (2003).
3. Ceplecha, Z. Geometric, dynamic, orbital and photometric data on meteoroids from photographic
fireball networks. Bull. Astron. Inst. Czech. 38, 222–234 (1987).
4. Borovička, J., Spurný, P. & Keclı́ková, J. A new positional astrometric method for all-sky cameras.
Astron. Astrophys. Suppl. Ser. 112, 173–178 (1995).
5. Ceplecha, Z., Spurný, P., Borovička, J. & Keclı́ková, J. Atmospheric fragmentation of meteoroids.
Astron. Astrophys. 279, 615–626 (1993).
6. Borovička, J., Spurný, P. & Ceplecha, Z. The Morávka meteorite fall: Fireball trajectory, orbit and
fragmentation from video records. Meteorit. Planet. Sci. Suppl. 36, A25–A26 (2001).
7. Drummond, J. D. A test of comet and meteor associations. Icarus 45, 545–553 (1981).
8. Halliday, I., Griffin, A. A. & Blackwell, A. T. in Highlights of Astronomy Vol. 6 (ed. West, R. M.) 399–404
(D. Reidel, Dordrecht, 1983).
9. Halliday, I. Detection of a meteorite “stream”: Observations of a second meteorite fall from the orbit
of the Innisfree chondrite. Icarus 69, 550–556 (1987).
10. Halliday, I., Blackwell, A. T. & Griffin, A. A. Evidence for the existence of groups of meteoriteproducing asteroidal fragments. Meteorit. Planet. Sci. 25, 93–99 (1990).
11. Drummond, J. D. Earth-approaching asteroid streams. Icarus 89, 14–25 (1991).
12. Drummond, J. D. The D discriminant and near-Earth asteroid streams. Icarus 146, 453–475 (2000).
13. Dodd, R. T., Wolf, S. F. & Lipschutz, M. E. A H chondrite stream: Identification and confirmation.
J. Geophys. Res. 98, 15105–15118 (1993).
14. Treiman, A. H. An improbable concentration of basaltic meteorite falls (HED and mesosiderite) in the
mid-20th century. Meteorit. Planet. Sci. 28, 246–252 (1993).
15. Stauffer, H. & Urey, H. C. Multiple fall of Přı́bram meteorites photographed III. Rare gas isotopes in
the Velka stone meteorite. Bull. Astron Inst. Czech. 13, 106–109 (1962).
16. Jenniskens, P. et al. Orbits of meteorite producing fireballs. The Glanerbrug—a case study. Astron.
Astrophys. 255, 373–376 (1992).
17. Ostro, S. J. et al. Radar observations of asteroid 4486 Mithra. DPS meeting 32 (American Astronomical
Society, Pasadena, California, 2000).
18. Ceplecha, Z. Fireballs photographed in central Europe. Bull. Astron. Inst. Czech. 28, 328–340 (1977).
19. Oberst, J. et al. The “European Fireball Network”: Current status and future prospects. Meteorit.
..............................................................
Controlled vesicle deformation and
lysis by single oscillating bubbles
Philippe Marmottant & Sascha Hilgenfeldt
Faculty of Applied Physics, University of Twente, PO Box 217, 7500AE Enschede,
The Netherlands
.............................................................................................................................................................................
The ability of collapsing (cavitating) bubbles to focus and concentrate energy, forces and stresses is at the root of phenomena
such as cavitation damage, sonochemistry or sonoluminescence1,2. In a biomedical context, ultrasound-driven microbubbles have been used to enhance contrast in ultrasonic
images3. The observation of bubble-enhanced sonoporation4–6—
acoustically induced rupture of membranes—has also opened up
intriguing possibilities for the therapeutic application of sonoporation as an alternative to cell-wall permeation techniques
such as electroporation7 and particle guns8. However, these
pioneering experiments have not been able to pinpoint the
mechanism by which the violently collapsing bubble opens
pores or larger holes in membranes. Here we present an experiment in which gentle (linear) bubble oscillations are sufficient to
achieve rupture of lipid membranes. In this regime, the bubble
dynamics and the ensuing sonoporation can be accurately controlled. The use of microbubbles as focusing agents makes
acoustics on the micrometre scale (microacoustics) a viable
tool, with possible applications in cell manipulation and cellwall permeation as well as in microfluidic devices.
Most experiments assessing sonoporation have used strong
ultrasonic fields giving rise to a multitude of simultaneous phenomena. The micrographs of ref. 4 suggest that the formation of
microjets may be involved, a process that is very difficult to control
and model9–11. Other candidates for damaging agents during bubble
collapse involve shockwaves emitted from the bubble12 and/or
subsequent heating of the cell wall13. All of these processes are
significant only when the bubble undergoes a fast, inertial collapse.
By constrast, some experiments14 demonstrated cell transfection
with moderate, off-resonance driving which should have resulted in
much weaker bubble oscillations. Our experiments use single
microbubbles fixed on a substrate, and use far smaller ultrasound
driving amplitudes, replacing the nonlinear dynamics of the inertial
collapse with gentle, linear oscillations. Linear oscillations can be
sufficient to rupture single cells, because the bubbles’ response
concentrates ultrasonic energy on the microscale, whereas it could
conventionally only be focused to about an ultrasonic wavelength (a
few centimetres or millimetres).
dunkel@math.mit.edu
Chapter 3
Chapter 3
Membranes
Membranes
The discussion in this section builds on the review article [Sei97] and the textbook [OLXY99].
The discussion in this section builds on the review article [Sei97] and the textbook [OLXY99].
3.1 Reminder: 2D di↵erential geometry
3.1consider
Reminder:
2D di↵erential
geometry
We
an orientable surface
in R3 . Possible local
parameterizations are
s2 ) 2 R3 local parameterizations are
We consider an orientable surface inFR(s3 .1 ,Possible
(3.1)
3
where (s1 , s2 ) 2 U ✓ R2 . Alternatively,
Cartesian coordinates (s1 , s(3.1)
F (s1 , sif2 )one
2 Rchooses
2) =
(x, y), then it suffices to 2specify
where (s1 , s2 ) 2 U ✓ R . Alternatively, if one chooses Cartesian coordinates (s1 , s2 ) =
z = f (x, y)
(3.2a)
(x, y), then it suffices to specify
or, equivalently, the implicit representation
z = f (x, y)
(3.2a)
(x, y, z) = z
or, equivalently, the implicit representation
(3.2b)
f (x, y).
The vector representation (3.1) can
bez)related
they).‘height’ representation (3.2a) by
(x, y,
= z to
f (x,
(3.2b)
0
1
x the ‘height’ representation (3.2a) by
The vector representation (3.1) can be related to
A
0 y 1
F (x, y) = @
(3.3)
x
f (x, y)
@
y A
F (x, y) =
(3.3)
f (x, y) the surface metric tensor g = (gij ) by
Denoting derivatives by F i = @s1 F , we introduce
Denoting derivatives by F i = @s1 gFij, =
weFintroduce
the surface metric tensor g = (g(3.4a)
ij ) by
i · F j,
abbreviate its determinant by
gij = F i · F j ,
(3.4a)
dunkel@math.mit.edu
where (s1 , s2 ) 2 U ✓ R . Alternatively, if one chooses Cartesian coordinates (s1 , s2 ) =
(x, y), then it suffices to specify
(x, y), then it suffices to specify
z = f (x, y)
(3.2a)
Surface metric tensor
z = f (x, y)
or, equivalently, the implicit representation
or, equivalently, the implicit representation
(x, y, z) = z
f (x, y).
(3.2a)
(3.2b)
(x,related
y, z) =tozthe f‘height’
(x, y).representation (3.2a) by
(3.2b)
The vector representation (3.1) can be
0
1
x
The vector representation (3.1) can be related
to the ‘height’ representation (3.2a) by
F (x, y) = @ y0 A
(3.3)
1
f (x, y) x
F (x, y) = @
y A
Denoting derivatives by F i = @si F , we introduce the surface metric tensor g = (gij ) by
f (x, y)
gij = F i · F j ,
(3.3)
(3.4a)
Denoting derivatives by F i = @s1 F , we introduce the surface metric tensor g = (gij ) by
abbreviate its determinant by
gij = F i · F j ,
|g| := det g,
(3.4b)
abbreviate
itsassociated
determinant
by
and define the
Laplace-Beltrami
operator r2 by
p
1 |g| :=
1 det g,
r h = p @i (gij |g|@j h),
|g|
2
(3.4c)
and define the associated Laplace-Beltrami operator r2 by
51
p
1
1
2
r h = p @i (gij |g|@j h),
|g|
(3.4a)
(3.4b)
(3.4c)
for some function h(s1 , s2 ). For the Cartesian
parameterization (3.3), one finds explicitly
49
0 1
0 1
1
0
F x (x, y) = @ 0 A ,
F y (x, y) = @ 1 A
(3.5)
fx
fy
and, hence, the metric tensor
✓
◆
✓
◆
dunkel@math.mit.edu
(x, y, z) = z
f (x, y).
(3.2b)
The vector representation (3.1) can be related to the ‘height’ representation (3.2a) by
0
1
x
F (x, y) = @ y A
(3.3)
f (x, y)
Denoting derivatives by F i = @si F , we introduce the surface metric tensor g = (gij ) by
gij = F i · F j ,
(3.4a)
|g| := det g,
(3.4b)
abbreviate its determinant by
and define the associated Laplace-Beltrami operator r2 by
p parameterization (3.3), one finds explicitly
1
for some function h(s1 , s2 ). Forrthe
Cartesian
1
2
p
h = Cartesian
@i (gij |g|@
j h),
for some function h(s1 , s2 ). For the
parameterization
(3.3),(3.4c)
one finds explicitly
|g|
for some function h(s1 , s2 ). For the
Cartesian
parameterization
0 1
0 1 (3.3), one finds explicitly
01 1
00 1
0 1 1 51
001
@
A
@
F x (x, y) = @01 A ,
F y (x, y) = @10A
(3.5)
F x (x, y) = @ 0 A ,
F y (x, y) = @ 1 A
(3.5)
F x (x, y) = f0x
,
F y (x, y) = f1y A
(3.5)
fx
fy
fx
fy
and, hence, the metric tensor
and, hence, the metric tensor
and, hence, the metric tensor
✓
◆ ✓
◆
✓F x · F x F x · F y ◆ ✓1 + fx22 fx fy ◆
◆ ✓1 + fx2 fx fy ◆
g = (gij ) = ✓F
(3.6a)
x · Fx Fx · Fy =
g = (gij ) = FFyx··FFxx FFyx··FFyy = 1fy+fxfx 1 f+x fyy22
(3.6a)
g = (gij ) = F y · F x F y · F y = fy fx 1 + fy2
(3.6a)
fy fx 1 + fy
Fy · Fx Fy · Fy
and its determinant
and its determinant
and its determinant
|g| = 1 + fx22+ fy22,
(3.6b)
|g| = 1 + fx2 + fy2 ,
(3.6b)
|g| = 1 + fx + fy ,
(3.6b)
where fx = @x f and fy = @y f . For later use, we still note that the inverse of the metric
where fx = @x f and fy = @y f . For later use, we still note that the inverse of the metric
where isfxgiven
= @x fbyand fy = @y f . For later use, we still note that the inverse of the metric
tensor
tensor is given by
tensor is given by
✓
◆
2
✓
◆
1
1
+
f
f
x fy ◆
2
1
1
y
✓
1
1+f
f f .
(3.6c)
g 1 = (gij 1) =
(3.6c)
g 1 = (gij 1 ) = 1 + fx212+ fy22 1 f+y ffxyy2 1 +fxxffx2yy2 .
(3.6c)
g = (gij ) = 1 + fx2 + fy2
fy fx 1 + fx2 .
f
f
1
+
f
1 + fx + fy
y x
x
Assuming the surface is regular at (s1 , s2 ), which just means that the tangent vectors F 1
Assuming the surface is regular at (s1 , s2 ), which just means that the tangentdunkel@math.mit.edu
vectors F 1
Assuming
the
surface
is
regular
at
(s
,
s
),
which
just
means
that
the
tangent
vectors F 1
and F 2 are linearly independent, the local
1 2 unit normal vector is defined by
and F 2 are linearly independent, the local unit normal vector is defined by
where fx = @x f and fy = @y f . For later use, we still note that the inverse of the metric
tensor is given by
✓
◆
2
1
1 + fy
fx fy
.
(3.6c)
g 1 = (gij 1 ) =
fy fx 1 + fx2
1 + fx2 + fy2
Surface normal & curvature
Assuming the surface is regular at (s1 , s2 ), which just means that the tangent vectors F 1
and F 2 are linearly independent, the local unit normal vector is defined by
N=
F1 ^ F2
.
||F 1 ^ F 2 ||
(3.7)
In terms of the Cartesian parameterization, this can also be rewritten as
0
1
fx
r
1
@ fy A .
N=
=p
2
2
||r ||
1 + fx + fy
1
(3.8)
Here, we have adopted the convention that {F 1 , F 2 , N } form a right-handed system.
To formulate ‘geometric’ energy functionals for membranes, we still require the concept
of curvature, which quantifies the local bending of the membrane. We define a 2 ⇥ 2curvature tensor R = (Rij ) by
Rij = N · (F ij )
(3.9)
and local mean curvature H and local Gauss curvature K by
1
H = tr (g
2
1
· R) ,
K = det(g
Adopting the Cartesian representation (3.2a), we have
0 1
0 1
0
0
F xx = @ 0 A ,
F xy = F yx = @ 0 A ,
fxx
fxy
wiki
50
1
· R).
F yy
(3.10)
0
1
0
=@ 0 A
fyy
(3.11a)
dunkel@math.mit.edu
where fx = @x f and fy = @y f . For
use,
wefy2still
|g|later
=1+
fx2 +
, note that the inverse of the metric
(3.6b)
tensor is given by
where fx = @x f and fy = @y f . For later use, we
✓ still 2note that◆the inverse of the metric
1
1 + fy
fx fy
tensor is given by g 1 = (g 1 ) =
.
(3.6c)
ij
1 + fx2 + fy2 ✓ fy fx2 1 + fx2◆
1
1 + fy
fx fy
(3.6c)
g 1 = (gij 1 ) =
2 . the tangent vectors
Assuming the surface is regular at 1(s+
sx22),+which
just
F1
fy fxmeans
1 + fthat
fy2
1, f
x
and F 2 are linearly independent, the local unit normal vector is defined by
Assuming the surface is regular at (s1 , s2 ), which just means that the tangent vectors F 1
F unit
1 ^ Fnormal
2
and F 2 are linearly independent, the
local
vector is defined by
N=
.
(3.7)
||F 1 ^ F 2 ||
F1 ^ F2
N
=
.
(3.7)
In terms of the Cartesian parameterization,
||F 1 this
^ F 2can
|| also be rewritten as
0
1
fbe
In terms of the Cartesian parameterization,
this
x rewritten as
r
1 can also
@ fy 1
A.
N=
=p
(3.8)
0
2
2
||r ||
1 + fx + fy
1fx
r
1
@ fy A .
N=
=p
(3.8)
2
2
||r
||
1
+
f
+
f
Here, we have adopted the convention that {Fx1 , F 2y, N } 1form a right-handed system.
To formulate ‘geometric’ energy functionals for membranes, we still require the concept
Here,
we havewhich
adopted
the convention
that
{F 1 , Fof
a right-handed
system.
2, N
of curvature,
quantifies
the local
bending
the} form
membrane.
We define
a 2 ⇥ 2To formulate
‘geometric’
energy functionals for membranes, we still require the concept
curvature
tensor R
= (Rij ) by
of curvature, which quantifies the local bending of the membrane. We define a 2 ⇥ 2curvature tensor R = (Rij ) by
Rij = N · (F ij )
(3.9)
Surface normal & curvature
Rij Gauss
= N · (F
and local mean curvature H and local
curvature
K by
ij )
1 and local
and local mean curvature
curvature
H =H tr
(g 1 · R)Gauss
,
K = det(gK1 by
· R).
2
1
1
1
H
=
tr
(g
·
R)
,
K
=
det(g
· R).
Adopting the Cartesian representation
(3.2a), we have
2
0 1
0 1
0 1
Adopting the Cartesian
0 representation (3.2a), we 0have
0
A ,
@ 01
A
00 A
1,
0 01
F xx = @
F xy = F yx = @
F yy = 0
0
0
0
f
f
f
xx
xy
yy
F xx = @ 0 A ,
F xy = F yx = @ 0 A ,
F yy = @ 0 A
fxx
fxy
fyy
50
50
(3.9)
(3.10)
(3.10)
(3.11a)
(3.11a)
dunkel@math.mit.edu
Surface normal & curvature
yielding the curvature tensor
✓
◆
✓
◆
1
N ·F
N · F xy
fxx fxy
yielding the curvature
(Rij ) = tensor xx
=p
(3.11b)
2
2
✓
◆
✓
◆
N
·
F
N
·
F
f
f
yx
yy
1
+
f
+
f
yx
yy
yielding the curvature tensor
y
1x
N · F xx N · F xy ◆
✓
✓fxx fxy ◆
(Rij ) =
=1 p
(3.11b)
2
2
1
N
·
F
N
·
F
f
f
N
·
F
N
·
F
f
f
Denoting the eigenvalues ofyxxx
the matrix
· R1 +
byfx1+and
,xxwe yyobtain
for the mean
yyxy g
fy 2yx
xy
(Rij ) =
=p
(3.11b)
2
2
N · F yx N · F yy 1
f
f
curvature
1+f +f
yx
yy
Denoting the eigenvalues of the matrix g2 · R by 1x andy 2 , we
obtain
for the mean
2
(1 + fy )fxx
2fx fy fxy + (1 + fx )fyy
1
1
curvature
Denoting the eigenvalues
of
the
matrix
g
·
R
by 1 and 2 , we obtain for the(3.12)
mean
H = (1 + 2 ) =
2
2(1 + fx2 + fy2 )3/2
2
curvature
(1 + fy )fxx 2fx fy fxy + (1 + fx2 )fyy
1
H = curvature
(1 + 2 ) = (1 + f 2 )f
(3.12)
2 )+
3/2(1 + f 2 )fyy
and for the Gauss
xx +2f
21
fx2x f+y ffxy
y 2(1
x
y
H = (1 + 2 ) =
(3.12)
2
2 f+ f 2 )3/2
2
+
f
f2(1
f
and for the Gauss curvature K = 1 · 2 = xx yy x xy y.
(3.13)
2 + f 2 )2
(1
+
f
2y
and for the Gauss curvature
fxx fyyx fxy
K relates
= 1 · 
(3.13)
22 .
2 =
An important result that
curvature
and
topology
is
the
Gauss-Bonnet
theorem,
2
2
f
f
f
xx
yy
(1 + fx + fy xy
)
K
=

·

=
.
(3.13)
1
2
which states that any compact two-dimensional
Riemannian
manifold
M
with
smooth
2
2
2
(1 + fx + fy )
An important
result curvature
that relates
andcurvature
topology kisg the
Gauss-Bonnet
boundary
@M , Gauss
K curvature
and geodesic
of @M
satisfies thetheorem,
integral
which
that result
any compact
two-dimensional
manifold
M with theorem,
smooth
Anstates
important
that relates
curvature andRiemannian
topology is the
Gauss-Bonnet
equation
Z
boundary
@M ,that
Gauss
Ktwo-dimensional
andIgeodesic curvature
kg of manifold
@M satisfies
the integral
which states
anycurvature
compact
Riemannian
M with
smooth
equation
K dA
+ geodesic
kg ds =curvature
2⇡ (M ).kg of @M satisfies the integral
(3.14)
boundary @M , Gauss curvature
K and
Z M
I @M
equation
Z Kon
I the
dAM+, ds
kg ds
2⇡ (M along
).
Here, dA is the area element
line=element
@M , and (M ) the (3.14)
Euler
M
KisdA
+ @M
(Mwhere
).
(3.14)
characteristic of M . The latter
given
by k(M
2 2g,
g is the genus (number
of
g ds) = 2⇡
2
Here,
dA isofthe
elementM on
, ds @M
theMline
@M , and hence
(M ) the
handles)
M . area
For example,
theM
2-sphere
= Selement
has g =along
0 handles
(S2 )Euler
= 2,
dunkel@math.mit.edu
2(number
characteristic
of M
. The
latter
is given
by
(M
)g =
21 handle
2g, where
is the
genus
of
whereas
torus
T2the
has
=element
and gtherefore
(T
))=the
0. Euler
Here,
dA aistwo-dimensional
the
area
element
on
M ,=ds
line
along
@M
, and
(M
Gauss curvature
(1 + fx + fy )
2
f
f
f
xx
yy
xy
important result
that
and. topology is the Gauss-Bonnet
theorem,
K=
1 ·relates
2 = curvature
(3.13)
(1 + fx2 + fy2 )2 Riemannian manifold M with smooth
states that any compact two-dimensional
Gauss-Bonet Theorem
ary
, Gauss
curvature
K andand
geodesic
curvature
kg of @M satisfies
the integral
tant@M
result
that relates
curvature
topology
is the Gauss-Bonnet
theorem,
on
that any compact two-dimensional Riemannian manifold M with smooth
Z
I
M , Gauss curvature K and geodesic curvature kg of @M satisfies the integral
K dA +
kg ds = 2⇡ (M ).
(3.14)
M I
@M
Z
Gauss geodesic Euler
dA is the area element
element
along
(M ) the Euler
K dAcurvature
+on Mk,g ds
ds the
=curvature
2⇡line(M
).
(3.14)
characteristic@M , and
teristic of M . M
The latter is@Mgiven by (M ) = 2 2g, where g is the genus (number of
s)
M . element
For example,
M = S2along
has g @M
= 0, handles
hence
heofarea
on M ,the
ds 2-sphere
the line element
and (Mand
) the
Euler (S2 ) = 2,
sofa M
two-dimensional
torusbyM (M
= T) 2=has
= where
1 handle
therefore
(T2 )of= 0.
. The latter is given
2 g2g,
g isand
the genus
(number
M
. For example,
the 2-sphere
M=
S2 closed
has g =surface,
0 handles
hence over
(S2 )K= is2, always a
uation
(3.14) implies
that, for
any
theand
integral
o-dimensional
torus
M=
T2 has g = the
1 handle
and therefore
(T2 ) =represents
0.
nt.
That is, for
closed
membranes,
first integral
in Eq. (3.14)
just a
(3.14) implies
that, for
any closed surface, the integral over K is always a
(constant)
energetic
contribution.
at is, for closed membranes, the first integral in Eq. (3.14) represents just a
wiki
ant) energetic contribution.
Minimal surfaces
nimal
al
surfacessurfaces
are surfaces that minimize the area within a given contour @M ,
2
0
-2
Z
aces are surfaces that minimize
a given contour @M ,
A(M |@Mthe
) =area within
dA = min!
(3.15)
Z
M
|@M ) =
dAz ==min!
(3.15)
ing a Cartesian A(M
parameterization
f (x, y) and abbreviating f =dunkel@math.mit.edu
@ f as before,
i
i
K = 1 · 2 =
(1 +
fx2
+
fy2 )2
.
(3.13)
An important result that relates curvature and topology is the Gauss-Bonnet theorem,
ich states that any compact two-dimensional Riemannian manifold M with smooth
undary @M , Gauss curvature K and geodesic curvature kg of @M satisfies the integral
uation
Z
I
K dA +
kg ds = 2⇡ (M ).
(3.14)
Gauss-Bonet Theorem
M
@M
re, dA is the area element on M , ds the line element along @M , and (M ) the Euler
aracteristic of M . The latter is given by (M ) = 2 2g, where g is the genus (number of
ndles) of M .Convex
For example,
the 2-sphere M = S2 has g = 0 handles and hence (S2 ) = 2,
polyhedra
ereas a two-dimensional torus M = T2 has g = 1 handle and therefore (T2 ) = 0.
Equation (3.14) implies that, for any closed surface, the integral over K is always a
Vertices!
Edges!
Faces!
Euler characteristic:!
Name
Image
E in Eq.
F
V−E+F
nstant. That is, for closed membranes, the firstV integral
(3.14) represents
just a
vial (constant) energetic
contribution.
Tetrahedron
4
6
4
2
2
Hexahedron or cube
Minimal surfaces
Octahedron
8
12
6
2
6
12
8
2
nimal surfaces are surfaces
that minimize the area
within
a given
contour2 @M ,
Dodecahedron
20
30
12
Z
Icosahedron
12
30
20
2
A(M |@M ) =
dA = min!
wiki
(3.15)
M
suming a Cartesian parameterization z = f (x, y) and abbreviating fi = @i f as before,
dunkel@math.mit.edu
have
trivial (constant) energetic contribution.
3.2
Minimal surfaces
Minimal surfaces are surfaces that minimize the area within a given contour @M ,
Z
A(M |@M ) =
dA = min!
(3.15)
M
Assuming a Cartesian parameterization z = f (x, y) and abbreviating fi = @i f as before,
we have
q
p
(3.16)
dA = |g| dxdy = 1 + fx2 + fy2 dxdy =: L dxdy,
and the minimum condition (3.15) can be expressed in terms of the Euler-Lagrange equations
catenoid
A
@L
0=
= @i
.
(3.17)
f
@fi
51
Goldstein lab (Cambridge)
dunkel@math.mit.edu
whereas
a two-dimensional
torus M = T has g = 1 handle and therefore (T ) = 0.
trivial
(constant)
energetic contribution.
Equation (3.14) implies that, for any closed surface, the integral over K is always a
constant. That is, for closed membranes, the first integral in Eq. (3.14) represents just a
3.2
Minimal
surfaces
trivial (constant)
energetic
contribution.
Minimal surfaces are surfaces that minimize the area within a given contour @M ,
Z
3.2 Minimal surfaces
A(M |@M ) =
dA = min!
(3.15)
Minimal surfaces are surfaces that minimizeMthe area within a given contour @M ,
Assuming a Cartesian parameterization z Z
= f (x, y) and abbreviating fi = @i f as before,
A(M |@M ) =
dA = min!
(3.15)
we have
M
q
p
2
|g| dxdy = 1 z+=fx2f+
L dxdy, fi = @i f as before,
(3.16)
dA =parameterization
Assuming a Cartesian
(x,fyy)dxdy
and =:
abbreviating
we have
and the minimum condition
qbe expressed in terms of the Euler-Lagrange equap (3.15) can
tions
(3.16)
dA = |g| dxdy = 1 + fx2 + fy2 dxdy =: L dxdy,
A
@L
0
=
=
@
. in terms of the Euler-Lagrange (3.17)
and the minimum condition (3.15) can be expressed
equai
f
@fi
tions
A 51 @L
0=
= @i
.
(3.17)
f
@fi
p
51
Inserting the Lagrangian L = |g|, one finds
"
!
fx
p
0=
@x
+ @y
1 + fx2 + fy2
which may be recast in the form
fy
p
1 + fx2 + fy2
(1 + fy2 )fxx 2fx fy fxy + (1 + fx2 )fyy
0=
=
(1 + fx2 + fy2 )3/2
!#
2H.
(3.18)
(3.19)
Thus, minimal surfaces satisfy
H=0
,
1 =
2 ,
dunkel@math.mit.edu
(3.20)
whereas
a two-dimensional
torus M = T has g = 1 handle and therefore (T ) = 0.
trivial
(constant)
energetic contribution.
Equation (3.14) implies that, for any closed surface, the integral over K is always a
constant. That is, for closed membranes, the first integral in Eq. (3.14) represents just a
3.2
Minimal
surfaces
trivial (constant)
energetic
contribution.
Minimal surfaces are surfaces that minimize the area within a given contour @M ,
Z
3.2 Minimal surfaces
A(M |@M ) =
dA = min!
(3.15)
Minimal surfaces are surfaces that minimizeMthe area within a given contour @M ,
Assuming a Cartesian parameterization z Z
= f (x, y) and abbreviating fi = @i f as before,
A(M |@M ) =
dA = min!
(3.15)
we have
M
q
p
2
|g| dxdy = 1 z+=fx2f+
L dxdy, fi = @i f as before,
(3.16)
dA =parameterization
Assuming a Cartesian
(x,fyy)dxdy
and =:
abbreviating
p
we have
Inserting
the Lagrangian
|g|,can
one
finds
and
the minimum
condition
expressed in terms of the Euler-Lagrange equaqbe
pL =(3.15)
tions
(3.16)
dA =" |g| dxdy = 1 +!fx2 + fy2 dxdy =: L dxdy,
!#
fx A
@L p fy
p
0
=
@
+
@
(3.18)
x
y . in terms
0
=
=
@
(3.17)
and the minimum condition
(3.15)
can
be
expressed
of
the
Euler-Lagrange
equai
2
2
2
2
1 + fx +ffy
@fi 1 + fx + fy
tions
A 51 @L
which may be recast in the form
0=
= @i
.
(3.17)
f
@f
i
(1 + fy2 )fxx 2fx fy fxy + (1 + fx2 )fyy
0=
= 2H.
(3.19)
2
2
3/2
(1 + fx + fy51
)
Thus, minimal surfaces satisfy
H=0
,
1 =
2 ,
(3.20)
implying that each point of a minimal surface is a saddle point.
3.3
Thermal excitations of almost flat membranes
Assuming that a quasi-infinite membrane prefers a flat configuration, we postulate the
energy functional
Z
dunkel@math.mit.edu
kc
2
where LP is the persistence length, defined by the condition hfx + fy i = 1.
3.4
Helfrich’s model
Assuming that lipid bilayer membranes can be viewed as two-dimensional surfaces, Helfrich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
a closed membrane
kc
fc = (2H
2
c0 )2 + kG K,
(3.31)
where constants kc , kG are bending rigidities and c0 is the spontaneous curvature of the
membrane. The full free energy for a closed membrane can then be written as
Z
Z
Z
Fc = dA fc +
dA + p dV,
(3.32)
where is the surface tension and p the osmotic pressure (outer pressure minus inner
pressure). Minimizing F with respect to the surface shape, one finds after some heroic
manipulations the shape equation2
p
2 H + kc (2H
c0 )(2H 2 + c0 H
2K) + kc r2 (2H
c0 ) = 0,
(3.33)
where r2 is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
wiki
uses our earlier result
A
=
f
2H,
and the fact that the volume integral may be rewritten as3
Z
Z
1
V = dV = dA F · N ,
3
www.math.tamu.edu/~bonito/
2
The full derivation can be found in Chapter 3 of Ref. [OLXY99].
(3.34)
(3.35)
dunkel@math.mit.edu
ipid bilayer where
membranes
can be viewed as two-dimensional surfaces, HelLP is the persistence length, defined by the condition hfx22 + fy22 i = 1.
where
is the persistence
length,
defined byenergy
the condition
hfx area
+ fy i for
= 1.
posed in 1973
theLPfollowing
geometric
curvature
per unit
ne
3.4 Helfrich’s
Helfrich’s model
model
3.4
kc
2
fc = that
(2H lipidc0bilayer
) + kGmembranes
K,
(3.31) surfaces, HelAssuming
can
be
viewed
as
two-dimensional
Assuming2 that lipid bilayer membranes can be viewed as two-dimensional surfaces, Helfrich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
frich
[Hel73]
proposed and
in 1973
thethe
following
geometriccurvature
curvature energy
kc , kG are bending
rigidities
c0 is
spontaneous
of theper unit area for
a closed membrane
a closed membrane
full free energy
for a closed membrane can
kc then be2 written as
f Z
= kc (2H c0 )2 + kG K,
(3.31)
Z
Z
fcc =
(3.31)
2 (2H c0 ) + kG K,
2
Fc =
dA
f
+
dA
+
p
(3.32)curvature of the
c
where constants
kc , kG are bendingdV,
rigidities and c0 is the spontaneous
where constants kc , kG are bending rigidities and c0 is the spontaneous curvature of the
membrane. The full free energy for a closed membrane can then be written as
membrane. The full free energy
for a closedZ membrane Zcan then be written as
Z
urface tension and p the osmotic pressure
(outer
pressure
Z
Z
Z minus inner
F = dA fc +one dA
+ p dV,
(3.32)
mizing F with respect to the surface
finds
some heroic
Fcc = shape,
dA fc +
dA
+ after
p dV,
(3.32)
he shape equation2
where is the surface tension and p the osmotic pressure (outer pressure minus inner
where is the surface tension and p the osmotic pressure (outer pressure minus inner
pressure). Minimizing
with respect2to the surface shape, one finds after some heroic
2
2 H + kc (2H
c0 )(2H
+ c0 H FF 2K)
(2H
c0 ) = shape,
0,
(3.33)
pressure).
Minimizing
with +
respect
the surface
one finds
after some heroic
2kc r to
manipulations the shape equation2
manipulations the shape equation
Laplace-Beltrami operator
surface.
derivation
Eq. (3.33)
p 2 Hon
+ kthe
c0 )(2H22The
+ c0 H
2K) + kcof
r22 (2H
c0 ) = 0,
(3.33)
c (2H
p
2
H
+
k
(2H
c
)(2H
+
c
H
2K)
+
k
r
(2H
c
)
=
0,
(3.33)
c
0
0
c
0
result
where r22 is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
where r is A
the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
uses our earlier result
(3.34)
uses our earlier =
result2H,
f
A
A = 2H,
(3.34)
=
2H,
(3.34)
f
t the volume integral may be rewritten as3 f
and the
fact thatZthe volume integral may be rewritten as33
Z
and the fact that the volume
1 integral
Z mayZbe rewritten as
Z
1
V = dV = dA F · NZ,
(3.35)
1
V
=
dV
=
dA
F
·
N
,
(3.35)
3 V = dV = dA 3 F · N ,
(3.35)
3
2
tion can be foundThe
in Chapter
3 ofcan
Ref.
full derivation
be[OLXY99].
found in Chapter 3 of Ref. [OLXY99].
2
1 be found in Chapter 3 of 1Ref. [OLXY99].
3The full derivation can
Here,
we made
the volume formula dV = 13 h dAof
forheight
a cone h
or =
pyramid
use of the volume
formula
dV use
= of
F · Nof. height h = F · N .
3 h dA for a cone or pyramid
3
Here, we made use of the volume formula dV = 3 h dA for a cone or pyramid of height h =dunkel@math.mit.edu
F ·N .
hich gives
where LP is the persistence length, defined by the condition hfx22 + fy22 i = 1.
where LP is the persistence length, defined by the condition hfx + fy i = 1.
3.4
3.4
Helfrich’s model
model
Helfrich’s
Assuming that lipid bilayer membranes can be viewed as two-dimensional surfaces, HelAssuming that lipid bilayer membranes can be viewed as two-dimensional surfaces, Helfrich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
frich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
a closed membrane
a closed membrane
kc
fc = kc (2H c0 )22 + kG K,
(3.31)
fc = 2 (2H c0 ) + kG K,
(3.31)
2
where constants k , k are bending rigidities and c is the spontaneous curvature of the
where constants kcc, kGG are bending rigidities and c00 is the spontaneous curvature of the
membrane. The full free energy for a closed membrane can then be written as
membrane. The full free energy
Z for a closedZ membrane Zcan then be written as
Z
Z
Z
F = dA f +
dA + p dV,
(3.32)
Fcc = dA fcc +
dA + p dV,
(3.32)
where is the surface tension and p the osmotic pressure (outer pressure minus inner
where is the surface tension and p the osmotic pressure (outer pressure minus inner
pressure). Minimizing F with respect to the surface shape, one finds after some heroic
pressure). Minimizing F with respect
to the surface shape, one finds after some heroic
manipulations the shape equation22
manipulations the shape equation
p 2 H + kc (2H c0 )(2H22 + c0 H 2K) + kc r22 (2H c0 ) = 0,
(3.33)
p 2 H + kc (2H c0 )(2H + c0 H 2K) + kc r (2H c0 ) = 0,
(3.33)
where r22 is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
where r is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
uses our earlier result
uses our earlier result
A
A = 2H,
(3.34)
(3.34)
f = 2H,
f
and the fact that the volume integral may be rewritten as33
and the fact that the volume integral
Z mayZbe rewritten as
Z
Z
1
V = dV = dA 1 F · N ,
(3.35)
V = dV = dA 3 F · N ,
(3.35)
3
2
The full derivation
2
3The full derivation
Here, we made use
3
can be found in Chapter 3 of Ref. [OLXY99].
V informula
can
be volume
found
Chapter
of 1Ref.
of the
dV3 =
h dA[OLXY99].
for a cone or pyramid of height(3.36)
h=F ·N .
=
1.
13
Here, we made use of the volume
formula
dV
=
h
dA
for
a
cone
or
pyramid
of
height
h
=dunkel@math.mit.edu
F ·N .
3
f
where LP is the persistence length, defined by the condition hfx22 + fy22 i = 1.
where LP is the persistence length, defined by the condition hfx + fy i = 1.
3.4
3.4
Helfrich’s
model
Helfrich’s model
Assuming that lipid bilayer membranes can be viewed as two-dimensional surfaces, HelAssuming that lipid bilayer membranes can be viewed as two-dimensional surfaces, Helfrich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
frich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
a closed membrane
a closed membrane
kc
fc = kc (2H c0 )22 + kG K,
(3.31)
fc = 2 (2H c0 ) + kG K,
(3.31)
2
where constants kc , kG are bending rigidities and c0 is the spontaneous curvature of the
where constants kc , kG are bending rigidities and c0 is the spontaneous curvature of the
membrane. The full free energy for a closed membrane can then be written as
membrane. The full free energyZ for a closedZmembrane Zcan then be written as
Z
Z
Z
Fc = dA fc +
dA + p dV,
(3.32)
Fc = dA fc +
dA + p dV,
(3.32)
where is the surface tension and p the osmotic pressure (outer pressure minus inner
where is the surface tension and p the osmotic pressure (outer pressure minus inner
pressure). Minimizing F with respect to the surface shape, one finds after some heroic
pressure). Minimizing F with respect
to the surface shape, one finds after some heroic
manipulations the shape equation22
manipulations the shape equation
p 2 H + kc (2H c0 )(2H22 + c0 H 2K) + kc r22 (2H c0 ) = 0,
(3.33)
p 2 H + kc (2H c0 )(2H + c0 H 2K) + kc r (2H c0 ) = 0,
(3.33)
where r22 is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
where r is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
uses our earlier result
uses our earlier result
A
(3.34)
A = 2H,
(3.34)
f = 2H,
f
and the fact that the volume integral may be rewritten as33
and the fact that the volume integral
Z may Zbe rewritten as
1
Z
Z
V = dV = dA 1 F · N ,
(3.35)
V = dV = dA 3F · N ,
(3.35)
3
2
The full derivation can be found in Chapter 3 of Ref. [OLXY99].
2
3The full derivation can be found in Chapter 3 of 1Ref. [OLXY99].
www.math.tamu.edu/~bonito/
Here, we made use of the volume
formula dV = h dA for a cone
3
Here, we made use of the volume formula dV = 133h dA for a cone
or pyramid of height h = F · N .
or pyramid of height h =dunkel@math.mit.edu
F ·N .
where LP is the persistence length, defined by the condition hfx2 + fy2 i = 1.
3.4
Helfrich’s model
Assuming that lipid bilayer membranes can be viewed as two-dimensional surfaces, Helfrich [Hel73] proposed in 1973 the following geometric curvature energy per unit area for
a closed membrane
fc =
kc
(2H
2
c0 )2 + kG K,
(3.31)
where constants kc , kG are bending rigidities and c0 is the spontaneous curvature of the
membrane. The full free energy for a closed membrane can then be written as
Z
Z
Z
Fc = dA fc +
dA + p dV,
(3.32)
where gives
is the surface tension and p the osmotic pressure (outer pressure minus inner
which
pressure). Minimizing F with respect to the surface shape, one finds after some heroic
V
manipulations the shape equation2
= 1.
(3.36)
f
p 2 H + kc (2H c0 )(2H 2 + c0 H 2K) + kc r2 (2H c0 ) = 0,
(3.33)
For open membranes with boundary @M , a plausible energy functional is given by
where r2 is the Laplace-Beltrami operator on the surface. The derivation of Eq. (3.33)
Z
Z
I
uses our earlier result
Fo = dA fc +
dA +
ds,
(3.37)
A
@M
= 2H,
(3.34)
f
where is the line tension of the boundary. In this case, variation yields not only the
3
corresponding
shape
but also may
a non-trivial
set ofasboundary
conditions.
and the fact that
theequation
volume integral
be rewritten
Z
Z
1
V = dV = dA F · N ,
(3.35)
3
2
3
The full derivation can be found in Chapter 3 of Ref. [OLXY99].
Here, we made use of the volume formula dV = 13 h dA for a cone or pyramid of height h = F · N .
dunkel@math.mit.edu