PHYSICAL REVIEW E 84, 021905 (2011)

Diffusion coefficient of an inclusion in a liquid membrane supported bya solvent of arbitrary thickness
Kazuhiko Seki,1 Sanoop Ramachandran,2 and Shigeyuki Komura2
1

National Institute of Advanced Industrial Science and Technology (AIST) AIST Tsukuba Central 5, Higashi 1-1-1, Tsukuba 305-8565, Japan2
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan(Received 18 May 2011; revised manuscript received 12 July 2011; published 4 August 2011)
The diffusion coefficient of an inclusion in a liquid membrane is investigated by taking into account theinteraction between membranes and bulk solvents of arbitrary thickness. As illustrative examples, the diffusion
coefficients of two types of inclusions, a circular domain composed of fluid with the same viscosity as thehost membrane and that of a polymer chain embedded in the membrane, are studied. The diffusion coefficientsare expressed in terms of the hydrodynamic screening lengths, which vary according to the solvent thickness.
When the membrane fluid is dragged by the solvent of finite thickness, via stick boundary conditions, multiplehydrodynamic screening lengths together with the weight factors to the diffusion coefficients are obtained fromthe characteristic equation. The conditions for which the diffusion coefficients can be approximated by theexpression including only a single hydrodynamic screening length are also shown.
DOI: 10.1103/PhysRevE.84.021905

PACS number(s): 87.16.D−, 82.35.Lr, 68.05.−n

I. INTRODUCTION

Recent advances in experimental techniques have made thedirect observation of the Brownian motion of μm-sized objects
in membranes using microscopy and imaging a routine process[1–7]. As a result, diffusion coefficients can be measured
accurately and it is now possible to address the issue ofthe differences between Brownian motion of macromolecules
embedded in membranes in various environments.
Vesicles with sizes of the order of 10 μm are frequently
used in experiments while the typical distance of a supported
membrane from the substrate is of the order of 20 Å [5,6].
Obviously, in both these general cases the coupling of amembrane with its environment is very different and hence
it will influence the Brownian motion of inclusions. In thispaper, we investigate the influence of solvent environmentson diffusion of an inclusion embedded in a membrane. In the
biological context, there are many examples of membranes
coming in contact with a solvent of various depths such as intissues.
Biological membranes can be regarded as two-dimensional(2D) viscous fluids. An important feature of membranes as atransport media is that they are not purely isolated [12–14].
Liquid membranes are coupled to surrounding solvents by interaction of polar head groups of lipid molecules with solvents;
they form quasi-2D systems coupled to three-dimensional(3D) solvents. The coupling to the surrounding environments
induces the momentum exchange between the membrane andthe solvents. The influence of the momentum exchange on
the Brownian dynamics has been theoretically investigatedby introducing a phenomenological coupling constant or
simplifying the solvents flow [8–17]. These studies have also
been extended to investigate the concentration fluctuations[18,19,22].
Despite the large number of studies, the Brownian motionof an object in liquid membranes has not yet been fully understood. In a hydrodynamic description, 2D flow in a bilayer
membrane can be regarded as viscous and the interaction
between liquid membranes and surrounding solvents can betaken into account by the stick boundary condition between
1539-3755/2011/84(2)/021905(10)them. Diffusion coefficients of macroscopic inclusions embedded in membranes were analytically investigated for a planar
membrane surrounded by solvent layers of infinite [8–11] orvery small thickness [12–17]. These studies revealed that thehydrodynamic flow in a membrane is screened by the solvent
drag force and is characterized by a hydrodynamic screening
length. When a planar membrane is surrounded by an infinite
thickness of solvent, it is called the Saffman and Delbrück(SD) hydrodynamic screening length, ν −1 , and is given by
the ratio between the 2D membrane viscosity η and the 3D
solvent viscosity ηs , ν −1 = η/ηs [8,9]. (As we shall see below,
the dimension of the 2D membrane viscosity is that of 3D
solvent viscosity times a length.) In the opposite limit of a thinsolvent layer of the thickness h, the Evans and Sackmann√(ES) hydrodynamic screening length given by h/ν isappropriate [14]. In both limits, the diffusion coefficientsdepend logarithmically on the size of the inclusions as longas the size is smaller than the hydrodynamic screening length.
On the other hand, the diffusion coefficients depend on thesize of the inclusions very differently when the size of the
inclusions exceeds the hydrodynamic screening length. Thesestudies naturally lead to the interest in the hydrodynamic
screening length and its influence on the diffusion coefficients
when the solvent layer has a finite thickness.
The solvent flow can be varied by changing the solvent
thickness. The flow of solvents influences the membrane flow
through the stick boundary condition imposed between themembrane and the solvents. As a result, the diffusion coefficients depend on the solvent thickness. The influence of thefinite solvent thickness has been recently studied for diffusionof a disk [20], concentration fluctuations [21–23], correlated
diffusion [24–27], and polymer diffusion in a membrane[28]. Diffusion coefficients of other types of inclusions onmembranes [17,29–32] or on Langmuir monolayers [33] havealso been theoretically calculated. However, the investigationon the influence of finite thickness of solvent was limited
to numerical evaluation of the diffusion coefficients, where
the dependence of the hydrodynamic screening length on
the solvent thickness was not completely elucidated [20–28].
In this paper, the relation between the diffusion coefficients

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SEKI, RAMACHANDRAN, AND KOMURAPHYSICAL REVIEW E 84, 021905 (2011)

and the hydrodynamic screening lengths are thoroughly
investigated for an arbitrary thickness of the solvent layers
on the basis of the analytical expression on the hydrodynamic
screening lengths.
The relation between the diffusion coefficients and thehydrodynamic screening lengths can be shown in a straightforward manner for a polymer embedded in a membraneby the Zimm model, where the equilibrium average of thehydrodynamic interactions is performed in two dimensions[17,28,31]. The multiple hydrodynamic screening lengths arethen found for the finite solvent thickness. The diffusion
coefficients are expressed by the weighted sum; each termin the sum is a product of the weight factor and the functionof the dimensionless size of the polymer normalized by eachhydrodynamic screening length. On the basis of the analyticalexpression, the condition that the diffusion coefficient isapproximately represented solely by the ES hydrodynamic
screening length can be discussed in detail. We show thatthe diffusion coefficient cannot be approximated by the ES
hydrodynamic screening length when both ν −1 = η/ηs andthe size of the macromolecule are smaller than the solvent
thickness.
Essentially the same relation between the diffusion coefficients and the hydrodynamic screening lengths is obtained for
diffusion of a circular liquid domain with the same viscosity
as that of the host membrane. The diffusion coefficient of acircular liquid domain embedded in a membrane has beenstudied in relation to the recently proposed raft model, whererafts are formed by sphingomyelin and cholesterol-rich liquiddomains [1,6,7,34–38]. It is believed that rafts undergo lateral
Brownian motion within a bilayer membrane and act as platforms for protein association and signaling [35]. Previously,
the diffusion coefficient of a circular liquid domain of arbitrary
size was derived in the limit of infinite depth of solvent layeror the limit of small depth of solvent layer [7,16,39,40]. In thispaper, the results are generalized for the arbitrary thicknessof solvent layers. The diffusion coefficient is obtained as asimple integral that can be expressed again as the sum of theterms given by functions of the same hydrodynamic screening
lengths multiplied by the same weight factors as those for thepolymer diffusion coefficients.
In Sec. II, the membrane hydrodynamics is reviewed. The
diffusion coefficient of a polymer embedded in a membrane
is obtained in Sec. III. In Sec. IV, the relation between thehydrodynamic screening length and the solvent thickness is
discussed. The diffusion coefficient of a liquid domain in amembrane is obtained in Sec. V. Finally, Sec. VI is devoted toconclusions.
II. HYDRODYNAMIC FLOW IN A MEMBRANE
AND SOLVENTAs shown in Fig. 1, we consider the situation where theliquid membrane is supported by a bulk solvent on the solid
substrate. The situation where the membrane is also supported
by a solvent from above will be considered in Sec. VI. We
denote the 2D flow in the membrane by v(r) where r = (x,y)
represents a position within the plane of the membrane. The
membrane is regarded to be incompressible,∇ · v(r) = 0.

(1)

z

η

0

ηs
-h
substrate
FIG. 1. Schematic picture showing a planar liquid membrane with
2D viscosity η located at z = 0. It is supported on a solvent with 3D
viscosity ηs . A substrate is located at z = −h bounding the solvent.

Here ∇ is a differential operator in the 2D Euclidean space.
The viscous flow in the membrane can be expressed by the
Stokes equation in two dimensions,η∇ 2 v(r) − ∇p(r) + fs (r) = 0,(2)where η is the 2D membrane viscosity, p(r) the in-planepressure, and fs (r) the in-plane force exerted on the membranefrom the solvent. The last quantity can be obtained when
the solvent fluid velocities are determined. The stress tensor
of the liquid membrane is given by
σαβ (r) = −p(r)δαβ + η[∂α vβ (r) + ∂β vα (r)],(3)

where δαβ is the Kronecker delta, and α, β are x, y. Then
Eq. (2) can be represented in terms of the stress tensor as
div σ + f s (r) = 0,(4)
where (div σ )α = β ∂σαβ /∂xβ .
As shown in Fig. 1, the membrane is located in theplane at z = 0. The solvent velocities v(3) (r,z), satisfy the
incompressibility condition∇˜ · v(3) (r,z) = 0,(5)where ∇˜ represents a differential operator in the 3D Euclidean
space. We denote the 3D viscosity of the solvent as ηs , and thesolvent flow also obeys the 3D Stokes equation
˜ (3) (r,z) = 0,ηs ∇˜ 2 v(3) (r,z) − ∇p(6)where p(3) (r,z) represents the pressure of the solvent. The
solvent is supported on the substrate, which is located atz = −h. The no-slip boundary condition is imposed at z = −has well as between the membrane flow and the solvent flow.
Through this boundary condition, the surrounding solvent
exerts a drag force on the liquid membrane.
The drag force in Eq. (2) can be expressed as fs = −(I −êz êz ) · σ (3) (r,0) · êz , where êz is the unit vector along the zaxis. The tensorial component of I is given by δij , and I − êz êzdenotes the projection to the in-plane space. The stress tensor
of solvent is given by
σij(3) (r,z) = −p(3) (r,z)δij + ηs ∂˜i vj(3) (r,z) + ∂˜j vi(3) (r,z) ,(7)
where i, j denote x, y, z.

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PHYSICAL REVIEW E 84, 021905 (2011)

Using the stick boundary conditions at z = 0 and z = −h,we solve the hydrodynamic equations from Eq. (5) to Eq. (6) toobtain fs . In the Fourier space, fs is calculated to be [22,33,41]
fs [k] = −ηs k coth(kh)v[k],(8)where k = (kx ,ky ) and k = |k|. The real space velocity fieldof the membrane flow v(r) can be expressed asd 2kv(r) =
v[k] exp(ik · r).
(9)(2π )2The Fourier space mobility tensor G[k] associated with thevelocity field is given by [22,33,41]


1
kα kβ
Gαβ [k] =. (10)
δαβ − 2
η[k 2 + νk coth(kh)]kIn order to calculate diffusion coefficients, the mobility
tensor in Fourier space should be transformed into real space.
Previously, the inverse Fourier transform of the mobility tensor
was analytically performed in the limits of infinite or zerothicknesses of a solvent layer. In the next section, the inverse
Fourier transformation of the mobility tensor is analytically
performed for an arbitrary thickness of a solvent.
III. DIFFUSION COEFFICIENT OF A
TWO-DIMENSIONAL POLYMER CHAIN

As an illustrative example for the influence of finite
thickness of solvent on the diffusion coefficient of a macromolecule embedded in a 2D planar membrane, we consider
the diffusion of a polymer chain confined in the membrane[17,28,31]. Previously, the influence of the solvent on diffusion
coefficients is analytically investigated in the limits of verythin or infinite thicknesses of solvent layers. In these works,
the hydrodynamic screening length is a key quantity incharacterizing the screening of the flow of membrane by thepresence of solvent layers. The influence of finite thicknessof solvent was investigated by numerically evaluating theinverse Fourier transform of the mobility tensor, where thehydrodynamic screening length was not even defined. In thissection, the hydrodynamic screening lengths are obtained froman analytical equation for arbitrary thickness of solvent layer.
The conformation of a 2D polymer chain embedded in a
2D membrane is represented by N beads with position vectors,{Rn } = (R1 , . . . ,RN ), under the potential energy
kB T 
(Rn − Rn−1 )2 ,b2 n=2NU=(11)where b is the Kuhn length [42]. The mobility tensor associatedwith the beads is given by the inverse Fourier transform ofEq. (10) asd 2kGαβ (Rn − Rm ) =Gαβ [k] exp [ik · (Rn − Rm )] .
(2π )2(12)
Within the pre-averaging approximation [42], the polymer
diffusion coefficient is expressed as N
dn N dmg(n − m),(13)
Dpoly = kB T
N 0 N
0where g(n − m) is the isotropic component of mobility
tensor Gαβ (Rn − Rm ) [28]. By using Eq. (12), two analyticalexpressions for the diffusion coefficients have been derived
from Eq. (13) in the limits of very thin or infinite thicknessof solvent layers [17,28]. Here, we investigate the diffusion
coefficient by keeping the finite depth of the solvent layer
without taking the limits. By expanding 1/[k + ν coth(kh)] in
partial fractions, we note the general relation [43]
 ∞
 ∞
∞

f (k)
kf (k)dk
Cj
dk 2, (14)=k+ν
coth(kh)k + κj2
00
j =1
where f (k) is an arbitrary function, κj and Cj will be later
given by Eqs. (16) and (17), respectively. By introducing
Eq. (14) into Eqs. (12) and (13), we obtain,


∞

1 2 2
1
exp
b κj |n − m|g(n − m) = −
Cj
8π η
4
j =1


1
(15)
× Ei − b2 κj2 |n − m| ,4where Ei(−z) is the exponential integral [44].
In the real space, the mobility tensor is expressed in termsof an infinite number of characteristic lengths, κj−1 , where κjis determined by the following characteristic equationκjcot(κj h) = .
(16)νAll the roots of the equation are given by κ = ±κj with
j = 1,2, . . .. The characteristic lengths relative to h, 1/(κj h),depend on νh given by the viscosity ratio ν = ηs /η, andrepresent the screening of hydrodynamic flow in a 2D
membrane due to the presence of the solvent. The contributionof each screening length is weighted by the factor
Cj =

2ν
.
hκj2 + hν 2 + ν(17)Using Eq. (15), the diffusion coefficient is obtained as
Dpoly =∞
kB T 1 
1 + j2 (2 ln j + γ )
4
4π
η
j
j =1

2
− j − exp j2 Ei − j2 ,
Cj

(18)

where γ = 0.5772 . . . is Euler’s constant. In the above,wepolymer size as j ≡√ have defined the dimensionless√
N bκj /2 = Rg κj , and Rg = N b/2 is the radius of gyration
for the 2D Gaussian polymer chain.
The limiting expression for 1  1 is


3
kB T
γ
− ln 1 − +.
(19)Dpoly ≈ C1
4π η
2
4As will be discussed in the next section, the above expression
is close to the exact result under the additional condition ofνh < 1, which is needed to replace the sum in Eq. (18) withthe term related to 1 . When 1  1, Eq. (18) reduces toDpoly ≈ C1

kB T 1(2 ln 1 + γ − 1) .
4π η 12(20)

This expression holds regardless of the value of h as long as it
is finite. The sum in Eq. (18) can be represented by the single

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SEKI, RAMACHANDRAN, AND KOMURAPHYSICAL REVIEW E 84, 021905 (2011)

dominant term as long as 1  1. However, the additional
condition of νh < 1 is required when 1  1, about which weshall discuss in the next section.
IV. HYDRODYNAMIC SCREENING LENGTH VERSUSSOLVENT THICKNESS

If the approximated diffusion coefficient obtained by taking
into account only the smallest positive value of κj (denotedby κ1 ) reproduces the exact results, then κ1−1 can be regarded
as the effective hydrodynamic screening length.
First, we consider the value of κ1 , which is the inverse of
the effective hydrodynamic screening length as long as thehigher-order (j  2) terms can be ignored. We first note theseries expansion,x cot x = 1 + 2

∞


x2

n=1

x 2 − n2 π 2

(21)

.

Since the lowest-order term can be estimated as 1 + 2x 2 /(x 2 −π 2 ) ≈ x 2 /(νh), the approximate expression for κ1 turns out tobeκ1 ≈3νh + π 2 −hydrodynamic screening length rapidly decreases as shown in
Fig. 2.
Next we consider the condition for which the diffusion
coefficient can be characterized by a single hydrodynamic
screening length as a good approximation for the exact expression including multiple hydrodynamic screening lengths
associated with higher-order κj . Judging from Eq. (16) and
Fig. 3(a), κj takes discrete values that are almost equally
separated. When κ1 Rg is well separated from κ2 Rg andthe diffusion coefficient is given by the weighted sum ofmonotonically decreasing functions of κj Rg multiplied by the
rapidly decreasing weights, the sum can be well represented by
the term associated with κ1 Rg alone. Below, we show that κ1 Rg
is well separated from κ2 Rg when Rg > h and the weights
rapidly decay when h < 1/ν.
Since we have κj ≈ κ1 + π (j − 1)/ h, the hydrodynamic
screening lengths are separated by the factor 1/ h. Hence κ2 Rg
is well separated from κ1 Rg when Rg / h > 1. It is convenient
to define the cutoff size Rg∗ = h over which the expression
with κ1 Rg could be very different from that with κ2 Rg .
In Fig. 3(b), the weight factors Cj are shown against
κj /ν. The weight factors Cj in Eq. (17) decrease with

1/2

(3νh)2 + 2νhπ 2 + π 4
2h2

(a)

. (22)In the limit of νh/π 2 < 1, κ1 can be further approximated as


νh
(23)
κ1 ≈ κ 1 − 2 ,π√where κ = ν/ h is the inverse of the ES hydrodynamic
screening length defined in the limit of h → 0. In Fig. 2,
the smallest positive values for the inverse of the characteristic
lengths are presented against the solvent layer thickness, h.
By increasing the solvent layer thickness h, the inverse of theκ3hcot(x) or x/(νh)

5
κ1h

κ2h

0

-5

0

2

4

6

8

x
(b)
0

0

10

10

Cj

κ1/ν

10

10

-1

10

10

-1

-2

νh=0.1-3

νh=10.0

10

-1

0

1

10

10

νh=1.0

2

10

0

νh

50

100

150

200

κj/ν

FIG. 2. (Color online) The smallest positive values for the
inverse of the characteristic lengths against the solvent thickness.
Both quantities are normalized by ν = ηs /η. The red thick line
represents the smallest positive root of the characteristic equation,
Eq. (16), √calculated numerically. The long dashed line represents
κ/ν = 1/ νh. The short dashed line represents the result of Eq. (23).
The thin solid line represents the result of Eq. (22).

FIG. 3. (Color online) (a) The pictorial solution of the characteristic equation, Eq. (16); cot(x) against x and x/(νh) against x
for νh = 1. The cross points of lines are xj = κj h. The smallest
positive value is x1 . κ1 is obtained by κ1 = x1 / h (b) Cj against
κj /ν. Cj represents the weight associated with each hydrodynamic
screening length, Eq. (17). Blue squares, red circles, and dots
represent νh = 0.1, νh = 1.0, and νh = 10.0, respectively.

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PHYSICAL REVIEW E 84, 021905 (2011)

increasing κj . The ratio of C2 /C1 is an important factor
in estimating whether the term related to κ1 is dominant
over other terms. Fig. 3(b) shows that the difference between C1 and C2 increases by decreasing the thicknessof solvent layer. Specifically, we have κj ≈ κ1 + π (j −1)/ h and C2 /C1 ≈ κ12 /κ22 ≈ νh/π 2 . Hence the diffusion
coefficient can be approximated by the expression involving κ1 alone when νh/π 2 < 1. When the condition νh<1
is satisfied, C2 /C1 < 1 and the diffusion coefficients can beapproximated by those obtained by Evans and Sackmann,
where C1 ∼ 1/(1 + νh/2) ∼ 1. It is then convenient to define
the critical thickness of solvent h∗ = 1/ν. If the solvent depth
exceeds h∗ , the weight of C2 is not much different from that of
C1 . It should be noticed, however, that the precise estimationof the contribution from the higher modes requires the wholeexpression besides the weights.
The expression of the diffusion coefficient depends on thekind of inclusions. As a representative example, we consider
the diffusion coefficient of a polymer to study conditions to
use a single effective hydrodynamic screening length given by1/κ1 . In Fig. 4, we show the polymer diffusion coefficients
against the size of the polymer Rg to study whether thepolymer diffusion coefficients can be approximated by an
expression without summation. When νh  1, the polymer
diffusion coefficients can be approximated by taking into
account only κ1 as shown in Fig. 4. It is consistent with thefact that the weight C2 is smaller than C1 when νh  1 since
C2 /C1 ∼ νh/π 2 . The situation corresponds to that consideredby Evans and Sackmann.
When νh > 1, C1 is close to C2 and the functional formof the diffusion coefficient should be carefully examined.
When νh > 1 and Rg > h holds, κ1 Rg > 1 is satisfied. Thenthe diffusion coefficient is well approximated by Eq. (20)showing 1/(κ1 Rg )2 dependence. Notice that 1/(κj Rg )2 decays
relatively fast by increasing j . When νh > 1 and the size ofthe polymer Rg exceeds the solvent thickness h, the diffusion
coefficient is approximated by the expression given in termsof κ1 Rg alone.
When νh > 1 and Rg < h, on the other hand, a significant
deviation is seen for the diffusion coefficients if the higherorder terms are ignored, as can be seen from Fig. 4. Thisdeviation originates from the fact that the weak logarithmic
dependence on κj Rg and κ2 Rg is not well separated from κ1 Rg
if Rg / h < 1. Also notice that C1 is close to the other valuesof Cj if νh > 1. In such a situation, multiple hydrodynamic
screening lengths should be taken into account.
To summarize, we have four length scales: the critical
thickness of the solvent h∗ = ν −1 , the cutoff size of thepolymer Rg∗ = h, the SD hydrodynamic screening length ν −1 ,and the ES hydrodynamic screening length κ1−1 [see Eq. (22)].
Although h∗ is identical to the SD hydrodynamic screening
length ν −1 , their physical meanings are different as explained
below.
The diffusion coefficient can be approximately expressedby using either ν −1 or κ1−1 . For the thin solvent layer,h < 1/ν, the weights rapidly decrease with increasing j andthe diffusion coefficient is given by κ1−1 regardless of themacromolecule size. This is the limiting case considered by
Evans and Sackmann. For thick solvent layers exceeding
the critical thickness h∗ = 1/ν, the diffusion coefficient canbe approximated by the expression including only the single hydrodynamic screening length κ1−1 , when the size ofthe macromolecule is larger than Rg∗ = h. In this case, wehave (κ2 − κ1 )Rg ∼ Rg / h > 1 and κ1 Rg is well separated
from κ2 Rg . When the solvent thickness exceeds h∗ = 1/ν andthe size of macromolecules is smaller than Rg∗ = h, on theother hand, the diffusion coefficient shows weak logarithmic
dependence on κj Rg and multiple hydrodynamic screening
lengths should be taken into account. The diffusion coefficientis expressed by the hydrodynamic screening length ν −1 in thelimit of h → ∞.
V. DIFFUSION COEFFICIENT OF A CIRCULAR DOMAIN

010

In this section, we consider the diffusion coefficient of acircular liquid domain in a membrane (see Fig. 5). Although

4πηD/kBTνh=10.0
10

-1

z
10

-2

νh=1.0
0

10

10

-3

νh=0.1
-4
-2

10

10

-1

0

10

h

1

10

νRg

ηs

-h

substrate

FIG. 4. (Color online) The diffusion coefficients of polymer
against size for various solvent thicknesses. The solid red lines
indicate the complete solution obtained from Eq. (18). The shortdashed black lines indicate the approximate solution by assuming acharacteristic length scale for hydrodynamic screening given by 1/κ1 .
The short dashed line for νh = 0.1 overlaps with the exact solution.

FIG. 5. Schematic picture showing a liquid domain embedded ina planar liquid membrane located at z = 0. Both a liquid domain anda membrane have the same 2D viscosity η. It is supported on a solventwith 3D viscosity ηs . A substrate is located at z = −h bounding thesolvent in the lower region.

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the characteristic equation, Eq. (16) associated with thesolvent and the membrane flow should not be altered bychanging the inclusion from polymers to liquid domains, thecondition that the diffusion coefficient can be approximatedby truncating the infinite sum to a single expression depends
on the size dependence of the diffusion coefficients. The size
dependence can differ between polymers and liquid domains.
For simplicity, we consider the case when the viscosity of thecircular liquid domain is the same as that of the host membranedenoted by η. Previously, simple expressions for the diffusion
coefficients were obtained for either infinite or very thin limitsof the solvent layers [16,39]. Here we generalize the results toarbitrary thickness of solvent layers.
We consider the situation for which the center of the circular
object moves with the velocity U, and its edge is assumed tokeep a circular shape without any deformation. The velocity
field inside and outside the circular domain satisfy [45]δ(r − R)= 0, (24)
2π R
and the incompressibility condition given by Eq. (1) for all r.
Here fs was defined before in Eq. (8), and F( ) is the force
exerted at the periphery of the circle in the direction normal
to the circular boundary [39]. If we take the origin of thecoordinates at the center of the circular domain and choose thex coordinate in the direction of U, F( ) should vary according to
the velocity U at the periphery of the circle. From the symmetry
with respect to U, F( ) can be expressed as [39]η∇ 2 v(r) − ∇p(r) + fs (r) + F( ) (r)F( ) (r) = F ( n) cos θ n,

(25)

where n = r/r is the outward normal unit vector at the surfaceof the circle of radius R, and θ is the angle between U and r.
Our task is to calculate the total force exerted on the circulardomain in the steady stateF=d σ ·n+d r f s,
(26)
r=R

rR

where σ is the stress tensor of the liquid membrane given
by Eq. (3) and d denotes the line integration. The first term
represents the force exerted from the membrane flow field,
and the second term represents the direct friction force exerted
from the solvent surrounding the membrane to the circulardomain.
By using Gauss’s theorem, we findF=d σ ·n+dr f s
r=R
rR


=
d r div σ +dr f srRrRδ(r − R)=−d r F ( ) (r)
2π R
rR
 2π
dθ
F ( n)
cos2 θ = −= −F ( n) êx
(27)
êx ,
2π
2
0
where Eq. (4) and Eq. (24) are used to obtain the third
equality, and êα denotes the unit vector along the α direction.
Equation (27) shows that it is sufficient to calculate F ( n) toobtain the total force exerted on the circular object from themembrane flow field and the solvent.

The velocity field can be formally expressed asδ(r − R).
(28)vα (r) = dr Gαβ (r − r )Fβ( ) (r )2π R
In real space, the mobility tensor is expressed by the Fouriertransform of Eq. (10) asd 2kGαβ (r) =Gαβ [k] exp(ik · r).
(29)(2π )2Equation (28) can be rewritten as
d 2kvα (r) =dr exp(ik · r)Gαβ [k] exp(−ik · r )(2π )2δ(r − R).
(30)
× Fβ( ) (r )2π R
Let ϕ denote the angle between r and U. Then we obtainδ(r − R)
dr exp(−ik · r )Fβ( ) (r )
2π R
 2π
rβ
dϕ
exp[−i(kx cos ϕ + ky sin ϕ)R]F ( n) cos ϕ ,
=
2π
r
0
(31)
where r /r is the unit orientational vector. Equation (30) canbe rewritten by using Eq. (31) and the relation
kα kβ rβrkα
δαβ − 2= α − 2 (kx cos ϕ + ky sin ϕ).
(32)
k
r
r
k
The integration with respect to ϕ can be performed (seeAppendix for the useful relations to perform this integration),and the result becomes
ky2
exp(ik · r)
d 2k
J1 (kR)êxv(r) =(2π )2 η[k 2 + νk coth(kh)] k 3 R kykx k y+ 2 J2 (kR) − 3 J1 (kR) êy F ( n) .
(33)
k
k R
Finally, we note k = (k cos φ,k sin φ) and r =(r cos θ,r sin θ ) as well as the relation

 2π

d 2k
dφ ∞ dkkexp(ik·r)=(2π )22π 0 2π0× exp[−ikr (cos θ cos φ+ sin θ sin φ)].
(34)
Then the integration with respect to φ can be performed toobtain ∞J1 (kR)J1 (kr)v(r) = êx
F ( n) .
dk(35)
2 + νk coth(kh)]2πηkRr[k0
By identifying the velocity at the periphery of the domain to
be U and using Eq. (27), we obtain ∞J1 (kR)2.
(36)
dk
U = −Fπ ηkR 2 [k 2 + νk coth(kh)]0
The friction coefficient is given by ζ = −F/U. Following
the Einstein relation D = kB T /ζ , we obtain the diffusion
coefficient of a domain as ∞J1 (kR)2. (37)
dk
Ddom = kB T
π ηkR 2 [k 2 + νk coth(kh)]0021905-6

DIFFUSION COEFFICIENT OF AN INCLUSION IN A . . .

PHYSICAL REVIEW E 84, 021905 (2011)

This is the generalization of the result obtained by De Koker
to the case of finite solvent depth [39].
A. Limit of infinite thickness of the solvent layer

In this limit, the diffusion coefficient of the circular object
was first calculated for the solid circular disk by Saffman andDelbrück [8,9]. For a circular liquid domain that has the same
viscosity as the outside of the domain, the diffusion coefficient
was obtained by De Koker [39]. The similar expression was
obtained for the domain shape relaxation times [46].
By taking the limit of kh  1 in Eq. (10), the mobility
tensor can be written as [22,24]


1
kα kβ
.
(38)
δ
−
Gαβ [k] =αβη(k 2 + νk)k2
If the above mobility tensor is used in Eq. (37), it reduces tothat derived by De Koker [39]. In this case, the integration canbe performed by using MATHEMATICA with the use of Meijer
G functions [47]

2
kB T
−
−1
Ddom =2(νR)2
2π η(νR)
 1/2,3/2
1. (39)− 3/2 G32 24 (νR)2 
0,1,3/2, − 1πThis expression is useful to take the limits with respect to νR.
In the case of νR  1, the above expression reduces to
 


2
1
kB T
Ddom ≈
ln
−γ +
.
(40)
4π η
νR
4
The difference from the result by Saffman and Delbrück is the
additional factor 1/4 in the right-hand side of Eq. (40) [8,9].
This means that the diffusion coefficient of a circular domain
is slightly larger than that of the disk, since the flow induced
inside the domain reduces the friction between the membrane
flow and the domain periphery compared to that between themembrane flow and the solid edge. In the opposite limit ofνR  1, the diffusion coefficient is obtained as
4kB T
Ddom ≈,(41)
3π 2 ηs R
which is inversely proportional to the domain radius, R. The
obtained diffusion coefficient is again slightly larger than thatof the disk in the same limit [10,11]
Ddisk ≈

kB T
.
8ηs R

(42)

The fact that Ddom is inversely proportional to R is consistent
with the result of the 2D polymer chain in the membrane [28].
B. The limit of thin solvent layer

The diffusion in supported membranes in the νh  1 limit
was originally considered by Evans and Sackmann for the solid
disk immersed in the membrane [14]. The diffusion coefficientof a circular viscous domain embedded in the membrane wasrecently studied by the present authors [16]. In this case,
Eq. (10) takes the following form


1
kα kβ
,
(43)
δαβ − 2Gαβ [k] =η(k 2 + κ 2 )kwhere κ ≡ (ν/ h)1/2 . The above mobility tensor was previously
used by the present authors [15–19].
We replace νk coth(kh) κ 2 for h → 0 in the integrand of
Eq. (37). A rigorous condition of small h needs some care sinceκ = (ν/ h)1/2 diverges in the limit of h → 0 when ν is finite.
In the previous section, we have discussed the condition indetail and shown that the results are valid under the condition
given by νh < 1. With this replacement, we obtain


1
kB T
Ddom =
−
I
(κR)K
(κR)
,
(44)
1
1
π η(κR)2 2
which coincides with our previous result [16]. However, it
should be noted that the diffusion coefficient was obtainedby taking into account the hydrodynamic force from themembrane alone in Ref. [16]. In order to compare the present
result with our previous result, the direct friction between
the solvent and the domain, π η(κR)2 , should be added
to the previous result. This leads to adding kB T /π η(κR)2 tothe diffusion coefficient. For comparison, we also write theresult by Evans and Sackmann [14]
4K1 (κR) −1
kB T
2
+
DES =
,
(45)
π η(κR)2
κRK0 (κR)
where the direct friction between the solvent and the domain
is added. As pointed out before, Eq. (44) is slightly larger
than Eq. (45) [16]. This is because the fluid flow in thedomain reduces the friction between the domain and the hostmembrane at the edge.
In the limit of κR  1, the previous result is reproduced
[16]
 


2
kB T
1
Ddom ≈
ln
−γ +
.
(46)
4π η
κR
4
In the opposite limit of κR  1, the diffusion coefficient isobtained as
kB T
.
(47)
Ddom ≈
2π η(κR)2
In this limit, Ddom decays as 1/R 2 as pointed out before [16].
C. Finite thickness of solvent layer

In the case of finite h, the integration of Eq. (37) can betransformed into the summations as employed before


∞

1
kB T
−
I
Cj
(κ
R)K
(κ
R)
, (48)
Ddom =1 j
1 j
π η(κj R)2 2
j =1
where Cj is the weight factor given by Eq. (17) and κj isdetermined by Eq. (16).
When νh < 1, Cj decreases rapidly as j increases as
already shown in the previous section. In this case, Eq. (48)can be approximated by the lowest order expression,


1
kB T
−
I
(κ
R)K
(κ
R)
, (49)
Ddom ≈ C1
1 1
1 1
π η(κ1 R)2 2
where κ1 is the smallest positive value of κj . For κ1 R  1,
Eq. (49) reduces to
 


2
1
kB T
Ddom ≈ C1
ln
−γ +
,
(50)
4π η
κ1 R
4

021905-7

SEKI, RAMACHANDRAN, AND KOMURAPHYSICAL REVIEW E 84, 021905 (2011)

whereas for κ1 R  1, it reduces to
Ddom ≈ C1kB T.
2π η(κ1 R)2(51)

When νh > 1, Eq. (49) approximates the exact expression,
Eq. (48), only when the expression multiplied to Cj rapidly
decreases with increasing j as we have already stated in theprevious section. Since the hydrodynamic screening lengths
and the weights factors are common, the difference between
the polymer and the circular liquid domain originates fromthe nondimensional size dependence. However, the size dependence is very similar between the polymer and the circulardomain (i.e., weak logarithmic dependence for relatively small
sizes and the algebraic dependence at large sizes). As aconsequence, essentially the same results as those shown in
Fig. 4 are obtained for the two cases. The condition that the
diffusion coefficient can be approximated by the expression
with a single hydrodynamic screening length is essentially thesame for the polymer and the circular liquid domain. Whenνh > 1, Eq. (51) is a good approximate expression of Eq. (48)but Eq. (50) is not.
Before closing the section, we compare in Fig. 6 thegeneralized solution of De Koker given by Eq. (37) with theresults in the two limits; the original solution of De Koker
obtained in the limit of h → ∞ and the results of Eq. (44)obtained in the limit of h → 0. The results of Eq. (37) withνh = 0.1 overlap with the results of Eq. (44). The diffusion
coefficient of Eq. (44) is slightly larger than that of the solid
disk, Eq. (45). By increasing νh the results shift toward
the original solution of De Koker. In the asymptotic limit,

010

4πηD/kBT-110

νh=10.0-210νh=1.0-310

νh=0.1
-4

10

-2

10

-1

10

0

10

1

10

νR
FIG. 6. (Color online) The diffusion coefficients of a liquiddomain against size for various solvent thicknesses. The solid redlines represent the generalized solution of De Koker given by
Eq. (37). νh = 10, νh = 1, and νh = 0.1 from right to left. The
result in the limit of thin solvent layer, Eq. (44), overlaps with thatof νh = 0.1. The red dashed-dotted line represents the asymptotic
solution for νh = 10, Eq. (51). The red thin dashed line indicatesthe Evans-Sackmann expression, Eq. (45). The blue long dashedline represents the original solution of De Koker obtained by taking
h → ∞ limit in Eq. (37). The blue dots denote the results of Eq. (40)and the blue dashed-dotted line indicates the asymptotic results,
Eq. (41).

the diffusion coefficient scales with 1/R 2 in the generalized
solution of De Koker, while the diffusion coefficient scaleswith 1/R in the the original solution of De Koker. In the
opposite limit of R → 0, all the results of Eq. (37) as wellas the original solution of De Koker show the logarithmic
dependence on R as represented by Eq. (40).
VI. CONCLUSIONS

The diffusion coefficient of an inclusion in a membrane
is strongly influenced by the presence of solvents due tothe stick boundary condition between the membrane and thesolvent. The thickness of the solvent layer is a key parameter
controlling the diffusion of an inclusion in a membrane. In
this work, the diffusion coefficient of a polymer confined in amembrane is obtained for an arbitrary thickness of solvents.
We also study the influence of a finite thickness of solvent on
the diffusion coefficient of a circular liquid domain with thesame viscosity as that of the host membrane. Previously, thediffusion coefficient of a circular liquid domain was expressedby a single integral in the limit of infinite solvent thickness [39].
In this work, the integral expression is generalized to the caseof finite solvent depth. The various analytical expressions are
obtained from the integral expression.
In general, the diffusion coefficient of inclusions decreases
as the solvent thickness decreases. The solvent induces dragagainst the membrane flow, and as a result diffusion issuppressed. The drag increases as the distance between themembrane and the substrate decreases. More precisely, the influence of the solvent on the membrane flow is characterized by
the hydrodynamic screening lengths. Multiple hydrodynamic
screening lengths can be obtained from the characteristic
equation, Eq. (16), when the solvent thickness is nonzero.
The largest hydrodynamic screening length characterizes thelength scale of momentum dissipation from the membrane tothe solvent. The membrane flow dissipates the momentum tothe solvent through the stick boundary condition. For finite
solvent depth, the largest √hydrodynamic screening length isapproximately given by h/ν. The diffusion coefficientsare expressed by the weighted sum of the functions of
the nondimensional size of the inclusion normalized by thehydrodynamic screening lengths. The weights are given by
Eq. (17).
By examining both the weights and the nondimensional
size dependence, we find that the diffusion coefficient canbe approximated by the expression given by the largest
hydrodynamic screening length alone except when both ν −1
and the size of the inclusion are smaller than the solvent
thickness. (The results are summarized in Table I.) Whenthe solvent thickness is larger than ν −1 , h > ν −1 , the weights
decrease slowly with increasing j . Hence ν −1 can be viewed as
the critical solvent depth below which there is a dominant mode
given by the largest hydrodynamic screening length. If the sizeof the inclusion is smaller than the solvent thickness, κ1 R isclose to κ2 R since we have κ2 R − κ1 R ∼ R/ h. Besides, if
the diffusion coefficient depends weakly on κj R, the diffusion
coefficient expressed by κ1 R alone cannot represent the exact
summed result. In particular, the sum of logarithmic functionsof screening lengths weighted by the similar factors cannot be
represented by one of the terms. The situation is met when021905-8

DIFFUSION COEFFICIENT OF AN INCLUSION IN A . . .

PHYSICAL REVIEW E 84, 021905 (2011)

TABLE I. Summary of the size dependence of the diffusion coefficient when the solvent depth is finite.
amplitude separation

screening lengths separation(h < 1/ν)(h < R)solvent depth

domain size

size dependencecompletecompleteincomplete
incomplete–a–aincomplete
completeh < 1/κ1 < 1/ν
h < 1/κ1 < 1/ν
1/ν < 1/κ1 < h
1/ν < 1/κ1 < h

R < 1/κ1
1/κ1 < R
R<hh<Rln(κ1 R)1/(κ1 R)2–b
1/(κ1 R)2a

The summation in the expression of the diffusion coefficient can be approximated by the dominant term as long as amplitude separation iscomplete no matter about the separation of screening lengths.
b
The summation in the expression of the diffusion coefficient cannot be approximated by a single dominant term. Even in the regime of1/κ1 < R < h, 1/(κ1 R)2 spatial dependence is not obtained.

both ν −1 and the size of the inclusion are smaller than thesolvent thickness.
The results summarized above are obtained for the finite
solvent thickness. In the limit of h → ∞, a new lengthscale appears as discussed by Diamant [25]. The diffusion
coefficient is given by the new hydrodynamic screening
length ν −1 .
For relatively small inclusions, the diffusion coefficientscan be approximated by the logarithmic function of thesize√ normalized by the ES hydrodynamic screening lengthh/ν when the solvent depth is small (i.e, 1/ν > h). In the
opposite limit of h → ∞, the diffusion coefficient is givenby the logarithmic function of the size normalized by ν −1 .
In the intermediate solvent depth, the diffusion coefficientsare expressed by the sum of multiple terms and the diffusion
coefficient cannot be represented by the logarithmic functionof the largest hydrodynamic screening length. However, thediffusion coefficient may be approximated by a logarithmic
function. This directs us to define an empirical interpolationof the effective hydrodynamic
screening lengths by the inverse√of κ ∗ = ν/(1 + νh) for any value of h when inclusions are
small.
The size dependence of diffusion coefficients is influenced
by the solvent depth. In the case of a supported membrane,
the typical value of h is 20 Å and νh ∼ 10−2 can be estimatedby introducing typical values of membranes; ηs = 10−2 poiseand η given by 1 poise multiplied by the membrane thickness
5 × 10−3 μm [5]. This is the case when the ES hydrodynamic
screening length Eq. (45) or its modification Eq. (44) arerelevant. By using relatively large size of inclusion, R > h =20 Å , the asymptotic 1/R 2 dependence of diffusion coefficient
can be observed.
In the case of vesicles of 10-μm size, νh 1 can be estimated by interpreting vesicle radius as solvent thickness [6].
Since the inclusion is normally smaller than the vesicle radius,
we haveh < 1. In this case, we can estimate as κR ∼√ R/√
νR/ νh < νh ∼ 1. If the vesicle radius is regarded as asolvent thickness, it may be difficult to observe the asymptotic
1/R 2 dependence of the diffusion coefficient. However, thisestimation is not rigorous but is done just for the purpose of
indicating the boundary effect caused by the finite radius of avesicle. The real flow inside a vesicle should be different fromthat in the presence of the solid substrate. It should be also
reminded that there is an additional difficulty to differentiate
the translational diffusion of a domain from the rigid rotationof the vesicle [7].

For simplicity, we have considered the situation where themembrane is floated on a solvent layer. In general, both sides
of a membrane are surrounded by solvents. We consider thecase that the solvent layer on the membrane is covered by a
substrate and is not a freestanding film. We denote the 3D
viscosity of solvent and the thickness in the upper domain as
ηs+ and h+ , respectively, and those in the lower domain as ηs−and h− , respectively.
Equations (10) and (37) are valid if we replace k +ν coth(kh) with k + ν + coth(kh+ ) + ν − coth(kh− ), whereν + = ηs+ /η and ν − = ηs− /η [23]. Correspondingly, the characteristic equation becomesν + cot(κj h+ ) + ν − cot(κj h− ) = κj .

(52)

The influence of solvents on both sides of the membrane canbe investigated by studying the roots of Eq. (52). By using
cot x ≈ 1/x, we obtain

κ1 =

ν+
ν−
+
,
h+
h−

(53)

when κ1 h+ < 1 and κ1 h− < 1.
In the simple situation where the membrane is sandwichedby the same depth of solvent layers h = h+ = h− , both thecharacteristic equation, Eq. (16) and the weights given by
Eq. (17) still hold by using the renewed definition of ν =ν + + ν − . The diffusion coefficients in this particular case canbe obtained from those presented in this paper by substitutingν = ν + + ν − [16].
If both h+ and h− are infinite, the screening length changes
from κ1−1 to (ν + + ν − )−1 . Our study on the influence of thefinite solvent thickness indicates that the diffusion coefficient
can be approximately expressed in terms of κ1 except whenboth the SD screening length and the size of the inclusion aresmaller than both h+ and h− .

ACKNOWLEDGMENTS

We would like to thank Youhei Fujitani for fruitful
discussions. This work was supported by KAKENHI (Grantin-Aid for Scientific Research) on Priority Area “SoftMatter Physics” and Grant No. 21540420 from the Ministry of
Education, Culture, Sports, Science and Technology of Japan.

021905-9

SEKI, RAMACHANDRAN, AND KOMURAPHYSICAL REVIEW E 84, 021905 (2011)

 2π

APPENDIX: USEFUL RELATIONS

dϕ
exp[−i(kx cos ϕ + ky sin ϕ)a] cos2 ϕ
2π
1 ∂2
k2
J1 (ka),= − 2 2 J0 (ka) = − x2 J2 (ka) +
a ∂kx
k
ka
 2π
dϕ
exp[−i(kx cos ϕ + ky sin ϕ)a] cos ϕ sin ϕ
2π
0
kx ky
1 ∂2
J0 (ka) = − 2 J2 (ka).
= − 2a ∂kx ∂kyk0

We have used the relations,
 2π
0

dϕ
exp[−i(kx cos ϕ + ky sin ϕ)a] = J0 (ka)
2π

(A1)

and

[1] P. Cicuta, S. L. Keller, and S. L. Veatch, J. Phys. Chem. B 111,
3328 (2007).
[2] Y. Kaizuka and J. T. Groves, Biophys. J. 86, 905 (2004).
[3] E. A. J. Reitz and J. J. Neefjes, Nature Cell Biol. 3, E145 (2001).
[4] Y. Gambin, R. Lopez-Esparza, M. Reffay, E. Sierecki, N. S. Gov,
M. Genest, R. S. Hodges, and W. Urbach, Proc. Natl. Acad. Sci.
USA 103, 2098 (2007).
[5] M. Tanaka, J. Hermann, I. Haase, M. Fischer, and S. G. Boxer,
Langmuir 23, 5638 (2007).
[6] M. Yanagisawa, M. Imai, T. Masui, S. Komura, and T. Ohta,
Biophys. J. 92, 115 (2007).
[7] S. Aliaskarisohi, P. Tierno, P. Dhar, Z. Khattari, M. Blaszczynski,
and Th. M. Fischer, J. Fluid Mech. 654, 417 (2010).
[8] P. G. Saffman and M. Delbrück, Proc. Natl. Acad. Sci. USA 72,
3111 (1975).
[9] P. G. Saffman, J. Fluid Mech. 73, 593 (1976).
[10] B. D. Hughes, B. A. Pailthorpe, and L. R. White, J. Fluid Mech.
110, 349 (1981).
[11] E. P. Petrov and P. Schwille, Biophys. J 94, L41 (2008).
[12] T. Izuyama, in Dynamics of Ordering Processes in Condensed
Matter, edited by S. Komura and H. Furukawa (Plenum,
New York, 1988), p. 505.
[13] Y. Y. Suzuki and T. Izuyama, J. Phys. Soc. Jpn. 58, 1104 (1989).
[14] E. Evans and E. Sackmann, J. Fluid Mech. 194, 553 (1988).
[15] K. Seki and S. Komura, Phys. Rev. E 47, 2377 (1993).
[16] S. Ramachandran, S. Komura, M. Imai, and K. Seki, Eur. Phys.
J. E 31, 303 (2010).
[17] S. Komura and K. Seki, J. Phys. II 5, 5 (1995).
[18] Y. Tserkovnyak and D. R. Nelson, Proc. Natl. Acad. Sci. USA
103, 15002 (2006).
[19] K. Seki, S. Komura, and M. Imai, J. Phys. Condens. Matter 19,
072101 (2007).
[20] H. A. Stone and A. Ajdari, J. Fluid Mech. 369, 151 (1998).
[21] M. Haataja, Phys. Rev. E 80, 020902(R) (2009).
[22] K. Inaura and Y. Fujitani, J. Phys. Soc. Jpn. 77, 114603 (2008).
[23] S. Ramachandran, S. Komura, K. Seki, and M. Imai, Soft Matter
7, 1524 (2011).
[24] N. Oppenheimer and H. Diamant, Biophys. J. 96, 3041
(2009).

(A2)

(A3)

[25] H. Diamant, J. Phys. Soc. Jpn. 78, 041002 (2009).
[26] N. Oppenheimer and H. Diamant, Phys. Rev. E 82, 041912(2010).
[27] S. Ramachandran, S. Komura, and G. Gompper, Europhys. Lett.
89, 56001 (2010).
[28] S. Ramachandran, S. Komura, K. Seki, and G. Gompper, Eur.
Phys. J. E 34, 11046-3 (2011).
[29] A. J. Levine, T. B. Liverpool, and F. C. MacKintosh, Phys. Rev.
Lett. 93, 038102 (2004).
[30] A. J. Levine, T. B. Liverpool, and F. C. MacKintosh, Phys. Rev.
E 69, 021503 (2004).
[31] M. Muthukumar, J. Chem. Phys. 82, 5696 (1985).
[32] A. Naji, A. J. Levine, and P. A. Pincus, Biophys. J. 93, L49(2007).
[33] Th. M. Fischer, J. Fluid Mech. 498, 123 (2004).
[34] K. Simons and E. Ikonen, Nature (London) 387, 569 (1997).
[35] D. A. Brown and E. London, Annu. Rev. Cell Dev. Biol. 14, 111(1998).
[36] J. F. Klingler and H. M. McConnell, J. Phys. Chem. 97, 6096
(1993).
[37] G. Orädd, P. W. Westerman, and G. Lindblom, Biophys. J. 89,
315 (2005).
[38] A. K. Kenworthy, B. J. Nichols, C. L. Remmert, G. M. Hendrix,
M. Kumar, J. Zimmerberg, and J. Lippincott-Schwartz, J. Cell.
Biol. 165, 735 (2004).
[39] R. De Koker, Ph.D. thesis, Stanford University, 1996.
[40] Y. Fujitani, J. Phys. Soc. Jpn. 80, 074609 (2011).
[41] D. K. Lubensky, R. E. Goldstein, Phys. Fluids 8, 843 (1996).
[42] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics(Clarendon, Oxford, 1986).
[43] F. John, Commun. Pure Appl. Math. 1, 45 (1950).
[44] M. Abramowitz and I. A. Stegun, Handbook of Mathematical
Functions (Dover, New York, 1972).
[45] H. A. Stone and H. M. McConnell, Proc. R. Soc. London A 448,
97 (1995).
[46] B. A. Camley, C. Esposito, T. Baumgart, and F. L. H. Brown,
Biophys. J. 99, L44 (2010).
[47] Wolfram Research Inc., MATHEMATICA 8.0 (Wolfram Research,
Champaign, Illinois, 1988).

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