Soft Matter
PAPER
Dynamics of passive and active membrane tubes
Cite this: Soft Matter, 2020,
16, 9319

Sami C. Al-Izzi, abcd Pierre Sens,
Shigeyuki Komura *g

cd

Matthew S. Turneref andUtilising Onsager’s variational formulation, we derive dynamical equations for the relaxation of a fluid
membrane tube in the limit of small deformation, allowing for a contrast of solvent viscosity across themembrane and variations in surface tension due to membrane incompressibility. We compute therelaxation rates, recovering known results in the case of purely axis-symmetric perturbations and making
new predictions for higher order (azimuthal) m-modes. We analyse the long and short wavelength limitsof these modes by making use of various asymptotic arguments. We incorporate stochastic terms to ourdynamical equations suitable to describe both passive thermal forces and non-equilibrium active forces.
Received 15th July 2020,
Accepted 8th August 2020

We derive expressions for the fluctuation amplitudes, an eﬀective temperature associated with active

DOI: 10.1039/d0sm01290d

experimental assay that might enable measurement of these fluctuations to infer the properties of thefluctuations, and the power spectral density for both the thermal and active fluctuations. We discuss anactive noise. Finally we discuss our results in the context of active membranes more generally and give

rsc.li/soft-matter-journal

an overview of some open questions in the field.

1 Introduction
Membrane tubes, formed by bilayers of phospholipid molecules, are structures that are ubiquitous in cells. They are vitalto the function of many organelles including the peripheral
Endoplasmic Reticulum (ER).1 So-called membrane nanotubes
have been identified more recently and implicated as a pathway
in inter-cellular signalling.2 Membrane tubes can be formed
from a patch of membrane by the action of a localised normal
force on the membrane, e.g., from molecular motors such as
myosin or kinesin, or from the formation of a coat of intrinsically curved proteins on the membrane.3–5
From a statistical mechanics perspective, there alreadyexists a significant body of work on the thermal fluctuationsof membrane tubes.6,7 A striking prediction from these theories
is that the bending modes of the tube are critical in the longwavelength limit, meaning that fluctuations are predicted
to diverge at the linear level. Anharmonic terms in the freeaSchool of Physics & EMBL-Australia node in Single Molecule Science,
University of New South Wales, Sydney, AustraliabDepartment of Mathematics, University of Warwick, Coventry CV4 7AL, UK
cInstitut Curie, PSL Research University, CNRS, Physical Chemistry Curie, F-75005,
Paris, FrancedSorbonne Université, CNRS, UMR 168, F-75005, Paris, FranceeDepartment of Physics & Centre for Complexity Science, University of Warwick,
Coventry CV4 7AL, UKf
Department of Chemical Engineering, University of Kyoto, Kyoto 615-8510, Japang
Department of Chemistry, Graduate School of Science, Tokyo Metropolitan
University, Tokyo 192-0397, Japan. E-mail: komura@tmu.ac.jp

This journal is © The Royal Society of Chemistry 2020

energy then control the excess area and associated length
fluctuations.6 These studies have gained contemporary relevance with the development of fluorescence microscopy andoptical tweezers techniques able to infer the power spectral
density of fluctuations on tubes pulled from Giant Unilamelar
Vesicles (GUVs).8 Such studies may also have some relevanceto the statistical mechanics of tubular networks9,10 and on thelength fluctuations of tubes held by a fixed force.11
Work on the dynamics of membrane tubes has focused on
the simplified axisymmetric case, in particular the dynamics
of the pearling instability of membrane tubes,12–15 and the
dynamics of tether pulling from a GUV or cell.16–18 A further
area of study is that of particle lateral mobility within themembrane.19–21 These examples demonstrate how the curved
geometry of the membrane tube can lead to rich physicalphenomena, in the form of visco-elastic couplings21 and nonNewtonian rheological behaviour.17,18
Recently there has been an increased interest in quantifying
the dynamics and fluctuations of membrane tubes in a biological setting1 and in understanding the transport dynamics
of cargo within membrane tubes.2,22,23 Such scenarios are,in general, driven far from equilibrium by active forces from
cytoskeletal interactions24 or proteins such as proton pumpschanging conformation when consuming ATP.25,26 In order tohave a complete physical understanding of such processes one
needs to develop a theory of active membrane tubes.
In this paper, we will focus on the dynamics of membranetubes, deriving equations of motion from Onsager’s variational
principle in the manner of ref. 27–29, and analysing theSoft Matter, 2020, 16, 9319--9330 | 9319

Paperrelaxation dynamics in Fourier space. We then consider thecase where stochastic forces act on the membrane and derive
the statistical properties of the shape undulations, in particularfocusing on the case where active noise dominates. Here theterm active refers to a noise term which breaks detailed balance.
Such active membrane systems have been studied extensively forthe case of flat membranes25,30,31 and spherical vesicles.24,29
These descriptions also have relevance for experiments incorporating active proteins into GUVs32 and in the analysis of red blood
cell flicker.24,33,34
In Section 2 we derive hydrodynamic equations of motion
for a membrane tube utilising the Osager’s variational principle.
We decompose these equations of motion in Fourier space,allowing the equations to be reduced to an overdamped equation
for the perturbations in the radial displacement of the tube
surface. The relaxation dynamics of the radial displacement are
discussed in Section 3. The onset of a pearling instability is
discussed in Section 4. In Section 5 we analyse the fluctuationsof membrane tubes due to stochastic forces of two types; the firstcorresponding to thermal fluctuations and the second corresponding to a simple form of active noise that breaks detailedbalance. We derive the fluctuation spectra for these ‘‘active’’ tubes
and calculate a wave length-dependent eﬀective temperature ofsuch fluctuations, to be compared to thermal fluctuations.
Finally in Section 6 we discuss possible ways to quantify theparameters in our active fluctuations model from experiment,
the relation of our work to previous studies and someopen problems in the study of membrane tubes and active
membranes more generally.

2 Membrane tube dynamics
Here we introduce the geometry required to build our model,
and the notation we will use. We then derive equations of
motion for the relaxation dynamics of the membrane tube in
the linear response regime.
2.1

Geometry

We treat the membrane as a two-dimensional manifold,
M C R3. Vectors in the ambient space will be denotedx A R3 and vectors in the tangent bundle to the manifold as
x A T(M). We parametrise the manifold, M, with the vectorr = (r cos y, r sin y,z) where r(y,z,t) = r0[1 + u(y,z,t)], see Fig. 1(a).
We will consider the small deformation limit where u { 1.
Local tangent vectors can be induced on the surface by taking
derivatives with respect to y and z, giving ey = qyr and ez = qzr,
- - respectively. A complete triad can be defined by {ey,ez,n} wheren = (ey  ez)/|ey  ez| is the normal vector to the surface. The
metric and second fundamental (bilinear) forms are then- defined as g = gijdXidX j = eiejdXidX j and b = bijdXidX j whereibij = nqjei and dX are the coordinate basis of the cotangent
bundle. By raising the indices of the second fundamental form
with the metric and taking the trace and determinant, we
define the mean curvature, H = bii/2, and Gaussian curvature,
K = det bij.

9320 | Soft Matter, 2020, 16, 9319--9330

Soft Matter

Fig. 1 (a) Cross section of a membrane tube with time-dependent
undulation of its radial position r = r0[1 + u(y,z,t)] about the equilibrium
radius r0. Here Z+ and Z are the viscosity of the exterior and interiorambient fluid, respectively. (b) Surface plots of the Fourier decompositions,Pu¼
uqm eiqzþimy for qr0 = 1.
q;m

The membrane is assumed to behave as a fluid in thetangential direction and we define a vector flow field of thelipids in the membrane as v A G(T(M)), where G(T(M)) is asection of the tangent bundle.
2.2

Elastic free energy

The elastic free energy of an incompressible fluid membrane
can be written using the Helfrich–Canham–Willmore energy35–37
ð
h
i
k
(1)
Fel ¼
dA s þ ð2HÞ2 ;2M
where s is the surface tension, k the bending rigidity and the area
pﬃﬃﬃﬃﬃ
element is given by dA ¼ jgjdydz. We have integrated out
the contribution from the Gaussian curvature and saddle splay
modulus by assuming no changes in topology and treat the tube
as having infinite length. We also neglect spontaneous curvature(set it equal to zero), assuming that we are considering a symmetric bilayer.
The area element and the mean curvature squared are given,
up to second order, by
1dA  r0 1 þ u þ r02 @z2 u þ @y2 u dydz;
(2)
2
H2 


1 1
1
 r02 @z2 u þ @y2 u þ u þ f2uðr02 @z2 u þ 3@y2 uÞ
r02 2
2

 r02 ð@z uÞ2 þ ðr02 @z2 u þ @y uÞ2 þ ð@y uÞ2 þ 3u2 g :(3)ÐIf we take the full elastic free energy F ¼ Fel  DP dV,
where DP = P  P+ is the hydrostatic pressure diﬀerence acrossThis journal is © The Royal Society of Chemistry 2020

Soft MatterPaper

the membrane, then the ground state r = r0 must satisfy themodified Laplace equationsk DP ¼ 0;
r0 2r03(4)

in order to minimise the elastic free energy.
For the Onsager’s formulation of membrane dynamics,
we need to know the rate of change of the free energy. This isgiven by
 
ð
k 3
1
5

F ¼ dydz 2 u þ r02 @z2 u þ r04 @z4 u þ @y2 u
r0 2
2
2
(5)



2 2 2
4
2 2
2
_
þ 2r0 @y @z u þ @y u  s u þ r0 @z u þ @y u r0 u;
where a dot () indicates a time derivative and we have made useof eqn (4), or equivalently the constraint that total volume isÐÐ
preserved dydzu ¼  dydzu2 =2. Note that F is a functional:
only of the normal velocity vn = r0u + O(u2) and not thetangential components of membrane velocity v. This is also
true for arbitrarily large shape perturbations.38
2.3

Dissipation and constraints

We will consider only the dissipation due to the ambient fluidas this is the dominant dissipative mechanism at large lengthscales.39 At the scale of cell membranes (10 nm–100 mm),viscous dissipation dominates the dynamics of the fluid. Hence
we neglect the contribution from inertia and assume zero
Reynolds number.40
We define the velocity in the ambient fluid as V = V aea,3
where we use Greek indices to denote coordinates in R , andsummation over repeated indices is assumed. The dissipation
functional for the bulk fluid is given by41
ð
ab
P ¼
dV  Z D
;
(6)
ab D
V

where Z is the viscosity and dV  is the volume element of theexterior (V+) and interior (V) regions respectively, as shownin Fig. 1(a). Moreover, D
ab = (raV b + rbV a )/2 is the rate-ofstrain tensor where ra is the ambient covariant derivative in
R 3.
Our system has several constraints which, in the Onsager’s
formulation, will be imposed using Lagrange multipliers.42
Firstly, the membrane and ambient fluid are incompressible
so must satisfy the following conditions
raV a = 0,

riv  2vnH = 0,(8)

for the membrane. Further constrains come in the form of noslip and no-permeation boundary conditions on the bulk fluidat the membrane: a i:
V |r0 = vi,(9)
V r|r0 = r0u
where the Latin indices denote the projection of the velocities
in R3 onto the tangent basis of the membrane.

This journal is © The Royal Society of Chemistry 2020

Rayleighian and equations of motion

To derive the full equations of motion using the Onsager’s
formulation, we must first write down the Rayleighian.27,28,41,43
The full Rayleighian for the system is found by taking the sumof the rate-of-change of free energy for the system, eqn (5), andthe energy dissipations, eqn (6), and adding in the constraints
on the system using Lagrange multipliers. This formulation isequivalent to Onsager’s kinetic equation with reciprocal coeﬃcients, but recast as a variational formalism.43–45
Thus our Rayleighian reads
ð
ð

_ 
dV þ Pþ ra V aþ
R ¼ Pþ þ P þ F þ dAzðri vi þ uÞ
Vþ

M



ð
V

h

dV  P ra V a þ

ð
M

h
i
aþ
i
i
dA½mþ
i ðV jr0 Þ  v

i

a
i
i
þ
þ m
V rþ jr0  r0 u_ þ l V r jr0  r0 u_
i ðV jr0 Þ  v þ l

i

;

(10)

where z, P, m
i and l are the Lagrange multipliers imposing ourconstraints. Note that we choose the sign for P and z so that theycorrespond to pressure and surface tension variation, respectively.
We now proceed to use Onsager’s principle and minimise theRayleighian to find the equations of motion for the membrane.42
Taking variations of eqn (10) with respect to V a|r0 yields
8ZD
ri |r0  mi = 0,

(11)



8ZD
rr|r0  P  l = 0,

(12)showing that m
i and l correspond to the traction forces acting

on the membrane.
Extremising with respect to vi gives
riz  m+i  mi = 0.

(13)

By eliminating the Lagrange multipliers, we further haveriz + Z+D+ri  ZDri = 0,(14)which is simply tangential force balance on the membrane.
:
Taking variations with respect to r0u and eliminating l,
we obtain normal force balance on the membrane

k 3
1
5
0 ¼ 3 u þ r02 @z2 u þ r04 @z4 u þ @y2 u þ 2r02 @y2 @z2 u þ @y4 u
r0 2
2
2


(7)

for the bulk fluid and
i

2.4

s
z
þ
 

ðu þ r02 @z2 u þ @y2 uÞ þ  Zþ Dþ
rr þ P þ Z Drr  P :
r0
r0
(15)

Varying with respect to z simply gives the membrane incompressibility condition, eqn (8). Varying with respect to V a andP gives the usual Stokes equations and incompressibility
condition, respectively,Zr2V a = raP,
2.5

raV a = 0.

(16)

Fourier mode decomposition

Next we solve the equations for the bulk fluid and calculate thetraction forces on the membrane. Here we make use of theSoft Matter, 2020, 16, 9319--9330 | 9321

PaperSoft Matterknown solution to the Stokes equations in cylindrical coordinates given by ref. 40
~ ¼ rf þ r  c~ez þ r@r rx þ @z x~
ez ;
(17)
V
P = 2Zqz2x,(18)

where (f,c,x) are scalar functions that each satisfy the Laplace
equation. We decompose these functions in Fourier space in yand z in terms of the coordinate systems harmonic basis
0  1
0 1
Fqm
f
C
B
C XB
B
C 
B C
iqzþimy
B C
C
;
(19)
Bc C ¼qm CPqm ðrÞe
B
@
A
@
A
q;m
x
X
qm

where the bulk velocity terms can be expressed using F
qm,
:

z
C
qm and Xqm, and they are thus just functions of uqm and vqm.
zSolving for vqm and zqm allows us to write Fqm, Cqm, and Xqm in:terms of uqm.
Finally, by substituting back into eqn (15), we obtain thelinear response equation for the shape in Fourier space as:B Q,m uqm = A Q,m uqm,
(24)
where Q = qr0 and

2 1
5
3
AðQ; mÞ ¼ Q2 þ m2  Q2  m2 þ
2  2
2

 S 1  Q2  m2 ;BðQ; mÞ ¼

withP
qm ðrÞ ¼

8< Pþ
qm ðrÞ ¼ Km ðqrÞ;: P ðrÞ ¼ Im ðqrÞ:

(20)

qm

In the above, Im(qr) and Km(qr) are modified Bessel functions of
the first and second kind, respectively.
We now introduce the Fourier transform as defined byP
f ðy; zÞ ¼
fqm eiqzþimy . The form of surfaces given by theq;mm-mode perturbations is shown in Fig. 1(b). Applying theboundary conditions on the bulk flow in Fourier space allows: y
us to find F
qm, Cqm, and Xqm in terms of the variables u, v , andvz. The boundary conditions in Fourier space read01
r0 u_ qm
BC
B vy CB qm C
@
A
vzqm
0





2 
F
qm @r Pqm þ ðim=rÞCqm Pqm þ Xqm @r Pqm

1

B
C
B
C
B ðim=rÞF P  C @r P þ imX @r P  P =r C
¼B
qm qm
qm
qm
qm
qm
qm
C
B
C
@
A




iqF
qm Pqm þ irqXqm @r Pqm þ Pqm =r

;

r¼r0

(21)
where the right hand side is evaluated at r = r0. Then we canmake use of the continuity equation to eliminate vyqm =::
r0(qvzqm  iuqm)/m and find F
qm, Cqm, and Xqm in terms of uqmz
and vqm, which are given in Appendix A.
In Fourier space, the components of the tangential force
balance equation read


im
im rþ
yþ
zqm þ Zþ r@r ðVqm=rÞ þ Vqm
r0
r
r¼r0
(22)


im r

y
 Z r@r ðVqm =rÞ þ Vqm
¼ 0;
r
r¼r0




z+

z
iqV r
iqzqm + Z+ iqV r+
qm + @ rV qm r=r0  Z
qm + qrV qm r=r0 = 0(23)9322 | Soft Matter, 2020, 16, 9319--9330

r03 zqmrþþ ðPþ
qm  Zðw þ 1Þ@r Vqm
k r0
i 
1
r
 P
Þ
:
þ Zð1  wÞ@r Vqm
qm r¼r0 u_ qm

(25)

(26)

In the above, we have introduced the notations S = sr02/k,
Z = Z+ + Z, and w = (Z+  Z)/Z. The function B(Q,m) issometimes referred to as the Fourier transform of the inverseof the Oseen kernel.46 In the absence of hydrostatic pressure
diﬀerence across the tube membrane, i.e., DP = 0, the tube
radius is set by a balance between bending and tension
stresses, and we have S = 1/2.5 Note that zqm, P
qm andV r,y
qm are, after solving the tangential force balance equations,::simply proportional to uqm. Hence by dividing by uqm in eqn (26)
we get the friction coeﬃcient at the linear response level.
The exact form of B is in general too complex to write down
except for the m = 0 case for which it isBðQ;0Þ ¼

Zr03
ð1 þ Q2 Þkð1  wÞI12ðw þ 1ÞK12þ;
2QI1 I0  Q2 ðI02  I12 Þ 2QK1 K2  Q2 ðK12  K02 Þ(27)

where all the modified Bessel functions are evaluated at r = r0.
For larger values of m, we evaluate B numerically using
Mathematica (Wolfram Research, Champaign, IL). It is interesting to note that, compared with the spherical case, the m andQ modes are not constrained by the other as they are in the caseof spherical harmonics.28
We note that the behaviour of B in the limit of Q { 1 can becomputed for the m = 0 mode as


Zr03
wþ1
;
(28)

2ð1

wÞ
þ
BðQ; 0Þ 
2 logð2=QÞ  1  2g
kQ2
where g E 0.577 is the Euler–Mascheroni constant.

3 Relaxation dynamics
For some initial condition at t = 0, the solution to eqn (24) is
given by uqm(t) = uqm(0) exp[l(Q,m)t], where l(Q,m) = A(Q,m)/
B(Q,m) describes the rate at which an undulation in the tubes
radius decays back to the ground state. In this section, weThis journal is © The Royal Society of Chemistry 2020

Soft Matter
analyse the form of l to understand the stability and relaxation
dynamics of the Fourier modes in the shape of themembrane tube.
The decay rates l(Q,m) are plotted as a function of dimensionless wavenumber Q = r0q in Fig. 2 and 3. Throughout, we fix
the total viscosity Z = Z+ + Z and vary the relative viscosityw = (Z+  Z)/Z. The plots are shown in units associated with thetime-scale of the total viscosity given by t = Zr03/k. As the
dynamics remains unchanged on changing the sign of bothm and Q, we will restrict our discussion to m, Q Z 0.

Paper
Let us first discuss the m = 0 modes. For the value of S = 1/2,
corresponding to an equilibrium ground state with no netpressure, the relaxation rate is given by
AðQ; 0ÞBðQ; 0Þ
1
3¼ Q4  Q2 þ  Sð1  Q2 Þ
2
2
 3

Zr0 ð1 þ Q2 Þ
ð1  wÞI12

2QI1 I0  Q2 ðI02  I12 Þkl0 ðQÞ ¼

þFig. 2 Plot of dimensionless decay rate lt as a function of dimensionlesswavenumber Q for the modes uqm when (a) m = 0 and (b) m = 1 for varying
relative viscosity w = (Z+  Z)/Z. We keep Z fixed as the rate l has been
non-dimensionalized by the viscous time associated with the total viscosityt = Zr03/k. The dimensionless surface tension is given by S = 1/2 such thatthe ground state has no hydrostatic pressure discontinuity.

This journal is © The Royal Society of Chemistry 2020ðw þ 1ÞK12
2QK1 K2  Q2 ðK12  K02 Þ

(29)

1
;

which is positive and the undulations are always stable. The
above expression gives the scaling behaviour l0 B Q2 in thesmall Q regime, as shown in Fig. 2(a). As expected in the large Q
limit, the scaling behaviour coincides with that of a flat
membrane where l0 B Q3, and all relative viscosities converge
to a universal relaxation rate. This is a consequence of theapproximate symmetry between the interior and exterior atsuch small length scales.
The behaviour of l for m = 1 is evaluated numerically andshown in Fig. 2(b). For large values of Q, the scaling behaviour
is again like that of a flat membrane for the same reason as them = 0 modes. However, at small values of Q, some interesting
phenomena is encountered, which is strongly dependent on therelative viscosity. As Q - 0, the external dissipation due tothe tube being dragged through the fluid dominates therelaxation rate.
When |m| = 1, the internal motion simply corresponds tolocally translating the cross section of the tube. In the longwavelength limit these gradients in velocity become small
and thus the dissipation associated with the interior fluid
decreases. In this long wavelength limit, the tube behaves likean elastic rod immersed in a viscous fluid in terms of its
relaxation, and tends slowly towards l B (g + log Q)Q2 as
the interior dissipation becomes less dominant. This scaling
behaviour for a continuous Zimm model of an elastic rod under
tension is discussed in Appendix B.
Notice that the case w = 1 corresponds to Z+ = 0. Hence, inthe long wavelength limit, there is essentially no friction, andthe m = 1 mode relaxation rate diverges as Q - 0. The crossover
between interior to exterior dominant dissipation means that,in the limit of Z+ { Z, the relaxation rate can be nonmonotonic in Q. Hence l first decreases and then increases
at intermediate Q before being screened by the exterior viscosity
at long wavelengths, as seen in the case w = 0.95 in Fig. 2(b).
For higher modes of |m| Z 2, the dissipation is dominatedin the long-wavelength regime by the gradients in velocity
coming from the cross-sectional deformations of the tube.
Thus, as Q decreases, the relaxation rate becomes constant,
as shown in Fig. 3. This constant increases with m because eachsuccessive mode costs more bending energy to excite, so will
relax faster. In the high Q limit, the relaxation rate scales likethat of a flat membrane with l B Q3 for all m. We plot theSoft Matter, 2020, 16, 9319--9330 | 9323

PaperFig. 3 Plot of dimensionless decay rate lt as a function of dimensionlesswavenumber Q for the modes uqm when m = 2, 3, 4, 5. We keep Z fixed as
the rate l has been non-dimensionalized by the viscous time associatedwith the total viscosity t = Zr03/k. The dimensionless surface tension is
given by S = 1/2 such that the ground state has no hydrostatic pressure
discontinuity.

|m| Z 2 modes only for w = 0 because changing w does notnoticeably alter the relaxation rates for these modes as thehigher m modes behave like a flat membrane, and hence theirrelaxation depends only on the constant total viscosity, Z.

4 Pearling instability
For the m = 0 mode, there is an instability when the tube isplaced under high surface tension.14 The growth rate ordispersion relation of such an instability is given by thenegative of eqn (29). The threshold for the instability at Q = 0
is given by S = 3/2, which corresponds to the point when A(0,0)
changes sign.12,15
This instability is analogous to the Rayleigh–Plateau
instability in a column of fluid,47,48 where forces arising fromthe interface surface tension act to minimise the total interfacearea-to-volume ratio, and thus the fluid breaks up into spherical droplets. Similar forces arise in the case of membranetubes although these are counteracted by the presence of
membrane bending rigidity, k. The exact form of this instability
growth rate was found previously in ref. 14, where only axisymmetric perturbations were considered, and was shown to coincide with earlier works when variations in surface tension were
neglected.12,15,49
For large surface tension S and similar values of viscosity(w E 0), the wavenumber with the maximum of the growth rate,
argmaxQ{l0(Q)}, is a monotonic function of S which rapidly
approaches Q E 0.6.14,15 The growth rate, l0, is plotted inFig. 4 for diﬀerent values of the relative viscosity w. Note that9324 | Soft Matter, 2020, 16, 9319--9330

Soft Matter

Fig. 4 Plot of dimensionless growth rate lt for the pearling instability as
a function of dimensionless Q for varying values of relative viscosity w. We
keep Z fixed as the rate l has been non-dimensionalized by the viscoustime associated with the total viscosity t = Zr03/k. The dimensionlesssurface tension is set here to S = 100.

short wavelength perturbations, Q Z 1, are always stable as thesurface tension terms in A(Q,0) are always positive for Q Z 1.

5 Fluctuations of membrane tubes
We now consider the relaxation dynamics of the tube underthermal and active fluctuations. This is given by adding
thermal and active forces to eqn (24)

:


ac
B Q,m uqm = A Q,m uqm + xth(30)
qm + xqm,
acwhere xthqm and xqm denote the passive (thermal) and active
forces respectively. The statistical properties of the thermal
noise are given in the standard way

hxthqmðtÞi = 0,
th
0
hxth
qm ðtÞxq0 m0 ðt Þi ¼

2kB T
BðQ; mÞdqq0 dmm0 dðt  t 0 Þ;
k

(31)
(32)

where kB is Boltzmann constant, T is the temperature and thestar, (*), denotes the complex conjugate.
For the active fluctuations, we write
hxacqmðtÞi = 0,
ac
0
hxac
qm ðtÞxq0 m0 ðt Þi ¼

(33)

2

½FðQ; mÞ jtt 0 j=tacedqq0 dmm0 ;
2tac

(34)

where tac is the correlation time of the active forces, and thephysics of the active processes will be captured in our choiceof active force density, F(Q,m).30 We will consider only thesimplest case where direct forces acting on the membrane isconstant, i.e., F(Q,m) = F, although more realistic models could
be considered.24,32

This journal is © The Royal Society of Chemistry 2020

Soft Matter
5.1

Paper

Thermal fluctuationsFirst we consider the case when there are no active fluctuations,
i.e., xac
qm = 0. Solving eqn (30) by Fourier transform in time(assuming any initial conditions have decayed away) yields thefollowing covariancehuqm ðtÞuq0 m0 ðt 0 Þith ¼

kB T lðQ;mÞjtt 0 j
dqq0 dmm0 :
e
kA

(35)

where th identifies these as thermal (passive) fluctuations. If we
Fourier transform in time with the convention uðtÞ ¼Ð
do=ð2pÞuðoÞeiot we can also find the frequency domain covariancehuqm ðoÞuq0 m0 ðo0 Þith ¼

2kB TBdqq0 dmm0 dðo  o0 ÞkðA2 þ B2 o2 Þ

(36)

the tube. The equilibrium properties of such fluctuations, suchas excess area and length fluctuations, are discussed in ref. 6.
Due to the one-dimensional character of these modes, it isexpected that the criticality will be preserved even in theanharmonic regime.6
5.2

Active fluctuations

Turning our attention to the case of active fluctuations, we willfind the statistical properties of the shape fluctuations due topurely active noise. We assume that the active and thermal
noise terms are uncorrelated, hence the total shape fluctuations can be found by simply adding the active and passive
contributions.
To find the covariance, we Fourier transform in time to findF 2 dqq0 dmm0 dðo  o0 Þwhich we will make use of later.
huqm ðoÞuq0 m0 ðo0 Þiac ¼ 2:(38)
ðA þ B2 o2 Þð1 þ tac 2 o2 ÞThen the equal time variance is given by"#1 Inverting the Fourier transform for o and o 0 gives the covarikB T
1
53
ðQ2 þ m2 Þ2  Q2  m2 þ  Sð1  Q2  m2 Þ
hjuqm j2 ith ¼; ance in time, which after some algebra gives
k
2
2
2
0
0
F 2 Atac ejtt j=tac  BeðA=BÞjtt j(37)
huqm ðtÞuq0 m0 ðt 0 Þiac ¼
dqq0 dmm0 :2A
A2 tac 2  B2
6
in accordance with the equipartition theorem. The equal time(39)
covariance is plotted against Q in Fig. 5 for m = 0, 1, 2, 3. Herethe dimensionless surface tension is chosen to be S = 1/2 and Hence the equal time variance becomes
we choose a typical order of magnitude for the bending rigidity,
F2
hjuqm j2 iac ¼:(40)
k/(kBT) = 10.50
2AðAtac þ BÞ
A striking prediction is the divergence of the m = 1 modes,
i.e., criticality, with a power-law scaling h|uq1|2ith B Q2, in the
As this quantity depends on the dissipation in the system
limit Q - 0. This criticality is due to the m = 1 modes being through the factor B, it is obvious that the fluctuations are
one-dimensional Goldstone modes in the long wavelength nonequilibrium. If we assume that the activity correlation time
limit. In other words, for small Q, they only locally translate is an order of magnitude more than the viscous time scale,the cross-section of the tube which does not alter the energy of t /t E 10, and that the forces exerted to the membrane isac

Fig. 5 Equal time variance for thermal fluctuations as a function ofdimensionless wavenumber Q for modes m = 0, 1, 2, 3. The dimensionlesssurface tension is S = 1/2 and the bending rigidity is k/(kBT) = 10.

This journal is © The Royal Society of Chemistry 2020

about 1 pN over an area r02, then one can estimate F B 1–10.29
Using these parameters along with w = 0, F2 = 2.5 we plot in
Fig. 6 the active fluctuations given by eqn (40). A peak in them = 0 mode is apparent, the position of which depends on therelative value of the active time-scale tac/t. The decay in active
fluctuations of the m = 0 mode as Q - 0 is due to the viscousdamping that suppresses such non-equilibrium fluctuations.
This does not appear in the thermal case as the thermal force
fluctuation scales like B(Q,m).
The divergence at small Q observed in the m = 1 modes isretained but with an exponent that diﬀers from the thermalcase according to h|uq1|2iac B Q2 log Q. It is interesting tobriefly consider the more general case when F is not constant,and its eﬀect on the critical nature of the m = 1 mode.
In particular if, say by a curvature sensitive coupling, F(Q,m) B
Qn/2 in the small Q limit, then the fluctuations would scale ash|uq1|2iac B Qn2 log Q. For example, if the active noise is coupled
to the change in curvature from the ground state of the tube, thenone might expect F(Q,m) B 1/(1  m2  Q2), or n = 4(see Appendix C), which leads to bending mode fluctuations thatscale like h|uq1|2iac B Q6 log Q.
As the m = 1 mode is the softest on the tube, it dominates
the real space fluctuations and, by Parseval’s Theorem, we have

Soft Matter, 2020, 16, 9319--9330 | 9325

PaperSoft Matter

Fig. 6 Equal time variance for active fluctuations of the membrane tube
as a function of dimensionless wavenumber Q for modes m = 0, 1, 2, 3.
The dimensionless parameters are S = 1/2, w = 0, F2 = 2.5 and tac/t = 10.

Fig. 7 Eﬀective temperature of the membrane tube with both thermal
and active fluctuations, eqn (41), plotted as a function of dimensionlesswavenumber Q for modes m = 0, 1, 2, 3. The dimensionless parameters are
S = 1/2, k/(kBT) = 10, w = 0, F2 = 2.5, and tac/t = 10.

Ð1hjuðy; zÞj2 i  2 2p=L dq=ð2pÞhjuq1 j2 i. This implies that by measuring the real space fluctuations of the tube, either throughflorescence microscopy or other indirect means,8,51 and thenvarying the length of the tube, and thus the cut-oﬀ wavelength,
one could infer the long-wavelength form of the active noise.
This experimental assay would provide a simple test of ourpredictions and might help to diﬀerentiate between diﬀerent
types of active noise. A similar method has been used toquantify active noise in neurites.52
Another possible observable is the eﬀective temperature as afunction of Fourier parameters:
2 achjuqm j i
Teff ðQ; mÞ¼1þ
:Thjuqm j2 ith(41)This quantity is plotted in Fig. 7 for the same parameters of
F2 = 2.5, tac/t = 10, w = 0, k/(kBT) = 10 and S = 1/2. The plot showsthat, for long tubes, the highest eﬀective temperature is foundin the m = 1 modes and that these are likely to dominate thespectrum.
Measuring the temperature of fluctuations of long tubes, forexample those pulled from GUVs,8 and varying the viscosity ofthe exterior fluid may provide a way to quantify the magnitude
and time constant of such active correlations in experiment.
Fig. 8 shows the eﬀective temperature of the m = 1 modes
for varying relative viscosity w, along with the asymptotic
result predicted using a Zimm model for such modes (seeAppendix B). As the m = 1 modes dominate much of thefluctuation spectrum, and given the small size of most tubes
formed in real membrane systems, it might prove diﬃcult toresolve the other modes directly. Hence the m = 1 modes areprobably the best candidate for such a direct measurementwith varying viscosities.

9326 | Soft Matter, 2020, 16, 9319--9330

Finally, we compute the Power Spectral Density (PSD) for theactive and thermal cases from the frequency domain correlation functions, h|uqm(o)|2ith,ac, given by eqn (36) and (38). The
PSD is given by
PSDth;ac ðoÞ ¼ 2

X

hjuqm ðoÞj2 ith;ac(42)

Q 4 Qmin ;m 0

where the factor of 2 comes from counting negative q and mmodes, and Qmin is the cutoﬀ set by the length of the tube.
We plot the PSD in Fig. 9 for the same parameters as before andusing discrete values of Q = 2pnr0/L to perform the summation
where we choose n A [1,50], m A [0,3] and the tube length
L = 100r0.
The combined PSD, PSDth(o) + PSDac(o), displays three
regimes; a plateau at small o mostly dominated by active noise,a transitional region governed by the active noise which scales
like PSD(o) B o4, and finally a high frequency regime
dominated by the thermal behaviour. This final regime scales ina similar way to that for a flat membrane, PSD(o) B o5/3.53,54
Crucially, the low frequency regime, which is the only region thatcan be probed by current experimental techniques, shows adramatic diﬀerence between the passive and active cases.
This could be measured directly using current experimental
methods,8,55 using an ATP-depleated system as a reference forthe passive thermal case. A similar technique has been used toanalyse the active ‘‘flicker’’ of red blood cells,34 and could provide a
direct way of quantifying active behaviour in membrane tubes.
Varying the external viscosity could also provide an additional
probe of the form of the active forcing.

This journal is © The Royal Society of Chemistry 2020

Soft Matter

Fig. 8 Eﬀective temperature of the membrane tube with both thermal
and active fluctuations, eqn (41), plotted as a function of dimensionlesswavenumber Q for modes m = 1 for diﬀerent values of relative viscosity w.
The dimensionless parameters are S = 1/2, k/(kBT) = 10, F2 = 2.5, andtac/t = 10. The dotted line shows the asymptotic approximation obtainedby using a Zimm model for a rod in a viscous fluid.

Fig. 9 The power spectral density, PSD(o), plotted against frequency, o,in the thermal, active and combined cases. The dimensionless parameters
are S = 1/2, k/(kBT) = 10, w = 0, F2 = 2.5, and tac/t = 10. We choose a tube oflength L = 100r0.

6 Summary and discussion
In this paper, we have investigated the passive and active
dynamics of a fluid membrane tubes. Using Onsager’s variational formalism, we have calculated the full relaxation
dynamics for the Fourier modes in the shape of the membraneThis journal is © The Royal Society of Chemistry 2020

Paper
tube, assuming a small deformation limit. This work accounts
for variations in surface tension through the use of the
Lagrange multiplier, z, imposing local incompressibility, previously only considered in the axis-symmetric case.14 We also
consider the viscosity contrast between the interior and exteriorambient fluid.
The scaling behaviour of the relaxation modes is analysedand characterised in both the long and short wavelength limits.
In the short wavelength limit, we recover the scaling behaviourof a flat membrane for all angular modes. More interesting
behaviour is found in the long wavelength limit, particularly inthe case of the bending modes (m = 1), where we find arelaxation rate that scales like that of the normal modes of anelastic rod in a viscous fluid. We also reproduce the pearling
instability growth rate found by Boedec et al.,14 which isrecovered when we set m = 0 and choose a suﬃciently highvalue of the surface tension (S 4 3/2). These relaxation
dynamics are significantly diﬀerent from those found for flat
membranes27,39 or for spherical vesicles.28 In the case of thespherical vesicles, the system can be written purely in terms of
one Fourier mode due to the coupling imposed by sphericalsymmetry. This does not happen in the case of the tube as Q
and m are independent of one another.
We then make use of these relaxation equations to computethe fluctuation spectra for passive thermal fluctuations and asimple minimal model of active fluctuations.29,30 The active
noise breaks the fluctuation dissipation theorem, as is apparent
in the presence of dissipative terms in the mean square fluctuations, see eqn (40). The active noise for a ‘‘direct force’’ term also
shows a modified critical behaviour of the bending modes(m = 1) in the long-wavelength limit found in thermal
fluctuations.6 It is likely that more complex curvature couplingsin the long-wavelength limit could also modify the criticalexponent. We have proposed a possible experimental assay basedon measuring the real space fluctuations of the tube and varying
its length. This could be used to infer the functional form of
the active noise experimentally in both in vivo and in vitro
systems.1,8,51 We also compute the eﬀective temperature of thesystem with both thermal and active fluctuations and show that,for long tubes, the clearest signature of the active noise is in thebending modes. This should be a measurable prediction withcurrent experimental setups, e.g., using a similar approach usedby Valentino et al.8 and changing the external viscosity.
Finally we compute the power spectral density for thermal,
active and combined fluctuations of membrane tubes, a quantitythat is directly measurable with current experimental setups
using optical tweezers,8,55 and show that the active fluctuations
dominate the measurable low frequency regime. This could be
used to directly infer information about the size of forces andactivity time scales for diﬀerent sources of activity. It could also
be interesting to consider the eﬀect of diﬀerent forms of theactive force, F(Q,m), particularly in the case where this includes alength-scale associated to curvature coupling (as might be thecase with proteins changing conformation).
In this work we have assumed that the bilayer is symmetric,
such that there is no spontaneous curvature term in theSoft Matter, 2020, 16, 9319--9330 | 9327

PaperHelfrich–Canham energy. However if the tubes were formed bya coat of proteins with preferred curvature (BAR-domain proteins, for example), then an additional spontaneous curvatureterm would appear in the elastic free energy.56,57 This is known
to have an effect on the relaxation rates and criterion for the
pearling instability.58
More complex boundary conditions allowing for slip andpermeation of the membrane with the ambient fluid could also
be considered by adding the corresponding dissipative terms to
the Rayleighian.32,59 These eﬀects would be most relevant atvery short wavelengths and thus only modify the high (Q,m)
part of the relaxation rates and fluctuation spectra.59 Permeation may also play a role for the m = 0 mode at very longwavelengths and thus be important in tubes undergoingvolume changes, by osmotic swelling for example ref. 26. It isalso possible that in some scenarios the membrane may allow
slip freely but support no shear stress across the membrane. In
this case a change in boundary condition from no-slip to noshear-stress would need to be considered.46
Perhaps the most pressing open question in the field ofactive membranes is what functional form is best used torepresent the active fluctuations and if all current descriptions
can be unified in some manner. The simple model of a direct
force used in this paper has been used successfully throughout
the literature to describe real systems.29–31,34 However, it is notclear how well motivated this is at the microscopic level. The
model we have used involves the assumption that the work
done by the active forces per unit time is constant, althoughanother possible coarse-grained model might assume constant
applied force.30 More complex models of activity have been
proposed for specific situations, for example using dipole
forces and allowing fluid permeation of the membrane.32
However, a general framework is lacking and the effect onobservable phenomena is not yet well understood.
For future work, it would be interesting to consider theeﬀects of diﬀerent formulations of activity, both in tubes andother geometries. An example might be active pumps thattransport ions across the membrane, and thus increasing
the tube’s osmotic pressure.26 A further study could focus oncoupling the active fluctuations to a field associated with thedensity of some spontaneous curvature-inducing proteinson the membrane. This has already been studied for a flat
membrane in ref. 25 where a curvature induced instability was
shown to arise. The interplay between this instability and thehighly curved geometry of a membrane tube could be quite
rich. It may also be of interest to consider the eﬀect of aviscoelastic ambient fluid, as this may give a better approximation to the cytoplasm in cells. Not only would this givepotentially richer dynamics, due to the presence of an additional time scale, but it could also be useful in understanding
more realistic biological processes.1,2,60

Soft Matter
A Expressions for U
qm, Wqm, Nqm
Here we give expressions for the scalar Laplace function
decompositions for the Stokes equations after imposing theboundary condition, eqn (21), and making use of the continuity
equation to eliminate vyqm. This gives
h
z
z
2
Fþ
qm ¼ 2r0 Q 4ivqm 3Qr0 u_ qm Km1 4m Qr0 u_ qm 2ivqmÞKm1 Km
i h
þ 4 r0 u_ qm þivzqm Km2 Q2 r0 u_ qm Kmþ12  7Q3 Km13þ 2ð9m8ÞQ2 Km12 Km þ4Q mðm8Þ2Q2 Km1 Km2i1 8m m2 þQ2 Km3 þQ3 Kmþ13 ;Cþ
qm ¼ r0

h

8Km2

vzqm



4

2

m þ2ðmþ1ÞmQ þQ

4



(43)



 iQr0 u_ qm mð3mþ2ÞþQ2
þ 8QKm1 Km
m3 þðm2ÞQ2 vzqm iðm2ÞQr0 u_ qm

i

  2

m þQ2 vzqm iQr0 u_ qm
þ 2Q2 3Km12 þKmþ12h mQð8 m3 þmQ2 Km3 7Q3 Km13 Q3 Kmþ13i1
þ2ð89mÞQ2 Km Km12 þ 4Q 2Q2 ðm8Þm Km2 Km1 Þ ;Xþ
qm ¼ r0h(44)
8iKm2vzqm

 2
m þQ2 iðmþ1ÞQr0 u_ qm

þ 6iQ2 Km12 þ2iQ2 Kmþ12 vzqm 8QKm Km1i
 Qr02 u_ qm imvzqmh Qð8 m3 þmQ2 Km3 þ7Q3 Km13 þQ3 Kmþ13i1þ 2ð9m8ÞQ2 Km Km12 þ4Q ðm8Þm2Q2 Km2 Km1 Þ ;Fqm ¼r0

h

Im2



(45)

z
2
m 1 r0 u_ qm iQvqm þQIm1 Qr0 @uqm ivzqm
22mIm Im1 Qr0 u_ qm ivzqm

i h QðQ2 Im13 þmQIm3

i1

þ 2ðm2ÞmQ2 Im2 Im1 þð23mÞQIm Im12 Þ ;
(46)
h
3 z
2 z
C
qm ¼r0 2Im Im1 m vqm þðm1ÞQ vqm iðm1ÞQr0 u_ qm
þ QIm12
m2 þQ2 vzqm Qir0 u_ qm Þ




 i
þ iIm2 r0 u_ qm 2mðmþ1ÞþQ2 þiQvzqm mðmþ2ÞþQ2
 mQðQ2 Im13 þð2ðm2ÞmQ2 ÞIm2 Im1

Conflicts of interest
There are no conflicts to declare.

9328 | Soft Matter, 2020, 16, 9319--9330

þ ð23mÞQIm Im12 þmQIm3 Þ

1

;
(47)

This journal is © The Royal Society of Chemistry 2020

Soft Matter

Paper

h
z
z
2
2
X
qm ¼ r0 Im Im1 Qr0 u_ qm þ2imvqm þIm ðmþ1Þr0 u_ qm þiQvqm
i h iQvzqm Im12  QðQ2 Im13 þð2ðm2ÞmQ2 ÞIm2 Im1
þ ð23mÞQIm Im12 þmQIm3 Þi1

C Possible direct curvature coupling;(48)

where the modified Bessel functions Km and are evaluatedat r = r0.

B Relaxation dynamics of linear Zimm
model
Here we consider the relaxation dynamics of small planarnormal perturbations to a thin elastic rod whose position is
given by
r = xðtÞcos qz,0,z ,(49)
and has geodesic curvature kg = q2x cos qz. We are motivatedto study the relaxation dynamics of this system as, on lengthscales much larger than the tube radius, the m = 1 mode of amembrane tube can likely be thought of as the dynamics of athin elastic rod. Here membrane flows are small and most of
the friction comes from the drag of the tube through the bulk
fluid. We want to see if, at least at a scaling level, this can give a
simple understanding of the long wavelength relaxation
dynamics.
If we assume qx(t) { 1, the elastic force per unit length on
the rod is given by 
f~ ¼ Kr2 kg þ Skg ; 0; 0 ¼ Kq4 x cos qz  Sq2 x cos qz; 0; 0 ;(50)where K is the bending rigidity of the rod and S is the tension.61
We can write the dynamics of this rod as a continuous
Zimm modelð~
r_ ¼  d~sLð~r ~
sÞf~ð~
sÞ;
(51)
-

-

where L(r  r 0 ) is the Oseen tensor62


ð~
r ~
sÞ ð~
r ~
sÞ
1
:
Lð~
r ~
sÞ ¼
I
j~
r ~
sj2
8pZj~
r ~
sj

(52)

ea ~
eb .
Here # is the tensor product and I ¼ dab~
At linear order and in the long-wavelength limit, this givesx_ 

the effective temperature of fluctuations in the long wavelength
limit, dotted line in Fig. 8.

ðKq4 þ Sq2 ÞCiðqr0 Þ
x;
4pZ

(53)

Ð1
where Ciðqr0 Þ ¼  r0 dx cosðqxÞ=x is the cosine integral function, and we have chosen a short-wavelength cutoﬀ of the rod
radius, r0. Then we obtain a relaxation rate that scales asl B (g + log Q)Q2 in the small Q limit, where g is the Euler–
Mascheroni constant. This result agrees, at the scaling level,with relaxation dynamics of a membrane tubes bending modein the long wavelength limit. It also gives the correct scaling forThis journal is © The Royal Society of Chemistry 2020

Here we briefly outline a possible scheme to give active fluctuations with direct curvature coupling. If instead of writing theshape equation in terms of the local deviation from theequilibrium radius, r0, we write it as a local deviation fromthe equilibrium mean curvature, H = 1/(2r0). The we can write
an equation of the form




:
dHqmB Q,m = A Q,m dHqm + xac,H(54)
qm
where dH = (1  m2  Q2)uqm and xac,Hqm gives the active direct
curvature coupling noise with the following statisticalproperties
hxac,Hqm ðtÞi = 0
ac;H 0hxac;Hqm ðtÞxq0 m0 ðt Þi ¼

f 2 jtt 0 j=tacedqq0 dmm0 ;
2tac

(55)
(56)

where f is a constant. In the shape equation for uqm, eqn (30),this corresponds to F(Q,m) = f/(1  m2  Q2) which is discussed
briefly in the main text.

Acknowledgements
S. C. A.-I. would like to acknowledge funding from the following
sources; the EPSRC under grant number EP/L015374/1 CDT in
Mathematics for Real-World Systems, the Labex CelTisPhyBio(ANR-11-LABX-0038, ANR-10-IDEX-0001-02), the EMBL-Australia
program, and Tokyo Metropolitan University under the Graduate
Short-term Inbound and Outbound Program. S. C. A.-I. would also
like to thank Prof. S. Komura for his generous hospitality during
his stay at TMU. M. S. T. acknowledges the generous support of
the Japan Society for the Promotion of Science (JSPS), via a longterm fellowship, and the hospitality of both Prof. R. Yamamoto(Kyoto University) and Prof. S. Komura (Tokyo Metropolitan
University). S. K. acknowledges support by a Grant-in-Aid for
Scientific Research (C) (Grant No. 18K03567 and Grant No.
19K03765) from the JSPS, and support by a Grant-in-Aid for
Scientific Research on Innovative Areas ‘‘Information Physics of
Living Matters’’ (Grant No. 20H05538) from the Ministry of
Education, Culture, Sports, Science and Technology of Japan.

Notes and references
1 J. Nixon-Abell, C. J. Obara, A. V. Weigel, D. Li, W. R. Legant,
C. S. Xu, H. A. Pasolli, K. Harvey, H. F. Hess, E. Betzig,
C. Blackstone and J. Lippincott-Schwartz, Science, 2016,
354, 6311.
2 S. Abounit and C. Zurzolo, J. Cell Sci., 2012, 125(5), 1089.
3 A. Yamada, A. Mamane, J. Lee-Tin-Wah, A. Di Cicco, C. Prévost,
D. Lévy, J.-F. Joanny, E. Coudrier and P. Bassereau, Nat.
Commun., 2014, 5, 3624.

Soft Matter, 2020, 16, 9319--9330 | 9329

Paper
4 D. Cuvelier, I. Derényi, P. Bassereau and P. Nassoy, Biophys. J.,
2005, 88, 2714.
5 I. Derényi, F. Jülicher and J. Prost, Phys. Rev. Lett., 2002,
88, 238101.
6 J.-B. Fournier and P. Galatola, Phys. Rev. Lett., 2007, 98,
018103.
7 S. Komura and R. Lipowsky, J. Phys. II, 1992, 2(8), 1563.
8 F. Valentino, P. Sens, J. Lemière, A. Allard, T. Betz,
C. Campillo and C. Sykes, Soft Matter, 2016, 12, 9429.
9 T. Tlusty, S. A. Safran and R. Strey, Phys. Rev. Lett., 2000,
84, 061244.
10 T. Tlusty and S. A. Safran, J. Phys.: Condens. Matter, 2000,
12, A253.
11 C. Barbetta and J.-B. Fournier, Eur. Phys. J. E: Soft Matter
Biol. Phys., 2009, 29, 183.
12 K. L. Gurin, V. V. Lebedev and A. R. Muratov, J. Exp. Theor.
Phys., 1996, 83, 321.
13 R. Bar-Ziv and E. Moses, Phys. Rev. Lett., 1994, 73, 101392.
14 G. Boedec, M. Jaeger and M. Leonetti, J. Fluid Mech., 2014,
743, 262.
15 P. Nelson, T. Powers and U. Seifert, Phys. Rev. Lett., 1995,
74, 173384.
16 P. Nassoy, D. Cuvelier, R. Bruinsma and F. Brochard-Wyart,
EPL, 2008, 84, 18004.
17 F. Brochard-Wyart, N. Borghi, D. Cuvelier and P. Nassoy,
Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 7660.
18 E. Evans and A. Yeung, Chem. Phys. Lipids, 1994, 73, 39.
19 D. R. Daniels and M. S. Turner, Langmuir, 2007, 23, 6667.
20 M. L. Henle and A. J. Levine, Phys. Rev. E: Stat., Nonlinear,
Soft Matter Phys., 2010, 81, 011905.
21 M. Rahimi, A. DeSimone and M. Arroyo, Soft Matter, 2013,9, 11033.
22 S. Marbach, D. S. Dean and L. Bocquet, Nat. Phys., 2018,14, 1108.
23 D. Holcman, P. Parutto, J. E. Chambers, M. Fantham, L. J.
Young, S. J. Marciniak, C. F. Kaminski, D. Ron andE. Avezov, Nat. Cell Biol., 2018, 20, 1118.
24 H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov,
C. Sykes, J.-F. Joanny, G. Gompper and T. Betz, Nat. Phys.,2016, 12, 513.
25 S. Ramaswamy, J. Toner and J. Prost, Phys. Rev. Lett., 2000,
84, 153494.
26 S. C. Al-Izzi, G. Rowlands, P. Sens and M. S. Turner, Phys.
Rev. Lett., 2018, 120, 138102.
27 J.-B. Fournier, Int. J. Non Linear Mech., 2015, 75, 67.
28 T. V. Sachin Krishnan, R. Okamoto and S. Komura, Phys.
Rev. E, 2016, 94, 062414.
29 T. V. Sachin Krishnan, K. Yasuda, R. Okamoto and S. Komura,
J. Phys.: Condens. Matter, 2018, 30, 175101.
30 N. Gov, Phys. Rev. Lett., 2004, 93, 268104.
31 J. Prost and R. Bruinsma, EPL, 1996, 33, 321.

9330 | Soft Matter, 2020, 16, 9319--9330

Soft Matter
32 J.-B. Manneville, P. Bassereau, S. Ramaswamy and J. Prost,
Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2001, 64,
021902.
33 N. S. Gov and S. A. Safran, Biophys. J., 2005, 88(3), 1859.
34 R. Rodrı́guez-Garcı́a, I. López-Montero, M. Mell, G. Egea,
N. S. Gov and F. Monroy, Biophys. J., 2015, 108(12), 2794.
35 P. B. Canham, J. Theor. Biol., 1970, 26, 61.
36 W. Helfrich, Z. Naturforsch., C: J. Biosci., 1973, 28, 11693.
37 T. J. Willmore, Anal. Stunt. ale Univ. Iasi Sect. I. a Math, 1965,
11B, 493.
38 M. Arroyo and A. DeSimone, Phys. Rev. E: Stat., Nonlinear,
Soft Matter Phys., 2009, 79, 031915.
39 U. Seifert and S. A. Langer, Biophys. Chem., 1994, 49, 113.
40 J. Happel and H. Brenner, Low Reynolds number hydrodynamics: with special applications to particulate media, M.
Nijhoﬀ, Distributed by Kluwer Boston, 1983.
41 L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Elsevier, 2013.
42 M. Doi, Soft Matter Physics, Oxford University Press, Oxford,
2013.
43 M. Doi, J. Phys.: Condens. Matter, 2011, 23, 284118.
44 L. Onsager, Phys. Rev., 1931, 37, 405.
45 L. Onsager, Phys. Rev., 1931, 38, 2265.
46 R. Shlomovitz, N. S. Gov and A. Roux, New J. Phys., 2011,
13, 065008.
47 S. Tomotika, Proc. R. Soc. A, 1935, 150, 322.
48 L. Rayleigh, Philos. Mag., 1892, 34, 145.
49 T. R. Powers, Rev. Mod. Phys., 2010, 82, 1607.
50 S. A. Rautu, D. Orsi, L. Di Michele, G. Rowlands, P. Cicuta
and M. S. Turner, Soft Matter, 2017, 13, 3480.
51 P. Georgiades, V. J. Allan, G. D. Wright, P. G. Woodman,
P. Udommai, M. A. Chung and T. A. Waigh, Sci. Rep., 2017,
7, 16474.
52 F. Gárate1, T. Betz, M. Pertusa and R. Bernal, Phys. Biol.,
2015, 12, 066020.
53 A. G. Zilman and R. Granek, Phys. Rev. Lett., 1996, 77, 234788.
54 A. G. Zilman and R. Granek, Chem. Phys., 2002, 284, 195.
55 A. Allard, F. Valentino, C. Sykes, T. Betz and C. Campillo,
2020, arXiv:2003.07917.
56 D. Kabaso, N. Bobrovska, W. Góźdź, N. Gov, V. Kralj-Iglič,P. Veranič and A. Iglič, J. Biomech., 2012, 45(2), 231.
57 A. Iglič, H. Hägerstrand, P. Veranič, A. Plemenitaš and
V. Kralj-Igliče, J. Theor. Biol., 2006, 240(3), 368.
58 U. Jeleri and N. S. Gov, Phys. Biol., 2015, 12, 066022.
59 K. Yasuda, R. Okamoto, S. Komura and J.-B. Fournier, Soft
Mater., 2018, 16(3), 186.
60 S. Komura, K. Yasuda and R. Okamoto, J. Phys.: Condens.
Matter, 2015, 27, 432001.
61 B. Audoly and Y. Pomeau, Elasticity and Geometry: From Hair
Curls to the Non-linear Response of Shells, OUP Oxford, 2010.
62 M. Doi and S. F. Edwards, The Theory of Polymer Dynamics,
Clarendon Press, 1986.

This journal is © The Royal Society of Chemistry 2020