Materials 2012, 5, 1923-1932; doi:10.3390/ma5101923
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materials
ISSN 1996-1944
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ArticleLateral Dynamics in Polymer-Supported Membranes
Shigeyuki Komura 1,∗ , Sanoop Ramachandran 2 and Kazuhiko Seki 3
1

Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan
University, Tokyo 192-0397, Japan
2
Polymer Physics Group, Department of Physics, Free University of Brussels, Campus Plaine, CP 223,
Brussels 1050, Belgium; E-Mail: sramacha@ulb.ac.be3
National Institute of Advanced Industrial Science and Technology (AIST), Ibaraki 305-8565, Japan;
E-Mail: k-seki@aist.go.jp* Author to whom correspondence should be addressed; E-Mail: komura@tmu.ac.jp;
Tel.: +81-42-677-2537; Fax: +81-42-677-2525.
Received: 11 September 2012; in revised form: 30 September 2012 / Accepted: 12 October 2012 /Published: 19 October 2012

Abstract: We investigate the lateral dynamics in a purely viscous lipid membrane which
is supported by a thin polymer sheet (polymer-supported membrane). The generalized
frequency-dependent mobility tensor of the polymer-supported membrane is obtained bytaking into account the viscoelasticity of the polymer sheet. Due to its viscoelasticity, thecross-correlation functions of two particles embedded in the membrane exhibit an anomalous
diffusion. A useful relation for two-point microrheology connecting the cross-correlation
function and the modulus of the polymer sheet is provided.
Keywords: supported membranes; lipids; viscoelasticity; diffusion; microrheology

1. Introduction
Biomembranes are thin two-dimensional (2D) fluids which separate inner and outer environmentsof organelles in cells [1]. The fluidity of biomembranes is guaranteed mainly due to the lipid
molecules, which are in the liquid crystalline state at physiological temperatures. Proteins and othermolecules embedded in biomembranes undergo lateral diffusion, which plays an important role inbiological functions [2,3]. In order to study the lateral dynamics in biomembranes, lipid-bilayer
membranes supported on solid substrates have been widely used [4]. The standard method is toMaterials 2012, 5

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place membranes, either biomembranes or artificial membranes, directly on solids, which are called
as solid-supported membranes. However, experiments using solid-supported membranes have somefundamental difficulties. This is because the membrane-substrate distance is usually not large enough to
avoid direct contact between membrane proteins and the solid surface. Such a direct contact leads to africtional coupling between proteins and the solid support, which may cause protein denaturation.
One way to avoid this problem is to put thin soft polymeric materials between the membrane andthe solid substrate [5]. For such polymer-supported membranes, the typical thickness of a polymer
cushion is 10–100 nm. Moreover, polymer-supported membranes can be used as cell-surface models
that connect biological and artificial materials [6]. In fact, it is known that cytoplasm of eukaryotic cells
contains proteins and organelles, including a thick sub-membrane layer of actin-meshwork forming apart of the cell cytoskeleton. Also the extra-cellular fluid consists of extracellular matrix or hyaluronic
acid gel. It is important to note that hydrated polymer cushions are generally viscoelastic rather thanpurely viscous because they are essentially polymer solutions. Recently, it was theoretically shown by
us that the mean square displacement of a single inclusion embedded in a viscous membrane exhibits
an anomalous diffusion due to the viscoelasticity of the surrounding solvent [7]. This means that thepresence of a polymer sheet could have a significant influence on the membrane lateral dynamics.
In this paper, we discuss the dynamics and response of polymer-supported membranes by taking into
account the viscoelastic property of the thin polymer sheet. Focusing on the two-particle tracking case,we show that the viscoelasticity of the polymer cushion leads to an anomalous (subdiffusive) dynamics
in the 2D viscous membrane. We also give a relation connecting the cross-correlation function and themodulus of the polymer sheet, which is useful for two-point microrheology experiments.
In the next section, we shall first discuss the hydrodynamic mobility tensor of polymer-supported
membranes. Using this mobility tensor, we obtain in Section 3 the cross-correlation functions of two
distinct particle positions, and argue the effects due to the viscoelasticity of the ambient polymer sheet.
In Section 4, we give a useful relation for two-point microrheology as well as some discussion.
2. Mobility Tensor of Polymer-Supported Membranes
Our model system is similar to those in [7–9] and is schematically depicted in Figure 1. A flat thinfilm of purely viscous membrane with 2D viscosity η lies on the xy-plane a distance h away from asolid substrate. The space between the thin film and the substrate is filled with a viscoelastic polymer
sheet with frequency dependent 3D viscosity ηp [ω]. The space on the other side of the film is occupied
by a semi-infinite fluid with 3D viscosity ηf . All fluids including the membrane are assumed to beincompressible and their inertial effects are neglected here. The latter assumption is always justified aslong as we are concerned with a long-time behavior.
A convenient way to characterize the membrane dynamics as a fluid medium is through thehydrodynamic mobility tensor G, which relates the membrane velocity v to a point force F. Within
the linearized hydrodynamics, they are related byv[k, ω] = G[k, ω] · F[k, ω].

(1)Here the 2D Fourier transform in space and the Fourier–Laplace transform in time for any function∫∞∫∞of space and time f (r, t) is defined by f [k, ω] = −∞ d2 r 0 dt f (r, t) exp[−i(k · r + ωt)], whereMaterials 2012, 5

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k = (kx , ky ) is the 2D wavevector and ω the angular frequency. The mobility tensor for a supported
membrane is given by the Equation (80) in [9] when the surrounding fluids are purely viscous, and how totake into account the viscoelasticity of the ambient fluid is discussed in [7]. Using these previous results,
we obtain the frequency-dependent mobility tensor for a polymer-supported membrane (see Figure 1) as
1Gαβ [k, ω] = 2
ηk + ηf k + ηp [ω]k coth(kh)

(

kα kβ
δαβ − 2
k

)
,

(2)

where α, β = x, y and k = |k|. We note that the case when h → ∞ and ηp [ω] = ηf (without polymer)
was originally considered by Saffman and Delbrück [10–12].
Figure 1. Schematic picture showing a planar viscous membrane with 2D viscosity η. It
is supported by a viscoelastic polymer cushion of 3D frequency-dependent viscosity ηp [ω].
The distance between the membrane and the solid substrate (or the thickness of the polymer
sheet) is h. The upper region of the film is occupied by a semi-infinite fluid with 3D viscosity
ηf . The two correlated point particles “1” and “2” are separated by a distance r.

ηf
h

2
η

r

1

ηp[ω]

Since we are interested in polymer-supported membranes, we follow here the discussion in [13], andconsider the limit of a vanishingly small h → 0, as opposed to the Saffman–Delbrück case. Then theabove expression is approximated as
)
(
1
kα kβ
Gαβ [k, ω] ≈,(3)
δαβ − 2
η(k 2 + νk + κ2 [ω])kwhere ν = ηf /η and(κ[ω] =

ηp [ω]
ηh

)1/2
.

(4)

√
Notice that κ[ω] does not depend on k. Equation (3) has two poles at k± = ν[−1 ± 1 − 4(κ/ν)2 ]/2,
which in the limit h → 0 turn into k± = ±iκ. Hence, for sufficiently small h, the mobility tensor takes
the form of
)
(
1
kα kβ
Gαβ [k, ω] ≈.
(5)δαβ − 2
η(k 2 + κ2 [ω])kMaterials 2012, 5

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This expression indicates that the presence of the upper viscous fluid becomes irrelevant in the limit of
h → 0. In the limit of ω → 0, Equation (5) reduces to the static mobility tensor discussed by Evans andSackmann [14,15] and later generalized by us [16]. In the following, we shall use Equation (5) for thefrequency-dependent mobility tensor of polymer-supported membranes.
Concerning the viscoelasticity of the hydrated polymer sheet, we assume that its complex
modulus obeys a power-law behavior Gp [ω] = G0 (iω)α with α < 1, as generally argued byGranek [17]. This behavior is commonly observed for various polymeric solutions at high frequencies.
Examples are α = 1/2 and α = 2/3 for Rouse and Zimm dynamics, respectively [18], andα = 3/4 for semi-dilute solutions of semi-flexible polymers such as actin filaments [19]. For
particle-tracking experiments [20,21], it is more convenient to work in the Laplace domain defined by∫∞f˜(s) = 0 dt f (t)e−st . One can easily convert from the Fourier–Laplace domain to the Laplace domain
by substituting s = iω. Hence G̃p (s) = G0 sα , and the Laplace transform of the polymer sheet
viscosity is given by η̃p (s) = G̃p (s)/s = G0 sα−1 . Therefore Equation (4) can be written in the Laplacedomain as()1/2
G0 sα−1κ̃(s) =.
(6)ηh
3. Two-Particle Correlated Dynamics
Using Equations (5) and (6), we have discussed in [7] the Brownian motion of a circular disk
immersed in a membrane. Here we discuss the correlated dynamics of two distinctive particles
immersed in a polymer supported membrane. This situation is relevant to two-point microrheology
experiments [22] as further discussed in the next section. Compared to single-particle microrheology,
there are several advantages to perform multi-particle microrheology [21]. For example, long-time
convective drift can be automatically subtracted in this method so that measurements of probe
self-diffusivities become possible over longer times. Multi-particle techniques can be also used toinvestigate heterogeneous materials.
Consider a pair of point particles embedded in the membrane undergoing Brownian motion separated
by a 2D vector r. The quantity of interest is the cross-correlation function (CCF) of the particle
displacements ⟨∆rα1 (0)∆rβ2 (t)⟩r , where ∆rαi is the displacement of the particle i (= 1, 2) along theaxis α (= x, y). We also define the x-axis to be along the line connecting the two particles, i.e., r = rêx .
According to the fluctuation dissipation theorem, the CCF is related to the time-dependent coupling
mobility Mαβ (t) in the Laplace domain as [21]⟨∆r̃α1 ∆r̃β2 (s)⟩r =for sufficiently large r.
time-dependent CCF:

2kB T
M̃αβ (r, s),s2(7)

The inverse Laplace transform of Equation (7) provides us with the⟨∆rα1 (0)∆rβ2 (t)⟩r =1
2πi∫ c+i∞
ds
c−i∞

2kB T
M̃αβ (r, s)est .
s2(8)Since Mxy = 0 by symmetry, it is sufficient to consider the longitudinal coupling mobility Mxx and thetransverse one Myy .

Materials 2012, 5

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Notice that the coupling mobility Mαβ (r) is directly related to the inverse Fourier transform (in space)of the mobility tensor Gαβ (r). Since Gαβ (r) can generally be expressed as Gαβ (r) = C1 (r)δαβ +C2 (r)rα rβ /r2 , the longitudinal and the transverse coupling mobilities are given by Mxx (r) = C1 (r) +C2 (r) and Myy (r) = C1 (r), respectively.
3.1. Longitudinal Coupling
By utilizing Equation (5) and the results in References [9,13,23], the coupling mobilities forpolymer-supported membranes can be obtained analytically. First the Laplace transform of thelongitudinal coupling mobility is given by[]11
K1 [κ̃(s)r]M̃xx (r, s) =−,(9)2πη (κ̃(s)r)2
κ̃(s)rwhere K1 [z] is the modified Bessel function of the second kind, order one. In Figure 2, we plot theresult of numerical inverse Laplace transform of Equation (8) with the longitudinal coupling mobility
Equation (9) when α = 1/2 corresponding to the Rouse dynamics. Here the dimensionless CCF andtime are defined by Cxx (t̄) = (πη/kB T )(ηh/G0 r2 )1/(α−1) ⟨∆rα1 (0)∆rβ2 (t)⟩r and t̄ = (ηh/G0 r2 )1/(α−1) t,
respectively.
Figure 2. Scaled longitudinal cross-correlation function Cxx (t̄) as a function of scaled timet̄ when α = 1/2. These dimensionless quantities are defined in the text. The dotted anddashed lines correspond to the asymptotic expressions given by Equations (11) and (13),respectively.
6

10

3

xx

C
(t̄)
Cxx(t)

10

0

10

-3

10

-6

10

10

-6

10

-4

10

-2

0

10

2

t̄t

10

4

10

6

10

8

10

In the limit of a large distance r → ∞ (κ̃(s)r ≫ 1), the above expression can be approximated asM̃xx (r, s) ≈11.
2πη (κ̃(s)r)2(10)

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When the viscosity of the polymer cushion obeys the power-law behavior η̃p (s) = G0 sα−1 , as assumed
in the previous section, we use Equation (6) for κ̃(s) and perform the inverse Laplace transform given in
Equation (8). Then we obtain⟨∆rx1 (0)∆rx2 (t)⟩r ≈kB T htα,πG0 Γ[1 + α] r2(11)where Γ[z] is the gamma function. This result shows that the viscoelasticity of the polymer sheet leads
to a subdiffusive time dependence of the CCF. Since α < 1, the viscoelasticity slows down the normal
diffusion process. The 1/r2 -dependence in Equation (11) arises from the mass conservation in 2D rather
than the momentum conservation [24].
In the limit of a small distance r → 0 (κ̃(s)r ≪ 1), on the other hand, Equation (9) asymptoticallybehaves as
[ (
)
]
1
2
1
ln
−γ+
,
(12)
M̃xx (r, s) ≈4πηκ̃(s)r2
where γ = 0.5772 · · · is the Euler’s constant. Following the same process as above, the longitudinal
CCF asymptotically behaves as[ ()]
4ηhtα−1
kB T
12⟨∆rx (0)∆rx (t)⟩r ≈t ln+ (α − 3)γ − α + 2 ,(13)
4πη
G0 r 2
which grows like t ln(1/t). Such a logarithmic correction leads to a time-dependent diffusivity. It should
be noted, however, that this asymptotic expression is valid only when G0 r2 t1−α /ηh ≪ 1. The asymptotic
expressions given by Equations (11) and (13) are also plotted in Figure 2, which shows an agreementwith the numerical evaluation.
3.2. Transverse Coupling
Next we discuss the transverse coupling motion. The transverse coupling mobility is analyticallygiven by [9,13,23][]
1
K1 [κ̃(s)r]1M̃yy (r, s) =K0 [κ̃(s)r] +
−
,
(14)
2πη
κ̃(s)r(κ̃(s)r)2
where the K0 [z] is the modified Bessel functions of the second kind, order zero. The numericallyevaluated dimensionless transverse CCF is shown in Figure 3. Since the transverse CCF becomes
negative for large t̄, we have taken its absolute values in the log-log plot.
In the large distance limit of r → ∞, Equation (14) asymptotically behaves asM̃yy (r, s) ≈ −11,2πη (κ̃(s)r)2(15)

which is indeed negative. Hence the time dependence of transverse CCF is simply given by⟨∆ry1 (0)∆ry2 (t)⟩r ≈ −kB T htα.
πG0 Γ[1 + α] r2

In the small distance limit of r → 0, on the other hand, Equation (14) becomes
[ (
)
]
1
2
1
M̃yy (r, s) ≈
ln
−γ−
,
4πη
κ̃(s)r
2

(16)

(17)

Materials 2012, 5

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and its inverse Laplace transform yields the following transverse CCF
)
]
[ (
kB T
4ηhtα−112+ (α − 3)γ − α .
⟨∆ry (0)∆ry (t)⟩r ≈t ln
4πη
G0 r 2(18)

Similarly as before, the asymptotic expressions in Equations (16) and (18) are also plotted in Figure 3.
Figure 3. Scaled transverse cross-correlation function |Cyy (t̄)| as a function of scaled time t̄when α = 1/2. Notice that Cyy (t̄) can be negative. The dotted and dashed lines correspondto the asymptotic expressions given by Equations (16) and (18), respectively.
6

10

3

t̄)|
||CCyyyy((t)|

10

0

10

-3

10

-6

10

10

-6

10

-4

-2

10

0

10

t̄ t

2

10

4

10

6

10

8

10

4. Discussion
In this paper, we have discussed the lateral dynamics in a purely viscous lipid membrane supported bya viscoelastic polymer sheet. Using the generalized frequency-dependent mobility tensor for the polymer
supported membrane, we obtained the cross-correlation function (CCF) of two point particles embeddedin the membrane. The obtained CCF exhibits a subdiffusive time dependence reflecting the viscoelastic
property of the polymer cushion.
Since the subdiffusive behavior is expected for t̄ ≫ 1, we shall estimate the value of the crossovertime t∗ from the condition t̄ ∼ 1. Here we use the values η ∼ 10−10 Pa·s·m [25], h ∼ 10−7 m [6],and r ∼ 10−6 m. Some consideration is required for the value of G0 because its dimension depends onthe value of α due to our definition Gp [ω] = G0 (iω)α . Following the description of polymer solutionsin [18], we expect that G0 roughly scales as G0 ∼ Gτ α ∼ G(ηs /G)α ∼ G1−α ηsα where G is theconstant modulus, τ is the slowest relaxation time, and ηs is the solvent viscosity. Using the values
G ∼ 10 Pa [26] and ηs ∼ 10−3 Pa·s, the crossover time is roughly estimated to be t∗ ∼ 10−8 s when
α = 1/2. Since this crossover time is rather short, one would observe the subdiffusive behavior inmost of the realistic experiments such as microrheology. When r is of the order of a protein size, sayr ∼ 10−8 m, the crossover time is much longer and becomes t∗ ∼ 1 s. In this case, the lateral dynamics
is almost diffusive up to t∗ . However, the subdiffusive behavior becomes always dominant when t → ∞.

Materials 2012, 5

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So far, we have assumed that the viscoelasticity of the polymer sheet obeys a simple power-law
behavior, i.e., Gp [ω] = G0 (iω)α . This is certainly an oversimplified assumption, and the generalfrequency dependence of the polymer solution is more complex [18]. In fact, the technique of
microrheology enables us to measure the local mechanical response of various soft matter over awide frequency range [27,28]. For two-point microrheology, the measurement of the CCF gives therheological information of the hydrated polymer cushion. This can be most easily demonstrated by thelarge distance behavior of the longitudinal coupling mobility given by Equation (10). With the use of
Equation (7), we havekB T hkB T h⟨∆r̃x1 ∆r̃x2 (s)⟩r = 2 2=.
(19)πr s η̃p (s)πr2 sG̃p (s)This equation relates the observed transverse CCF to the modulus of the underlying polymer sheet (rather
than the membrane). In other words, we can extract the 3D bulk information by using the 2D dynamics
taking place in the membrane. Once G̃p (s) is obtained from the experiment, the frequency dependenceof the storage and the loss moduli can be deduced by identifying Gp [ω] = G′p [ω] + iG′′p [ω] = G̃s(s = iω). Notice that these two representations are equivalent because G′p [ω] and G′′p [ω] are related by
the Kramers–Kronig relation [20,21].
Some caution is required when applying our theory to experiments. For two-particle tracking, thedistance between the two point particles (which can be small membrane proteins) should be larger than
the network mesh size. Otherwise, the viscoelastic effect is not reflected in the lateral dynamics. In
order to obtain the full time behavior of the particle motion, one should use a viscoelastic modulus thatis dependent both on wavevector and frequency.
In this paper, we have intentionally treated the membrane as a purely viscous 2D fluid in order toemphasize the role of the ambient polymer sheet. In general, however, lipid membranes themselves
can also be viscoelastic. Recently, viscoelasticity of phospholipid Langmuir monolayers in a liquidcondensed phase was measured using active microrheology [29,30]. The viscoelasticity of membrane
itself was theoretically taken into account in [31,32].
Acknowledgments
We thank C.-Y.D. Lu and E. Sackmann for useful discussions. This work was supported by
Grant-in-Aid for Scientific Research (grant No. 24540439) from the MEXT of Japan. SK also
acknowledges the supported by the JSPS Core-to-Core Program “International research network for
non-equilibrium dynamics of soft matter”.
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