Long-lived anomalous thermal diffusion induced by elastic cell membranes on nearby particles

1 MB·

by  Abdallah Daddi-Moussa-Ider

Long-lived anomalous thermal diffusion induced by elastic cell membranes on nearby
particles
Abdallah Daddi-Moussa-Ider, Achim Guckenberger and Stephan Gekle
Biofluid Simulation and Modeling, Fachbereich Physik, Universita¨t Bayreuth
(Dated: January 22, 2016)
The physical approach of a small particle (virus, medical drug) to the cell membrane represents
the crucial first step before active internalization and is governed by thermal diffusion. Using a
fully analytical theory we show that the stretching and bending of the elastic membrane by the
approaching particle induces a memory in the system which leads to anomalous diffusion, even
though the particle is immersed in a purely Newtonian liquid. For typical cell membranes the
transient subdiffusive regime extends beyond 10 ms and can enhance residence times and possibly
6
binding rates up to 50%. Our analytical predictions are validated by numerical simulations.
1
0
PACSnumbers: 47.63.-b,87.16.D-,47.63.mh,47.57.eb
2

n
a I. INTRODUCTION membrane possessing shear and bending resistance with
J fluid on both sides. As the typical sizes and velocities

0 are small, the theory is derived in the small Reynolds
Endocytosis, the uptake of a small particle by a liv-
2 number regime neglecting the non-linear term, but in-
ingcellisoneofthemostimportantprocessesinbiology cluding the unsteady contribution in the Navier-Stokes
] [1–3]. Currentresearchisfocusedmainlyonthebiophys-
n equations. Our most important finding is that there
ical and biochemical mechanisms which govern endocy-
y exists a long-lasting subdiffusive regime with local ex-
tosis when particle and cell are in direct physical con-
d ponents as low as 0.87 extending over time scales be-
tact. Much less investigated, yet equally important, is
- yond 10ms. Such behavior is qualitatively different from
u the approach of the particle to the cell membrane before
diffusion near hard walls where the diffusion, albeit be-
fl physical contact is established [4]. In many physiolog- ing slowed down, still remains normal (i.e. the mean-
. ically relevant situations, e.g., inside the blood stream,
s square-displacement increases linearly with time). Re-
c the cell and the particle are both suspended in a sur-
markably, our system exhibits subdiffusion in a purely
i
s roundingliquidandtheapproachisgovernedbythermal Newtonian liquid whereas most commonly subdiffusion
y diffusion of the small particle. The thermal diffusion of
is observed for particles in viscoelastic media. The sub-
h smallparticles(fibrinogen)naturallyoccurringinhuman
p diffusive regime increases the residence time of the par-
blood has furthermore been suggested as the root cause
[ ticle in the vicinity of the membrane by up to 50% and
of red blood cell aggregation [5–7].
is thus expected to be of important physiological signif-
2
Thermaldiffusionofasphericalparticleinabulkfluid icance. Our analytical particle mobilities are quantita-
v
1 iswellunderstoodandgovernedbythecelebratedStokes- tivelyverifiedbydetailedboundary-integralsimulations.
0 Einstein relation. This relation builds a bridge between Power-spectraldensitieswhichareamenabletodirectex-
1 the particle mobility when an external force is applied perimental validation using optical traps are provided.
2 to it and the random trajectories observed when only
0 thermal fluctuations are present. Particle mobilities and
1. thermal diffusion near solid walls have been thoroughly II. RESULTS
0 investigatedboththeoretically[8–13]andexperimentally
6 [14–26] finding a reduction of the particle mobility due
A spherical particle with radius R=100nm is located
1 to the proximity of the wall. Some theoretical works
at a distance z = 153nm above an elastic membrane
: 0
v have investigated particle mobilities and diffusion close and exhibits diffusive motion as illustrated in the in-
i tofluid-fluidinterfacesendowedwithsurfacetension[27–
X set of Fig. 1. The membrane has a shear resistance
30]orsurfaceelasticity[31–33]withcorrespondingexper- κ =5·10−6N/mandbendingmodulusκ =2·10−19Nm
r iments [34–41]. For the case of a membrane with bend- s b
a which are typical values of red blood cells [43]. The area
ing resistance transient subdiffusive behavior has been
dilatation modulus is κ = 100κ . The fluid properties
a s
observed in the perpendicular direction [19]. Regarding
correspond to blood plasma with viscosity η =1.2mPas.
biological cells, recent experiments have measured parti-
Figure1showsthemean-square-displacement(MSD)for
cle mobilities near different types of cells as well as giant
parallelaswellasperpendicularmotionasobtainedfrom
unilamellar vesicles (GUVs), both of which possess an
our fully analytical theory to be described below. For
elastic membrane separating two fluids, and found that
short times (t < 50µs) the MSD follows a linear behav-
the mobility near the cell walls does decrease but not as
ior with the normal bulk diffusion coefficient D since
0
strongly as near a hard wall [4].
the membrane does not have sufficient time to react on
Here we derive a fully analytical theory for the diffu- these short scales. This is in agreement with a sim-
sion of a small particle in the vicinity of a realistic cell ple balance between viscosity and elasticity for shear,
2
FIG. 2. (a) Minimum of the local exponent plotted against
particle-membrane separation. Significant subdiffusion is ob-
served up to distances roughly ten times the particle radius.
(b) The time required to diffuse one particle radius increases
due to the presence of the membrane thus leading to an en-
hanced residence time of the particle in the vicinity of the
membrane which may increase the probability of triggering
endocytosis.
FIG. 1. Mean-square displacement (red line) of a particle
withradiusR=100nmdiffusingz =153nmaboveared-blood
0
cell membrane in lateral (top) and perpendicular (bottom)
direction as predicted by our theory at T =300K. For short
times t(cid:46)50µs the MSD follows bulk behavior (black dashed
line)whileforlongtimestheMSDfollowshard-wallbehavior
(blue dash-dotted line). In between, a subdiffusive regime is
evident extending up to 10ms and beyond. Insets show the
local exponent which goes down until 0.87 for perpendicular
diffusion.
τ = ηR/κ ≈ 37µs, and bending, τ = ηR3/κ ≈ 22µs.
s s b b
For t > 50µs we observe a downward bending of the FIG. 3. Predicted power-spectral density of position fluctu-
MSD which is a clear signature of subdiffusive behavior. ations if the same bead as in Fig. 1 is confined by a typical
Indeed, as shown in the insets of Fig. 1, the local ex- optical trap of strength K = 10−5N/m [4]. Similar as in
ponent α = ∂log(cid:104)x2(cid:105) diminishes from 1 down to 0.92 in the MSD of Fig. 1 a transition from hard-wall-like behavior
∂logt (bluedash-dottedline)forlowfrequenciestobulk-likebehav-
theparalleland0.87intheperpendiculardirection. The
ior (black dashed line) at high frequencies is seen.
subdiffusive regime extends up to 10ms in the parallel
and even further in the perpendicular direction, which is
long enough to be of possible physiological significance. its own radius which gives an approximate measure of
Finally, for long times, the behavior turns back to nor- the ”diffusion speed”. As expected based on the data
mal diffusion with α ≈ 1. Compared to the short-time from Fig. 1, diffusion in the perpendicular direction is
regime, however, the diffusion coefficient is now signif- slowed down significantly more than for lateral motion,
icantly lower and approaches the well-known behavior see Fig. 2 (b), in agreement with recent experimental
near a solid hard wall with Dwall,(cid:107) = D0(1−9/16R/z0) observations [4].
in the parallel and Dwall,⊥ = D0(1 − 9/8R/z0) in the Experimentally,longMSDscanbedifficulttomeasure
perpendicularcase, respectively. Diffusionforlongtimes astheparticlemaymoveoutofthefocalplaneduringthe
therefore turns out to depend only on the particle dis- recording time. A commonly used technique is therefore
tanceandtobeindependentofthemembraneproperties. to confine the particle to its position using optical traps.
In Fig. 2 (a) we show the minimum of the local expo- One then records the power spectral density (PSD) of
nent for different particle-membrane separations. Even particlefluctuationsarounditsequilibriumposition. The
for distances ten times the particle radius, a significant PSDs predicted by our theory for a typical optical trap
deviation of the local exponent from 1 is still observable. with spring constant K = 10−5N/m [4] as a function of
FromtheMSDsitisstraightforwardtoestimatethetime frequency f = ω/2π are shown in Fig. 3. The general
T required by the particle to diffuse a distance equal to behavioroftheunconstrainedsystemisnotqualitatively
D
3
altered by the optical confinement: for high frequencies where the membrane deformation is u and the notation
the behavior is bulk-like (mirroring the bulk-like MSD u denotespartialspatialderivatives. Themoduliareκ
,· s
at short times) while for low frequencies the PSD ap- for shear resistance and κ for bending resistance while
b
proaches that expected near a solid wall (mirroring the the ratio between shear and area dilatation modulus is
hard-wall like MSD at long times). The frequency range C =κ /κ . The no-slip condition at the membrane sur-
a s
ofthetransitionliesmainlybelow1kHzandshouldthus face relates the surface deformation to the local fluid ve-
be experimentally accessible. locity
du
=v| . (6)
III. THEORY dt z=0
Togetherwithequations(4),(5)and(6)thisrepresentsa
Our theoretical development leading to figures 1
closedmathematicalproblemforthevelocityfieldv. For
through 3 proceeds via the calculation of particle mo-
its solution, the Stokes equations (4) are first Fourier-
bilities and the fluctuation-dissipation theorem and can
transformed into frequency space. The dependency on
be sketched as follows (a detailed derivation is given in
thexandy coordinatesisFourier-transformedintowave
Appendices A-C). We consider a spherical particle of ra-
vectors q and q which subsequently allows us to con-
dius R driven by an oscillating force F (t) = F eiωt x y
ω 0 siderthecontributionsofthelongitudinalandtransversal
in a fluid with density ρ and dynamic viscosity η whose
velocity components separately [28]. After eliminating
complex mobility µ(ω) for a fixed ω is defined as
the pressure, this leads to three differential equations for
V (t)=µ(ω)F (t) (1) thethreevelocitycomponentsforwhichananalyticalso-
ω ω
lution can be found. From the velocity field the mobility
and can be separated into the three contributions correctionoftheparticleisdirectlyobtained. Thedetails
are given in Appendix B.
µ(ω)=µ0+µu0(ω)+∆µ(ω,z0). (2) The mobility correction is a tensorial quantity which
in the present case has two components for the mobil-
Here, µ = 1/(6πηR) is the usual steady-state bulk mo-
0 ity parallel ∆µ (ω,z ) and perpendicular ∆µ (ω,z ) to
(cid:107) 0 ⊥ 0
bility,
the membrane. Furthermore, the mobility correction in
(cid:16) √ (cid:17) eachdirectioncanbesplitintoacontribution∆µbdueto
µu0(ω)=µ0 e−Rλ −i−1 (3) bending resistance and a contribution ∆µs due to shear
resistance and area dilatation. The final results are con-
with λ2 =ρω/η is the correction due to fluid inertia [44] veniently expressed in terms of the dimensionless num-
and ∆µ(ω,z ) is the correction due to the elastic mem- bers:
0
brane at distance z . In order to derive the mobility
0 12z ηω
corrections, we employ the commonly used approxima- β = 0 ,
κ +κ
tion of a small particle (R/z (cid:28) 1). Using numerical s a
0
simulations of a truly extended particle, we will show (cid:18)4ηω(cid:19)1/3
β =2z ,
below that this approximation is surprisingly good even b 0 κ
b
for R/z =0.65. The problem is thus equivalent to solv-
0 (cid:18)ρω(cid:19)1/2
ingtheunsteadyStokesequationswithanarbitrarytime
σ =z , (7)
dependent point force F located at r 0 η
0
∂v where β captures the effect of shear resistance and area
−ρ +η∇2v−∇p+Fδ(r−r )=0
∂t 0 dilatation, β the effect of bending resistance and σ the
b
∇·v =0 (4) effect of fluid inertia on the mobility corrections.
The mobility corrections are
with the fluid velocity v, the pressure p and the point
force position r0. The elastic membrane is located at ∆µ(cid:107),s 3i R (cid:90) ∞ s3(re−r−se−s)2
= ds
z = 0, has infinite extent in x and y directions and is µ σ2z 4(r−s)s2−βσ2
0 0 0
surrounded by fluid on both sides. Following the usual
3i R (cid:90) ∞ s3e−2r
approximationofsmalldeformations,weimposethetrac- + ds, (8)
tionjumpatz =0whichfollowsfromtheSkalak[45]and 4 z0 0 r(cid:0)1+2Cβr−is2(cid:1)
Helfrich[46]lawsfortheshearandbendingresistanceas ∆µ 3i R (cid:90) ∞ 4rs7(e−r−e−s)2
(cid:107),b
detailed in Appendix A = ds, (9)
µ σ2z 16s5(r−s)−rβ3σ2
0 0 0 b
κ
∆fx =− s (2(1+C)u +u +(1+2C)u ) ∆µ 6i R (cid:90) ∞ s5(e−s−e−r)2
3 x,xx x,yy y,xy ⊥,s = ds, (10)
∆fy =−κ3s (uy,xx+2(1+C)uy,yy+(1+2C)ux,xy) ∆µµ⊥0,b = σ6i2zR0 (cid:90)0∞ 4(r4s−7(ss)es−2r−−σr2eβ−s)2 ds,(11)
∆fz =κb(uz,xxxx+2uz,xxyy+uz,yyyy) (5) µ0 σ2z0 0 r(16s5(r−s)−rβb3σ2)
4
√
with r = s2+iσ2. The integrals are well-behaved and ation treated in figure 1, the contribution of fluid inertia
thus amenable to straightforward numerical integration. iscompletelynegligibleinthefrequencyrangethatisaf-
The effect of inertia on the diffusion has recently been fected by membrane elasticity which is the focus of this
investigated in bulk systems [47–51]. However, as shown work.
in the Supporting Information [52], for the realistic situ- In the following, we will thus consider the case σ =0,
for which an analytical solution is possible:
∆µ 3 R(cid:32) 5 β2 3iβ (cid:18) β2 iβ (cid:18) β2(cid:19)(cid:19) (cid:33)
(cid:107),s = − + − +iβ(1+C)eiβ(1+C)E (iβ(1+C))+ − + 1− eiβE (iβ) ,(12)
µ 8z 4 8 8 1 2 2 4 1
0 0
∆µµ(cid:107),b = 634zR (cid:18)−2+ iβ3b3 (cid:0)φ++e−iβbE1(−iβb)(cid:1)(cid:19), (13)
0 0
∆µ⊥,s = 9i R 1 (cid:0)1−4eiβE (iβ)(cid:1), (14)
µ 16z β 5
0 0
∆µ 3iβ R(cid:32)(cid:18)β2 iβ 1(cid:19) √3 5i (cid:18)β2 iβ 1(cid:19)(cid:33)
µ⊥,b = 8bz 12b + 6b + 6 φ++ 6 (βb+i)φ−+ 2β +e−iβbE1(−iβb) 12b − 3b − 3 , (15)
0 0 b
with and to assess the accuracy of the point-particle approx-
imation for finite-radius particles. BIMs are a standard
φ± =e−izbE1(−izb)±e−izbE1(−izb) (16) method for solving the steady Stokes equations [1] in-
cluding elastic surfaces [56]. Some details on our imple-
where zb = jβb and j = e2iπ/3. Bar denotes complex mentationaregivenintheSI.Comparedwithmostother
conjugate. En denotes the exponential integral En(x) = flowsolvers, BIMshavetheadvantagethattheyareable
(cid:82)∞e−xt/tndt [53]. to treat a truly inifinite fluid domain thus excluding ar-
1
From the frequency-dependent mobilities the mean- tifacts due to periodic replications of the system.
square displacement in a thermally fluctuating system
We simulate a spherical particle driven by an oscil-
can be computed using the fluctuation-dissipation theo-
lating force with frequency ω. By recording the instan-
rem with the velocity autocorrelation function φ (t) as
v taneous particle velocity, the mobility correction ∆µ(ω)
an intermediate step [54] as detailed in Appendix C
can be obtained from the amplitude ratio and the phase
k T (cid:90) ∞ shift between force and velocity as illustrated in the SI.
φ (t)= B µ(ω)eiωtdω (17)
v 2π In Fig. 4 we compare our theoretical prediction to the
−∞
result of BIM simulations with R/z = 0.1 and find ex-
(cid:90) t 0
(cid:10)x(t)2(cid:11)=2 (t−s)φ (s)ds. (18) cellent agreement. Splitting the mobility correction into
v
thecontributionsduetoshear/arearesistance(greenline
0
in Fig. 4) on the one hand and bending resistance (red
Using the mobilities from Eqs. (12) - (15), the MSD can
line) on the other, we find that bending resistance man-
beanalyticallycomputedandtheresultingequationsare
ifests itself at significantly lower frequencies than shear
given in Appendix C. In order to compute the MSDs
resistance. Asmightintuitivelybeexpected, theparallel
shown in figure 1 mobilities are calculated using the ini-
mobility is mainly determined by shear resistance, while
tialparticle-membranedistancez ,whichisequivalentto
0 for the perpendicular mobility bending resistance domi-
assuming a not too large deviation of the particle from
nates. Yet, we note that for both directions, shear/area
its initial position.
resistance and bending resistance are important. This
Similarly, the power spectral densities of the position
becomes apparent especially at low frequencies: neither
fluctuationsasshowninfigure3canbecalculatedas[12]
shear/area resistance nor bending resistance alone are
abletorecoverthehard-walllimit. AsshownintheSI,a
2k TRe(cid:2)µ(ω)−1(cid:3)
S(ω)= B . (19) similar effect appears in the limit of infinitely stiff mem-
(ωRe[µ(ω)−1])2+(ωIm[µ(ω)−1]+K)2 branes: only if shear and bending stiffness both tend to
infinity does one recover the hard-wall limit.
Finally,weinvestigatethevalidityofthepoint-particle
IV. MOBILITY SIMULATIONS approximation for particles close to the interface. For
this,weusetheparametersasinFig. 1. EvenforR/z =
0
Weuseboundary-integral(BIM)simulationstoobtain 0.65 the agreement is still surprisingly good as shown in
a direct validation of the frequency-dependent mobilities Fig. 5.
5
V. CONCLUSION
We have presented a fully analytical theory for the
thermal diffusion of a small spherical particle in close
vicinity to an elastic cell membrane. The frequency-
dependent particle mobilities predicted by the theory
are in excellent agreement with boundary-integral sim-
ulations, even for surprisingly large particles where the
point-forceapproximationmadeinthetheoryisnolonger
strictly valid. Independent of the membrane properties,
themean-squaredisplacementisshowntobebulk-likeat
short and hard-wall-like at very long times. In between,
however,thereexistsasignificanttimespanduringwhich
the particle shows subdiffusion with exponents as low as
0.87. For membrane parameters corresponding to a typ-
ical red blood cell the subdiffusive regime extends up to
and beyond 10ms and may thus be of possible physio-
logical significance, e.g., for the uptake of drug carriers
or viruses by a living cell. Our results can be directly
FIG. 4. The complex mobility of a spherical particle driven verified experimentally by comparing the power-spectral
by a sinusoidal force with frequency ω situated a distance z0 densities of the position fluctuations in Fig. 3.
abovethemembrane. TheoreticalpredictionsfromEqs.(12)-
In living cells the membrane elastic properties depend
(15) are shown as black dashed lines (real part) and black
onthelocalcholesterollevel[57]whichcanleadtolocal-
solidlines(imaginarypart)andcomparedtoBIMsimulations
ized patches of varying stiffness. According to our calcu-
shownascircles(realpart)andsquares(imaginarypart). The
lations,adjustingtheshear/bendingrigiditywouldallow
green and red lines show the contributions due to shear and
the cell to specifically influence the endocytosis proba-
bendingresistance,respectively. ForR/z =0.1(withC =1,
0
κ R2/κ =2)theagreementbetweentheoryandsimulations bility: An enhanced bending stiffness combined with re-
s b
is excellent. For very low frequencies the hard wall behavior duced shear elasticity would reduce perpendicular diffu-
is obtained (blue dashed line). sion–keepingtheapproachingparticleclosetothemem-
brane for a longer time – and at the same time enhance
parallel diffusion – allowing the particle to survey more
quicklythecellsurfaceforfavorablebiochemicalbinding
sites.
ACKNOWLEDGMENTS
The authors gratefully acknowledge funding from the
Volkswagen Foundation and the KONWIHR network
as well as computing time granted by the Leibniz-
Rechenzentrum on SuperMUC.
Appendix A: MEMBRANE MECHANICS
In this appendix, we give the derivation of the lin-
earized tangential and normal traction jumps as stated
in Eq. (5) of the main text. Initially, the interface is
described by the infinite plane z = 0. Let the position
vector of a material point before deformation be A, and
a after deformation. In the undeformed state, we have
FIG.5. Therealpart(dashedlines)andtheimaginarypart A(x,y) = xex + yey, where ei, with i ∈ {x,y,z} are
(solid lines) of the complex mobility for a particle moving the Cartesian base vectors. Hereafter, we shall reserve
parallel (a) or perpendicular (b) to a realistically modeled the capital roman letters for the undeformed state. The
red blood cell membrane with parameters corresponding to membrane can be defined using two covariant base vec-
Fig. 1. Even for R/z0 = 0.65 as used here the agreement is tors a and a , together with the normal vector n. a
1 2 1
still good.
and a are the local non-unit tangent vectors to coordi-
2
nate lines. In the Cartesian coordinate system, a =a
1 ,x
6
√
anda =a ,wherethecommadenotesaspatialderiva- where J := 1+I ≈ 1+e is the Jacobian determi-
2 ,y s 2
tive. The unit normal vector to the interface reads nant, representing the ratio between the deformed and
undeformed local surface area.
a ×a
n= 1 2 . (A1) Severalmodelshavebeenproposedinordertodescribe
|a ×a |
1 2 the mechanics of elastic membranes. The neo-Hookean
model is characterized by a single parameter containing
Itcanbeseenthatthecovariantbasevectorsintheunde-
themembraneelasticshearandareadilatationmodulus,
formedstateareidenticaltothoseoftheCartesianbase.
whiletheSkalakmodel[45]usestwoseparateparameters
Thedisplacementvectorofapointonthemembranecan
forshearandareadilatationresistance,respectively. The
be written as
strain energy in the Skalak model reads [61]
u=a−A=u e +u e +u e . (A2)
x x y y z z WSK = κs (cid:0)(I2+2I −2I )+CI2(cid:1), (A10)
12 1 1 2 2
The covariant base vectors are therefore
where C = κ /κ is the ratio between area dilation and
a =(1+u )e +u e +u e , (A3) a s
1 x,x x y,x y z,x z
shear modulus. By taking C =1, the Skalak model pre-
a =u e +(1+u )e +u e , (A4)
2 x,y x y,y y z,y z dictsthesamebehaviorastheneo-Hookeanforsmallde-
formations [18]. The calculations yield to the first order
and the linearized normal vector reads
a stress tensor in the form of
n≈−uz,xex−uz,yey+ez. (A5) 2κ (cid:18)u +Ce (cid:15) (cid:19)
ταβ ≈ s x,x . (A11)
The components of the metric tensor in the deformed 3 (cid:15) uy,y+Ce
statearedefinedbytheinnerproducta =a .a . Note
αβ α β
that A is then nothing but the second order identity
αβ
tensor δ . From Eqs. (A3) and (A4), a can straight- b. Bending resistance
αβ αβ
forwardly be computed. The contravariant tensor (con-
jugate metric) is the inverse of the covariant tensor. We Undertheactionofanexternalload,theinitiallyplane
directly have to the first order membrane bends. For small membrane curvatures, the
bending moment M can be related to the curvature ten-
(cid:18) (cid:19)
aαβ ≈ 1−2ux,x −2(cid:15) , (A6) sor via the linear isotropic model [58, 62]
−2(cid:15) 1−2u
y,y
Mβ =−κ (bβ −Bβ), (A12)
α b α α
where 2(cid:15) = u +u is the engineering shear strain.
x,y y,x
In the following, we will first derive the in-plane stress where κ is the bending modulus, having the dimension
b
tensor. A resistance to bending will be added indepen- ofenergy. Herebα isthemixedversionofthesecondfun-
β
dentlybyassumingalinearisotropicmodelequivalentto damental form which follows from the curvature tensor
the Helfrich model for small deformations [58]. (second fundamental form)
b =n.a for α,β ∈{1,2} (A13)
αβ α,β
a. In-plane stress tensor
viabβ =b aδβ ≈u . Asthesurfacereferenceisaflat
α αδ z,αβ
membrane, Bβ therefore vanishes. The bending moment
Here we use the Einstein summation convention, in α
reads
which a covariant index followed by the identical con-
travariantindex(orviceversa)isimplicitlysummedover
Mβ ≈−κ u . (A14)
the index. The two invariants of the transformation are α b z,αβ
given by Green and Adkins [59]
The surface transverse shear vector Q is obtained from
I =Aαβa −2, (A7) a local torque balance with the exerted moment by [62]
1 αβ
I2 =detAαβdetaαβ −1, (A8) ∇αMαβ −Qβ =0, (A15)
where Aαβ = δ , is the contravariant metric tensor of
αβ where ∇ is the covariant derivative defined for a con-
the undeformed state. The two invariants are found α
travariant tensor Mαβ by
to be equal and they are given by I = I = 2e =
1 2
2(ux,x+uy,y), where e denotes the dilatation. The con- ∇ Mαβ =∂ Mαβ +Γα Mηβ +Γβ Mαη, (A16)
travariantcomponentsofthestresstensorταβ arerelated λ λ λη λη
tothestrainenergyW(I ,I )viaaconstitutivelaw. We
1 2 whereΓλ aretheChristoffelsymbolsofthesecondkind,
have [18] αβ
defined by Γλ = a .aλ, and aλ are the contravariant
αβ α,β
2 ∂W ∂W basis vectors, which are related to those of the covariant
ταβ = Aαβ +2J aαβ, (A9)
Js ∂I1 s∂I2 basisviathecontravariantmetrictensorbyaα =aαβaβ.
7
To first order, only the partial derivative in Eq. (A16) where q := |q|. Note that the inverse transformation
remains. is given also by Eq. (B2). Since the membrane shape
The raising and lowering indices operation on the sec- depends on the history of the particle motion we also
ond order tensor M implies that Mαβ = aαγaβδM , perform a Fourier analysis in time which for a function
γδ
which, to the first order, is the same as Mβ given by Eq. f(t) is
α
(A14). The contravariant component of the transverse
(cid:90)
shear vector is therefore F{f(t)}=f(ω)= f(t)e−iωtdt. (B3)
R
Qβ ≈−κ u . (A17)
b z,αβα In the following, the Fourier-transformed function pair
f(t) and f(ω) are distinguished only by their argument
whilethetildeisreservedtodenotethespatial2DFourier
c. Equilibrium Equation
transforms. The unsteady Stokes equations (4) thus be-
come
The membrane equilibrium condition including both
the shear and the bending forces reads [62] −(iρω+ηq2)v˜ +ηv˜ −iqp˜+F δ(z−z )=0(B4)
l l,zz l 0
∇αταβ −bβαQα =−∆fβ, (A18) −(iρω+ηq2)v˜t+ηv˜t,zz+Ftδ(z−z0)=0(B5)
ταβbαβ +∇αQα =−∆fz, (A19) −(iρω+ηq2)v˜z+ηv˜z,zz−p˜,z+Fzδ(z−z0)=0(B6)
iqv˜ +v˜ =0(B7)
where ∆fβ, with β ∈ {x,y} is the tangential traction l z,z
jump at the elastic wall, and ∆fz is the normal traction The pressure in Eq. (B4) can be eliminated using Eq.
jump. The second term on the left-hand side (LHS) of (B6). Since the continuity equation (B7) gives a direct
Eq. (A18) is irrelevant in the first order approximation. relationbetweenthecomponentsv˜ andv˜,thefollowing
l z
The same is true for the first term on the LHS of Eq. fourth-order differential equation for v is obtained
z
(A19).
Finally, the linearized traction jump across the mem- v˜z,zzzz−(2q2+iλ2)v˜z,zz+q2(q2+iλ2)v˜z =
brane is q2 iqF (B8)
F δ(z−z )+ lδ(cid:48)(z−z ),
κs(cid:0)∆ u +(1+2C)e (cid:1)=−∆fβ, η z 0 η 0
3 (cid:107) β ,β
where δ(cid:48) is the derivative of the delta Dirac function,
κ ∆2u =+∆fz, (A20)
b (cid:107) z satisfying the property xδ(cid:48)(x) = −δ(x) for a real x, and
where ∆ f = f + f is the horizontal Laplace- λ2 =ρω/η.
(cid:107) ,xx ,yy
Beltramiofagivenfunctionf. Eqs.(A20)areequivalent
to Eqs. (5) of the main text.
2. Boundary conditions
Appendix B: DERIVATION OF PARTICLE a. Velocity boundary conditions
MOBILITIES
At the interface z = 0, the velocity components are
1. Hydrodynamic equations in Fourier space continuous
[v˜ ]=0, (B9)
We start by transforming Eqs. (4) of the main text to α
Fourier space. The spatial 2D Fourier transform for a
where α ∈ {l,t,z} and [f] = f(z = 0+)−f(z = 0−)
given function f is defined as
denotes the jump of a quantity f across the interface. In
(cid:90) addition, the no-slip condition Eq. (6) gives
F{f(ρ)}=f˜(q)= f(ρ)e−iq.ρd2ρ, (B1)
R2 v˜α(q,z =0,ω)=iωu˜α(q,ω). (B10)
whereρ=(x,y)istheprojectionofthepositionvectorr
onto the horizontal plane, and q =(q ,q ) is the Fourier
x y b. Tangential stress jump
transformvariable. SimilarlyasinBickel[28],allthevec-
torfieldsaresubsequentlydecomposedintolongitudinal,
The presence of the membrane leads to elastic stresses
transversal and normal components. For a given quan-
tity Q˜, whose components are (Q˜ ,Q˜ ) in the Cartesian which,inequilibrium,arebalancedbyajumpinthefluid
x y stress across the membrane:
coordinate base, its components in the new orthogonal
base (Q˜l,Q˜t) are given by the following transformation [σ ]=[η(v +v )]=∆fα, (B11)
zα z,α α,z
(cid:18)Q˜ (cid:19) 1(cid:18)q q (cid:19)(cid:18) Q˜ (cid:19) where α∈{x,y}. The tangential traction jump ∆fα for
x = x y l , (B2)
Q˜y q qy −qx Q˜t an elastic membrane experiencing a small deformation is
8
given by Eq. (5). We mention that only the resistance for α,β ∈{l,t,z}. For computing the particle mobilities
to shear and area dilatation is relevant to the first order the relevant quantities are the diagonal components G˜ ,
tt
approximation for the tangential traction jump. G˜ , and G˜ which can be derived by solving first the
zz ll
Usingthetransformationsgivenby(B2)togetherwith independent Eq. (B5) for G˜ , then Eq. (B8) for G˜ and
tt zz
the no-slip condition Eq. (B10), we straightforwardly finallyobtainingG˜ fromsolvingEq. (B8)andemploying
ll
express the first and second derivatives of ux and uy in the incompressibility condition (B7) as detailed in the
our new orthogonal basis. After some algebra, the two following.
tangential conditions are
[v˜ ]=−iα q2v˜| , (B12)
t,z s t z=0 a. Transverse-transverse component
[iqv˜ +v˜ ]=−4iαq2v˜| , (B13)
z l,z l z=0
LetusdenotebyKtheprincipalsquarerootofq2+iλ2,
where
i.e.
αs =κs/3ηω (B14) (cid:115) (cid:115)
(cid:112) (cid:112)
q2+ q4+λ4 −q2+ q4+λ4
is a characteristic length for shear and K = +i . (B21)
2 2
α=α /B =(κ +κ )/6ηω (B15)
s s a Note that for the steady Stokes equations, λ = 0 and
therefore K = q. The general solution of Eq. (B5) for
with B =2/(1+C).
the transverse velocity component is
Eq. (B12) gives the jump condition at the interface
for the transverse velocity component v˜. Note that the 
t Ae−Kz for z >z ,
latter is independent of area-dilatation, whereas both κ  0
s v˜ = BeKz+Ce−Kz for 0<z <z , (B22)
and κ are involved in the longitudinal and the normal t 0
velocitaies. Eq. (B13) can be written by employing the DeKz for z <0.
incompressibility equation (B7) together with the conti-
The integration constants A-D are determined by the
nuity of the normal velocity across the interface as
boundaryconditions. v˜ iscontinuousatz =z ,whereas
t 0
[v˜ ]=−4iαq2v˜ | . (B16) thefirstderivativeisdiscontinuousduetothedeltaDirac
z,zz z,z z=0
function,
F
c. Normal stress jump v˜ | −v˜ | =− t. (B23)
t,z z=z0+ t,z z=z0− η
The normal-normal component of the jump in the
In order to evaluate the four constants, two additional
stress tensor reads
equations must be provided. By applying the continu-
ity of the transverse velocity component at the interface
[σ ]=[−p+2ηv ]=∆fz. (B17)
zz z,z together with the tangential traction jump given by Eq.
Only the bending effect is present in ∆fz to the first (B12), we find that the transverse-transverse component
of the Green function is given by
order, as it can be seen from Eq. (5). Using the in-
compressibility equation (B7) and the continuity of the 1 (cid:18) iα q2 (cid:19)
longitudinal velocity component across the interface, the G˜ = e−K|z−z0|+ s e−K(z+z0) ,
tt 2ηK 2K−iα q2
normal stress jump in Fourier space reads s
(B24)
[v˜ ]=4iα3q6v˜ | , (B18) for z ≥0 and by
z,zzz b z z=0
1 1
where G˜ = e−K(z0−z), (B25)
tt η2K−iα q2
(cid:114) κ s
αb = 3 4ηbω, (B19) for z ≤ 0. For the steady Stokes equations, the solution
reads
is a characteristic length for bending.
(cid:18) (cid:19)
1 iα q
G˜ = e−q|z−z0|+ s e−q(z+z0) , (B26)
tt 2ηq 2−iα q
s
3. Green functions
for z ≥0 and
The Green’s functions are tensorial quantities which 1 1
G˜ = e−q(z0−z), (B27)
describe the fluid velocity in direction α tt ηq2−iα q
s
v˜ =G˜ F˜ , (B20) for z ≤0.
α αβ β
9
b. Normal-normal component c. Longitudinal-longitudinal component
As we are interested here in G˜ we set F˜ = 0 in Eq. When the normal force F is set to zero in Eq. (B8),
zz l z
(B8). Thegeneralsolutionofthisfourthorderdifferential and only a tangential force F is applied, the derivative
l
equation is of the Dirac function imposes the discontinuity of the
second derivative at z =z , whereas the third derivative
 0
Ae−qz+Be−Kz for z >z , is continuous. We have
 0
v˜ = Ceqz+De−qz+EeKz+Fe−Kz for 0<z <z ,
z 0 iqF
Geqz+HeKz for z <0. v˜z,zz|z=z0+ −v˜z,zz|z=z0− = η l. (B33)
(B28)
At the singularity position, i.e. at z = z0, the velocity After solving Eq. (B8) for the normal velocity v˜z, the
v˜z and its first two derivatives are continuous. However, longitudinal velocity v˜l can directly be obtained thanks
the delta Dirac function imposes the discontinuity of the to the incompressibility equation (B7). We find that the
third derivative longitudinal-longitudinal component G˜ is
ll
v˜z,zzz|z=z0+ −v˜z,zzz|z=z0− = q2ηFz. (B29) G˜ll = 2η1ZS(cid:18)Ke−K|z−z0|−qe−q|z−z0|
2iαq2(KP −qQ)(cid:16) (cid:17)
At the membrane, v˜z and its first derivative are con- + qe−q|z|−Ke−K|z|
tinuous. However, shear and bending impose a discon- ZPQ(2iαq2−S)
tinuity in the second and third derivatives respectively 2iα3q6K(P −Q) (cid:16) (cid:17)(cid:19)
+sgn(z) b e−q|z|−e−K|z| .
(Eqs. (B16) and (B18)). The system can readily be ZPQ(2iα3q5−KS)
b
solved in order to determine the constants. The calcula- (B34)
tions are straightforward but lengthy and thus omitted
here. We find that the normal-normal component of the When the steady Stokes equations are considered, one
Green function is given in a compact form by simply gets
q (cid:18) 1 (cid:18)
G˜zz = 2µKZS Ke−q|z−z0|−qe−K|z−z0| G˜ll = 4ηq (1−q|z−z0|)e−q|z−z0|
2iαq3K(P −Q) (cid:16) (cid:17) (cid:18)iαq(1−qz )(1−qz) izz α3q5 (cid:19) (cid:19)
+sgn(z) e−q|z|−e−K|z| + 0 + 0 b e−q(z+z0) ,
ZPQ(2iαq2−S) 1−iαq 1−iα3q3
b
2iα3q5(qP −KQ) (cid:16) (cid:17)(cid:19) (B35)
+ b Ke−q|z|−qe−K|z| .
ZPQ(2iα3q5−KS)
b for z ≥0 and
(B30)
(cid:18)
1
Here P =eqz0, Q=eKz0, S =K+q and Z =K−q. G˜ll = 4ηq 1−q(z0−z)
For the steady Stokes equations, i.e. by taking the
iαq(1+qz)(1−qz ) izz α3q5 (cid:19)
limits when K →q and Q→P, one gets + 0 + 0 b e−q(z0−z),
1−iαq 1−iα3q3
b
(cid:18)
1 (B36)
G˜ = (1+q|z−z |)e−q|z−z0|
zz 4ηq 0
for z ≤0.
(cid:18)iαzz q3 iα3q3(1+qz )(1+qz)(cid:19) (cid:19)
+ 0 + b 0 e−q(z+z0) ,
1−iαq 1−iα3q3
b
(B31) 4. Particle mobilities
for z ≥0 and
We now obtain the mobility corrections defined in
(cid:18) Eq. (1) and given specifically in Eqs. (8)-(11) (includ-
1
G˜ = 1+q(z −z) ing the inertial term) and Eqs. (12)-(15) (without fluid
zz 4ηq 0
inertia) of the main text. For this, using Eq. (B2) on
iαzz q3 iα3q3(1+qz )(1−qz)(cid:19) Eq. (B20), one derives the transformation of the tenso-
+ 0 + b 0 e−q(z0−z),
1−iαq 1−iα3q3 rial Green’s functions back to Cartesian directions:
b
(B32) q2 q2
G˜ (q,z,ω)= yG˜ (q,z,ω)+ xG˜ (q,z,ω),(B37)
xx q2 tt q2 ll
for z ≤ 0. Note that both the shear and the bending
moduli are involved in the normal-normal component of q2 q2
G˜ (q,z,ω)= xG˜ (q,z,ω)+ yG˜ (q,z,ω).(B38)
the Green functions. yy q2 tt q2 ll
10
We then subtract the infinite space Green’s functions a. Parallel mobility
in the Fourier domain which can be obtained via the
above derivation with the membrane moduli set to zero, Shear effect. Considering only the part due to shear
i.e. resistance in Eqs. (B35) and (B26) and using Eq. (B40)
with (B39), we find after passing to polar coordinates:
∆G˜(0)(q,z,ω)=G˜ (q,z,ω)−G˜ (q,z,ω)| ,
γγ γγ γγ α,αb=0
where γ ∈{x,y,z}. This defines the wave-vector d(eBp3e9n)- ∆G˜(cid:107),s(q,φ,ω)|z=z0 = iz0e2−η2qz0(cid:18)T ωsin−2iφqz
dent corrections s 0 (C1)
(1−qz )2cos2φ(cid:19)
+ 0 ,
∆G˜ (q,z,ω)=G˜ (q,z,ω)−G˜(0)(q,z,ω), BT ω−2iqz
(cid:107) xx xx s 0
=G˜ (q,z,ω)−G˜(0)(q,z,ω),
yy yy where T = 6z η/κ is a characteristic time for shear.
s 0 s
∆G˜ (q,z,ω)=G˜ (q,z,ω)−G˜(0)(q,z,ω). (B40) The temporal inverse Fourier transform reads
⊥ zz zz
toDobuteatino tthheeflpuoiindtv-pealorctiitcyleaatptphreoxpiamrtaitciloenpiotsiistiosunffiwchieicnht ∆G˜(cid:107),s(q,φ,t)|z=z0 =−z0e−22ηqTz0θ(t)(cid:18)e−qTz0st sin2φ
s
(C2)
ifsulelqiunavletrosethFeouvreileorcittryanofsftohrempaorfttihcleeGitrseeelfn.’sInfsutnecatdioonfsthtoe + (1−qz0)2e−B2qTzs0t cos2φ(cid:19).
B
real space coordinates (ρ, z), we can thus limit ourselves
to evaluate the inverse Fourier transform of Eqs. (B40)
An exact expression of the time dependent mobility
at(ρ=0, z =z ). Bypassagetopolarcoordinatesq =
0 x correction due to shear in the parallel case can then be
qcosφ and q = qsinφ, the correction in the particle
y obtained by spatial inverse Fourier transform
mobility to the first order of R/z can be obtained
0
∆µ (τ) 3 R θ(τ) N (τ)
1 (cid:90) 2π(cid:90) ∞ (cid:107),s =− B (C3)
∆µ (ω)= ∆G˜ (q,φ,z =z ,ω)qdqdφ µ 32z T (2+τ)2(τ +B)4
(cid:107) (2π)2 (cid:107) 0 0 0 s
0 0
1 (cid:90) ∞ where τ = t/T , and again B = 2/(1+C). θ(t) denotes
∆µ (ω)= ∆G˜ (q,z =z ,ω)qdq, s
⊥ 2π ⊥ 0 the Heaviside step function, with θ(0)=1/2 and
0
(B41)
N (τ)=4B3(1+2B)+36B3τ +B(B2+48B+8)τ2
B
which directly lead to Eqs. (8)-(11) of the main text. +40Bτ3+2(B+4)τ4.
A similar procedure can be followed for the steady case (C4)
where the fluid inertia is neglected leading to Eqs. (12)-
(15). Bending effect. Considering the part due to bending
resistance we obtain
cos2φ iq4z5
Appendix C: COMPUTING ∆G˜ (q,φ,ω)| = 0 e−2qz0, (C5)
MEAN-SQUARE-DISPLACEMENTS FROM (cid:107),b z=z0 4η Tbω−iq3z03
PARTICLE MOBILITIES
togiveafterapplyingthetemporalinverseFouriertrans-
form
1. Time dependent mobility corrections
A crucial step in order to compute the mean-square- ∆G˜(cid:107),b(q,φ,t)|z=z0 =−q4z05θ4(ηtT)cos2φe−2qz0−tqT3bz03.
b
displacements as described in the following section is (C6)
to transform the frequency-dependent particle mobilities The time dependent mobility can immediately be ob-
back to the time domain. As shown in the Supporting tained after applying the inverse Fourier transform
Information,theinertialcontributiontothemobilitycor-
rfoercetiroenstirsicnteogulirgsiebllveesfofrrormeanliostwicosncetonatrhieoscaasnedσw=e0th.eFroer- ∆µµ(cid:107),b(t) =−38za θT(t)(cid:90) ∞u5e−2u−Ttbu3du, (C7)
0 0 b 0
thesakeofsimplicity,wedonotstartfromthereal-space
particlemobilitiesgiveninEqs.(12)-(15),butinsteadde- where T = 4ηz3/κ . The presence of u3 in the ex-
b 0 b
part from the wave-vector-dependent Green’s functions ponential argument makes the analytical evaluation of
in Eq. (B40) to perform first an inverse Fourier trans- this integral impossible. To overcome this difficulty, we
form in time followed by an inverse Fourier transform evaluate the integral numerically and fit the result (as a
in space. Note that the inverse order is possible for the function of t) with an analytical empirical form which is
shear-related part, but the calculations are much more necessarytoproceedfurther. Thisprocedureisknownas
complicated. theBatchelorparametrization[20]. Itcanbeshownthat