Journal of Physics: Condensed Matter
J. Phys.: Condens. Matter 27 (2015) 432001 (7pp)

doi:10.1088/0953-8984/27/43/432001

Fast Track Communication

Dynamics of two-component membranes
surrounded by viscoelastic media
Shigeyuki Komura, Kento Yasuda and Ryuichi Okamoto
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Tokyo 192-0397, Japan
E-mail: komura@tmu.ac.jp
Received 9 July 2015, revised 18 August 2015
Accepted for publication 1 September 2015
Published 8 October 2015
Abstract

We discuss the dynamics of two-component fluid membranes which are surrounded by
viscoelastic media. We assume that membrane-embedded proteins can diffuse laterally andinduce a local membrane curvature. The mean squared displacement of a tagged membranesegment is obtained as a generalized Einstein relation. When the elasticity of the surrounding
media obeys a power-law behavior in frequency, an anomalous diffusion of the membranesegment is predicted. We also consider the situation where the proteins generate active
non-equilibrium forces. The generalized Einstein relation is further modified by an effective
temperature that depends on the force dipole energy. The obtained generalized Einstein
relations are useful for membrane microrheology experiments.
Keywords: membrane, microrheology, diffusion(Some figures may appear in colour only in the online journal)

I. Introduction

calculated the frequency-dependent out-of-plane (transverse)mean squared displacement (MSD) of a membrane segment
and the linear response to external forces.
In this work, we discuss the dynamics and responses of
two-component membranes containing proteins such as ionchannels, ion pumps, or photo-active proteins like bacteriorhodopsin. These proteins can diffuse laterally within themembranes. We use a model that incorporates curvatureconcentration coupling as well as hydrodynamics interactions. We calculate the MSD of a tagged membrane segment
by taking into account the viscoelasticity of the surrounding
media as well as the diffusivity of the proteins. Our aim is
to derive a generalized Einstein relation for the membraneprotein system, which is useful for membrane microrheology
experiments.
Furthermore, the state of the membrane proteins can beeither passive or active [3]. In the latter case, the proteinsconsume the chemical energy and drive the membrane out of
equilibrium. It was experimentally shown that active forces
due to ion pumps enhance membrane fluctuations [4–7].

Biomembranes are thin two-dimensional fluids which separate inner and outer environments of organelles in cells. The
fluidity of biomembranes is guaranteed mainly due to the lipid
molecules which are in the liquid crystalline state at physiological temperatures. Proteins and other molecules embedded in biomembranes undergo lateral diffusion which plays
important roles for biological functions [1]. It should be noted,
however, that biomembranes are not isolated 2D systems, but
are coupled to the surrounding polar solvent such as water.
Moreover, the solvent surrounding biomembranes is viscoelastic rather than purely viscous. This is a common situation in all eukaryotic cells whose cytoplasm is a soup ofproteins and organelles, including a thick sub-membrane layerof actin-meshwork forming a part of the cell cytoskeleton[1]. The extra-cellular fluid can also be viscoelastic because
it is filled with extracellular matrix or hyaluronic acid gel.
Recently, Granek discussed the dynamics of an undulating
bilayer membrane surrounded by viscoelastic media [2]. He
0953-8984/15/432001+7$33.00

1© 2015 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter 27 (2015) 432001

There exist two important theoretical models for active membranes; (i) Prost–Bruinsma (PB) model which takes into
account the stochastic nature of the pumps [8, 9], and (ii)
Ramaswamy–Toner–Prost (RTP) model which considers thecoupling between the protein density and membrane curvature[10]. In this paper, we use the simplified RTP model proposedby Sankararaman et al [11], and investigate the effects of thesurrounding viscoelastic media on the membrane dynamics.
As discussed in [11], one of our important assumptions isthat the membrane is impermeable to the solvent both for theactive and passive states.
In the next section, we describe the free energy of themembrane-protein system. In section III, we present dynamic
equations which take into account the viscoelasticity of thesurrounding media. On the basis of the model, we calculate
the membrane MSD and obtain a generalized Einstein relation in section IV. The case of active membranes will be discussed in section V. Finally, the obtained MSD is related to the
response function in the last section.

G(s)
G(s)
Figure 1. Asymmetric proteins (triangles) embedded in a fluid
bilayer membrane. Accumulation of the proteins induces a local
curvature of the membrane. The membrane is surrounded by aviscoelastic medium with a frequency-dependent modulus G(s).

k T⟨h(q, t )h(−q, t )⟩ = B 4 ,(3)κeff qk TκH¯〈h(q, t )ψ(−q, t )〉 = − B −1 2 ,(4)κeff χ0 qk Tκ
⟨ψ(q, t )ψ(−q, t )⟩ = B −1 ,(5)κeff χ0

II. Free energy

where kB is the Boltzmann constant, T the temperature, and κeff the effective bending rigidity given byκeff = κ(1 − κH¯ 2 /χ−0 1). Notice that κeff < κ irrespective of
the sign of H̄ because χ−0 1 > 0. The stability of the free energy
requires that the coupling parameter H̄ needs to be smallenough to satisfy the condition κH¯ 2 /χ−0 1 < 1.

A two-component fluid membrane is regarded as an infinitesimally thin two-dimensional (2D) surface embeddedin three-dimensional (3D) space. In order to describe themembrane deformation, we use the Monge gauge which isvalid for nearly flat surfaces. Here the membrane surface is
specified by its height above the flat xy-plane, h(ρ, t ), whereρ = (x, y ) and t is time. In this representation, the mean curvature of the surface is given by H = (∇2h )/2 to the lowest
order.
Next we denote the number density of the embedded
membrane proteins by ψ(ρ, t ). As shown in figure 1, theseintercalated protein molecules are assumed to induce a local
curvature of the membrane surface [12, 13]. To leading orderin gradients of h, the free energy functional of the membraneprotein system is given by [5, 10, 11]III. Dynamics
Next we discuss the dynamics of two-component membranes. We follow the argument in [11] and assume thatthe membrane is impermeable to the surrounding solvent.
This is valid over the length scales of concern in the experiments [5]. When the membrane is surrounded by viscoelastic
media, as considered by Granek [2], a generalized Langevinequation for the motion of the displacement field h(ρ, t ) is
given by1
F [h , ψ ] =d2ρ [κ(∇2h )2 − 2κH¯ ψ(∇2h ) + χ−0 1ψ 2],
(1)
2

∫

where κ is the bending rigidity, H̄ψ the protein density dependent spontaneous curvature, χ−0 1 the susceptibility which isassumed to be positive here. The above model is limited to aliner level, and we do not address nonlinear effects [9]. For the
clarity of our presentation, we only discuss tensionless membranes and do not include the surface tension energy proportional to (∇h )2. Effects of surface tension have been discussed
in detail in the literature [14].
In the following, we introduce the 2D spatial Fourier transform of h(ρ, t ) defined ast∂h(ρ, t )δF=−
dt ′ d2ρ′ Λ(ρ − ρ ′, t − t ′)0∂tδh(ρ ′, t ′)(6)+ ξ ( ρ , t ),∫∫where Λ(ρ, t ) is the retarded Oseen mobility. Its Laplace transform is given by∞1
Λ(ρ, s ) =dt Λ(ρ, t )e−st =,(7)08πη (s )ρ

∫where ρ = | ρ| and η (s ) is the frequency-dependent viscosityof the surrounding viscoelastic media (s being the frequency
in the Laplace domain). The 2D Fourier transform of equation (7) is∫h(q, t ) = d2ρ h(ρ, t )e−iq ⋅ ρ,(2)and similarly for ψ(q, t ). Using the free energy equation (1),one can easily show that the static (equal-time) correlation
functions are given by1
Λ(q, s ) =,(8)4η (s )q2

J. Phys.: Condens. Matter 27 (2015) 432001

the time correlation function ⟨h(q, t )h(−q, 0)⟩ can be obtained
from equation (13) aswith q = | q|. The average of the thermal noise in equation (6)is ⟨ξ (ρ, t )⟩ = 0, whereas its correlation obeys the following
fluctuation-dissipation theorem (FDT) [15, 16]

kBTκeff q 4
(16)(Dq 2 + s ) + Λ(q, s )κ2H¯ 2q 4 /χ−0 1
×.
(Λ(q, s )κq 4 + s )(Dq 2 + s ) − Λ(q, s )Lκ2H¯ 2q 6⟨h(q, s )h(−q, t = 0)⟩ =⟨ξ (ρ, t )ξ (ρ ′, t ′)⟩ = 2kBT Λ(ρ − ρ ′, t − t ′).
(9)Since the proteins diffuse freely on the membrane surface,
the conserved quantity ψ(ρ, t ) should obey the continuity
equation of the form

Here we have used equations (3) and (4) for the equal-timecorrelation functions. Another important assumption to derive
the above correlation function is that the stochastic thermal
noise ξ and ζ are uncorrelated with the initial condition of theheight h(q, t = 0) = h 0, which is somewhat non-trivial [15].
Using equations (3) and (16) in equation (15), the Laplacetransformed MSD of a membrane segment can be conveniently written as [2]∂ψ(ρ, t )δF
= L ∇2+ ∇ ⋅ ζ (ρ, t ),(10)∂tδψ(ρ, t )where L is the transport coefficient which is assumed tobe constant. In this diffusion, however, the hydrodynamic
interaction is neglected [17]. The last term is a conserving
Gaussian noise with ⟨ζ (ρ, t )⟩ = 0 and its correlations are
given by⟨ζi(ρ, t )ζj (ρ ′, t ′)⟩ = 2kBTLδijδ (ρ − ρ ′)δ (t − t ′).
(11)φ (s ) = 2

Equations (6) and (11) are the set of equations to beinvestigated.
The above equations can be conveniently solved by using
the Laplace transform of h(q, t ) and ψ(q, t ) (see equation (7)).
Then equations (6) and (11) can be written in the matrix form as

Hereafter we use the general viscoelasticity relation for
the frequency dependent modulus G (s ) = sη (s ). Using equation (8) for Λ(q, s ), one can rearrange the integrand in equation (17) as

where D = L χ−0 1, h 0 = h(q, t = 0) and ψ0 = ψ(q, t = 0). Aftersome calculations, we obtain the solution as

1 kBT
π 4sG (s )−1
(18)⎛ κq 3
2 ¯2
∞
q5 ⎞
D
H
κ
⎟⎟ .
dq ⎜⎜
×
+ 1 − −1
0
4χ0 G (s ) Dq 2 + s ⎠⎝ 4G (s )φ (s ) =¯ 2ψ0h(q, s ) = [(Dq 2 + s )h 0 − Λ(q, s )κHq∫

¯ 2q ⋅ ζ (q, s )]+(Dq 2 + s )ξ (q, s ) − iΛ(q, s )κHq



(13)

Although it is impossible to perform the above integral analytically, equation (18) can be regarded as a generalized Einstein
relation for a segment of two-component membranes.
We recall here that the coupling parameter H̄ should besmall enough to satisfy the stability condition of the freeenergy equation (1). When κH¯ 2 /χ−0 1  1, equation (18) canbe expanded in powers of H̄ , and the approximated form canbe obtained as¯ 4h 0 + (Λ(q, s )κq 4 + s )ψ0
ψ(q, s ) = −[LκHq¯ 4ξ (q, s ) + i(Λ(q, s )κq 4 + s )q ⋅ ζ (q, s )]−LκHq×[(Λ(q, s )κq 4 + s )(Dq 2 + s )−Λ(q, s )Lκ2H¯ 2q 6]−1(14)We use equation (13) to calculate the membrane MSD in thenext section.

⎞−1
1 kBT ⎡⎢ ∞ ⎛ κq 3dq⎜
+ 1⎟
π 4sG (s ) ⎢⎣ 0
⎝ 4G (s )⎠(19)∞⎛ κq 3⎞−2 q 5 ⎤Dκ2H¯ 2
⎥.
dq ⎜
+ −1
+ 1⎟
⎝ 4G (s )⎠ Dq 2 + s ⎥⎦4χ0 G (s ) 0∫

φ (s ) ≈IV. Membrane mean squared displacement∫

The important quantity from the experimental point of view is
the MSD of a tagged membrane segment defined asHere both of the integrals can be performed analytically. Especially, by defining dimensionless quantitiesqˆ 3 = κq 3 /4G (s ) and sˆ = (s /D )(κ /4G (s ))2/3, the second integral can be written in the formφ (t ) = 〈[h(ρ, t ) − h(ρ, 0)]2 〉
=2

kBT

⎛1
⎞
(Dq 2 + s ) + Λ(q, s )κ2H¯ 2q 4 /χ−0 1⎟.
× ⎜⎜ −(Λ(q, s )κq 4 + s )(Dq 2 + s ) − Λ(q, s )Lκ2H¯ 2q 6 ⎟⎠
⎝s
(17)

⎛ Λ(q, s )κq 4 + s Λ(q, s )κH q 2 ⎞⎛ h(q, s ) ⎞ ⎛ ξ (q, s ) + h 0 ⎞
⎜
⎟⎜
⎟=⎜
⎟,
LκH q 4
Dq 2 + s ⎠⎝ ψ(q, s ) ⎠ ⎝ iq ⋅ ζ (q, s ) + ψ0 ⎠⎝(12)×[(Λ(q, s )κq 4 + s )(Dq 2 + s )−Λ(q, s )Lκ2H¯ 2q 6]−1d2q

∫ (2π)2 κeff q4

d2q∫ (2π )2 [〈h(q, t )h(−q, t )〉 − 〈h(q, t )h(−q, 0)〉].

(15)⎛ κq 3⎞−2 q 5
κ ⎞−4/3 I (sˆ)
dq ⎜
,
+ 1⎟
= ⎛⎜
(20)
⎟
⎝ 4G (s )⎠ Dq 2 + s ⎝ 4G (s ) ⎠0
D∫

The first term in the above integrand is the equal-time correlation function given by equation (3). The Laplace transform of3∞

J. Phys.: Condens. Matter 27 (2015) 432001

G (s ) = G 0s α,(25)where 0 ⩽ α ⩽ 1. This behavior is commonly observed forvarious polymeric solutions at high frequencies. Examples are
α = 1/2 and α = 2/3 for Rouse and Zimm dynamics, respectively [20], α = 3/4 for semi-dilute solutions of semi-flexible
polymers such as actin filaments [21]. The limits of α = 0 and
1 correspond to the purely elastic and purely viscous cases,
respectively.
For such power-law viscoelastic media, the membrane partof the time-dependent MSD can be readily obtained by performing the inverse Laplace transform [2]

I(s^)

0.4

0.2

1
kBT 2α /3t ,φmem(t ) = 3/2 1/3
(26)
1/3
3 2 Γ(1 + 2α /3) κ G2/3
0

0.0
0

12

where Γ(z ) is the gamma function. In the purely elastic caseof α = 0, we have

Figure 2. The plot of the integral I (sˆ) defined in equation (21).

1k T
φmem(t ) = 3/2 1/3 1/3B 2/3 ,(27)3 2 κ G0

where

which is independent of time. On the other hand, in the purely
viscous case of α = 1, equation (26) reduces to

3

6

s^

9

Here the variable is sˆ = (s /D )(κ /4G (s ))2/3.

qˆ 5
0(qˆ 3 + 1)2(qˆ 2 + sˆ)⎡ 2π
12πsˆ ⎛ 1ln sˆ ⎞ 2 14πsˆ 3⎟sˆ +=⎢ 5/2 − 5/2 − ⎜ +32⎝3(1 + sˆ ) ⎣ 3
2 ⎠
3
35/2
10πsˆ 4 ⎛ 1
ln sˆ ⎞ 5 ⎤⎟sˆ ⎥.
−πsˆ 7/2 + 5/2 − ⎜ −
⎝3
2 ⎠ ⎦
3
(21)
The behavior of I (sˆ) is illustrated in figure 2. Performing thefirst integral in equation (19) and using equation (20), we
obtain the Laplace transformed MSD as
I (sˆ) =∫∞1
kBT 2/3t
1/3
3 2 Γ(5/3) κ G2/3
0
(28)
k T
= 0.169 B 2/3 t 2/3,
κ1/3η 0dqˆφmem(t ) =3/2 1/3

where we have replaced G0 with η0 in the last expression. This
result was previously obtained by Zilman and Granek for single component membranes [22, 23].
Next we discuss the time-dependence of the diffusive MSD
φdiff (t ) in the presence of power-law fluid media as given byequation (25). For this purpose, we first consider the asymptotic behaviors of I (sˆ) in equation (21). In the limit of sˆ → 0,
we have

4/3
2 kBT ⎛ κ ⎞ 1 ⎡⎢
33/2 κH¯ 2I (sˆ) ⎤⎥
,
s
1
φ
(
)
≈
+
⎜
⎟
(22)
33/2 Dκ ⎝ 4G (s ) ⎠ sˆ ⎢⎣2π χ−0 1 ⎥⎦

2π
2π
I (sˆ) ≈ 5/2 − 5/2 sˆ,
(29)
3
3

which is the (approximated) generalized Einstein relation formembrane-protein systems in the weak coupling limit.
We denote the first and the second terms in equation (22)as φmem(s ) and φdiff (s ), respectively, so that the total MSD isexpressed as φ(s ) = φmem(s ) + φdiff (s ). The first term φmem(s )due to the membrane itself can be rewritten as

whereas in the opposite limit of sˆ → ∞, we get⎛1ln sˆ ⎟⎞ 1
I (sˆ) ≈ −⎜ −
.
(30)
⎝3
2 ⎠ sˆ

In order to calculate φdiff (t ), it is convenient to introduce acharacteristic time scale defined by1k T
φmem(s ) = 3/2 1/3 1/3 B 2/3 ,(23)3 2 κ sG (s )κ
⎞2/ (3 − 2α),τ = ⎛⎜⎟
(31)3/2⎝ 4G 0D ⎠which coincides with the result by Granek [2], as it should.
The second term φdiff (s ) is our new contribution due to the diffusion of proteins in the membrane;
4/3
1 kBTH¯ 2 ⎛ κ ⎞ I (sˆ)
,
s
φ
(
)
=
⎜
⎟
(24)
diff
sˆ
π Dχ−0 1 ⎝ 4G (s ) ⎠which is dependent on α. (Notice that the dimension of G0 also
depends on α.) In what follows, we shall use a dimensionlesstime defined by t˜ = t /τ. Using the above asymptotic expressions in equation (24) and performing the inverse Laplacetransform, we obtain in the long time limit of t˜ → ∞ as

where sˆ = (s /D )(κ /4G (s ))2/3 as before. We note here that φdiff
vanishes in the limit of H¯ → 0.
As a working example, we consider the situation where
both sides of the membrane are occupied by the same viscoelastic media with a frequency-dependent modulus that obeys
a power-law behavior [2, 18, 19];

1 kBTH¯ 2 ⎛ κ ⎞⎜⎟ τ 4α /3 − 1
π Dχ−0 1 ⎝ 4G 0 ⎠
(32)
2π ⎡
t˜ 2α /3t˜ 4α /3 − 1 ⎤× 5/2 ⎢−⎥.
Γ[4α /3] ⎦3 ⎣ Γ(1 + 2α /3)φdiff (t˜ ) ≈4

4/3

J. Phys.: Condens. Matter 27 (2015) 432001
3

10

(a)

f
G(s)

0

10

~

~

φdiff (t )

w+
w−

-3

10

G(s)
-6

10

-6

10

-3

10

0

10
~

t

3

10

6

Figure 4. The asymmetric dipole model for a membrane protein.
The force centers are located at distances w+ and w− from thebilayer midpoint. The magnitude of the active force is denoted by
f . The membrane is surrounded by a viscoelastic medium with afrequency-dependent modulus G(s).

10

3

10

f

(b)

it with the asymptotic expressions equations (32) and (33)
which are in good agreement with the numerical result. In
both cases, we see a clear crossover from the almost liner
behavior to the power-law behavior of t˜ 2α /3. In the case ofpurely elastic media of α = 0, both the membrane and diffusive contributions to MSD become independent of time
for large t˜ [2].

0

~

~

φdiff (t )

10

-3

10

V. Active membranes
-6

10

-6

10

-3

10

0

10
~

t

3

10

6

We now discuss the dynamic properties of a two-component membrane with active pumps which exert non-equilibrium forces on the surrounding fluid. As considered byManneville et al [5], each pump is represented as a force
dipole, i.e. two force centers of opposite sign but equal magnitude separated by a distance w. This is justified because
there should be no external force on the combined system ofpump/membrane/solvent whose overall momentum is conserved. If the positive and negative force centers are locatedasymmetrically with respect to the midpoint of the membrane, the pumps exert nonzero active forces on the membrane and the surrounding media. When the force centers
are located at z  =  w+ and z  =  −w−, as shown in figure 4,
the active force is proportional to the protein density andcan be written as1

10

Figure 3. Dimensionless MSD due to the diffusion as afunction of time when (a) α = 0 and (b) α = 1/2. Hereφ˜diff = φdiff /[(kBTH¯ 2 /πDχ−0 1)(κ /4G 0 )4/3τ 4α /3 − 1] andt˜ = t /τ = t /(κ /4G 0D3/2 )2/ (3 − 2α). The dotted and the dashed lines are
asymptotic expressions given by equations (32) and (33).

Here the first term is proportional to t 2α /3 as in equation (26),and can be also expressed as (κH¯ 2 /3χ−0 1)φmem(t ), whereφmem(t ) is given by equation (26). The same result would havebeen obtained simply by replacing κ in equation (26) by κeff .
The second term, on the other hand, is proportional to t 4α /3 − 1showing a different exponent.
In the short time limit of t˜ → 0, we obtain

Fpump = fψ(ρ, t )[δ (z − w+) − δ (z + w−)]ˆz ,
(34)

4/3
1 kBTH¯ 2 ⎛ κ ⎞φdiff (t˜ ) ≈⎜⎟ τ 4α /3 − 1
π Dχ−0 1 ⎝ 4G 0 ⎠(33)t˜
× [(2α − 3) ln t˜ + (2α − 3)γ − 2α + 1],
6

where f is the magnitude of the active force (taken to be constant) and ẑ is the unit vector along the z-direction.
Here we assume that pumps are always in the active
state over the time scales of interest, and the membrane
feels the active forces only via the surrounding fluid [11].
The presence of the active force due to the pumps modifies equation (6) and can be conveniently expressed in the
Fourier-Laplace domain. Corresponding to equation (12),we then havewhere γ = 0.5772  is Euler’s constant. Since α ⩽ 1, thisMSD essentially grows like t ln(1/t ). Such a logarithmic correction leads to a time-dependent diffusivity. Equations (32)and (33) are the important results of this paper.
In figure 3, we plot the dimensionless MSD φdiff due
to the diffusion as a function of t˜ = t /τ by performing thenumerical inverse Laplace transform of equation (24) withfull I (sˆ) when (a) α = 0 and (b) α = 1/2. We also compare1

In [5, 11], the difference between the densities of pumps transferring ionsin the up and down direction is denoted by ψ.
5

J. Phys.: Condens. Matter 27 (2015) 432001⎛ Λ(q, s )κq 4 + s Λ(q, s )[κHq¯ 2 − f Ω(q)]⎞⎛ h(q, s ) ⎞ ⎛ ξ (q, s ) + h 0 ⎞
⎜
⎟⎜
⎟=⎜
⎟,
4
¯
LκHq
Dq 2 + s⎝⎠⎝ ψ(q, s ) ⎠ ⎝ iq ⋅ ζ (q, s ) + ψ0 ⎠


where

(35)

while they neglected these diffusive terms in favor of higherorder terms.

Ω(q) = −(1 + qw−)e−qw− + (1 + qw+)e−qw+
(36)
q2 2
q2 3
2
3
≈
(w− − w+
) + (w−
− w+
)+
2
3

VI. Summary and discussionis the ‘structure factor’ for the force dipole calculated in[5, 11]. Here we take into account only the first quadratic termin q and approximate as f Ω(q) ≈ −Pwq 2, where w = w+ + w−
22
− w−)/2w represents theis the size of the pump and P = f (w+force dipole energy.
The calculation of the membrane MSD closely follows
that of the previous section. In the small coupling limit ofκH¯ 2 /χ−0 1  1, we obtainIn this paper, we have discuss the dynamics of two-component fluid membranes that are surrounded by viscoelastic
media. We have assumed that membrane proteins diffuse
laterally and induce a local curvature of the membrane. We
obtained the MSD of a tagged membrane segment by taking into account the viscoelasticity of the surrounding media.
When the elasticity of the surrounding media obeys a powerlaw behavior, G ∼ (iω )α, the MSD due to protein diffusionshows a crossover from t ln(1/t ) to t 2α /3 behaviors. We havealso discussed the situation when the proteins generate active
non-equilibrium forces. The generalized Einstein relation isfurther modified by an effective temperature that depends onthe force dipole energy. The generalized Einstein relations
that we obtained for two-component membranes (see equations (22) and (37)) are useful to measure the viscoelastic
properties of cytoplasm and/or extracellular matrix.
The obtained membrane MSD (in the Laplace domain) canbe used to express the response of a membrane to transverse
forces when it is surrounded by viscoelastic fluids. Within thelinear response theory, one can write

2 kBTeff ⎛ κ ⎞ 1
⎜
⎟
3/2
3
Dκ ⎝ 4G (s ) ⎠ sˆ(37)⎡33/2 κH¯ 2I (sˆ) ⎛⎜Pw ⎞⎟⎤⎥
1,
× ⎢1 ++
2π χ−0 1 ⎝κH¯ ⎠⎥⎦⎢⎣
4/3

φ(s ) ≈

where Teff is the effective temperature defined by

⎛
⎞
¯
κHPw
⎟.
Teff = T ⎜⎜1 −
(38)
−1 ⎟
κeff χ0 ⎠⎝¯ > 0, while
This effective temperature decreases when HP¯ < 02. The other difference between
it increases when HP
equations (22) and (37) is that the second term in equation (37) has an additional correction due to the pumps.
Notice that these non-equilibrium contributions vanish
when w+ = w−, implying that an asymmetry in the positionsof the force centers (w+ ≠ w−) is necessary for finite active
forces.
Nevertheless, since the s-dependence is the same between
equations (22) and (37), the MSD of membranes containing
passive pumps and those containing active pumps exhibit thesame scaling behavior. A similar result was also obtained byLacoste and Lau [24] who showed that these two cases lead toa sub-diffusive behavior when membrane permeation is negligible and the surrounding fluid is purely viscous. For highly
permeable membranes, on the other hand, a super-diffusive
behavior was predicted by Granek and Pierrat [25] for membranes described by the PB model, and later by Lacoste andLau [24] for those represented by the RTP model. Hence thepermeability is crucial for the membrane dynamics, whereas
the stochastic nature of the pumps could also lead to shorttime super-diffusion, as discussed before [8, 24–26]. The latter
effect in the presence of viscoelastic media will be discussed
in a separate publication. We also note that the effective temperature in equation (38) is different from that obtained byManneville et al in [5]. This is because we have calculated
equation (38) from the lowest-oder diffusive terms as in [11],hω = α*(ω )Fω,(39)where hω is the Fourier transform of the mean membrane
deformation profile under the action of an external point force
F(t) at the origin ρ = 0, Fω is the Fourier transform of F(t), andα*(ω ) = α′(ω ) + iα″(ω ) is the complex response function. In
terms of the response function, the FDT can be written as[15, 16]2k T(h2 )ω = B α″(ω ),(40)ωwhere (h2 )ω is the power spectral density. Hence the membrane response function can be related to the membrane MSD
byiω
α*(ω ) =φ*(ω ),
(41)2kBT
where φ*(ω ) is obtained from φ(s ) by substituting s = iω, i.e.
an analytic continuation. Since φ*(ω ) = φ*mem(ω ) + φ*diff (ω ),the mechanical response of two-component membranes differs from that of single-component membranes.
In our future work, we shall consider an active membrane
containing proteins with two internal conformational states[27, 28], and the effects of viscoelasticity of the surrounding
media. This can be a natural model for ion channels because
they undergo random transitions between ‘on’ and ‘off’ states.
The case of two-component membranes in a quasisphericalshape (vesicles) [29] is also worth considering in order tostudy the rheology of cells.

2

Our effective temperature differs from that in [11] because the correction
term has an opposite sign.
6J. Phys.: Condens. Matter 27 (2015) 432001

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