Journal of Physics: Condensed Matter
J. Phys.: Condens. Matter 30 (2018) 175101 (9pp)

https://doi.org/10.1088/1361-648X/aab6c7

Thermal and active fluctuationsof a compressible bilayer vesicle
T V Sachin Krishnan1,2, Kento Yasuda2, Ryuichi Okamoto3
and Shigeyuki Komura2
1

Department of Physics, Indian Institute of Technology Madras, Chennai, 600036, India
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University,
Tokyo 192-0397, Japan
3
Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan2

E-mail: komura@tmu.ac.jp
Received 30 November 2017, revised 8 March 2018
Accepted for publication 15 March 2018
Published 5 April 2018
Abstract

We discuss thermal and active fluctuations of a compressible bilayer vesicle by using theresults of hydrodynamic theory for vesicles. Coupled Langevin equations for the membrane
deformation and the density fields are employed to calculate the power spectral density matrixof membrane fluctuations. Thermal contribution is obtained by means of the fluctuation
dissipation theorem, whereas active contribution is calculated from exponentially decaying timecorrelation functions of active random forces. We obtain the total power spectral density as asum of thermal and active contributions. An apparent response function is further calculated inorder to compare with the recent microrheology experiment on red blood cells. An enhanced
response is predicted in the low-frequency regime for non-thermal active fluctuations.
Keywords: vesicles, hydrodynamics, thermal fluctuations, active fluctuations, microrheology(Some figures may appear in colour only in the online journal)

1. Introduction

dissipation theorem (FDT) due to non-thermal active forces.
This breakdown of FDT seems to be caused by metabolic processes taking place inside the RBC such as the pumping actionof ion channels or the growth of cytoskeletal networks [9].
Recent experiments have further clarified the functional significance of ATP-driven non-thermal fluctuations in cells [10].
It was shown that the activity of motor protein F1 F0 -ATPaseand actomyosin cytoskeleton affect the local organization
as well as the elastic properties of the lipid membrane [11].
Moreover, membrane fluctuations act as a generic regulatory
mechanism in cell adhesion by controlling the interaction
between cadherin proteins [12].
So far, several theoretical models have been suggested todescribe the statistics of membrane fluctuations in the presence of active forces. Prost and Bruinsma proposed a model
for membrane ion pumps and showed that active fluctuations
dominate at large wavelengths [13]. Later, Ramaswamy et alconsidered a model that accounts for the coupling between
the ion pump density and the membrane curvature, which
can destabilize the membrane shape [14]. For membranes
containing ion pumps, their time-dependent fluctuation or

Vesicles idealized as closed two-dimensional (2D) elastic
sheets serve as a simple model for complex biological cells
such as the red blood cells (RBCs). Studies of vesicle fluctuations have offered significant insights into the physical properties of RBCs. In thermal equilibrium, local deformationsof lipid membranes obey well-known statistics prescribed byseveral physical parameters [1]. Out-of-plane membrane fluctuations were measured by flicker analysis [2, 3] to obtain both
static and dynamic properties such as the membrane bending
modulus [4] or the wavelength-dependent relaxation rates [5].
Since the deformation energy of a lipid membrane is comparable to thermal energy, earlier theoretical works assumed
that vesicle fluctuations are driven by thermal forces [6].
However, recent experiments have revealed the existence ofactive non-thermal fluctuations in addition to thermal fluctuations [7, 8]. Using both active and passive microrheology
techniques, Turlier et al measured the power spectral density(PSD) as well as the response function of a single RBC [9].
Moreover, they demonstrated a violation of the fluctuation
1361-648X/18/175101+9$33.00

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T V Sachin Krishnan et al

J. Phys.: Condens. Matter 30 (2018) 175101

the mean square displacement (MSD) was later obtained byLacoste and some of the present authors [15, 16]. Apart fromion pumps in membranes, cytoskeletal proteins also act assources of active forces which affect the membrane fluctuation spectrum in a significant manner [17–20].
One important aspect that has been often neglected in the
dynamics of a bilayer membrane is the role of inter-monolayer
friction and hence the membrane lateral compressibility. As
a consequence of the bilayer structure, lipid monolayers areinevitably compressible in order to allow out-of-plane membrane deformations. It is known that the compression modedue to inter-monolayer friction drastically affects the bilayer
dynamics [21–24]. Recently, the present authors have investigated the relaxation dynamics of a compressible bilayer
vesicle with an asymmetry in the viscosity of the inner andouter fluid medium [25]. Using the framework of Onsager’s
variational principle [26], we showed that different relaxation
modes are coupled to each other due to the bilayer structureof a spherical vesicle. We have also discussed the dynamics ofa bilayer membrane coupled to a 2D cytoskeleton, for which
the slowest relaxation is governed by the intermonolayer friction [27]. It was predicted that forces applied at the scale ofcytoskeleton for a sufficiently long time can cooperatively
excite large-scale modes.
In this paper, we discuss both thermal and active fluctuationsof a compressible bilayer vesicle. Using the results of hydrodynamic theory that explicitly takes into account the intermonolayer friction [25], we consider the standard Langevinequation for the membrane deformation and the lipid density
fields neglecting hydrodynamic memory whose effects will beseparately discussed in the final section. We particularly calculate the PSD matrix and discuss its frequency dependence.
Thermal contribution to PSD is estimated by using the FDT,
whereas active contribution is obtained by assuming that thetime correlation of active random forces decays exponentiallywith a characteristic time scale. The total PSD for a bilayer
vesicle is naturally given by the sum of thermal and active
contributions. We also calculate an apparent response function that can be compared with a recent microrheology experiment on RBCs [9]. We shall argue that the apparent responseis enhanced in the low-frequency regime in accordance withthe experimental result.
In section 2, we review the relaxation dynamics of a compressible bilayer vesicle based on our previous work [25].
In section 3, we start with a coupled Langevin equation ofa bilayer vesicle and discuss the properties of thermal fluctuations by using the FDT. The effects of active fluctuations
are further discussed in section 4 by assuming exponentiallydecaying time correlations of active random forces. The summary and several discussions are finally given in section 5.

Figure 1. (a) Schematic picture of a vesicle showing the reference
sphere (dashed line) of radius r0 with fluids of viscosity η − inside andη + outside. Solid red line represents a deformed vesicle configuration
whose shape is parametrized with u(θ, ϕ). (b) Cross-section of acurved region in the bilayer showing the lipid molecules in eachmonolayer. Local densities in the monolayers are indicated as φ+ andφ− which are defined on the bilayer midsurface represented by thedashed line. The inter-monolayer friction w also acts at the bilayer
midsurface.

we define a dimensionless deviation in radius u by using theradius r0 of a reference sphere as [4, 5]r(θ, ϕ)u(θ, ϕ) =
− 1.
(1)
r0

For bilayer membranes with finite thickness, bending
deformation always accompanies stretching of one monolayer
and compression of the other. Hence the membrane mono­
layers should be weakly compressible, and the local lipid density in each monolayer is allowed to vary. Let us denote thenumber of lipids in the outer and the inner monolayers as N+
and N−, respectively. Since a spherical vesicle is stable only
if the outer monolayer has more lipids than the inner one, we
impose a condition N + > N −. We denote the reference lipid
density by ρ0 = (N + + N − )/(2A0 ) where A0 = 4πr02. Let
ρ+ (θ, ϕ) and ρ− (θ, ϕ) be the variables representing the local
lipid densities in the two monolayers. We then define dimensionless local density deviations in each monolayer as

2. Vesicle relaxation dynamics

ρ± (θ, ϕ)φ± (θ, ϕ) =
− 1.
(2)
ρ0

A vesicle is composed of two opposing layers of lipid molecules whose tails meet at the bilayer midsurface. For a nearly
spherical vesicle, the membrane surface can be described byan angle-dependent radius field r(θ, ϕ). As shown in figure 1,

Further calculations can be simplified by using the difference
and the sum of the densities given by2

T V Sachin Krishnan et al

J. Phys.: Condens. Matter 30 (2018) 175101

whereφ+ − φ−φ+ + φ−(3)φ∆ (θ, ϕ) =, φΣ (θ, ϕ) =
,
2
2



d = 4w (n − 1)(2n2 + 5n + 5)E + (n + 2)(2n2 − n + 2)

respectively.
In order to describe the configuration of a bilayer vesicle,
we make use of the three independent variables u, φ∆ , andφΣ . The free energy of a compressible bilayer vesicle is given
by [22, 25]κ
F = dA σ + (2H)2 + 2λHφ∆2  2  2 
+ k φ∆ + φΣ
+ dV ∆P,(4)+ 2(n − 1)(n + 1)(2n2 + 4n + 3)E2
+ 8n2 (n + 1)2 − 5 E + 2n(n + 2)(2n2 + 1).

(11)
In the above matrices, the dimensionless parameters aredefined as
σr2
λr0
kr2
(12)
σ  = 0 , λ =, k = 0 ,
κ
κ
κ
η−
wr0
w = + , E = + .
(13)
η
η

where H is the local mean curvature that can be expressed interms of u, σ is the membrane surface tension, κ is the bending
rigidity, λ is the strength of coupling between local density
and curvature, k is the monolayer compressibility, and ∆P is
the pressure difference between the inside and the outside of
the vesicle. The first and the second integrals are performed
over the area and the volume of the vesicle, respectively.
In general, the above three variables are all time-dependent.
Since they are all functions of θ and ϕ, we expand them interms of surface spherical harmonics Ynm (θ, ϕ) as [22, 25]

The symmetric matrix a represents the inverse susceptibility
and is composed of the static quantities. On the other hand,
the matrix c is constructed by the dynamic quantities and corresponds to the kinetic coefficient matrix. Notice that both aand c are symmetric in accordance with Onsager’s reciprocal
relation [26] and depend only on n.
In our previous work [25], we showed that the two relaxation modes are coupled to each other as a consequence ofthe bilayer nature and the spherical structure of the vesicle.
We investigated the effect of viscosity contrast E = η − /η +on the relaxation rates, and found that it linearly shifts the
crossover n-mode between the bending and the slipping
relaxations. As E is increased, the relaxation rate of thebending mode decreases, while that of the slipping moderemains almost unaffected. For parameter values close to theunstable region, some of the relaxation modes are dramatically reduced [25].

∞ nu(θ, ϕ, t) =unm (t)Ynm (θ, ϕ),(5)n=0 m=−nandδφ∆ (θ, ϕ, t) = φ∆ − φ∆
0
∞ 
n

=
φ∆
nm (t)Ynm (θ, ϕ),(6)n=2 m=−n+−+−where φ∆0 = (N − N )/(N + N ). It can be shown thatΣ
the mode associated with φ is decoupled from the other twomodes and will not be considered hereafter.
The dynamics of a vesicle is affected by the viscosities ofthe outside and the inside bulk fluids, η ± [28–30], and thefriction constant between the two monolayers, w [21–25]. We
have shown that the relaxation equations for u and φ∆ aregiven by the set of coupled equations [25]
c · a unm
∂ unm
=
−
,
(7)
φ∆
∂t φ∆
τ
nm
nm ,

3. Thermal fluctuations

η + r03(8)τ=κis the bending relaxation time and the dimensionless matricesa and c are given by

and the matrices

In this section, we discuss thermal fluctuations of a compressible bilayer vesicle. In the presence of random forces, thecoupled memoryless Langevin equation for the two variables
u and φ∆ can be written as [26]∂
ζ · x(t) = −γ · x(t) + ξ(t),
(14)
∂t

where we have used the vector notationunm
x = r0
,
(15)
φ∆
nm

for each (n, m)-mode. Here

a=







κ
ζ = η + r0 c−1 , γ = 2 a,
(16)
r0


 ∆ 2 (n − 1)(n + 2) n(n + 1) + σ  − 2λ φ∆
0 + k φ0n(n + 1)c=d4k φ∆
0 − 4λ − λ (n − 1)(n + 2)



4w + (2n + 1)(1 + E)−(n + 2) + (n − 1)E4k φ∆
0 − 4λ − λ (n − 1)(n + 2)

2k




−(n + 2) + (n − 1)E,(n + 2)(2n2 − n + 2) + (n − 1)(2n2 + 5n + 5)E
3

,

(9)

(10)

T V Sachin Krishnan et al

J. Phys.: Condens. Matter 30 (2018) 175101

where c−1 represents the inverse matrix. The vector x has thedimension of length in order to assign the proper dimension
for each term in the above Langevin equation.
The vector ξ represents random forces acting directly onu and φ∆ : u
ξnm (t).
ξ(t) = φ(17)ξnm (t)In thermal equilibrium, these forces have zero mean and obey
the following FDT [26]:ξ(t) = 0,
(18)
ξ(t1 )ξ T (t2 ) = 2kB Tζδ(t1 − t2 ),(19)where ‘T’ indicates the transpose operator, kB is the Boltzmann
constant, and T is the temperature. Notice that the friction
matrix ζ (or c−1) contains off-diagonal elements, implyingthat thermal fluctuations of the two fields u and φ∆ are correlated with each other. This is a unique feature of sphericallyclosed vesicles because such a coupling effect does not exist
for planar bilayer membranes [25].
The Fourier components of the 2 × 2 correlation function
matrix can be obtained by
 ∞
 T
xx
=
dt x(t)xT (0)eiωt ,(20)ωFigure 2. Dimensionless components of thermal PSD matrix asa function of dimensionless frequency ωτ . The characteristic
bending relaxation time is τ = η + r03 /κ which is chosen here asτ = 10 s (see table 1). The dimensionless parameter values are−4
σ  = 10−2, λ = 104 , k = 108, w = 107, φ∆
0 = 3 × 10 , E  =  1
(these values can be obtained from table 1). Three independentth
components are plotted; (u2 )thω (black), (uφ)ω (blue), andth(φ2 )th(uφ)(red).
The
component
becomes
negative for smallωωfrequencies as represented by open blue circles. The slope of  
−5/3 is drawn for comparison.

−∞

where ω is the frequency. With the use of equations (18) and(19), the thermal contribution to the correlation function
matrix is given by [31]Table 1. Typical values for the physical quantities.

−1−1(xxT )th· ζ · (γ + iωζ) ,(21)ω = 2kB T (γ − iωζ)

whose components depend on n. Then the PSD matrix is
obtained by summing over all the n-modes as 2 th(u )ω (uφ)th
ω
th
Pω =
(φu)th(φ2 )th
ω
ω

nmax

=
(2n + 1)(xxT )th(22)ω.

Bending modulus: κ10−19 J

Compressibility: kSurface tension: σOutside solvent viscosity: η +Viscosity contrast: E = η − /η +Inter-monolayer friction: w
Vesicle radius: r0

10−1 J m−2
10−11 J m−2
10−3 J·s m−3
1, 100109 J·s m−4
10−5 m
10 s
80 kB T
10−1 sMembrane relaxation time: τ = η + r03 /κ
Activity strength: Su /(η + r0 )Activity relaxation time: τun=2Notice that the components of the correlation function matrix
do not depend on m but are (2n + 1)-fold degenerated. In
the following numerical estimate, we set nmax = 1000 which
covers sufficient length scales for the present analysis.
In figure 2, we plot the frequency dependence of eachcomponent of the numerically calculated PSD matrix inequation (22) for a vesicle of size r0  =  10 μm. The otherparameter values and the relevant quantities are summarized in table 1. The three curves correspond to the threecomponents of the symmetric PSD matrix (u2 )thω (black),
2 th(uφ)th(φ)(blue),
and(red).
Withinthepresent
scaled
ω
ω
2 th
th
(φ
)
(uφ)
unit, (u2 )thismuch
larger
than
and
.
Hence
ω
ω
ω
the density field fluctuations are relatively small. The PSD
2 th
2 th
comp­onent (uφ)thω lies in between (u )ω and (φ )ω , andtakes negative values in the small frequency region (shownby open blue circles).
In general, all the PSD components are almost constant in the low-frequency regime and decay in thehigh-frequency regime. The important crossover frequencyω ∗ is set by the membrane relaxation time τ = η + r03 /κ andis given by ω ∗ ≈ 1/τ . For the parameter values in table 1,
the characteristic relaxation time is chosen as τ = 10 s andω ∗ = 0.1 s−1. In the low frequency regime of ωτ  1 (orω  ω ∗), all the PSD components become independent of thefrequency, which is a characteristic feature of a vesicle with afinite size. In the high frequency limit of ωτ  1 (or ω  ω ∗),the PSD (u2 )thω shows a power-law decay with an exponent  −1.60
which is close to the exponent  −5/3 obtained by Zilman andGranek for a flat membrane without any bilayer structure[32, 33]. We also note that the frequency dependence of (φ2 )thω
is very weak in the entire region. Hence, we shall mainly
focus on the behavior of displacement fluctuations (u2 )ω in
the following discussion.
4

T V Sachin Krishnan et al

J. Phys.: Condens. Matter 30 (2018) 175101

µunm (t1 )µφn m (t2 ) = 0.
(26)In the above, active fluctuations operate directly on thedisplacement and the density fields. They can be used tomodel any of the several metabolic processes that modify themembrane dynamics. For instance, nonequilibrium fluctuations acting on the displacement field mimic processes that
affect the membrane dynamics in the radial direction such as
pumping of ion channels or cytoskeletal growth. On the other
hand, fluctuations acting on the density field can represent theaction of flippase proteins which consume ATP and facilitate
exchange of lipid molecules between the two monolayers.
In general, the membrane dynamics can be affected by several active processes with different characteristic timescales.
For simplicity, however, we consider the case when there is
only one dominant active process characterized by a single
timescale for all the modes. Then the correlations of active
random forces are assumed to have the forms

Figure 3. Dimensionless thermal PSD (u2 )thω as a function ofdimensionless frequency ωτ for E  =  1 (black) and E  =  100 (green).
The other parameters are the same with those in figure 2. The slopeof  −5/3 is drawn for comparison.

Su −|t1 −t2 |/τu
µunm (t1 )µun m (t2 ) =eδnn δmm ,
(27)
2τu

To see the effect of viscosity contrast between the inside
and the outside of a vesicle, we plot in figure 3 the PSD comp­−+onent (u2 )thω for E = η /η = 1 (the same as in figure 2) andE  =  100 (large inside viscosity [34]). When the inside viscosity becomes larger, the crossover frequency ω ∗ becomes
smaller and the fluctuation amplitude is enhanced in the
low-frequency region. This is because the effective viscosity
for E = 1 is essentially given by the combination of the two
viscosities such that ηeff ≈ η + + η −. In the high frequency
regime, on the other hand, the PSD for E  =  100 is smaller
than that of E  =  1 because the displacement fluctuations aresuppressed when the inner viscosity is larger. However, thepower-law behavior in the large frequency limit (ω  ω ∗) ofthe PSD with an exponent −1.6 ≈ −5/3 is the same for thetwo cases of E  =  1 and 100.

Sφ −|t1 −t2 |/τφ
µφnm (t1 )µφn m (t2 ) =eδnn δmm ,
(28)
2τφ

where τu and τφ are the characteristic time scales, while Su andSφ denote the strengths of the activity. Here we also assume
that these quantities depend neither on n nor on m.
The Fourier transform of the above active force correlations can be expressed in the matrix form asSu0 1 + (ωτu )2,(µµT )ω = 


S
φ
(29)
0
1 + (ωτφ )2

where the diagonal elements are Lorentzian functions. Using
this expression, we can write the active contribution to thecorrelation function matrix as

4. Active fluctuations(xxT )ac· (µµT )ω · (γ + iωζ) ,(30)ω = (γ − iωζ)−1−1whose components depend only on n. Then the active PSD
matrix can be calculated by 2 ac

0
(u )ω
ac
Pω =
0
(φ2 )ac
ω

nmax

=
(2n + 1)(xxT )ac(31)
ω,In this section, we discuss the influence of nonequilibrium random
forces on the power spectrum of fluctuations. In real cells, these
active forces are generated as a consequence of metabolic processes such as the pumping action of ion channels or the growthof cytoskeletal networks [9]. The modified coupled Langevinequation in the presence of active random forces is given by∂ζ · x(t) = −γ · x(t) + ξ(t) + µ(t),
(23)
∂t

n=2

similar to the thermal contribution in equation (22). Finally,
the total PSD is given by the sum of thermal and active
contributions;where we have added the following vector of active random
forces to equation (14); uµnm (t).
µ(t) =
(24)µφnm (t)acPω = Pth(32)ω + Pω .

Here we have implicitly assumed that thermal and active
random forces are not correlated with each other.
In figure 4(a), we plot both the thermal PSD comp­onent (u2 )thω (black, the same as in figure 2) and the active
PSD comp­onent (u2 )acω (red) as a function of ωτ when

We assume that these random forces have zero mean and theircross correlations vanish, i.e.
µ(t) = 0,(25)5

T V Sachin Krishnan et al

J. Phys.: Condens. Matter 30 (2018) 175101

Figure 5. Dimensionless thermal response function

ω(u2 )thω /(2kB T) (black) and total apparent response function2 ac
ω(u2 )ω /(2kB T) = [(u2 )thω + (u )ω ]/(2kB T) (orange) as a functionof dimensionless frequency ωτ . The other parameters are the samewith those in figure 4. The slopes of  −2/3 and 1 are drawn forcomparison.

To present our result in a different way, we plot in figure 5
an apparent total response function defined by ωPω /(2kB T)(orange) as a function of ωτ . We use the same parameters
as in figure 4. In thermal equilibrium, the response function
ωPthω /(2kB T) (black) should satisfy the relation [31]ωPthω= χ (ω),
(33)
2kB T

where χ (ω) is the imaginary part of the complex response
function matrix given by
nmax

−1
χ(ω)
=
(2n + 1) (γ − iωζ) .
(34)Figure 4. (a) Dimensionless thermal PSD (u2 )thω (black) and active
PSD (u2 )acω (red) as a function of dimensionless frequency ωτ . The
parameters for the active contribution are Su /(η + r0 ) = 80 kB T ,τu /τ = 10−2, Sφ = 0 (see table 1). The other parameters are thesame with those in figure 2. (b) Dimensionless thermal PSD (u2 )thω
2 ac(black) and total PSD (u2 )ω = (u2 )thω + (u )ω (orange) as a function

n=2

Here the response function matrix can be obtained from equation (23) when both thermal and active fluctuations are absent,
i.e. ξ = µ = 0. In nonequilibrium case, the apparent response
function ωPω /(2kB T) (orange) is enhanced. Moreover, it
increases linearly with ω in the low frequency regime, while
it exhibits a power-law decay with an exponent  −2/3 in thehigh frequency regime. As we shall discuss in the next section,these results are consistent with the experimental findings by
Turlier et al [9] who used a more detailed model to take into
account the activity.
Generally speaking, equation (33) is an alternative expression of the FDT [31]. It states that the thermal PSD matrix Pthωand the imaginary part of the response function χ (ω) are intimately related in equilibrium. In the presence of active fluctuations, the total PSD of fluctuations includes both thermalacand active contributions Pthω + Pω as in equation (32), whereas
an experimentally measured response function should not bealtered even in nonequilibrium situations. Therefore, the FDT
is inevitably violated in the presence of active fluctuations,and the extent of violation is generally characterized by Pacω.

of dimensionless frequency ωτ . The slopes of  −5/3, −2, and  −4 are
drawn for comparison.

Su /(η + r0 ) = 80 kB T , τu /τ = 10−2, and Sφ = 0 neglecting
active fluctuations of the density field φ∆ . Since the membrane relaxation time is chosen as τ = 10 s for a vesicle ofsize r0  =  10 μm, the above parameter means that the activity
time scale is roughly τu = 10−1 s [9]. The active contrib­
ution decays as  ∼ω −2 in the intermediate frequency regime,
τ −1  ω  τu−1, and further decays as  ∼ω −4 in the high frequency region, ωτu  1. In figure 4(b), we plot the total PSD
component (u2 )ω (orange) given by equation (32) as well as
the thermal PSD (black, the same as in (a)) in order to compare between the equilibrium and nonequilibrium situations.
We note that active contrib­ution becomes important in the low
frequency region ωτu  1 and enhances the total fluctuationsin the nonequilibrium active case.
6

J. Phys.: Condens. Matter 30 (2018) 175101

T V Sachin Krishnan et al

5. Summary and discussions

membrane, Zilman and Granek predicted that a membrane segment undergoes anomalous out-of-plane diffusion for which
the MSD increases as  ∼t2/3 [32, 33]. In the frequency domain,−5/3
this behavior is consistent with our result of (u2 )thasω ∼ωshown in figure 2. Therefore, we confirm that the scaling relation for a flat membrane also holds for a compressible bilayer
vesicle. Moreover, we find that this scaling behavior is notaffected by the viscosity contrast E as shown in figure 3. The
thermal PSD component (u2 )thω is independent of ω at low frequencies (ωτ  1 or ω  ω ∗) because the slowest bending
relaxation time τ is set by the finite size of a vesicle.
In our theory, both the inertia of the membrane segment andthe inertia of the surrounding fluid are neglected when we calculate the hydrodynamic response of a bilayer vesicle. Within
this approximation, we have employed the standard memoryless Langevin equation in the overdamped form. On the other
hand, it is known that the Brownian motion of a sphericalparticle in an incompressible fluid should be described by thegeneralized Langevin equation [37] with the Boussinesq force[38]. For a spherical particle in a 3D fluid, the correction to the
Stokes friction due to the fluid inertia (called the Basset force)leads to a long-time tail behavior of the velocity autocorrelation function (typically  ∼t−3/2) [39, 40].
It is beyond the scope of the present paper to calculate thehydrodynamic memory effect of fluid inertia on the membrane friction matrix ζ (see equation (16)). However, we canroughly estimate the crossover frequency ω ∗∗ above which
the fluid inertia plays a role. The minimum of the crossover
frequency is roughly given by ω ∗∗ = η + /(ρr02 ), where η + isthe fluid viscosity, ρ is the fluid density, and r0 corresponds
to the vesicle radius which is the largest size in the problem.
With the value ρ ≈ 103 kg m−3 for water and the other values
quoted in table 1, we obtain ω ∗∗ ≈ 104 s−1 and ω ∗∗ τ ≈ 105 ,where τ ≈ 10 s is the bending relaxation time of the vesicle(see equation (8)). Generally, inertial effects of the fluid
become strong at frequencies higher than ω ∗∗. In other words,
effects of fluid inertia and stress propagation are negligible forfrequencies ω  ω ∗∗. Furthermore, an analogous crossover
frequency for a membrane segment (rather than a whole vesicle) can be even larger because its size is much smaller thanthe vesicle size r0.
The frequency range considered in figures 2–5 is10−2  ωτ  104 for which we are safely allowed to neglect
inertial effects of the surrounding bulk fluid. Although our
calcul­ation relies on the Langevin equation without any hydrodynamic memory effect, the ‘large frequency’ regime in thegraphs (1  ωτ  104 ) corresponds to the ‘small frequency’
regime when compared with the above estimated crossover
frequency ω ∗∗ τ ≈ 105 related to the fluid inertia. This argument justifies the usage of the standard Langevin equation andallows us to discuss the long time behavior of membrane fluctuations. We also comment that the frequency range 10−3–
103 s−1 plotted in the graphs are easily accessible by the cur­rent experimental techniques.
Here we shall briefly mention that both the inertia of thefluid membrane and the surrounding fluid were considered byCamley and Brown [41] and by Komura et al [42, 43] for an
infinitely large and flat membrane without any out-of-plane

In this paper, we have discussed the statistical properties ofthermal and active fluctuations of a compressible bilayer
vesicle. We have analyzed the coupled Langevin equation forthe membrane deformation and the lipid density variables. In
particular, we have calculated the PSD matrix of these fluctuations and discussed the frequency dependence of each comp­onent. Thermal contribution to PSD is estimated by means of
the FDT, whereas active contribution is obtained by assumingthat the time correlation functions of active random forces
decay exponentially with a characteristic time scale. We haveassumed that the total PSD is given by the sum of thermal
and active contributions. Our results show that the nonequilibrium active contribution affects fluctuations below a characteristic frequency set by the activity timescale. To comparewith the recent microrheology experiment on RBCs [9], wehave further obtained an apparent response function which isenhanced in the low-frequency regime.
Recently, Turlier et al reported a FDT violation andargued that the cell membrane fluctuations in healthy RBCs
are actively driven [9]. The experimentally measured fluctuations and response function were in agreement with their
theoretical calculations obtained from the spectrin networkmodel of active membranes. They showed that the timescaleof switching between active and inactive states of the phosphorylation sites on the spectrin network is roughly 0.15 s[9]. In order to make a comparison with their experiment,
we have set the activity timescale τu = 0.1 s to calculate theapparent response function in figure 5. We have shown that theenhanced response function increases linearly at low frequencies ωτ  1 and decays as  ∼ω −2/3 at very high frequenciesωτu  1. These results are in accordance with the exper­
imental result by Turlier et al [9].
Most of the existing works on membrane fluctuations
describe only the dynamics of the displacement field u. In thiswork, we have taken into account the compressibility of themembrane and have also discussed the dynamics of the lipid
density field φ∆ . In order to write down the coupled Langevinequation in equation (14), we have used the fact that the
dynamics of these two variables is expressed by the productof the two symmetric matrices, i.e. the inverse susceptibility
matrix a in equation (9) and the kinetic coefficient matrix cin equation (10). The obtained PSD matrix characterizes thefrequency dependencies of fluctuation amplitudes for themembrane displacement and the lipid density. Although someworks have reported the effects of bilayer nature on the static
fluctuation spectrum of the membrane displacement [35, 36],
we are not aware of any experimental work where the lipid
density fluctuation is investigated explicitly.
The thermal PSD component (u2 )thω is related to the
Fourier transform of the mean squared displacement (MSD)
of a tagged membrane segment. This is because the MSD
and the displacement correlation function are related by[u(t) − u(0)]2  = 2[u2 (0) − u(t)u(0)] and the timedependencies of the MSD and the correlation function are
the same except for the numerical prefactor (as long as theequal time correlation function u2 (0) is finite) [26]. For a flat
7

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J. Phys.: Condens. Matter 30 (2018) 175101

References

deformations. Especially, Camley and Brown showed in theformer paper that fluid inertia creates a long-time tail in themembrane velocity correlation function. In a viscous membrane, this tail crosses over from  ∼t−1 (as in a 2D fluid) atintermediate times to  ∼t−3/2 (as in a 3D fluid) at long times.
An analogous study for membranes undergoing out-of-planefluctuations can be interesting. However, it was also pointed outthat a long-time tail behavior in a membrane becomes important only when the correlation function reaches 10−6 of its initial value, which would be effectively unobservable for vesicles[41].
On the other hand, the memory effect due to the viscoelasticity of the surrounding media is worth considering. Whenthe outer fluid is viscoelastic, the constant viscosity η in the
Stokes equation should be replaced by a frequency dependent
viscosity η(ω) in equation (16) according to the correspondence principle of linear viscoelasticity [44, 45]. Granek useda generalized Langevin equation with a memory kernel tocalculate the membrane MSD [46]. He also showed that,
if the complex shear viscosity of the fluid is described by apower-law behavior η(ω) ∼ (iω)α−1, a tagged membranesegment would exhibit an anomalous subdiffusive behavior
and its MSD should grow as  ∼t2α/3. For α = 1, we recover
the scaling behavior for a purely viscous medium as described
before. Thus, we expect the PSD to scale as  ∼ω −(2α+3)/3
when the vesicle is surrounded by a viscoelastic medium
which can mediate mechanical memory effects.
Finally, it is worth mentioning that we have chosen simple
forms for active noise correlations that capture only limited features of active processes such as their timescale and strength. As
an extension of the present work, we shall consider a coupling
effect between active fluctuations and membrane local curvature. Such a coupling can be taken into account through a wavenumber dependent activity strength [17], and it will alter thefrequency dependence of active fluctuations. In general, the frequency dependency of PSD can be used to characterize and distinguish between different active noise sources in experiments.

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Acknowledgments
TVSK acknowledges Tokyo Metropolitan University for the
support provided through the co-tutorial program. SK and
RO acknowledge support by Grant-in-Aid for Scientific
Research on Innovative Areas ‘Fluctuation and Structure’(Grant No. 25103010) from the Ministry of Education,
Culture, Sports, Science, and Technology (MEXT) of Japanand by Grant-in-Aid for Scientific Research (C) (Grant No.
15K05250) from the Japan Society for the Promotion of
Science (JSPS).
ORCID iDs
Shigeyuki Komura

https://orcid.org/0000-0003-3422-5745

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