March 2012
EPL, 97 (2012) 68007
doi: 10.1209/0295-5075/97/68007

www.epljournal.org

Anomalous lateral diﬀusion in a viscous membrane surroundedby viscoelastic media
S. Komura1(a) , S. Ramachandran2 and K. Seki3
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University
Tokyo 192-0397, Japan2
Physique des Polymères, Université Libre de Bruxelles - Campus Plaine, CP 223, B-1050 Brussels, Belgium, EU
3
National Institute of Advanced Industrial Science and Technology (AIST) - Ibaraki 305-8565, Japan
received 4 December 2011; accepted in ﬁnal form 17 February 2012published online 21 March 2012
PACS

83.60.Bc – Linear viscoelasticity
87.16.D- – Membranes, bilayers, and vesicles
PACS 87.16.dp – Transport, including channels, pores, and lateral diﬀusion
PACS

Abstract – We investigate the lateral dynamics in a purely viscous lipid membrane surroundedby viscoelastic media such as polymeric solutions. We ﬁrst obtain the generalized frequencydependent mobility tensor and focus on the case in which the solvent is sandwiched by hard walls.
Due to the viscoelasticity of the solvent, the mean square displacement of a disk embedded in
the membrane exhibits an anomalous diﬀusion. A useful relation which connects the mean squaredisplacement and the solvent modulus is provided. We also calculate the cross-correlation of theparticle displacements which can be applied for two-particle tracking experiments.
c EPLA, 2012
Copyright Introduction. – Biomembranes are thin twodimensional (2D) ﬂuids which separate inner and outerenvironments of organelles in cells. The ﬂuidity ofbiomembranes is guaranteed mainly due to the lipid
molecules which are in the liquid crystalline state at
physiological temperatures. Proteins and other molecules
embedded in biomembranes undergo lateral diﬀusion
which plays an important role for biological functions [1].
It should be noted, however, that biomembranes are notisolated 2D systems, but are coupled to the surrounding
polar solvent such as water. Indeed the presence ofwater is essential for amphiphilic lipid molecules tospontaneously form bilayers by self-assembly.
In the last few decades, it has been recognized thatthe outer solvent has a signiﬁcant eﬀect on the membrane
dynamics which takes place in 2D. Saﬀman and Delbrück
considered the Brownian motion of a 2D disk conﬁned
in a ﬂuid membrane [2,3]. In their hydrodynamic model,
the transfer of membrane momentum to the outer threedimensional (3D) ﬂuid was taken into account through theboundary conditions at the membrane surfaces. The translational diﬀusion coeﬃcient in the weak-coupling (smalldisk) limit was shown to exhibit a logarithmic dependence
on the disk size. This dependence has been repeatedly(a) E-mail: komura@tmu.ac.jp

tested for various lipid molecules and proteins [4]. In thestrong-coupling (large-disk) limit [5], on the other hand,the diﬀusion coeﬃcient is inversely proportional to thedisk size showing the analogy to the 3D Stokes-Einstein
relation [6]. Such a 3D-like behavior in 2D membrane
is caused by the back ﬂow eﬀect mediated by the bulk
solvent.
In this letter, we discuss the dynamics and responses ofmembranes when their surrounding solvent is viscoelastic
rather than purely viscous. This is a common situation inall eukaryotic cells whose cytoplasm is a soup of proteins
and organelles, including a thick sub-membrane layer ofactin-meshwork forming a part of the cell cytoskeleton [1].
The extra-cellular ﬂuid can also be viscoelastic because it
is ﬁlled with extracellular matrix or hyaluronic acid gel.
In addition to the surrounding solvents, lipid membranes
themselves can be viscoelastic [7]. Although it turned outthat membranes are purely viscous in the latest report [7],their experimental technique using particle tracking
microrheology provides us with a new clue to investigate
the dynamical responses of lipid bilayers coupled to thesurrounding environments under controlled conditions.
Recently, viscoelasticity of phospholipid Langmuir monolayers in a liquid-condensed phase was measured using
active microrheology [8]. Being motivated by these works,
we discuss the mean square displacement (MSD) of a

68007-p1

S. Komura et al.
We also work in the low-Reynolds-number regime so thatthe inertial eﬀects can be neglected. Then the Stokes
equation for the ﬂuid membrane is given byρ

Fig. 1: Schematic picture showing a planar viscous membraneat z = 0 with 2D constant viscosity η. It is sandwiched by aviscoelastic solvent of 3D frequency-dependent viscosity ηs [ω].
Two impenetrable walls are located at z = ±h bounding thesolvent.

circular disk embedded in a 2D sheet by taking into
account the viscoelasticity of the surrounding media. Thisquantity is experimentally measurable in single-particle
tracking microrheology [9,10]. We further calculate thetwo-particle MSD which is useful for two-point microrheology experiments [11]. For both cases, we show thatthe viscoelasticity of the surrounding media leads to ananomalous diﬀusion in the 2D viscous membrane.
Recently, Granek discussed the dynamics of an undulating bilayer membrane surrounded by viscoelastic
media [12]. He calculated the frequency-dependent transverse (out-of-plane) MSD of a membrane segment and thelinear response to external forces. In our theory, we treat
the membrane as an inﬁnitely large ﬂat sheet with a 2D
viscosity η, and consider its lateral dynamics. A similar
problem was considered in refs. [13,14] in which theauthors have taken into account the viscoelasticity of themembrane itself because they were originally motivatedby the earlier experiment of ref. [7]. A more general theory
for the dynamics of viscoelastic membranes was given by
Levine and MacKintosh [15]. In these works, however,the surrounding solvent is assumed to be purely viscous.
Similar to Granek’s work, the main purpose of our work
is to point out the importance of the viscoelasticity ofthe bulk solvent on the membrane lateral dynamics suchas diﬀusion or linear viscoelastic response. To make thispoint clear enough, we intentionally treat the membrane
as a purely viscous 2D ﬂuid.
Hydrodynamic model. – We ﬁrst establish thegoverning hydrodynamic equations for our model. As
shown in ﬁg. 1, the ﬂuid membrane, ﬁxed in the xy-planeat z = 0, is embedded in a bulk solvent that is further
bounded by hard walls at z = ±h. Let v(r, t) be the 2D
velocity of the membrane ﬂuid at position r = (x, y) and attime t. We assume that the membrane is incompressible,∇ · v = 0.

∂v
= η∇2 v − ∇p + fs + F,∂t(2)

where ρ is the membrane 2D density, η the membrane 2D
constant viscosity, p the in-plane pressure, fs the force dueto the solvent described later, and F is any other force
acting on the membrane. Notice that ∇ stands for the 2D
diﬀerential operator.
Next we give the equations for the surrounding
viscoelastic solvent. The upper (z > 0) and the lower(z < 0) regions of the solvent are denoted by “+” and “−”,respectively. The velocities and pressures in these regions
are written as v± (r, z, t) and p± (r, z, t), respectively.
Similar to the ﬂuid membrane, the solvent is also assumed
to be incompressible,
˜ · v± = 0.
∇

(3)

The Stokes equation for the viscoelastic solvent is writtenas∂v±
˜ · σ ± − ∇p
˜ ±,
=∇
(4)
ρs
∂t
where ρs is the solvent 3D density (assumed to be the
˜ indicates the 3D diﬀerential
same for both solvents), and ∇operator. Within the linear viscoelasticity approximation,
the stress tensor σ ± in the above equation is given by thefollowing constitutive relation [16]: t±dt ηs (t − t )D± (t ),(5)σ (t) = 2−∞

where ηs (t) is the time-dependent solvent viscosity(assumed to be the same for both solvents). The rate-of˜ ± + (∇v
˜ ± )T ]/2, where
strain tensor is given by D± = [∇v
the superscript “T” denotes the transpose. The surrounding viscoelastic solvent exerts force fs on the membrane
as considered in eq. (2). It is given by the projectionof (σ + − σ − )z=0 · êz on the xy-plane, where êz is the unit
vector along the z-axis. We employ the stick boundary
conditions at any time, i.e., the equalities of the velocities,
at z = 0 and z = ±h. When the solvent is purely viscous,
this model reduces to that considered by the present
authors [17].
It is convenient to perform the 2D Fourier transformin space and the Fourier-Laplace (or one-sided Fourier)transform in timefor any function f (r, t) as deﬁned by∞∞
f [k, ω] = −∞ d2 r 0 dt f (r, t) exp[−i(k · r + ωt)], wherek = (kx , ky ) is the 2D wave vector and ω the angular frequency. Assuming that ﬂuids are at rest at
t = 0, we calculate fs and obtain the membrane velocity as v[k, ω] = G[k, ω] · F[k, ω], where G[k, ω] is thefrequency-dependent mobility tensor. Following thesimilar procedure described in refs. [13,14], we obtain(1)
68007-p2Gαβ [k, ω] =

δαβ − kα kβ /k 2,
ηk 2 + 2ηs [ω]k  coth(k  h) + iωρ(6)Anomalous lateral diﬀusion in a viscous membrane surrounded by viscoelastic media
where k  = k(1 + iωρs /ηs [ω]k 2 )1/2 , k = |k| and α, β = x, y.
In the limit of h → ∞, which we call as the “freemembrane case”, the above mobility tensor reduces toGfree
αβ [k, ω] =

δαβ − kα kβ /k 2
, (7)
ηk 2 + 2ηs [ω]k 1 + iωρs /ηs [ω]k 2 + iωρwhere we have dropped the inertial term at the end. Thisapproximation is justiﬁed when G0 sα−2 /ρh  1, whichis always valid for s → 0 corresponding to the long-time
behavior.

Single-particle tracking. – For the description of
the Brownian motion of a circular disk conﬁned in awhich was given in ref. [13]. The opposite limit of h→0 membrane, we basically follow the formulation in ref. [26].
is called as the “conﬁned membrane case” and the Let a and m be the radius and the mass of the disk,
corresponding mobility tensor becomes
respectively. The eﬀective generalized Langevin equation
is written as [27]2δαβ − kα kβ /k
Gcon
,
(8)
 t
αβ [k, ω] =22
η(k + κ [ω])ddt λ(t − t )U (t ) + R(t),(11)
m∗ U (t) = −dt−∞
where we have introduced κ[ω] whose square is deﬁned aswhere U (t) is the velocity of the disk, and λ(t) is the time2ηs [ω] iωρ+.
(9) dependent drag coeﬃcient given below. The random forceκ2 [ω] =ηhηR(t) satisﬁes the ﬂuctuation dissipation theorem (FDT)In the limit of ω → 0, eq. (6) reduces to the static mobil- when averaged over the ensemble of molecular motions;
ity tensor obtained in ref. [17]. The free membrane case i.e., R(t) = 0 and R(t0 )R(t0 + t) = kB T λ(t), where kB
was originally considered by Saﬀman and Delbrück [2,3], is the Boltzmann constant and T the temperature. In
and the relevant length scale is the Saﬀman-Delbrück ref. [26], it was shown that the renormalized mass is given
length η/ηs beyond which the membrane feels the pres- by m∗ = m + πρa2 which takes into account the additional
ence of the outer solvent. On the other hand, the conﬁned inertia due to the drag from the ﬂuid membrane. Furthermembrane case corresponds to that of a supported more, the Laplace transform of λ(t) is calculated to bemembrane close to the substrate as considered by Evansκ̃(s)aK1 [κ̃(s)a]2π η̃s (s)a2
and Sackmann [18] and later generalized by us [19].
+ 4πη
,
(12)
λ̃(s) =
Here the corresponding hydrodynamic screening lengthhK0 [κ̃(s)a]is set by the geometric mean of the Saﬀman-Delbrück
where K0 [x] and K1 [x] are modiﬁed Bessel functions oflength η/ηs and the distance h between the membranethesecond kind, order zero and one, respectively.
and the wall, i.e., ηh/ηs [20]. Notice that the presence
Following
the standard procedure to obtain theof the second wall only doubles the screening eﬀect.
MSD
[27],
one
can relate its Laplace transform and λ̃(s)In the following, we shall mainly consider the conﬁned
throughmembrane case which allows us to treat most of the
4kB T
calculations analytically. This is mainly because κ[ω],(13)∆r̃2 (s) =s2 λ̃(s)does not depend on k. However, it should be notedthat the limiting expression of eq. (8) gives a reasonable where the number of degrees of freedom tracked in theapproximation to the full expression of eq. (6) even for MSD is chosen to be two. For general t, the application ofh ∼ η/ηs at least for a large enough moving object [20,21]. the inverse Laplace transform provides us with the timeConcerning the viscoelasticity of the surrounding dependent MSD,
solvent, we assume that its complex modulus obeys
 c+i∞
1
4kB T stthe power-law behavior such that Gs [ω] = G0 (iω)α with
e .
ds
(14)
∆r2 (t) =α < 1, as generally argued by Granek [12]. This behavior
2πi c−i∞s2 λ̃(s)is commonly observed for various polymeric solutions at
In order to demonstrate how to use the above relations,high frequencies. Examples are α = 1/2 and α = 2/3 forRouse and Zimm dynamics, respectively [22], α = 3/4 we consider here the limit of a large disk size a → ∞for semi-dilute solutions of semi-ﬂexible polymers such (κ̃(s)a  1) so that the drag coeﬃcient in eq. (12) is2
as actin ﬁlaments [23]. From the viewpoint of particle- dominated by the ﬁrst term, i.e., λ̃(s) ≈ 2π η̃s (s)a /h.
tracking microrheology experiment [24,25], it is more Then the Laplace transformed MSD in eq. (13) becomes
convenient ∞ to work in the Laplace domain deﬁned by
2kB T h2kB T hf˜(s) = 0 dt f (t)e−st . Since we have been working in=∆r̃2 (s) = 2 2
.
(15)
πa
s
η̃
(s)
πa2 sG̃s (s)sthe Fourier-Laplace domain, it is straightforward toconvert to the Laplace domain by substituting s = iω.
This is the equation which relates the observed 2D MSD
From G̃s (s) = G0 sα , the Laplace transform of the solvent
to the modulus of the surrounding bulk solvent (rather
α−1. Thenviscosity behaves as η̃s (s) = G̃s (s)/s = G0 s
than the membrane). In other words, we can extracteq. (9) simply becomes
the solvent 3D information by using the 2D information
α−1
α−1
due to the motion of a disk in the membrane. Once
ρs
2G
s
s
2G
0
0
+
≈
,
(10) G̃ (s) is obtained from the experiment, the frequencyκ̃2 (s) =s
ηhηηh
68007-p3

S. Komura et al.

2kB T htα ,∆r2 (t) = 2
πa G0 Γ[1 + α]

(16)

6

10

4

10

2

10

MSD

dependence of the storage and the loss moduli can be
deduced by identifying Gs [ω] = Gs [ω] + iGs [ω] = G̃s (s =iω). Notice that these two representations are equivalentbecause Gs [ω] and G s [ω] are related by the KramersKronig relation [24,25,27].
On the other hand, suppose the modulus of the bulk
solvent is known a priori to behave as G̃s (s) = G0 sαfrom independent experiments, the MSD can be simply
obtained from eqs. (14) and (15) as

0

10

-2

10
where Γ[x] is the gamma function. This calculation showsthat the viscoelasticity of the solvent results in a subd-4
iﬀusive time dependence of the MSD. Since α < 1, the10viscoelasticity slows down the normal diﬀusion process.
This is the main result of this letter. Compared with the-6
3D case [9,10], the above expression is unique because it
10 -6
-4
-2
0
2
4
6
8
2
2
is proportional to h/a . This 1/a -dependence arises from10 10 1010 10 10 10 10
the mass conservation in 2D rather than the momentum
conservation [17].
In the limit of a small disk size a → 0 (κ̃(s)a 1), Fig. 2: Scaled MSD as a function of scaled time t̄ when
the situation is more complicated. In this case, the drag α = 1/2. These dimensionless quantities are deﬁned in the text.
The dotted and dashed lines correspond to the asymptotic
coeﬃcient asymptotically behaves ast λ̃(s) ≈ 4πη ln2κ̃(s)a

expressions given by eqs. (16) and (18), respectively.

−1
−γ

,

(17)

where γ = 0.5772 · · · is Euler’s constant. By using eq. (10)for κ̃(s), a similar calculation yields
 2ηhtα−1kB T
2∆r (t) ≈t ln+ (α − 3)γ − α + 1 .
2πη
G0 a2
(18)
Since α < 1, this MSD grows like t ln(1/t). Such a logarithmic correction leads to a time-dependent diﬀusivity.
It should be noted, however, that this asymptotic expression is valid only when G0 a2 t1−α /ηh 1. We also mention
that all the above expressions recover our previous results
when the solvent is purely viscous, i.e., α → 1 [19].
In ﬁg. 2, we plot the result of numerical inverse
Laplace transform of eq. (14) with the full dragcoeﬃcient eq. (12) when α = 1/2, and compared itwith the asymptotic expressions; eqs. (16) and (18).
Here the dimensionless MSD and time are givenby (2πη/kB T )(ηh/2G0 a2 )1/(α−1) ∆r2 (t) and t̄ = (ηh/
2G0 a2 )1/(α−1) t, respectively. As mentioned above, eq. (18)
is valid only for short time (t̄ 1), while eq. (16) describes
the asymptotic long-time behavior (t̄  161/(1−α) ). Theintermediate time region is described by an apparent
power law t̄(1+α)/2 , which will be explained elsewhere.
Two-particle tracking. – So far, we have discussed
the motion of a single disk of ﬁnite radius a. As discussed
in ref. [25], there are several advantages to perform multiparticle microrheology. For example, long-time convective
drift can be automatically subtracted in this method
so that measurements of probe self-diﬀusivities become

possible over longer times. Multi-particle techniques canbe also used to investigate heterogeneous materials. Here
we discuss the eﬀects of the solvent viscosity on the crosscorrelation of two distinct probe positions, namely, twopoint microrheology [11]. We show below that the distance
between the two points essentially corresponds to the sizeof a disk in single-particle microrheology.
Consider a pair of point particles embedded in themembrane undergoing Brownian motion separated by
a 2D vector r. The quantity of interest is the crosscorrelation of the particle displacements ∆rα1 (0)∆rβ2 (t)r ,where ∆rαi is the displacement of the particle i (= 1, 2)
along the axis α (= x, y). We also deﬁne the x-axis to bealong the line connecting the two particles, i.e., r = rêx .
According to the FDT, this correlation function is related
to the coupling mobility Mαβ (t) in the Laplace domain
as [25]2kB T
M̃αβ (r, s),(19)∆r̃α1 ∆r̃β2 (s)r =s2
for suﬃciently large r. The above equation is the analog
of eq. (13) for the two-particle tracking. Since Mxy = 0 bysymmetry, we consider the longitudinal coupling mobility
ML = Mxx and the transverse one MT = Myy . Note thatthe coupling mobility is not the inverse of the coupling
resistance in the multi-particle case [25]. By utilizing theresults in refs. [17,28,29], one can obtain these mobilities
analytically.
First the Laplace transform of the longitudinal coupling
mobility turns out to be11
K1 [κ̃(s)r]−M̃L (r, s) =.
(20)
2πη (κ̃(s)r)2
κ̃(s)r

68007-p4

Anomalous lateral diﬀusion in a viscous membrane surrounded by viscoelastic media
In the limit of a large distance r → ∞ (κ̃(s)r  1), the anomalous diﬀusion reﬂecting the viscoelastic property
above expression can be approximated asof the bulk solvent. For single-particle microrheology
experiments, we presented a useful relation which connects
11.
(21) the MSD and the solvent modulus in the Laplace domainM̃L (r, s) ≈2πη (κ̃(s)r)2
when the size of the disk is large enough. We also obtained
the cross-correlation of the particle displacements which
As in the calculation of MSD of a single disk, we use
can be used for two-particle tracking experiments. Our
eq. (10) for κ̃(s) and perform the inverse Laplace transformtheory can be applied not only for lipid membranes butof eq. (19). Then we obtain
also for Langmuir monolayers.
α
It is worthwhile to point out the implicit assumptions
t
kB T h∆rx1 (0)∆rx2 (t)r ≈.
(22) which are used in the present theory [25]. First we have2
2πG0 Γ[1 + α] r
assumed that the system obeys the FDT which allows
2
The subdiﬀusive dependence on time and the 1/r - us to relate the thermal ﬂuctuations of the probe disk
dependence on distance is analogous to eq. (16), implying directly to the time-dependent drag coeﬃcient (see eq.
(13)) or the coupling mobility (see eq. (19)). However,that r corresponds to a.
In the limit of a small distance r → 0 (κ̃(s)r 1), on in non-equilibrium situations with membrane transport
proteins, for instance, the FDT can be violated and thethe other hand, eq. (20) asymptotically behaves asdeviation from the present theory may arise. Recently, it
 


2
1
1
was reported that an in vitro model system consistingM̃L (r, s) ≈
ln
−γ +
.
(23)
of a cross-linked actin network with embedded force4πηκ̃(s)r2
generating myosin motors strongly violates FDT [30]. The
Following the same process as above, the cross-correlation second crucial assumption is that the generalized dragfunction asymptotically behaves ascoeﬃcient or the coupling mobility are given precisely by their Newtonian analogs, but with the Newtonian viscosity
2ηhtα−1kB T
ηs replaced by the frequency-dependent complex viscosityt ln∆rx1 (0)∆rx2 (t)r ≈
4πη
G0 r2
ηs [ω] at all frequencies (see eq. (5)). This is not an obvious
assumption, but it has proved to be quite successful for+(α − 3)γ − α + 2 ,(24)
any probe motion in a simple linear viscoelastic material
under various conditions at least in 3D [25].
which is valid for G0 r2 t1−α /ηh 1. Notice that eq. (24)Some caution is required when applying our theory tois also analogous to eq. (18).
experiments if the viscoelastic solvent is, for instance, a
Next the transverse coupling mobility is given bysemi-dilute polymer solution. For single-particle tracking,
our continuum approach is valid for inclusion sizes much
1
1
K1 [κ̃(s)r]larger than the mesh size of the network. This can be−M̃T (r, s) =.
K0 [κ̃(s)r] +2πηκ̃(s)r(κ̃(s)r)2
relevant such as for a micron-size membrane domain and a(25) network correlation length of tens of nanometers. However
The large distance limit (r → ∞) and the small distance our theory may not be applicable for small membrane
limit (r → 0) of this expression areproteins which feel the solvent as purely viscous. For
two-particle tracking, the distance between the two point
11
M̃T (r, s) ≈ −,(26) particles (which can be small membrane proteins) should
22πη (κ̃(s)r)
be larger than the network mesh size. In order to obtainthe full time behavior of the particle motion, one should
and
 


2
1
1
use a viscoelastic modulus that is dependent both onM̃T (r, s) ≈
ln
−γ −
,
(27)
wave vector and frequency. On the other hand, any simple
4πη
κ̃(s)r2
ﬂuid can be eﬀectively viewed as viscoelastic at very highrespectively. Since these forms diﬀer from eqs. (21) andmolecular frequencies [31]. Our model can be also used to(23) only by a sign, we do not repeat here the same
study such a high-frequency dynamics by using a Maxwell
calculations. However, as far as the time dependence ofmodel for the solvent.
12
∆ry (0)∆ry (t)r is concerned, it is essentially given by eqs.
In the present work, we have mainly discussed the(22) and (24).
conﬁned membrane case in order to obtain analytical
Discussion. – In this letter, we have discussed the expressions. Unfortunately, a single analytic expression of
dynamics in a purely viscous lipid membrane surrounded the drag coeﬃcient for the whole range of the disk sizeby viscoelastic solvents such as polymeric solutions. Using is not known for the free membrane case. However, in thethe generalized frequency-dependent mobility tensor for limit of a large disk size a → ∞, the asymptotic expression
the conﬁned membrane case, we calculated the MSD of was obtained by Hughes et al. as λfree ≈ 16ηs a whicha disk embedded in the membrane and obtained some depends only on the solvent viscosity and is proportional
asymptotic expressions. The obtained MSD exhibits an to the size [5]. Assuming the replacement of the solvent
68007-p5

S. Komura et al.
viscosity with the time-dependent one as discussed above,
the MSD of the disk becomes∆r̃2 (s)free =

kB T
kB T
=
.
4as2 η̃s (s) 4asG̃s (s)(28)

When the modulus of the bulk solvent behaves as G̃s (s) =G0 sα as before, the MSD in the free membrane case isgiven bykB T
tα ,∆r2 (t)free =(29)
4aG0 Γ[1 + α]
which is again a subdiﬀusive behavior. In the opposite
limit of a → 0, the Laplace transformed drag coeﬃcient
can be written as [2,3] 
λ̃free (s) ≈ 4πη lnηη̃s (s)a



−1
−γ

.

(30)

It then follows that  α−1 

ηt
kB T
t ln
∆r (t)free =+ (α − 2)γ − α + 1 ,
πη
G0 a
(31)
for G0 at1−α /η 1. Although the above argument is
not rigorous, it essentially captures the anomalous
diﬀusion behavior in the free membrane case. For thetwo-particle tracking case, however, the corresponding
cross-correlation function can be obtained without any
ambiguity since both the longitudinal and transverse
coupling mobilities were analytically obtained for the freemembrane case [29].
Currently we are extending our model to the casein which the two solvents have asymmetric viscoelastic
properties or when the disk is actively driven. Additionaleﬀects such as the ﬁnite curvature of vesicles or the out-ofplane deformation of the membrane should also be taken
into account.
2∗∗∗
We thank C.-Y. D. Lu for useful discussions. This work
was supported by Grant-in-Aid for Scientiﬁc Research(grant No. 21540420) from the MEXT of Japan. SK
also acknowledges the support by the JSPS Core-toCore Program “International research network for nonequilibrium dynamics of soft matter”.
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