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JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 23 (2011) 072205 (5pp)

doi:10.1088/0953-8984/23/7/072205

FAST TRACK COMMUNICATION

Hydrodynamic coupling between two fluid
membranes
Sanoop Ramachandran and Shigeyuki Komura
Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan
University, Tokyo 192-0397, Japan
E-mail: komura@tmu.ac.jp

Received 21 December 2010, in final form 12 January 2011
Published 2 February 2011
Online at stacks.iop.org/JPhysCM/23/072205
Abstract
The coupled in-plane diffusion dynamics between point-particles embedded in stacked fluid
membranes is investigated. We calculate the contributions to the coupling longitudinal andtransverse diffusion coefficients due to particle motion within the different as well as the same
membranes. The stacked geometry leads to a hydrodynamic coupling between the two
membranes.
(Some figures in this article are in colour only in the electronic version)

Biological membranes are fundamental to the existence oflife with their ability to separate the ‘in’ and ‘out’ of acell. Typical membranes are composed of lipid molecules
which can spontaneously self-assemble into a fluid bilayer
structure [1].
Under normal physiological conditions,finite temperature induces Brownian motion of membrane
constituents, resulting in diffusive transport. In the seminal
paper by Saffman and Delbrück [2], the diffusion coefficientof a rigid protein in a membrane was calculated. Following
this work, there have been theoretical studies on diffusion ofa rigid disc on supported membranes [3], a liquid domain inmonolayers [4, 5], and a rod on immersed membranes [6, 7] oron Langmuir monolayers [8].
The present work focuses on the coupling of diffusion dynamics between two fluid membranes through an intervening
bulk fluid. It was reported that model experimental systems inwhich two lipid membranes are stacked on a substrate exhibit
correlated dynamics [9]. This planar geometry has becomefavorable to study the membrane dynamics which is otherwise
not possible in vesicles [10]. The coupling effect between
two membranes can be important in biological systems witha large concentration of cells such as tissues. Other examples
are Gram-negative bacteria which enclose a periplasmic spacewith an approximate width of about 15–20 nm between theirinner and outer lipid bilayers [1]. Highly folded membranousorganelles such as Golgi apparatus also correspond to asituation in which membranes come in close proximity to each0953-8984/11/072205+05$33.00

other. In all these cases, it is very relevant to consider thehydrodynamic coupling between two biomembranes.
Hydrodynamic models of membranes exploit their fluid
nature, in which the membranes are assumed to be twodimensional (2D) viscous fluid sheets embedded in a threedimensional (3D) solvent. One such investigation involved thecalculation of the correlated diffusion of proteins embedded
in a membrane immersed in an unbounded fluid [11]or a membrane adjacent to a solid support [12].
In
this communication, we report on the coupling diffusion
coefficients of particles embedded in such a stacked membrane
system.
We first establish the governing equations. It is assumed
here that the membranes are infinite planar sheets of fluid.
The persistence length is defined as the length over which thecorrelations of the normal vector decay to zero [13]. For a lipid
bilayer membrane with bending rigidity of about 10kB T , thepersistence length turns out to be much larger than the size of atypical cell. Hence we neglect out-of-plane fluctuations of themembrane for mathematical simplicity. The fluid membranes
are embedded in a bulk solvent such as water or suitable buffer
solution. As shown in figure 1, the membranes are fixed in thex y -plane at z = ±h/2. Let v(i) (r) be the 2D velocity of themembrane fluids. Here the index i = 1, 2 represents the two
membranes, and the 2D vector r = (x, y) represents a pointin the planes of the membranes. We work in the low-Reynolds
number regime of the membrane hydrodynamics so that the1© 2011 IOP Publishing Ltd Printed in the UK & the USA

J. Phys.: Condens. Matter 23 (2011) 072205

Fast Track Communication

In the above, I is the identity tensor and the superscript ‘T’indicates the transpose. Similarly, the force on membrane
2 is given by the projection of (σ (ii) − σ (iii) )z=−h/2 · êz on
the x y -plane. The general procedure is to first resolve (1)into components along k and perpendicular to it, where kis the 2D wavevector in Fourier space. We then solve
the resulting differential equations for the velocities. Stickboundary conditions at the membrane–solvent interfaces are
imposed. It is also assumed that the solvent velocities decay
to zero at sufficiently large distances from the membranes. A
similar calculation for a single membrane confined between
two walls has been previously performed [14, 15].
Owing to the linearity of the governing Stokes equations,the in-plane velocity in membrane 1 can be obtained in Fourier
space as11)12)vα(1) [k] = G (αβ[k]Fβ(1) [k] + G (αβ[k]Fβ(2) [k].

(11)

inertial effects can be neglected. This allows us to use the 2D
Stokes equations

η∇ v

−∇p

(i)

(i)
+ f(i)
= 0,
s +F

with

(1)

g(K , H ) = 1 + 2 K (1 + K ) + (1 + 2 K ) coth(K H ),along with the incompressibility condition∇ · v(i) = 0.

(3)

where ∇˜ represents a 3D differential operator. Similar to thefluid membrane, the solvents in all the regions are considered
to be incompressible∇˜ · v( j ) = 0.

(4)

The presence of the surrounding solvent is important because
it exerts force on the liquid membranes. The force on
membrane 1, indicated as f(s1) in (1), is given by the projectionof (σ (i) − σ (ii) )z=h/2 · êz onto the x y -plane of the membrane.
Here êz is the unit vector along the z -axis, and σ ( j ) are thestress tensors

σ ( j ) = − p( j )I + ηs [∇˜ v( j ) + (∇˜ v( j ) )T ].

(9)and ν ≡ 2ηs /η. Notice that ν −1 represents the hydrodynamic
screening length. We have also used the definitions H = hνand K = k/ν with k = |k|. By symmetry, a similar set of
expressions can be written down for the membrane 2 also. The
above mobility tensors represent the hydrodynamic coupling
between the two membranes.
Consider a pair of particles embedded in the membrane
undergoing Brownian motion separated by r. For sufficiently
large times, the particle displacements obey rα rβ  =2 Dαβ (r)t , where rα represents the displacement of the firstparticle and rβ represents that of the second particle. In
the above, the diffusion tensor for over-damped dynamics isgiven by the Einstein relation Dαβ = kB T G αβ , where kB
is the Boltzmann constant and T the temperature. We now
apply these definitions to a double-membrane system. Theline of centers connecting any two particles in the membranes
after projection onto the 2D plane can be taken to be along
the x -axis without loss of generality. Then we obtain the(11)coupling longitudinal diffusion coefficients as DL (r) =kB T G (xx11) (r êx ) for two particles within the same membrane,(12)
and DL (r) = kB T G (xx12) (r êx ) for two particles in different
membranes. The coupling transverse diffusion coefficients areDT(11) (r) = kB T G (yy11) (r êx ) and DT(12) (r) = kB T G (yy12)(r êx ).
The longitudinal coupling diffusion coefficient is associatedwith Brownian motion along the line of centers, while thetransverse one is associated with motion perpendicular to theline of centers [11, 12, 16]. In the following, we discuss theabove four diffusion coefficients sequentially.

(2)

Here ∇ is a 2D differential operator, η is the 2D membrane
viscosity (same for both the membranes), p (i) (r) the 2D inplane pressure, f(i)s (r) the force exerted on the membrane by
the surrounding fluid, and F(i) (r) is any other force acting on
the membrane.
The solvent regions are denoted by the index j = i–iii. The velocities and pressures in these regions are written
as v( j ) (r, z) and p ( j )(r, z), respectively. We assume that thesolvent in the three regions have the same 3D viscosity denotedby ηs . The solvent inertia is neglected and hence it also obeys
the 3D Stokes equationsηs ∇˜ 2 v( j ) − ∇˜ p ( j ) = 0,

(12)Here G αβ and G αβ (α, β = x, y ) are the mobility tensors
given by
kα kβ
1 1 + 2 K + coth(K H )(11)G αβ [k] = 2, (7)δαβ − 2
ην
K g(K , H )k
1 cosech(K H )
kα kβ
12)
δαβ − 2,G (αβ[k] = 2(8)
ην K g(K , H )kFigure 1. Schematic picture showing a set of stacked fluid
membranes (solid thick lines) having 2D viscosities η located atz = ±h/2. The two membranes are denoted by the labels 1 and 2.
The solvents of 3D viscosities ηs are labeled by the regions i, ii and
iii.

2 (i)

(6)

(5)
2

J. Phys.: Condens. Matter 23 (2011) 072205

Fast Track Communication

Figure 2. Dimensionless DL(11) (R, H ) as a function of R for various
values of H . The red circles, blue squares and green triangles
correspond to H = 0.1, 1 and 10, respectively. The solid line
corresponds to (11) representing the analytical expression for thelarge H limit. The dashed line corresponds to the small H limit.

Figure 3. Dimensionless DT(11) (R, H ) as a function of R for various
values of H . The red circles, blue squares and green triangles
correspond to H = 0.1, 1 and 10, respectively. The solid line
corresponds to (13) representing the analytical expression for thelarge H limit. The dashed line corresponds to the small H limit.

The real-space expressions for the mobility tensors can beobtained by inverse Fourier transform. Using the notation R =r ν , we obtain the longitudinal coupling diffusion coefficient
within the same membrane as ∞1 + 2 K + coth(K H ) J1 (K R),
DL(11) (R, H ) = 2
dKg(K , H )KR
0
(10)
where Jn (t) are the Bessel functions of the first kind of ordern . Hereafter, all the diffusion coefficients are scaled by afactor kB T /(4πη) in order to make them dimensionless. In(11)figure 2, we plot DL (R, H ) as a function of R for various
H following numerical integration. It is a monotonicallydecreasing function of R . The red circles ( H = 0.1), blue
squares ( H = 1) and green triangles ( H = 10) represent
the intermediate membrane separations respectively. Thesesymbols represent the same H values in the rest of the figuresin this communication.
For H  1, we can approximate coth(K H ) ≈ 1 so thatthe H -dependence drops out. In this limit, the integral can beanalytically performed to yieldOn recasting in dimensional form, we see that this limiting
expression is independent of the membrane viscosity η, anddependent only on the solvent viscosity ηs . When H
1,(11)
on the other hand, (10) becomes DL (R/2)/2. The vanishingthickness of the solvent region ii results in a rescaling of ν −1 bya factor of two, and hence the resultant expression. The solidand dashed lines in figure 2 represent the above limiting casesof large and small H limits, respectively. It is thus observedthat the presence of the second membrane also has a finite
contribution to the longitudinal coupling diffusion coefficient
within the same membrane.
Following the same argument, the transverse coupling
diffusion coefficient between two particles within the same
membrane is calculated according to ∞1 + 2 K + coth(K H )(11)DT (R, H ) = 2
dKg(K , H )0J1 (K R).
(12)
× J0 (K R) −KR

DL(11) (R) = −2π+ [H1 (R) − Y1 (R)],2
R
R

In figure 3, the variation of DT(11) (R, H ) as a function of R
for various H is plotted using numerical integration. Similar
to the longitudinal case, it is also a monotonically decreasing
function of R . For H  1, (12) can be analytically treated tohave the form

(11)

where Hn (t) are the Struve functions and Yn (t) are the
Neumann functions. Since the membranes are effectively
isolated from each other for large H , (11) coincides withthe analytical expression obtained for a single membranegeometry [11]. For R
1, (11) has a logarithmic behavior,(11)i.e., DL (R1) ≈ ln(2/R) − γ + 0.5, where γ =0.5772 . . . is Euler’s constant. In this regime, the outer solvent
surrounding the membrane is unimportant and the membrane
behaves effectively as a pure 2D fluid. In the opposite R (11)
1 limit, (11) decays algebraically DL (R  1) ≈ 2/R .

2π− [H1 (R) − Y1 (R)] + π[H0 (R) − Y0 (R)],
2
R
R
(13)
which also coincides with the expression for the transverse
coupling diffusion coefficient in a single membrane [11]. In
this case, the asymptotic behaviors for small and large R are(11)
DT(11) (R1) ≈ ln(2/R) − γ − 0.5 and DT (R  1) ≈2/R 2 , respectively. In the opposite limit of H
1, (12)(11)
becomes DT (R/2)/2. As before, the solid and dashed lines

DT(11) (R) =3

J. Phys.: Condens. Matter 23 (2011) 072205

Fast Track Communication

Figure 5. Dimensionless DT(12) (R, H ) as a function of R for various
values of H . The red circles, blue squares and green triangles
correspond to H = 0.1, 1 and 10, respectively. The dashed line
corresponds to the small H limit.

Figure 4. Dimensionless DL(12) (R, H ) as a function of R for various
values of H . The red circles, blue squares and green triangles
correspond to H = 0.1, 1 and 10, respectively. The dashed line
corresponds to the small H limit.

additional contributions to the coupling diffusion coefficients.
This is the main result of this work. Using typical values forthe solvent (water) and membrane (lipid bilayer) viscosities,
the hydrodynamic screening length ν −1 can be estimatedto be of the order of micrometers. This implies that an
adjacent membrane within this distance ( H < 1) can have astrong bearing on the diffusion dynamics, such as for typicalstacked supported membrane experiments [9]. Qualitatively,
the presence of the second membrane can enhance the effective
coupling diffusion.
Even though the model presented in this communication
captures the essential physics, it looks somewhat simplistic
in several respects. We have neglected the finite size effectsof the membrane inclusions which are known to modify themembrane response at small inter-particle distances [11, 12].
At distances much larger than the inclusion size, however,these effects become unimportant. The curvature effects of themembrane are significant when the radius of curvature is of theorder of the hydrodynamic screening length ν −1 [17, 18]. The
extension of our work to membranes in spherical configurationis possible. The out-of-plane fluctuations of the membranelead to a reduction in the diffusion coefficient of proteins insingle membranes [19, 20]. However, it is known that thepresence of a substrate or a second membrane would suppress
the out-of-plane membrane fluctuations [21–23]. Overall, we
expect that fluctuations and the presence of a substrate will not
qualitatively affect our results.
To summarize, we have calculated the longitudinal aswell as transverse coupling diffusion coefficients of particles
undergoing Brownian motion in a stacked double-membranegeometry. We obtained the additional contribution to thecoupling diffusion coming from the proximity to adjacentmembranes. As in biological systems such as tissues the cells
are rarely isolated, our results also imply that the proximity
to neighboring cells can affect the diffusion of objects in cell
membranes.

in figure 3 show the limiting cases of large and small H ,respectively.
Now we proceed to calculate the coupling diffusion
coefficients of two particles in different membranes. In thiscase, the longitudinal coupling diffusion coefficient expressedin dimensionless units is ∞cosech(K H ) J1 (K R).
DL(12) (R, H ) = 2
dK(14)g(K , H )
KR
0
(12)

The functional dependence of DL (R, H ) on R for various
H is shown in figure 4. Since cosech(K H ) ≈ 0 for H  1,
the above integral vanishes for large inter-membrane distances
as seen by the green triangles ( H = 10) in figure 4. This is areasonable result as the membranes are effectively independentof each other. In the limit of H
1, (14) results in
DL(11) (R/2)/2 which is plotted by the dashed line in figure 4.
As mentioned earlier, this function has an initial logarithmic
behavior followed by an asymptotic 1/R -decay.
The transverse coupling diffusion coefficient between two
particles in different membranes is given by ∞cosech(K H )DT(12) (R, H ) = 2
dKg(K , H )0J1 (K R)× J0 (K R) −
(15)
.
KR
(12)

In figure 5, the variation of DT (R, H ) as a function of R
for various H is plotted. The above integral also vanishes
when H  1, as expected for the decoupled membranes. In(11)the opposite limit of H
1, (15) results in DT (R/2)/2 asplotted by the dashed line in figure 5.
From figures 4 and 5, it can be observed that for large
H , the coupling diffusion coefficients vanish. This is dueto the exponential decay of the cosech(K H ) term. Up to
H = 1, the proximity to the second membrane leads to4

J. Phys.: Condens. Matter 23 (2011) 072205

Fast Track Communication

This work was supported by KAKENHI (Grant-in-Aid for
Scientific Research) on Priority Area ‘Soft Matter Physics’ andGrant No. 21540420 from the Ministry of Education, Culture,
Sports, Science and Technology of Japan.

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Hydrodynamic effects on concentration fluctuations in
multicomponent membranes Soft Matterat press doi:10.1039/c0sm00783h[15] Ramachandran S, Komura S, Seki K and Gompper G 2010
Dynamics of a polymer chain confined in a membrane
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and discrete-particle approach to biomembrane dynamics
with application to the mobility of curved integral membraneproteins Phys. Rev. Lett. 102 138102[20] Reister-Gottfried E, Leitenberger S M and Seifert U 2010
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