We argue that membrane viscosity, ηm, plays a prominent role in the thermal fluctuation dynamics of micron-scale lipid domains. A theoretical expression is presented for the timescales of domain shape relaxation, which reduces to the well-known ηm = 0 result of Stone and McConnell in the limit of large domain sizes. Experimental measurements of domain dynamics on the surface of ternary phospholipid and cholesterol vesicles confirm the theoretical results and suggest domain flicker spectroscopy as a convenient means to simultaneously measure both the line tension, σ, and the membrane viscosity, ηm, governing the behavior of individual lipid domains

Alexander

Benvegnu

Brian A. Camley

Chandler

Cicuta

Cinzia Esposito

Collins

Dietrich

Dimova

Esposito

Frank L.H. Brown

Frolov

Gennis

Goldstein

Harland

Honerkamp-Smith

Lubensky

Mann

Petrov

Risselada

Saffman

Safran

Samsonov

Stone

Tobias Baumgart

Veatch

PubMed

Crossref

Lipid Bilayer Domain Fluctuations as a Probe of Membrane Viscosity

AbstractWe argue that membrane viscosity, ηm, plays a prominent role in the thermal fluctuation dynamics of micron-scale lipid domains. A theoretical expression is presented for the timescales of domain shape relaxation, which reduces to the well-known ηm = 0 result of Stone and McConnell in the limit of large domain sizes. Experimental measurements of domain dynamics on the surface of ternary phospholipid and cholesterol vesicles confirm the theoretical results and suggest domain flicker spectroscopy as a convenient means to simultaneously measure both the line tension, σ, and the membrane viscosity, ηm, governing the behavior of individual lipid domains

Camley, Brian A.

Esposito, Cinzia

Baumgart, Tobias

Brown, Frank L.H.

Elsevier - Publisher Connector 

L44 Biophysical Journal Volume 99 September 2010 L44–L46Lipid Bilayer Domain Fluctuations as a Probe of Membrane ViscosityBrian A. Camley,†* Cinzia Esposito,‡ Tobias Baumgart,‡ and Frank L. H. Brown†§*
†Departments of Physics and §Chemistry and Biochemistry, University of California, Santa Barbara, California; and ‡Department of Chemistry,
University of Pennsylvania, Philadelphia, PennsylvaniaABSTRACT We argue that membrane viscosity, hm, plays a prominent role in the thermal fluctuation dynamics of micron-scale
lipid domains. A theoretical expression is presented for the timescales of domain shape relaxation, which reduces to the well-
known hm¼ 0 result of Stone and McConnell in the limit of large domain sizes. Experimental measurements of domain dynamics
on the surface of ternary phospholipid and cholesterol vesicles confirm the theoretical results and suggest domain flicker spec-
troscopy as a convenient means to simultaneously measure both the line tension, s, and the membrane viscosity, hm, governing
the behavior of individual lipid domains.Received for publication 27 April 2010 and in final form 9 July 2010.
*Correspondence: camley@physics.ucsb.edu or flbrown@chem.ucsb.eduEditor: Peter Tieleman.
 2010 by the Biophysical Society
doi: 10.1016/j.bpj.2010.07.007As a first step to understanding the biophysics of plasma
membranes (1), model membrane systems have been devel-
oped to mimic aspects of biomembranes under controlled,
simplified laboratory conditions (2). Much work has focused
on vesicles composed of ternary phospholipid/cholesterol
mixtures, where physical properties of lipid domains can
be characterized by fluorescence microscopy (3–6).
Among the most biologically important physical proper-
ties of inhomogeneousmembrane systems are the line tension
between coexisting phases s and the membrane viscosity hm.
Line tensions influence the distribution of domain sizes (7),
and viscosities set diffusion coefficients for lipid domains
(8) andmembrane proteins (9).Measurements of line tension
via microscopy are well known, particularly for lipid mono-
layers (10–12); recent ‘‘domain flicker spectroscopy’’ (6)
experiments were developed to measure the line tension on
the surface of ternary vesicles. The membrane viscosity is
not as simple to measure, though it may be estimated by
fitting diffusion coefficients to the Saffman-Delbru¨ck (SD)
form (8,13) or by microrheology (14).
In this letter,we show that flicker spectroscopymaybeused
to measure not only s, but also hm. Our theoretical work
exploits hydrodynamic analysis introduced by Stone and
McConnell (SM) (15), but extends their results to a physical
regime where membrane viscosity is relevant. Our experi-
ments show that domain relaxation times do deviate from
the hm¼ 0 SM predictions. By combining theory with exper-
iment, it becomes possible to directly measure hm.
Our analysis of domain fluctuations assumes an isolated
domain of constant area within a large flat membrane
(Fig. 1). We assume that the boundary energy is given by
E ¼ sL, with s the line tension and L the domain perimeter.
It is convenient, theoretically (15) and experimentally
(6,11), to express the domain shape in Fourier modes,
rðq; tÞ ¼ Rð1þ 1
2
P
ns0 unðtÞeinqÞ; with n from –N/2 to
N/2. To second order in un(t), the energy cost of deviations
from the minimum energy circle with radius R is (6)DE ¼ spR
2
XN=2
n¼ 2

n2  1~~unj2: (1)
The equipartition theorem (as applied to the Fourier compo-
nents of a real-valued physical quantity (16)) immediately
leads to the spectrum of equilibrium shape fluctuations (11),
~~unj2 ¼ 2kBT
spRðn2  1Þ; (2)
and a direct experimental route to the determination of s via
measurement of hjunj2i (6).
The time-dependence associated with fluctuations in un(t)
may be calculated within the hydrodynamic model intro-
duced by Saffman and Delbru¨ck (9), namely a single, thin,
flat fluid sheet with surface viscosity hm surrounded by
a bulk fluid of viscosity hf treated within the creeping-
flow approximation (Fig. 1). (17). This picture neglects
the dual leaflet structure of the bilayer and applies only to
symmetric bilayers with domains that are in registry across
both leaflets. The available experimental (19), theoretical
(20), and simulation (21) evidence suggests that domain
registry is nearly perfect in ternary model membrane
systems, with interleaflet domain mismatch confined to
areas of tens of lipids for an entire domain (20,21). The
picture from Saffman and Delbru¨ck (9) is expected to be
completely adequate to describe domain dynamics over
the optical length-scales observed experimentally.
Relaxation of a general domain shape is driven by the line
tension s, with the radially directed force per unit length
at the domain boundary given by the functional derivative
fr(q, t)¼ –R1d(DE(t))/dr(q, t) (22), which is, to linear order
in un(t),
100 101 102
10−4
10−3
10−2
10−1
100
Mode number n
R
el
ax
at
io
n 
tim
e 
τ n
 
(s)
 
 
η
m
 = 0 (SM)
η
m
 = 10−7 s.p.
η
m
 = 10−6 s.p.
η
m
 = 10−5 s.p.
1/n
1/n2
FIGURE 2 Relaxation times (Eq. 5) as a function of mode
number for several membrane viscosities assuming a domain
with R ¼ 2.5mm, s ¼ 0.1 pN, and hf ¼ 0.01 Poise (water). As
membrane viscosity is increased, the relaxation times increase,
and the scaling with n changes from tn ~ n
2 for R/n [ Lsd
(Eq. 7) to tn ~ n
1 for R/n  Lsd (Eq. 8).
FIGURE 1 The shape of a quasicircular lipid domain within
a thin,flatmembrane is specifiedby thedistance fromthedomain
center of mass to the boundary as a function of the polar angle q.
Both lipid phases are assumed to share the surface viscosity hm
(17). The membrane is surrounded by a bulk fluid of viscosity hf .
Biophysical Letters L45frðq; tÞ ¼  s
2R
XN=2
n¼N=2
ðn2  1unðtÞeinqh1
2
X
fnðtÞeinq: (3)
This force drives flowwithin the bilayer and in the bulk fluid.
In particular, the radial velocity at the domain boundary
vrðq; tÞ ¼ ddtrðq; tÞ ¼ R2
P
u
:
nðtÞeinq may be obtained through
application of the techniques of SM (15), or by use of the
more general formalism developed by Lubensky and Gold-
stein (22). The result is conveniently cast in terms of the Four-
ier modes (see the Supporting Material for further details)
vnðtÞ ¼ Ru: nðtÞ ¼ n
2R
hm
InðLÞfnðtÞ; (4)
where the integral InðLÞ ¼
RN
0
dxJ2nðxÞ=½x2ðx þLÞ ; JnðxÞ;
is a Bessel function of the first kind and L ¼ 2Rhf =hm:
Combining Eqs. 3 and 4 leads to u
:
nðtÞ ¼ unðtÞ=tn, with
the solution unðtÞ ¼ unð0Þet=tn , where
tn ¼ hmR
s
1
n2ðn2  1Þ
2
4Z
N
0
dx
J2nðxÞ
x2ðx þ LÞ
3
5
1
: (5)
The fluctuation-dissipation theorem (23) provides the
connection between the relaxation of un(t) and the equilib-
rium correlation functions measured in flicker spectroscopy,
hunðtÞunð0Þi ¼
unj2et=tn : (6)
Equation 5 is the primary theoretical result of this letter; the
expression is evaluated for a few representative parameters
in Fig. 2. Though there is no general closed-form solution
for the integral in Eq. 5, it reduces to two simple results in
appropriate limits. For large domains and sufficiently small
n (L [ n), dissipation in the bulk fluid dominates the
dynamics, hm may be neglected, and Eq. 5 approaches
a result generally attributed to SM (15,24),
t fluidn ¼
2pR2hf
s
n2  1=4
n2ðn2  1Þ: (7)
In the opposite limit (L  n), the membrane viscosity
dominates and hf may be neglected, recovering the result
of Mann et al. (24)
t membranen ¼
4hmR
ns
: (8)Because both Eq. 7 and Eq. 8 neglect a source of dissipation,
membrane fluidtn R tn , tn . The crossover between regimes
occurs where the wavelength of the fluctuations (~R/n)
becomes comparable to the SD length scale Lsd ¼ hm/2hf.
Membrane viscosities generally fall within (0.1–10) 
106 surface poise (poise-cm, or grams/s) (8,13,25), leading
to SD lengths Lsd ~0.1–10 mm. Recent experimental
measurements (6) on domains with radii of a few microns,
unlike the much larger domains originally studied by SM
and co-worker (10,15), are thus expected to deviate from
the SM result (see Fig. 2).
To test the above analysis, giant unilamellar vesicles of
a ternary mixture of phospholipids, namely, dipalmitoyl-
phosphatidylcholine (DPPC) and diphytanoylphosphatidyl-
choline (DiPhyPC) and cholesterol, were studied
experimentally using the flicker spectroscopy technique
(see (6) for details). Twenty-eight r(q, t) traces from indi-
vidual domains were analyzed, each from a vesicle with
25:55:20 molar ratios of DiPhyPC/Chol/DPPC at 20 5
1C (see the Supporting Material). Note that DiPhyPC
was chosen over dioleoylphosphatidylcholine (DOPC)
used in Esposito et al. (6) for its greater photostability
(26). Domain images were thresholded to find r(q, t), which
was Fourier-transformed to yield un(t). Line tensions (s)
were extracted from the variance in Fourier modes un via
Eq. 2 (with mean s ¼ 0.23 pN over all 28 traces) and relax-
ation times (tn) were determined by fitting hun(t)u–n(0)i to
single exponential decay. With s, R, and hf known, Eq. 5
has a single unknown parameter: hm. The relaxation times
over all measured n values were simultaneously fit to our
general result (Eq. 5) using hm as the fit parameter. A typical
fit is shown in Fig. 3. Applying this procedure to all
traces determined the mean membrane surface viscosity
hm ¼ (45 1)  106 s.p., consistent with the low-temper-
ature values observed from fitting diffusion constants. For
comparison, Petrov and Schwille (13) use the data of Cicuta
et al. (8) and find viscosities ofz2  106 s.p. at a similar
temperature, though for different lipids. A similar analysis
based on the SM expression for the tn (Eq. 7) was also
attempted (see the Supporting Material for further details).Biophysical Journal 99(6) L44–L46
2 3 4 510
−2
10−1
100
Mode number n
R
el
ax
at
io
n 
tim
e 
τ n
 
(se
co
nd
s)
 
Experimental data
Generalized theory
Stone−McConnell theory
FIGURE 3 DiPhyPC/Cholesterol/DPPC relaxation times for
asingledomain tracewithR¼3.8mm.(Errorbars) 95%confidence
intervals for the fit to Eq. 6. Thefitmembrane surface viscosityhm
is 3.25 106 s.p. Also plotted is the SM theory for the relaxation
times (Eq. 7). The theoretical results assume s ¼ 0.19 pN, as ex-
tracted from the variance in un (Eq. 2). (Dotted lines) Uncertainty
in the SMpredictions fromadjusting sby one standard deviation.
Uncertainty inscannot account for thedeviationbetweenSMand
experiment.
L46 Biophysical LettersBecause SM neglects hm, there are no free parameters in
tfluidn and relaxation times are predicted immediately from
s. The SM theory predicts relaxation times in clear
disagreement with the measurements (Fig. 3); additional
dissipation from the bilayer itself must be considered to
explain the data. We note that prior successful fits of the
SM theory to experimental results using DOPC/Choles-
terol/DPPC lipid mixtures (6) were only apparent. Equation
2 of this work corrects Eq. 3 of Esposito et al. (6). Also, the
extraction of tn from correlations in un(t) (via Eq. 6) corrects
the procedure of Esposito et al. (6), which was based upon
correlations in jun(t)j2. Experimental relaxation times that
appeared consistent with SM in Esposito et al. (6) are actu-
ally four-times longer than SM predictions when the anal-
ysis is carried out properly. This level of disagreement
between SM and experiment is similar to results summa-
rized in Fig. 3.
We have proposed a simple extension to the usual SM
theory for relaxation times of domain fluctuations (Eq. 5)
and have verified it against experimental data. The experi-
mental results suggest that membrane viscosity significantly
affects these relaxation times for the smallest wavelength
modes observable by microscopy. By combining equilib-
rium measurements of line tension (via Eq. 2) with the
measurement of dynamic relaxation, the viscosity of a lipid
bilayer may be determined.SUPPORTING MATERIAL
Details of the theoretical derivation and the comparison to experimental
data are available at http://www.biophysj.org/biophysj/supplemental/
S0006-3495(10)00851-9.
ACKNOWLEDGMENTS
This work was supported in part by grant No. CHE-0848809 from the
National Science Foundation and grant No. 2006285 from the US-Israel
Binational Science Foundation. B.A.C. acknowledges support from the
Fannie and John Hertz Foundation.Biophysical Journal 99(6) L44–L46REFERENCES and FOOTNOTES
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Lipid Bilayer Domain Fluctuations as a Probe of Membrane Viscosity 

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Lipid Bilayer Domain Fluctuations as a Probe of Membrane Viscosity

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