PHYSICAL REVIEW E 93, 052407 (2016)

Dynamics of a membrane interacting with an active wall
Kento Yasuda,* Shigeyuki Komura,† and Ryuichi Okamoto‡Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan(Received 8 February 2016; revised manuscript received 14 April 2016; published 13 May 2016)
Active motions of a biological membrane can be induced by nonthermal fluctuations that occur in the outerenvironment of the membrane. We discuss the dynamics of a membrane interacting hydrodynamically withan active wall that exerts random velocities on the ambient fluid. Solving the hydrodynamic equations of abound membrane, we first derive a dynamic equation for the membrane fluctuation amplitude in the presence ofdifferent types of walls. Membrane two-point correlation functions are calculated for three different cases: (i) astatic wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We focus on the mean squared
displacement (MSD) of a tagged membrane describing the Brownian motion of a membrane segment. For thestatic wall case, there are two asymptotic regimes of MSD (∼ t 2/3 and ∼ t 1/3 ) when the hydrodynamic decayrate changes monotonically. In the case of an active wall, the MSD grows linearly in time (∼ t) in the early stage,
which is unusual for a membrane segment. This linear-growth region of the MSD is further extended when theactive wall has a finite intrinsic time scale.
DOI: 10.1103/PhysRevE.93.052407
I. INTRODUCTION

The random slow dynamics of fluid membranes visible
as a flickering phenomenon in giant unilamellar vesicles(GUVs) or red blood cells (RBCs) has attracted many interestsin the last few decades [1]. These thermally excited shapefluctuations can be essentially understood as a Brownian
motion of a two-dimensional (2D) lipid bilayer membrane in athree-dimensional (3D) viscous fluid such as water. For spherically closed artificial GUVs, characteristic relaxation times
for shape deformations were calculated analytically [2–5].
Analysis of shape fluctuations can be used for quantitative
measurements of surface tension and/or bending rigidity ofsingle-component GUVs [6] or GUVs containing bacteriorhodopsin pumps [7].
Historically, investigations on fluctuations of cell membranes have started with RBCs whose flickering can beobserved under a microscope [8]. Brochard and Lennon
were among the first to describe quantitatively membrane
fluctuations as thermally excited undulations, mainly governedby the bending rigidity of the membrane [9]. Later experiments
showed that flickering in RBCs is not purely of thermal
origin but rather corresponds to a nonequilibrium situation
because the fluctuation amplitude decreases upon ATP depletion [10,11]. Here ATP hydrolysis plays an important role
to control membrane-spectrin cytoskeleton interactions [12].
More advanced techniques have demonstrated that, at longer
time scales (small frequencies), a clear difference exists between the power spectral density of RBC membranes measured
for normal cells and those ATP depleted; the fluctuation
amplitude turns out to be higher in the former [13,14]. At
shorter time scales, on the other hand, membranes fluctuate
as in the thermodynamic equilibrium. It should be noted,
however, that the role of ATP in flickering is still debatablebecause Boss et al. have recently claimed that the mean
fluctuation amplitudes of RBC membranes can be described*yasuda-kento@ed.tmu.ac.jp
komura@tmu.ac.jp‡
okamotor@tmu.ac.jp†

2470-0045/2016/93(5)/052407(12)by the thermal equilibrium theory, while ATP merely affects
the bending rigidity [15].
In order to understand shape fluctuations of RBCs, one
needs to properly take into account the effects of spectrin cytoskeleton network that is connected to the membrane by actin,
glycophorin, and protein 4.1R [1,12]. Gov et al. treated thecytoskeleton as a rigid wall (shell) located at a fixed distancefrom the membrane and assumed that its static and dynamic
fluctuations are confined by the cytoskeleton [16,17]. They
further considered that the sparse connection of membrane
and cytoskeleton gives rise to a finite surface tension forlength scales larger than the membrane persistence length.
The bending free energy for a membrane was extended toinclude a surface tension and a confinement potential withwhich the effects of ATP on the membrane fluctuations
was described. However, since an active component of themembrane fluctuations also depend on the fluid viscosity [18],they cannot be solely attributed to the static parameters suchas the surface tension or the potential. Gov and Safran later
estimated the active contribution to the membrane fluctuations
due to the release of stored tension in the spectrin filamentand membrane in each dissociation event [19,20]. In contrast
to static thermal fluctuations, they showed that the active
cytoskeleton may contribute to the membrane fluctuations atintermediate length scales.
Effects of membrane confinement are important not only
for shape fluctuations of RBCs but also for a hydrodynamic coupling between closely apposed lipid bilayer membranes [21,22], and dynamical transitions occurring in lamellar
membranes under shear flow [23,24]. After the seminal works
by Kramer [25] and by Brochard and Lennon [9], the wavenumber-dependent decay rate for the bending modes of amembrane bound to a wall was calculated by Seifert [26]and Gov et al. [17]. In particular, Seifert showed that thescale separation between the membrane-wall distance and thecorrelation length determined by the confinement potential canlead to various crossover behaviors of the decay rate. In thesehydrodynamic calculations, however, the wall that interacts
with the membrane was treated as a static object and does notplay any active role.

052407-1©2016 American Physical Society

KENTO YASUDA, SHIGEYUKI KOMURA, AND RYUICHI OKAMOTOηz¯plane. Within the Monge representation, which is valid fornearly flat surfaces, the membrane shape is specified by the
distance ℓ(ρ) = ℓ(x,y) between the membrane and the wall.
The free energy F of a tensionless membrane in a potential
V (ℓ) per unit area reads [27,28]!
"κ#(∇ 2 ℓ)2 + V (ℓ) ,(1)
F = d 2ρ
2

h(ρ)

η

PHYSICAL REVIEW E 93, 052407 (2016)(ρ)

x
FIG. 1. A fluctuating membrane interacting with an active wall.
The membrane separated by a distance ℓ(ρ) from the wall feels
a potential V (ℓ) per unit area. The average of ℓ(ρ) is ℓ̄, and themembrane fluctuation is defined by h(ρ) = ℓ(ρ) − ℓ̄. Between themembrane and the wall as well as above the membrane are filled witha fluid of viscosity η. Red arrows on the wall indicate active velocities
exerted on the fluid by the wall.

Quite generally, active motions of a membrane can be
induced by nonthermal fluctuations that occur in the outerenvironment of the membrane such as cytoskeleton or cytoplasm. In this paper, we consider the dynamics of a membraneinteracting with an active wall that generates random velocities
in the ambient fluid. These random velocities at the wall
can be naturally taken into account through the boundary
conditions of the fluid. We first derive a dynamic equation
for the membrane fluctuation amplitude in the presence ofhydrodynamic interactions. Then we calculate the membrane
two-point correlation functions for three different cases: (i) astatic wall, (ii) an active wall, and (iii) an active wall with an
intrinsic time scale. We especially focus on the mean squared
displacement (MSD) of a tagged membrane segment anddiscuss its asymptotic time dependencies for the above cases.
For the static wall case, the membrane fluctuates due to thermal
agitations, and there are two asymptotic regimes of MSD(∼ t 2/3 and ∼ t 1/3 ) if the hydrodynamic decay rate changes
monotonically as a function of the wave number. When the wallis active, there is a region during which the MSD grows linearlywith time (∼ t), which is unusual for a membrane segment.
If the active wall has a finite intrinsic time scale, the above
linear-growth regime of the MSD is further extended. As awhole, active fluctuations at the wall propagate through the surrounding fluid and greatly affects the membrane fluctuations.
This paper is organized as follows. In the next section, we
discuss the hydrodynamics of a bound membrane that interacts
with an active wall. We also derive a dynamic equation
for the membrane fluctuation amplitude in the presence ofhydrodynamic interactions. In Sec. III, we calculate themembrane two-point correlation functions for three different
cases of the wall as mentioned above. We investigate various
asymptotic behaviors of the MSD of a tagged membranein both the static and the active wall cases. Some further
discussions are provided in Sec. IV.
II. HYDRODYNAMICS OF A BOUND MEMBRANE
A. Free energy of a bound membrane

As depicted in Fig. 1, we consider a fluid membrane boundat an average distance ℓ̄ from a wall which defines the xy

where κ is the bending rigidity and d 2 ρ = dx dy. We use
a harmonic approximation for fluctuations h(ρ) = ℓ(ρ) − ℓ̄around the minimum of the potential at ℓ = ℓ̄ and obtain theapproximated form
!
κ
(2)
F ≈
d 2 ρ [(∇ 2 h)2 + ξ −4 h2 ],2where ξ = [κ/(d 2 V /dℓ2 )ℓ=ℓ̄ ]1/4 is the correlation length dueto the potential. Later we use a dimensionless quantity definedby & ≡ ξ/ℓ̄ in order to discuss different cases.
In the following, we introduce the 2D spatial Fouriertransform of h(ρ) defined as!
h(q) = d 2 ρ h(ρ)e−iq·ρ ,(3)

where q = (qx ,qy ). Then the static correlation function can beobtained from Eq. (2) as⟨h(q)h(−q)⟩ =

kB T
kB T
=
,
κ(q 4 + ξ −4 )E(q,ξ )

(4)

where kB is the Boltzmann constant, T the temperature, q =|q|, and we have introduced the notation E(q,ξ ) ≡ κ(q 4 +ξ −4 ).
In the present model, we assume that the wall is rigid
and does not deform. Even when the wall, mimicking thecytoskeleton network, is deformable, the above free energy[Eq. (1)] would not be changed if we regard ℓ as a local
distance between the membrane and the cytoskeleton. In thiscase, however, the bending rigidity κ should be replaced with
an effective one which is also dependent on the bending rigidityof the cytoskeleton network itself [29].
B. Hydrodynamic equations and boundary conditions

The dynamics of a membrane is dominated by the surrounding fluid, which is assumed to be incompressible andto obey the Stokes equation. We choose z as the coordinate
perpendicular to the wall located at z = 0 as in Fig. 1. Thenthe velocity v(ρ,z) and the pressure p(ρ,z) for z ̸= ℓ̄ satisfy
the following equations:∇ · v = 0,(5)

η∇ 2 v − ∇p − f = 0,

(6)where η is the viscosity of the surrounding fluid and f (ρ,z)
is any force acting on the fluid. The fluid velocity can beobtained from the above equations by supplementing themwith proper boundary conditions. In the Appendix, we show
a formal solution appropriate for the membrane-wall systemand obtain the fluid velocity v in terms of the force f . Without
loss of generality, we can choose the x and y coordinates as
the parallel (longitudinal) and the perpendicular (transverse)

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PHYSICAL REVIEW E 93, 052407 (2016)

DYNAMICS OF A MEMBRANE INTERACTING WITH AN . . .

directions to the in-plane vector q, respectively. Since thetransverse y component of the velocity is not coupled tothe other components, we are allowed to set vy = 0 in what
follows.
Let us denote the fluid regions 0 ! z ! ℓ̄ and ℓ̄ ! z withthe superscripts “−” and “+,” respectively. In general, we
consider time-dependent boundary conditions at z = 0 andtime-independent conditions at z → ∞:vx− (q,z = 0,t) = Vx0 (q,t),(7)vz− (q,z = 0,t) = Vz0 (q,t),(8)

vx+ (q,z → ∞,t) = vz+ (q,z → ∞,t) = 0.

(9)The statistical properties of Vx0 (q,t) and Vz0 (q,t) will be
discussed for different types of walls in the next section. As
described in the Appendix, the z component of the velocity isthen obtained asvz− (q,z,t) =A[sinh(qz) − qz cosh(qz)] + Bqz sinh(qz)− iqzVx0 (q,t)e−qz + (1 + qz)Vz0 (q,t)e−qz ,(10)vz+ (q,z,t) = Ce−q(z−ℓ̄) + Dq(z − ℓ̄)e−q(z−ℓ̄) ,(11)

where A, B, C, and D are the coefficients determined by theother boundary conditions at the membrane z = ℓ̄. Note thatboth vx and p can be also expressed in terms of these four
coefficients.
At z = ℓ̄ where the membrane exists, continuity of vx andvz yieldsvx− (q,z = ℓ̄,t) = vx+ (q,z = ℓ̄,t),(12)vz− (q,z = ℓ̄,t) = vz+ (q,z = ℓ̄,t),(13)

and incompressibility of the membrane requires that the inplane divergence of vx vanishes:iqvx− (q,z = ℓ̄,t) = 0.

(14)

Moreover, the forces are required to balance in the normal
direction at z = ℓ̄. This condition is written as
δF= −E(q,ξ )h(q,t),(15)−Tzz+ + Tzz− = −δh(q,t)where E(q,ξ ) was defined in Eq. (4). In the above, Tzz± is theFourier transformed zz component of the fluid stress tensor
Tij = −pδij + η(∂i vj + ∂j vi ),(16)

evaluated at z = ℓ̄ ± 0 and i,j = x,z. The above four boundary conditions in Eqs. (12)–(15) at z = ℓ̄ determine the solutionof v and p in the entire region of the fluid.
C. Dynamic equation of a bound membrane

Next we derive a dynamic equation for the membrane
fluctuation amplitude. The time derivative of the fluctuation
amplitude h(q,t) (membrane velocity) should coincide with
the normal velocity of the fluid at the membrane vz (q,z = ℓ̄,t)obtained from Eqs. (10) and (11) together with the four
coefficients (see also the Appendix). Using the result of theΞ = 10Ξ = 0.1

4

10

0

γ̃

10

-4

10

-8

10

-2

-1

10

10

0

10

q¯

1

10

2

10

FIG. 2. Dimensionless decay rate γ̃ ≡ γ t¯ [see Eq. (18)] witht¯ = 4ηℓ̄3 /κ as a function of dimensionless wave number q ℓ̄. The
solid and dashed lines represent & = ξ/ℓ̄ = 10 and 0.1, respectively.

above hydrodynamic calculation, we can write the dynamic
equation of h(q,t) as follows:∂h(q,t)= − γ (q,ℓ̄,ξ )h(q,t)∂t+ *x (q,ℓ̄)Vx0 (q,t) + *z (q,ℓ̄)Vz0 (q,t)+ ζ (q,t).

(17)

In the above, γ (q,ℓ̄,ξ ) is the hydrodynamic decay rate
γ (q,ℓ̄,ξ ) = ,(q,ℓ̄)E(q,ξ ),(18)

where the kinetic coefficient ,(q,ℓ̄) is given by,(q,ℓ̄) =1
sinh2 (q ℓ̄) − (q ℓ̄)2.
2ηq sinh2 (q ℓ̄)−(q ℓ̄)2 + sinh(q ℓ̄) cosh(q ℓ̄) + (q ℓ̄)(19)

The same expression was obtained by Seifert [26]. The second
and the third terms on the r.h.s. of Eq. (17) are due to the wallboundary conditions Eqs. (7) and (8). Our calculation yields−iq ℓ̄ sinh(q ℓ̄),sinh (q ℓ̄) − (q ℓ̄)2 + sinh(q ℓ̄) cosh(q ℓ̄) + (q ℓ̄)(20)sinh(q ℓ̄) + q ℓ̄ cosh(q ℓ̄).
*z (q,ℓ̄) =sinh2 (q ℓ̄) − (q ℓ̄)2 + sinh(q ℓ̄) cosh(q ℓ̄) + (q ℓ̄)(21)*x (q,ℓ̄) =2

The last term in Eq. (17) represents the thermal white noise;
its average vanishes ⟨ζ (q,t)⟩ = 0 while its correlation is fixedby the fluctuation-dissipation theorem (FDT) [30,31]⟨ζ (q,t)ζ (−q,t ′ )⟩ = 2kB T ,(q,ℓ̄)δ(t − t ′ ).

(22)

D. Hydrodynamic decay rate

We first introduce t¯ ≡ 4ηℓ̄3 /κ as a characteristic time. In
Fig. 2 we plot the scaled decay rate γ (q,ℓ̄,ξ )t¯ [see Eq. (18)]
as a function of the dimensionless wave number q ℓ̄ when052407-3

PHYSICAL REVIEW E 93, 052407 (2016)

KENTO YASUDA, SHIGEYUKI KOMURA, AND RYUICHI OKAMOTO& = ξ/ℓ̄ = 10 and 0.1. For our later discussion, it is usefulhere to discuss its asymptotic behaviors. We first note that the
kinetic coefficient ,(q,ℓ̄) in Eq. (19) behaves as$ 3 2ℓ̄ q /12η,,≈
1/4ηq,

q ≪ 1/ℓ̄q ≫ 1/ℓ̄.

1.0
0.8

Depending on the relative magnitude between ℓ̄ and ξ , twodifferent asymptotic behaviors of the decay rate can be
distinguished [26]. For ℓ̄ ≪ ξ (corresponding to & = 10 in
Fig. 2), the decay rate increases monotonically as

0.6⎧⎨κ ℓ̄3 q 2 /12ηξ 4 , q ≪ 1/ξ
1/ξ ≪ q ≪ 1/ℓ̄γ ≈ κ ℓ̄3 q 6 /12η,⎩ 3
κq /4η,1/ℓ̄ ≪ q.

(24)

0.0
0

2

4

6

8

10

ρ/ξ
FIG. 3. Dimensionless static correlator -̃ ≡ (4κ/kB T ξ 2 )- in thepresence of a static wall [see Eq. (31)] as a function of dimensionlessdistance ρ/ξ . Notice that -̃ takes a maximum value at ρ/ξ = 4.93.

and one can easily solve for h(q,t) ash(q,t) =h(q,0)e−γ (q,ℓ̄,ξ )t! t+
dt1 ζ (q,t1 )e−γ (q,ℓ̄,ξ )(t−t1 ) .

(27)

0

(25)

While the small-q and large-q behaviors are unchanged from
Eq. (24), here the decay rate decreases with increasing q in theintermediate range. This unusual decrease of the decay rate
clearly appears for 1 < q ℓ̄ < 10 in Fig. 2. Such an anomalous
behavior occurs due to the fact that the potential confines
the mean squared fluctuation amplitudes to ⟨h2 ⟩ ≈ kB T ξ 4 /κindependently of q [see Eq. (4)], while hydrodynamic damping
becomes less effective with increasing q [26]. We also notethat the absolute value of γ in the small-q region is sensitive
to the value of &, while it is independent of & in the large-q
region.
III. MEMBRANE TWO-POINT CORRELATIONFUNCTIONS

Using the result of the hydrodynamic calculation, we shalldiscuss in this section the two-point correlation functions ofbound membrane [33,34]. We separately investigate the casesof (i) a static wall, (ii) an active wall, and (iii) an active wallwith an intrinsic time scale.
A. Static wall

In the case of a static wall, the velocities at the wall vanish inEqs. (7) and (8), i.e., Vx0 (q,t) = Vz0 (q,t) = 0. Hence Eq. (17)reduces to∂h(q,t)= −γ (q,ℓ̄,ξ )h(q,t) + ζ (q,t),
∂t

0.4
0.2

The small-q behavior γ ∼ q 2 results from the conservationof the fluid volume between the membrane and the wall [32].
The dependence γ ∼ q 6 in the intermediate regime, where
the effect of potential becomes irrelevant, was predicted byBrochard and Lennon [9]. For large q, we recover the behaviorof a free membrane γ ∼ q 3 . All these asymptotic behaviors
are observed in Fig. 2.
For ξ ≪ ℓ̄ (corresponding to & = 0.1 in Fig. 2), on theother hand, γ changes nonmonotonically as [26]⎧⎨κ ℓ̄3 q 2 /12ηξ 4 , q ≪ 1/ℓ̄
1/ℓ̄ ≪ q ≪ 1/ξ
γ ≈ κ/4ηξ 4 q,
⎩ 3
κq /4η.
1/ξ ≪ q.

Φ̃

(23)

(26)

Using the above solution and Eq. (22), we calculate the membrane two-point correlation function which can be separated
into two parts [33,34]:⟨[h(ρ,t) − h(ρ ′ ,0)]2 ⟩ = -(ρ − ρ ′ ) + φ(ρ − ρ ′ ,t),(28)

where the translational invariance of the system has been
assumed. In the above, the first term is a purely static correlator,-(ρ − ρ ′ ) = ⟨[h(ρ) − h(ρ ′ )]2 ⟩!
d 2q′⟨h(q)h(−q)⟩[1 − eiq·(ρ−ρ ) ], (29)=2(2π )2

describing the static membrane roughness, while the second
term is a dynamical correlator,
!
d 2q
′
′
φ(ρ − ρ ,t) =2⟨h(q)h(−q)⟩eiq·(ρ−ρ )(2π )2× [1 − e−γ (q,ℓ̄,ξ )t ],(30)

describing the propagation of fluctuations with a distance |ρ −
ρ ′ |.
Using the static correlation function for h(q) in Eq. (4), we
first calculate the static correlator!
kB T ∞q′-(ρ − ρ ) =dq 4[1 − J0 (q|ρ − ρ ′ |)]πκ 0
q + ξ −4)(+,(|ρ − ρ ′ |/ξ )4 ** 1 1
1
kB T ξ 2
1 − G3,0,,0, (31)
=
*0,
4κ
π 0,4
256
2 2

where J0 (z) is the zero-order Bessel function of the first kind,and the Meijer G function is used in the last expression [35].
In Fig. 3 we plot the scaled static correlator -(ρ − ρ ′ ) as afunction of ρ/ξ where ρ = |ρ − ρ ′ |. Only in this plot, we use

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PHYSICAL REVIEW E 93, 052407 (2016)

DYNAMICS OF A MEMBRANE INTERACTING WITH AN . . .

ξ to scale the length because the above static correlator issolely determined by the free energy in Eq. (2), and Eq. (31)does not depend on ℓ̄. In the large distance ρ/ξ ≫ 1, the(route mean square) height difference between two points on
the bound membrane is proportional to ξ . It is interesting tonote that -(ρ − ρ ′ ) changes nonmonotonically and shows amaximum around ρ/ξ ≈ 4.93. A similar overshoot behaviorof the membrane profile was reported previously [36].
As for the dynamical correlator in Eq. (30), we perform theangular integration and obtain the expression!
kB T ∞qφ(ρ − ρ ′ ,t) =
dq 4
πκ 0
q + ξ −4
× [1 − e−γ (q,ℓ̄,ξ )t ]J0 (q|ρ − ρ ′ |).

3

10

10

(32)

10

1.0

In this regime, as discussed by Brochard and Lennon [9], theconservation of the enclosed incompressible volume between
the membrane and the wall is important, while the effect ofthe potential acting between them is irrelevant. The Fouriertransform of the above expression, i.e., the power spectral
density, was previously discussed by Gov et al. in Ref. [16].
In the long time regime (&6 t¯ ≪ t), the MSD saturates at the

10

(b)

10

6

Ξ = 10Ξ = 0.1

0.8

α

0.6
0.4
0.2

(34)

In the intermediate time regime (t¯ ≪ t ≪ &6 t¯), we have φ0 ∼t 1/3 , which stems from the intermediate-q behavior of γ in
Eq. (24):!
kB T ∞
1
kB T ℓ̄3 6
dq 3 [1 − e−(κ ℓ̄ q /12η)t ] ∼ 2/3 1/3 t 1/3 .
φ0 (t) ≈
πκ 0
q
κ η
(36)

0

-6

t/t̄

where we have used J0 (0) = 1. Instead of the correlation
length ξ , we hereafter use ℓ̄ to scale the length. Note thatthe hydrodynamic effect is manifested by the appearance ofthe length ℓ̄. In Fig. 4 we plot the dimensionless MSD φ0 as afunction of t/t¯ (recall that t¯ = 4ηℓ̄3 /κ) for & = 10 (monotonic
damping case) and & = 0.1 (nonmonotonic damping case),
respectively. In order to find out the asymptotic behaviors
clearly, we have also plotted an effective growth exponent α
defined by

For both & = 10 and 0.1, the MSD increases monotonicallyas a function of time. For & = 10 (ℓ̄ ≪ ξ ), there are threedifferent asymptotic regimes of the time dependence. In thesmall time regime (t ≪ t¯), the MSD behaves as φ0 ∼ t 2/3 ,which corresponds to the diffusion of a free membrane [33,34].
This scaling behavior can be obtained by using the large-q
behavior of the decay rate in Eq. (24):!
kB T ∞
1kB T
3
φ0 (t) ≈dq 3 [1 − e−(κq /4η)t ] ∼ 1/3 2/3 t 2/3 .
πκ 0
q
κ η
(35)

-3

-6

We first set ρ = ρ and discuss the MSD of a tagged membranesegment given by [33,34]!
kB T ∞q
φ0 (t) =dq 4[1 − e−γ (q,ℓ̄,ξ )t ],
(33)
πκ 0
q + ξ −4d ln φ0 (t).
d ln t010′α(t) =

Ξ = 10Ξ = 0.1

φ̃0

10

(a)

0.0
10

0

-6

10

10

6

t/t̄
FIG. 4. (a) Dimensionless MSD of a tagged membrane segmentφ̃0 ≡ (π κ/kB T ℓ̄2 )φ0 in the presence of a static wall [see Eq. (33)]
as a function of dimensionless time t/t¯ where t¯ = 4ηℓ̄3 /κ. The solidand dashed lines represent & = ξ/ℓ̄ = 10 and 0.1, respectively. (b)Effective exponent α of the MSDs in (a) as defined in Eq. (34).

value given by
φ0 (t → ∞) ≈(,kB T 2
kB T ℓ̄2 π &2
∼
ξ .
πκ
4
κ

(37)

For & = 0.1 (ξ ≪ ℓ̄), on the other hand, there are only two
asymptotic regimes. The MSD increases as φ0 ∼ t 2/3 in thesmall time regime (t ≪ &3 t¯), whereas in the long time regime(&3 t¯ ≪ t), it saturates at the value given by Eq. (37).
Let us consider then the case ρ ̸= ρ ′ . In Fig. 5 we plot
the scaled φ(ρ − ρ ′ ,t) in Eq. (32) as a function of ρ/ℓ̄ fordifferent times when & = 10. For all the cases, the dynamic
correlator changes nonmonotonically and exhibits a typicalundershoot behavior. The minimum of φ occurs for larger ρ
as time evolves. In the long time limit, t → ∞, φ(ρ − ρ ′ ,t)052407-5

PHYSICAL REVIEW E 93, 052407 (2016)

KENTO YASUDA, SHIGEYUKI KOMURA, AND RYUICHI OKAMOTO

Eq. (17). This equation can be also solved for h(q,t) as

80

60

φ̃

h(q,t) = h(q,0)e−γ (q,ℓ̄,ξ )t! t+ *x (q,ℓ̄)
dt1 Vx0 (q,t1 )e−γ (q,ℓ̄,ξ )(t−t1 )t/t̄ = 108t/t̄ = 106t/t̄ = 1040+ *z (q,ℓ̄)

40

+

20

0

0

dt2 Vz0 (q,t2 )e−γ (q,ℓ̄,ξ )(t−t2 )dt3 ζ (q,t3 )e−γ (q,ℓ̄,ξ )(t−t3 ) .

(38)

The random velocities generated at the wall are assumed tohave the following statistical properties:

0
0

! t

! t

20

40

ρ/ ¯

60

80

⟨Vx0 (ρ,t)⟩ = ⟨Vz0 (ρ,t)⟩ = 0,100

FIG. 5. Dimensionless dynamical correlator φ̃ ≡ (π κ/kB T ℓ̄2 )φin the presence of a static wall [see Eq. (32)] as a function ofdimensionless distance ρ/ℓ̄ for t/t¯ = 108 (solid black), 106 (dashedred), 104 (dotted blue) when & = ξ/ℓ̄ = 10. Here the characteristic
time is t¯ = 4ηℓ̄3 /κ.

(39)⟨Vx0 (ρ,t)Vx0 (ρ ′ ,t ′ )⟩ = 2Sx δ(ρ − ρ ′ )δ(t − t ′ ),(40)

⟨Vz0 (ρ,t)Vz0 (ρ ′ ,t ′ )⟩ = 2Sz δ(ρ − ρ ′ )δ(t − t ′ ),(41)

⟨Vx0 (ρ,t)Vz0 (ρ ′ ,t ′ )⟩ = 0,

(42)

⟨Vx0 (ρ,t)ζ (ρ ′ ,t ′ )⟩ = ⟨Vz0 (ρ,t)ζ (ρ ′ ,t ′ )⟩ = 0,(43)

in Eq. (32) coincides with the second term in Eq. (31) and isgiven by the Meijer G function.

where we have introduced the amplitudes Sx and Sz in Eqs. (40)and (41), respectively. With these statistical properties, we
can calculate the total two-point correlation function, which
consists of the static and the dynamical parts as before:

B. Active wall⟨[h(ρ,t) − h(ρ ′ ,0)]2 ⟩tot = -tot (ρ − ρ ′ ) + φtot (ρ − ρ ′ ,t).
(44)

We now investigate the case when the wall is active so that itexerts random velocities on the ambient fluid. The membrane
dynamics in the presence of an active wall is described by-tot (ρ − ρ ′ ) =

1
π

! ∞
0

dq q

(

In the above total correlation function, the static correlator
in the presence of the active wall becomes,kB T
Sz |*z (q,ℓ̄)|2
Sx |*x (q,ℓ̄)|2+[1 − J0 (q|ρ − ρ ′ |)]+κ(q 4 + ξ −4 )γ (q,ℓ̄,ξ )γ (q,ℓ̄,ξ )≡ -(ρ − ρ ′ ) + -x (ρ − ρ ′ ) + -z (ρ − ρ ′ ),

(45)

′

where *x and *z were obtained in Eqs. (20) and (21), respectively, while -(ρ − ρ ) was defined in Eq. (31) for the static wallcase. In the above equations, we have defined two correlators -x and -z . On the other hand, the dynamical correlator in Eq. (44)is given by
(
,
!
1 ∞
kB T
Sz |*z (q,ℓ̄)|2
Sx |*x (q,ℓ̄)|2′dq q+[1 − e−γ (q,ℓ̄,ξ )t ]J0 (q|ρ − ρ ′ |).
(46)
φtot (ρ − ρ ,t) =+π 0
κ(q 4 + ξ −4 )γ (q,ℓ̄,ξ )γ (q,ℓ̄,ξ )By setting ρ = ρ ′ , the total MSD of a tagged membrane segment in the presence of the active wall becomes(,!
Sx |*x (q,ℓ̄)|2
1 ∞kB T
Sz |*z (q,ℓ̄)|2+dq q+[1 − e−γ (q,ℓ̄,ξ )t ] ≡ φ0 (t) + φx0 (t) + φz0 (t),φtot (t) =π 0
κ(q 4 + ξ −4 )γ (q,ℓ̄,ξ )γ (q,ℓ̄,ξ )where the first term φ0 (t) was defined before in Eq. (33)for the static wall case, while φx0 and φz0 have been newly
defined here.
Before showing the result of MSD, we first discuss
the wave-number dependencies of the quantities |*x |2 /γ
and |*z |2 /γ appearing in Eqs. (45)–(47). These quantities
originating from the active wall are plotted in Fig. 6 as afunction of q ℓ̄ for & = 10 and 0.1. Using the asymptotic
behaviors of γ , as shown in Eqs. (24) and (25), we can obtain(47)

the limiting expressions for |*x |2 /γ and |*z |2 /γ as well.
When ℓ̄ ≪ ξ (corresponding to & = 10), we have⎧q ≪ 1/ξ
⎨3ηξ 4 /κ ℓ̄,2
1/ξ ≪ q ≪ 1/ℓ̄|*x | /γ ≈ 3η/κ ℓ̄q 4 ,(48)⎩
4ηℓ̄2 e−2ℓ̄q /κq, 1/ℓ̄ ≪ q,⎧⎨12ηξ 4 /κ ℓ̄3 q 2 , q ≪ 1/ξ
2
1/ξ ≪ q ≪ 1/ℓ̄|*z | /γ ≈ 12η/κ ℓ̄3 q 6 ,(49)⎩
4ηℓ̄2 e−2ℓ̄q /κq, 1/ℓ̄ ≪ q.

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PHYSICAL REVIEW E 93, 052407 (2016)

DYNAMICS OF A MEMBRANE INTERACTING WITH AN . . .

4

10

(a)

Ξ = 10Ξ = 0.1

(a)

60

0

Φ̃x

|Λx |2 /γ̃

10

40

-4

10

20
-8

10

-2

10

-1

10

q¯

10

0

0
0

1

10

20

40

20

40

60

80

100

60

80

100

ρ/ ¯

(b)

10

10

Ξ = 10Ξ = 0.1(b)5

6Φ̃z /104|Λz |2 /γ̃

10

0

10

4

2
-5

10

0
0

-10

10

-2

10

-1

10

q¯

10

0

1

10

FIG. 6. Plots of (a) |*x |2 /γ̃ [see Eq. (20)] and (b) |*z |2 /γ̃ [seeEq. (21)] as a function of dimensionless wave number q ℓ̄. Here γ̃ ≡γ t¯ [see Eq. (18)] with t¯ = 4ηℓ̄3 /κ is the dimensionless decay rate. The
solid and dashed lines represent & = ξ/ℓ̄ = 10 and 0.1, respectively.

For ξ ≪ ℓ̄ (corresponding to & = 0.1), on the other hand, we
obtain⎧q ≪ 1/ℓ̄⎨3ηξ 4 /κ ℓ̄,2|*x | /γ ≈ 4ηξ 4 ℓ̄2 q 3 e−2ℓ̄q /κ, 1/ℓ̄ ≪ q ≪ 1/ξ (50)⎩1/ξ ≪ q,4ηℓ̄2 e−2ℓ̄q /κq,
⎧
4
3 2
12ηξ
/κ
ℓ̄
q
,
q
≪ 1/ℓ̄
⎨2
4 2 3 −2ℓ̄q/κ, 1/ℓ̄ ≪ q ≪ 1/ξ
|*z | /γ ≈ 4ηξ ℓ̄ q e(51)⎩
1/ξ ≪ q.
4ηℓ̄2 e−2ℓ̄q /κq.

The static correlators -x and -z defined in Eq. (45) due tothe active wall can now be obtained by performing numericalintegrals. In Fig. 7 we plot the static correlators -x and -z as
a function of ρ/ℓ when & = 10. Here -x and -z are scaledby 4ηℓ̄Sx /πκ and 4ηℓ̄Sz /πκ, respectively. We notice that -x
behaves similarly to that of the static wall case - given in
Eq. (31) and plotted in Fig. 3. On the other hand, -z diverges
logarithmically for large ρ/ℓ̄ because the integral is found toρ/ ¯

FIG. 7. Dimensionless static correlators (a) -̃x ≡ (π κ/4ηℓ̄Sx )-x and (a) -̃z ≡ (π κ/4ηℓ̄Sz )-z (divided by 104 ) in
the presence of an active wall [see Eq. (45)] as a function ofdimensionless distance ρ/ℓ̄ when & = ξ/ℓ̄ = 10.

be infrared divergent. Such a logarithmic divergence is avoided
when we consider a finite membrane size, which gives rise to
a cutoff for small wave numbers in the integral of Eq. (45).
It should be noted that both -x and -z depend on ℓ̄ and ξ ,while - is solely determined by ξ . This means that -x and -z
include the geometrical as well as the hydrodynamic effects.
In Figs. 8 and 9, we plot the scaled membrane MSD φx0
and φz0 [see Eq. (47)], respectively, as a function of t/t¯ when
& = 10 and 0.1. For ℓ̄ ≪ ξ (corresponding to & = 10), there
are three different asymptotic regimes both for φx0 and φz0 . In
the small time regime (t ≪ t¯), we have φx0 ∼ t and φz0 ∼ t,showing a normal diffusive behavior. This is because φx0 canbe approximated as

052407-7!
4ηℓ̄2 Sx ∞3
φx0 (t) ≈dq e−2ℓ̄q [1 − e−(κq /4η)t ]
πκ
0
!
Sx
ℓ̄2 Sx t ∞dq e−2ℓ̄q q 3 ∼ 2 t.
≈
π
ℓ̄
0

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PHYSICAL REVIEW E 93, 052407 (2016)

KENTO YASUDA, SHIGEYUKI KOMURA, AND RYUICHI OKAMOTO
3

10

10

Ξ = 10Ξ = 0.106(a)Ξ = 10Ξ = 0.1

3

10

φ̃z0

φ̃x0

10

10

(a)

-3

10

10

0

-3

-6

10

-6

10
-9

10

10

-6

0

10

10

-9

10

6

10

0

-6

10

t/t̄
1.0
0.8

6

t/t̄
1.0

Ξ = 10Ξ = 0.1(b)10

Ξ = 10Ξ = 0.1

(b)

0.8

0.6

α

α

0.6

0.4

0.4

0.2

0.2

0.0

0.0
10

-6

0

10

10

6

10

0

-6

10

t/t̄

10

6

t/t̄

FIG. 8. (a) Dimensionless MSD of a tagged membrane segmentφ̃x0 ≡ (π κ/4ηℓ̄Sx )φx0 in the presence of an active wall [see Eq. (47)]
as a function of dimensionless time t/t¯ where t¯ = 4ηℓ̄3 /κ. The solidand dashed lines represent & = ξ/ℓ̄ = 10 and 0.1, respectively. (b)Effective exponent α of the MSDs in (a) as defined in Eq. (34).

FIG. 9. (a) Dimensionless MSD of a tagged membrane segment
φ̃z0 ≡ (π κ/4ηℓ̄Sz )φz0 in the presence of an active wall [see Eq. (47)]
as a function of dimensionless time t/t¯ where t¯ = 4ηℓ̄3 /κ. The solidand dashed lines represent & = ξ/ℓ̄ = 10 and 0.1, respectively. (b)Effective exponent α of the MSDs in (a) as defined in Eq. (34).

Notice that only small-q contributes to the integral, andthe same holds for φz0 . In the intermediate time regime(t¯ ≪ t ≪ &6 t¯), we have φx0 ∼ t 1/3 and φz0 ∼ t 2/3 , which canbe essentially obtained by the integrals in Eqs. (36) and (35),respectively. In the long time regime (&6 t¯ ≪ t), φx0 saturates
at the value,(
4ηℓ̄Sx 3&2
ηξ 2 Sx
φx0 (t → ∞) ≈
.
(53)
∼
πκ
8
κ ℓ̄

the value

On the other hand, φz0 diverges logarithmically for t → ∞,which can be seen in Fig. 9(a) and shown analytically. Sucha divergence in time occurs for small q and can be avoided
when the membrane size is finite as mentioned before.
For ξ ≪ ℓ̄ (corresponding to & = 0.1), on the other hand,
there are only two asymptotic regimes. The MSDs increase
both linearly as φx0 ∼ t and φz0 ∼ t in the small time regime(t ≪ &3 t¯). In the long time regime (&3 t¯ ≪ t), φx0 saturates at

φx0 (t → ∞) ≈(,4ηℓ̄Sx 3&4
ηξ 4 Sx
,
∼
πκ
8
κ ℓ̄3

(54)

while φz0 also diverges logarithmically as above.
C. Active wall with an intrinsic time scale

Finally we consider a situation in which the activity of
the wall occurs over a finite time scale τ . In this case, thestatistical properties of random velocities which have been
given in Eqs. (40) and (41) would be replaced by the following
exponential correlation function in time [19,20]:Sx′δ(ρ − ρ ′ )e−|t−t |/τ ,⟨Vx0 (ρ,t)Vx0 (ρ ′ ,t ′ )⟩ =
(55)
τ
Sz
′
(56)
⟨Vz0 (ρ,t)Vz0 (ρ ′ ,t ′ )⟩ = δ(ρ − ρ ′ )e−|t−t |/τ ,τ

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PHYSICAL REVIEW E 93, 052407 (2016)

DYNAMICS OF A MEMBRANE INTERACTING WITH AN . . .
3

10

(a)

10

-3

10

φ̃z0

φ̃x0

10

(a)

6

0

10
10

10

10

-6

10

0

-6

10

10

-3

τ /t̄ = 0
τ /t̄ = 102
τ /t̄ = 104

10

-9

10

-12

10

0

-6

τ /t̄ = 0
τ /t̄ = 102
τ /t̄ = 104

-9

10

3

6

10

0

-6

10

10

6

t/t̄

t/t̄
1.0

1.0

(b)

(b)
0.8

0.6

0.6

α

α

0.8

0.4

0.4

τ /t̄ = 0
τ /t̄ = 102
τ /t̄ = 104

0.2
0.0
10

0

-6

10

τ /t̄ = 0
τ /t̄ = 102
τ /t̄ = 104

0.2
0.0
10

6

10

0

-6

10

t/t̄

while the other velocity correlations remain the same. In
general, the intrinsic time scale τ can be different between
the x and z components. In the above relations, we have put afactor 1/τ so that the physical dimension of Sx and Sz is thesame as before.
Repeating the same procedure as before, we obtain the total
two-point correlation function, which can be also separated
into the static and dynamics parts as in Eq. (44). The
static correlators in the presence of the active wall now
become1-x (ρ − ρ ) =
π

! ∞
0

dq q

Sx |*x (q,ℓ̄)|2γ (q,ℓ̄,ξ )[γ (q,ℓ̄,ξ )τ + 1]

× [1 − J0 (q|ρ − ρ ′ |)],

6

t/t̄

FIG. 10. (a) Dimensionless MSD of a tagged membrane segmentφ̃x0 ≡ (π κ/4ηℓ̄Sx )φx0 in the presence of an active wall with an
intrinsic time scale [see Eq. (59)] as a function of dimensionlesstime t/t¯ where t¯ = 4ηℓ̄3 /κ. Different colors correspond to τ/t¯ = 0(solid black), 102 (dashed red), 104 (dotted blue), and we set & = 10.
(b) Effective exponent α of the MSDs in (a) as defined in Eq. (34).

′

10

(57)

FIG. 11. (a) Dimensionless MSD of a tagged membrane segment
φ̃z0 ≡ (π κ/4ηℓ̄Sz )φz0 in the presence of an active wall with an
intrinsic time scale [see Eq. (60)] as a function of dimensionlesstime t/t¯ where t¯ = 4ηℓ̄3 /κ. Different colors correspond to τ/t¯ = 0(solid black), 102 (dashed red), 104 (dotted blue), and we set & = 10.
(b) Effective exponent α of the MSDs in (a) as defined in Eq. (34).

-z (ρ − ρ ′ ) =

1
π

! ∞
0

dq q

Sz |*z (q,ℓ̄)|2γ (q,ℓ̄,ξ )[γ (q,ℓ̄,ξ )τ + 1]

× [1 − J0 (q|ρ − ρ ′ |)].

(58)

A similar static correlator was previously discussed by Gov
et al. [19,20]. Notice that the above expressions reduce tothose in Eq. (45) when τ → 0. This is reasonable because
the exponential function in Eqs. (55) and (56) reduce to a δfunction in the limit of τ → 0.
Similarly, the two MSD functions of a tagged membranesegment are given by!
1 ∞Sx |*x (q,ℓ̄)|2
φx0 (t) =dq q
π 0γ (q,ℓ̄,ξ )[γ (q,ℓ̄,ξ )τ + 1]

052407-9

× [1 − e−γ (q,ℓ̄,ξ )t ],(59)

PHYSICAL REVIEW E 93, 052407 (2016)

KENTO YASUDA, SHIGEYUKI KOMURA, AND RYUICHI OKAMOTO

1
φz0 (t) =
π

! ∞
0

dq q

Sz |*z (q,ℓ̄)|2γ (q,ℓ̄,ξ )[γ (q,ℓ̄,ξ )τ + 1]

× [1 − e−γ (q,ℓ̄,ξ )t ],(60)

which also reduce to those in Eq. (47) when τ → 0. In Figs. 10
and 11, we plot the scaled φx0 and φz0 , respectively, as afunction of t/t¯ for different τ values when & = 10. We first
notice that both φx0 and φz0 decrease when the intrinsic timescale τ is taken into account. Also the initial time regions
during which φx0 and φz0 grow linearly in time increase forlarger τ values, and the regions showing the scaling φx0 ∼ t 1/3
or φz0 ∼ t 2/3 become narrower. In the large τ limit, there will
be only two scaling regimes of the MSDs.
IV. SUMMARY AND DISCUSSION

In this paper, we have discussed the dynamics of a membrane interacting with an active wall that generates random
velocities. We have generally shown that the dynamics of abound membrane is significantly affected by active fluctuations
at the wall, and they propagate through the surrounding
fluid. Using the result of the hydrodynamic calculation ofa bound membrane, we have derived a dynamic equation
for the membrane fluctuation amplitude [see Eq. (17)]. As
noted by Seifert previously [26], there are two different
asymptotic behaviors of the hydrodynamic decay rate γ
depending on the relative magnitude between the average
membrane-wall distance ℓ̄ and the correlation length ξ [seeEqs. (24) and (25)]. We have obtained in Sec. III the membrane
two-point correlation functions for three different wall cases:(i) a static wall, (ii) an active wall, and (iii) an active wall withan intrinsic time scale.
As a dynamic part of the correlation function, we havemainly discussed the MSD of a tagged membrane andinvestigated its asymptotic time dependencies for the different
types of walls. For the static wall case, the membrane fluctuates
due to thermal agitations. When the decay rate γ changes
monotonically, the MSD given by Eq. (33) exhibits two
asymptotic behaviors φ0 ∼ t 2/3 and φ0 ∼ t 1/3 before it reaches
a constant value that is fixed by ξ (see Fig. 4). When the wallis active, on the other hand, the partial MSDs in Eq. (47) growlinearly in time, φx0 ∼ φz0 ∼ t, in the early stage. Comparedto the dynamics due to thermal fluctuations, this is a uniquebehavior of a membrane segment in the presence of an active
wall (see Figs. 8 and 9). When the active wall has a finite
intrinsic time scale τ as defined in Eqs. (55) and (56), thelinear-growth region of the MSD is further extended as τ isincreased.
The present work should be distinguished from thosedealing with the dynamics of “active membranes” [8]. These
membranes contain active proteins such as ion pumps which
consume the chemical energy and drive the membrane outof equilibrium. Being motivated by the theoretical predictions [37–43], some experiments have shown that active forces
enhance membrane fluctuations [44–46]. In our theory, we
have considered that the active components are incorporated
not in the membrane but in the wall, and we discussed theirhydrodynamic effects on the membrane fluctuations. Hence
our work is related to the recent work by Maitra et al. [47]

who discussed the dynamics of a membrane coupled to an
actin cortex consisting of filaments with active stresses andcurrents.
For our further discussion, we give here some numericalestimates of the quantities used in our calculations. As
an example, we consider the shape fluctuations of RBCs.
Previously, the data for a normal RBC [48] were well describedby using the following parameters [19,20]; κ ∼ 10−19 J, ℓ̄ ∼2–3 × 10−8 m, and ξ ∼ 2–3 × 10−7 m. Then the importantdimensionless parameter is roughly & = ξ/ℓ̄ ∼ 10 for RBCs,
and the decay rate γ is expected to increase monotonically as in
Eq. (24). Using the value of water viscosity η ∼ 10−3 J/m3 , weobtain the characteristic time scale as t¯ = 4ηℓ̄3 /κ ∼ 10−7 s.
Hence the second crossover time scale discussed in Eq. (36) isroughly given by &6 t¯ ∼ 10−1 s. Since t¯ and &6 t¯ are well
separated, the three different asymptotic regimes of φ0 (t)should be clearly observable.
The intrinsic time scale τ appearing in Eqs. (55) and (56)represents the duration of force production at the active walland can be roughly estimated as τ ∼ 10−3 s for the spectrin
network of RBCs [19,20]. Hence the choice of τ/t¯ ∼ 104
in Figs. 10 or 11 is reasonable. Moreover, the force balance
between the spectrin compression and the membrane bending
yields a characteristic length scale of the order of L ∼ 10−7 m.
From the viewpoint of dimensional analysis, the quantities Sxand Sz , which fix the amplitudes of the random velocities
in Eq. (40) and (41), respectively, can be evaluated as Sx ∼L4 /τ ∼ 10−24 m4 /s and similarly for Sz . With this value,
the amplitude of the MSD due to the active wall becomes
φx0 ∼ ηℓ̄Sx /κ ∼ 10−17 m2 . This value is comparable to thatof thermal fluctuations φ0 ∼ kB T ℓ̄2 /κ ∼ 10−16 m2 .
As mentioned in Sec. II, the decay rate γ changes
nonmonotonically when & ≪ 1. This situation may occur fora charged membrane pushed by an osmotic pressure [26].
When unscreened electrostatic interactions compete with anosmotic pressure, the condition & ≪ 1 is met whenever ℓ̄ ≫
κℓB /kB T ∼ 5 × 10−9 m where ℓB is the Bjerrum length. As
the unbinding transition point is approached [28], ℓ̄ becomesmuch larger than ξ .
Following the calculation by Seifert [26], we have shownin Sec. II that the hydrodynamic kinetic coefficient ,(q,ℓ̄)
is given by Eq. (19). In Ref. [17], Gov et al. used different
boundary conditions at the membrane and obtained an alternative expression for the kinetic coefficient:,G (q,ℓ̄) =e−2q ℓ̄ 2q ℓ̄
[e − 1 − 2q ℓ̄ − 2(q ℓ̄)2 ].
4ηq

(61)

As expressed in Eq. (14), Seifert and we have used anincompressibility condition for the fluid near the membrane,
whereas Gov et al. employed a zero-shear-stress condition,
which implies that the xz component of the shear on bothsides of the membrane are equal. Gov et al. insisted that thelatter condition is appropriate for a fluid membrane whichcannot support shear stress across its width [17]. We havequantitatively compared Eqs. (19) and (61) and confirmed thatthey only differ by a numerical factor of 4 in the small-q
regime, and the asymptotic scaling behaviors are completely

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PHYSICAL REVIEW E 93, 052407 (2016)

DYNAMICS OF A MEMBRANE INTERACTING WITH AN . . .

identical. Notice that ,G (q,ℓ̄) in Eq. (61) behaves as$ 3 2ℓ̄ q /3η, q ≪ 1/ℓ̄,G ≈1/4ηq,
q ≫ 1/ℓ̄,(62)

which can be compared with Eq. (23). In any case, thedifferences between Eqs. (19) and (61) are not significant as
far as the role of the wall is concerned, and various scaling
behaviors of the MSD discussed in this paper are valid.
In this paper, we have only dealt with a tensionlessmembrane whose shape is governed by Eq. (1). In Ref. [16],it was claimed that the attachment of the cytoskeleton to themembrane would induce an effective surface tension. For amembrane with a finite surface tension 2, the free energy is
modified to
,
(
!
κ 2 2
2
(63)
F = d 2ρ(∇ ℓ) + V (ℓ) + (∇ℓ)2 .
22

Science, and Technology of Japan, the Grant-in-Aid for
Scientific Research (C) (Grant No. 15K05250) from the Japan
Society for the Promotion of Science (JSPS), and the JSPS
Core-to-Core Program “International Research Network for
Non-equilibrium Dynamics of Soft Matter”.
APPENDIX: SOLUTIONS OF HYDRODYNAMIC
EQUATIONS

The incompressibility condition Eq. (5) and the Stokes
equation (6) can be formally solved for the fluid velocity v
in terms of the force f acting on the fluid [53]:! ∞′e−q|z−z |vx (q,z) =dz′[(1 − q|z − z′ |)fx (q,z′ )4ηq−∞+ iq(z′ − z)fz (q,z′ )],
(A1)
! ∞
′
e−q|z−z |2fy (q,z′ ),vy (q,z) =
dz′
(A2)
4ηq
−∞
! ∞
′
e−q|z−z |[(1 + q|z − z′ |)fz (q,z′ )vz (q,z) =
dz′
4ηq
−∞

Hence the quantity E in the static correlation function Eq. (4)should now be replaced withE(q,ξ ) = κ(q 4 + ξ −4 ) + 2q 2 .

(64)

The new term associated with the surface tension neither
modifies the small wave number nor the large wave number
asymptotic behaviors [26]. Although a more complicatedcrossover behavior can arise in the intermediate wave number
if 2 > κ/ξ 2 , we do not discuss it here because our main aim
is to see the effects of the active wall. In general, the presenceof the finite surface tension tends to suppress the membrane
fluctuations.
In our model, the outer fluid is assumed to be purely
viscous characterized by a constant viscosity η. It should
be noted, however, fluids surrounding biomembranes are
viscoelastic rather than purely viscous. This is a common
situation in all eukaryotic cells whose cytoplasm is a soupof proteins and organelles, including a thick submembrane
layer of actin-meshwork forming a part of the cell cytoskeleton [12]. The extra-cellular fluid can also be viscoelastic
because it is filled with extracellular matrix or hyaluronic
acid gel. In order to mimic the real situations, the dynamics
of undulating bilayer membrane surrounded by viscoelastic
media was considered before [49,50]. It was assumed thatboth sides of the membrane are occupied by viscoelastic media
with a frequency-dependent modulus that obeys a power-law
behavior G(ω) = G0 (iω)β with 0 ! β ! 1 [49–52]. Sucha frequency dependence is commonly observed for various
polymeric solutions at high frequencies. The limits of β = 0
and 1 correspond to the purely elastic and purely viscous cases,
respectively. Following the previous results, we argue thatthe time dependencies of MSD which has been expressed
as φ0 ∼ t α in the pure viscous case will be modified toφ0 ∼ t αβ for both the static and the active wall cases. Hencethe asymptotic exponent of the MSD is generally smaller thanthat for a purely viscous fluid.
ACKNOWLEDGMENTS

We thank T. Kato for useful discussions. S.K. acknowledges
support from the Grant-in-Aid for Scientific Research on
Innovative Areas “Fluctuation and Structure” (Grant No.
25103010) from the Ministry of Education, Culture, Sports,+ iq(z′ − z)fx (q,z′ )],(A3)

with q = |q|. Notice that vy in Eq. (A2) is not coupled to theother components and hence can be neglected.
For the fluid in the region 0 ! z ! ℓ̄, the forces are acting
both at z = 0 and z = ℓ̄ so that fx and fz can be written asfx (q,z) = fx (q,0)δ(z) + fx (q,ℓ̄)δ(z − ℓ̄),(A4)

fz (q,z) = fz (q,0)δ(z) + fz (q,ℓ̄)δ(z − ℓ̄).

(A5)

Substituting Eqs. (A4) and (A5) into Eq. (A3), we obtain
Eq. (10) where the the coefficients A and B are given by.
e−q ℓ̄ iq ℓ̄fx (q,ℓ̄) + (1 + q ℓ̄)fz (q,ℓ̄) ,2ηq.
e−q ℓ̄ −i(1 − q ℓ̄)fx (q,ℓ̄) + q ℓ̄fz (q,ℓ̄) .
B=
2ηq
A=

(A6)
(A7)

For the fluid in the region ℓ̄ ! z, the forces are acting only
at z = ℓ̄ so that fx and fz can be written asfx (q,z) = fx (q,ℓ̄)δ(z − ℓ̄),(A8)fz (q,z) = fz (q,ℓ̄)δ(z − ℓ̄).

(A9)

Substituting Eqs. (A8) and (A9) into Eq. (A3), we obtain
Eq. (11) where the coefficients C and D are given by
fz (q,ℓ̄)
C=
,
(A10)
4ηq
fz (q,ℓ̄) − ifx (q,ℓ̄)
.
(A11)
D=
4ηq
Using these four coefficients A, B, C, and D, the x component of the velocity vx± (q,z,t) and the pressure p± (q,z,t) are
obtained as follows:vx− (q,z,t) = − iAqz sinh(qz) + iB[sinh(qz) + qz cosh(qz)]052407-11+ (1 − qz)Vx0 (q,t)e−qz − iqzVz0 (q,t)e−qz ,(A12)

KENTO YASUDA, SHIGEYUKI KOMURA, AND RYUICHI OKAMOTO

vx+ (q,z,t) = −ie−q(z−ℓ̄) (C − D − Dq ℓ̄ + Dqz),

PHYSICAL REVIEW E 93, 052407 (2016)(A13)

p+ (q,z,t) = 2ηDqe−q(z−ℓ̄) .

(A14)

The four unknown coefficients are determined by the boundary
conditions (12), (13), (14), and (15) at z = ℓ̄.

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p− (q,z,t) = − 2ηq[A cosh(qz) − B sinh(qz)]+ 2ηq[Vz0 (q,t) − iVx0 (q,t)]e−qz ,052407-12(A15)