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Cite this: DOI: 10.1039/c2sm00001f

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Time resolved membrane fluctuation spectroscopy†

Timo Betz* and C�ecile Sykes

Received 1st January 2012, Accepted 20th March 2012

DOI: 10.1039/c2sm00001f

Probing the mechanical properties of phospholipid membranes is a fundamental characterization step

for biomimetic membrane systems as well as for living cellular systems. A common method relies on the

analysis of thermal membrane fluctuations, which has been implemented in video flicker spectroscopy.

Here we present a new optical method that directly measures the dynamics of membrane fluctuations

with nm and ms resolution, thus providing access to the bending modulus k and the membrane tension s

for measurement times of 10 s. Our method allows the observation of bilayer membrane fluctuations of

liposomes and the calculation of the power spectral density in yet unreported regimes of frequencies

>50 Hz and amplitudes <10 nm. The recorded data are in agreement with the Helfrich membrane

theory over 4 decades of frequencies (0.1 Hz–1000 Hz). However, we find a systematic overestimation

of the buffer viscosity, which can not be simply explained by measurement errors, but unveils an effect

that is not explained by the classical theory for membrane dynamics, and hence suggests that new

physics must be developed in the observed frequency and amplitude range. The experimental method is

easily reproducible on any optical tweezer setup equipped with a quadrant photodiode position

detection sensor.

1 Introduction

Biological membranes are of uttermost importance for the

proper function of living cells, since they spatially separate

intracellular compartments and present a boundary to the

extracellular environment. It is right at this interface where

signals are transduced,1 material is taken up and released by both

active and passive processes,2 and mechanical forces are passed

between the cell and its environment.3 For the proper under-

standing of biological cells, a physical understanding of the cell

bilayer membrane and its mechanics is necessary, and has

motivated a long series of investigations. In membrane bilayer, as

well as multilayer systems, the measurement of dynamic fluctu-

ations allows the derivation of the mechanical parameters of the

system. Nowadays, it is known that the mechanical properties of

a bilayer membrane, such as its bending rigidity k and tension s,

play a fundamental role in many cellular processes like motility,

proliferation and endo/exocytosis. It is generally accepted that

the bending rigidity depends on the lipid composition while

tension is determined by the experienced physical forces

presented, for example, by hydrostatic or osmotic pressure

differences.

Important model systems for biological membranes are arti-

ficial liposomes made up of controlled amphiphilic molecules

Institut Curie, Section de Recherche, UMR 168, 11, rue Pierre et MarieCurie, 75005 Paris, France. E-mail: timo.betz@curie.fr

† Electronic supplementary information (ESI) available. See DOI:10.1039/c2sm00001f

This journal is ª The Royal Society of Chemistry 2012

which self-organize in closed lipid bilayers. Depending on the

formation method, these liposomes can be of different sizes,

ranging from small unilamellar vesicles (SUV, radius 100 nm) to

giant unilamellar vesicles (GUV) with radii of up to 100 mm. In

the biological context it is advised to call these giant vesicles

‘‘liposomes’’, since the term vesicle is reserved for intracellular

vesicles that are of the size of SUVs. Such liposomes have been

used as membrane model systems and to mimic cells. Here we use

two different ways of preparing liposomes, the classical method

of electroformation4 and the inverted emulsion method.5,6 Both

methods have been used to mimic cell membranes and to

compare cell mechanics with controlled systems using different

lipid compositions.

The physical understanding of liposome mechanics is based on

the assumption of a liquid crystalline phase that exhibits bending

elasticity. This was used by Helfrich to describe the free energy

for bending and stretching a membrane.7 The Helfrich theory

was successfully used to model the static properties of

membranes, and during the past three decades several experi-

mental methods have been developed to measure membrane

mechanics. One class of method consists of static measurements,

such as micropipette aspiration,8 which measures the excess area

stored in thermal fluctuations, or optical tweezer-mediated

membrane tether pulling.9 To access membrane dynamics, video

microscopy has allowed the measurement of the time evolution

of the fluctuations with a temporal resolution of typically 20 ms,

which is usually limited by the video frame rate of z50 Hz. In

these reports, the membrane out-of-plane movement is directly

extracted from the images10,11 and fluctuation amplitudes are

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generally not accessible below 10 nm.11 Recently we introduced

a new optical method to measure membrane fluctuations of red

blood cells with sub nm and ms time resolution,12 thus allowing to

cover a timescale of up to 10 kHz. In this paper we apply this

technique to investigate liposome mechanics.

A theoretical description of membrane fluctuation amplitude

and dynamics was first developed by Brochard and Lennon for

the case of plane membranes.13 This was later extended to

spherical harmonics geometries by Schneider et al.14 and Milner

and Safran15 who included tension and spontaneous curvature in

the spherical description. Membrane dynamics is described

through an autocorrelation function S(q,t) of excited membrane

modes that represents the time course of relaxation for an excited

mode of wavenumber q. The classical theory predicts a single

exponential decay of the autocorrelation function due to energy

dissipation in the surrounding liquid S(q,t)f exp(�uqt) with the

mode dependent relaxation rate uq¼ kq3/(4h), where h represents

the medium viscosity. Other sources of dissipation like inter-

monolayer friction have been subsequently added by Evans and

Yeung16 and Seifert and Langer17 and lead to a relaxation rate

that depends on q2, in contrast to the classical relaxation process.

It should be noted that this friction is only expected to be

important at a high curvature corresponding to q > 107 m�1.

Further models have been proposed by Zilman and Granek, to

explain the experimental measurement of the stretched expo-

nential decay function in the structure factor function as found in

experimental studies.18,19 Recent experimental work used this

model to explain data on mode relaxation obtained with dynamic

light scattering (DLS) and flickering spectroscopy measure-

ments.20 This model fits well for semiflexible membranes

(k > kBT) with small fluctuation amplitudes. A very recent

experimental work investigated the decay of directly excited

fluctuation modes in GUVs, and presents a direct measurement

of stretched decay exponentials21 as S(q,t)f exp(�uqta), which is

not in agreement with classical membrane fluctuation theory.

Additional dissipative mechanisms like the mentioned inter-

monolayer friction or hybrid modes are evoked to explain such

a behavior.20,22,23 Hence, the dynamics of membrane fluctuations

remains a subject of investigation.

Besides these common descriptions of the lipid bilayer in terms

of the Helfrich Hamiltonian, there are also more complex

descriptions derived from smectic-A liquid crystal theory and

which have been successfully applied to multilayer stacks of lipid

membranes.24,25 The case of multilayer lipid membranes differs

from the one of a single bilayer by adding to the bending

modulus a compression modulus that acts across the layer

structure. However, this additional compression modulus is

generally irrelevant in the context of single bilayer membranes.

In both the single bilayer and the stacked multilayer, the inter-

monolayer friction is taken into account as a dissipative mech-

anism,17,25 but dissipation through the surrounding water is

usually not taken into account in the stacked multilayer systems,

while it turns out to be the dominant term for long wavelengths

in bilayer membranes immersed in water. For these reasons we

use the Helfrich model here to describe our experiments.

We provide new insights into the fluctuation dynamics of lipid

bilayer membranes by presenting the single point time resolved

fluctuation spectrum of liposomes under small fluctuation

amplitudes. Our data have a superior time resolution of 50 ms as

Soft Matter

compared to 20 ms in video microscopy and a better spatial

resolution of <1 nm as compared to 10 nm in video microscopy,

and we can hence test yet unexplored regimes. Further high

resolution measurements have been done in the context of

stacked multilayer systems with techniques like X-ray photon

correlation spectroscopy, neutron spin echo,24,25 X-ray scattering

and NMR,26 that however do not generally apply to individual

liposomes formed by single bilayer membranes due to an insuf-

ficient signal to noise ratio. A schematic overview of the current

techniques, their spatial and temporal resolution and their limi-

tations is given in the ESI† (see Fig. S1).

Here we show that we successfully recovered experimentally

the powerlaw dependence as predicted by the classical theory for

plane membranes13 and spherical harmonics.15 However, we

require the viscosity to be a fit parameter, eventually leading to

a viscosity of an order of magnitude higher than the viscosity of

the buffer, which we double checked by varying the viscosity of

the buffer used.

2 Materials and methods

2.1 Chemicals

Egg phosphatidycholine (EPC), 1,2-dioleoyl-sn-glycero-3-phos-

phocholine (DOPC), 1-stearoyl-2-oleoyl-sn-glycero-3-phos-

phocholine (SOPC), 1,2-dioleoyl-sn-glycero-3-([N(5-amino-1-

carboxypentyl)iminodiacetic acid]succinyl)(nickel salt) (DOGS-

NTA-Ni) and cholesterol are all purchased from Avanti Polar

Lipids, and are dissolved in chloroform at a stock concentration

of 10 mg ml�1. Casein is diluted in Millipore water at a concen-

tration of 5 mg ml�1. A 300 mM glucose buffer containing

0.5 mg ml�1 casein (GB), and 300 mM sucrose buffer (SB) are

prepared in Millipore water, and the osmolarity is adjusted to

300 mOs (GB) and 295 mOs (SB) using a freezing point

osmometer. All chemicals are purchased from Sigma Aldrich

unless otherwise stated.

2.2 Buffer viscosity

Buffer viscosity is increased by the addition of 35 mg ml�1, and

75 mg ml�1 Dextran 41000. The viscosities are measured by

trapping a 1 mm bead and recording the power spectral density

(PSD) of the bead position to determine the friction coefficient,

which is proportional to the medium viscosity.27 We measure the

viscosities to be 0.9 � 10�3 Pa s for buffers without Dextran,

1.3� 10�3 Pa s for 35 mg ml�1 Dextran 41000 and 2.7� 10�3 Pa s

for buffers containing 75 mg ml�1 Dextran 41000.

2.3 Liposome preparation

2.3.1 Electroformation.4 Briefly, lipids are diluted to a final

concentration of 1 mg ml�1 in chloroform and 10 ml of this

dilution are applied on an ITO glass chamber. The solvent is

evaporated for 1 h in vacuum, before 1 ml of SB is added and the

chamber is sealed. For electroformation, an alternating voltage

of 2.5 V with a frequency of 10 Hz is applied over night. After

formation, liposomes are diluted in GB to create a different

refractive index of the inside and outside solution. Liposomes are

used for up to one week.

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2.3.2 Inverted emulsion.5,6 Briefly, EPC, DOGS-NTA-Ni and

cholesterol are mixed at a ratio of 58 : 5 : 37, the solvent is

evaporated under nitrogen and 5 ml of mineral oil are added to

reach a total lipid concentration of 0.5 mg ml�1. This lipid mixture

is used to investigate the conditions of biomimetic liposomes

recently introduced.6 The mixture is sonicated for 30 min and

heated to 50 �C for 3 h to dissolve lipids in oil. An emulsion of

sucrose buffer in a lipid–oil mixture (ratio 1 : 200) is formed by

repeated syringe aspiration, and 50 ml of this emulsion is added

on top of 30 ml of glucose buffer in an Eppendorf tube. Lipids in

the oil phase self organize into a lipid-monolayer at the oil–water

interface through which the droplets of the emulsion are passed

through by mild centrifugation (100 g for 10 min).

2.4 Experimental setup

2.4.1 Fluctuation detection. To detect liposome membrane

fluctuations, we use an interferometric technique12,28 that was

recently introduced in the context of red blood cell membrane

mechanics. The setup is equivalent to optical tweezers, but

operates in the fluctuation detection mode at laser intensities of

50 mW, which is more than an order of magnitude below the

trapping regime. The near infra red fiber laser (l ¼ 1064 nm,

YLM-1-1064-LP, IPG, Germany) is intensity controlled by an

acousto optical modulator (AA-Optoelectronics, France),

coupled into the beam path of an inverted microscope (IX71,

Olympus) by an dichroic mirror (Thorlabs) and focused into the

object plane by a water immersion objective (60�, NA 1.2,

Olympus). The condenser is replaced by a long distance water

immersion objective (40�, NA 0.9, Olympus) to collect the light

and imaged by a 1 : 4 telescope on a InGaAs 4-quadrant

photodiode (QPD) (G6849, Hamamatsu). The resulting signal is

amplified by a custom built amplifier system (Oeffner Elec-

tronics, Germany) and digitized at a 200 kHz sampling rate and

16 bit using an analog input card (6353, National Instruments,

Austin, TX, USA). Liposomes are either slightly attached to the

substrate, or aspirated by a micropipette. To record a calibration

curve, the liposomes are scanned in the x-direction through the

laser focus by using a piezo stage (PiezoJena). As the refractive

index between the external and internal buffer increases, the light

passing through the liposome experiences a phase shift different

to the rest of the beam, and the resulting interference at the QPD

shows a linear regime with a slope k as described previously.12

Placing the liposome just at the center of this linear regime and

recording the signal q(t) on the QPD allows the calculation of the

position of the membrane m(t) ¼ q(t)/k. The accuracy of this

position detection is mainly limited by the signal-to-noise ratio

and is generally valid for membrane fluctuation amplitudes that

do not leave the linear regime (z �150 nm) of the calibration

curve. In every experiment, the slope k is measured first, followed

by a 10 s recording of the membrane fluctuation dynamics. This

scheme is repeated 10 times, the PSD for each run is calculated

and the average PSD is taken for further analysis.

2.4.2 Micropipette aspiration. To control the motion of the

liposomes and to apply a well defined tension, we aspirate lipo-

somes in a micropipette. Pipettes are pulled from glass capillaries

using a micropipette puller (P2000, Sutter Instruments, USA).

The opening of the pipette is cleaved to a diameter of 3–5 mm by

This journal is ª The Royal Society of Chemistry 2012

a microforge (MF-830, Narishige, Japan). Micropipettes are fil-

led with glucose buffer and mounted on the experimental

chamber. Before each experiment, the diameter is measured.

Micropipettes and experimental chambers are passivated by

applying a 5 mg ml�1 casein solution for 30 min. The micropi-

pette is connected to a water reservoir of variable height using

a tube system. The reservoir height is controlled in 1 mm steps by

a stepper motor (T-LA28A, Zaber, Canada). Before each

experiment, the point of zero pressure between the micropipette

and the chamber is measured by adjusting to the point of zero

flow of tracer particles aspirated in the micropipette. All height

changes Dd are recorded relative to the zero point. The tension of

the liposome is determined by the applied pressure Dp ¼ g � Dd

� r (r is the density of water), the radius of the liposome rv and of

the micropipette rp. Laplace’s law then yields the tension by:8

s ¼ rp

2�1� rp=rv

� D p (1)

2.5 Data analysis

The time-dependent function m(t) of the membrane position is

further processed using Matlab (The Mathworks, USA). First,

the Fast Fourier Transform (FFT) algorithm provided by Mat-

lab is used to determine the Fourier transform M ¼ FFT(m).

Then the PSD is calculated by:

P S D ¼ M �M�

p� s(2)

where the star denotes the complex conjugate, s is the sampling

frequency and p the number of measurement points of the time

series. Since the FFT values that are higher than the Nyquist

frequency do not contain additional information, the PSD is cut

at the Nyquist frequency s/2. The frequency resolution of these

data points is Df ¼ s/p. The data are fitted to the theory using

a Nelder–Mead Simplex Algorithm provided by Matlab.

3 Theoretical description of the PSD

3.1 Classical theory of membrane dynamics

3.1.1 Infinite plane membrane approximation. To explain the

temporal flickering spectrum of a lipid bilayer liposome we start

with the well known Helfrich Hamiltonian, quantifying the

energy required to bend and stretch a lipid bilayer:

F ¼ð d A

"1

2k�V2h�2þ 1

2s ðVhÞ2

#; (3)

where h is the extension of the membrane from its equilibrium

position. The integral presents the sum over the whole surface.

Spatial Fourier transformation gives the energy per q-mode,

which is known to be1

2kB T , due to the equipartition theorem.

Hence in the time average, the amplitude reads:Dh2q

E¼ kB T

k q4 þ s q2; (4)

In the presented experiments, however, we measure the

dynamics at a single point of the membrane, hence we have to

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include the relaxation dynamics. Solving the Navier–Stokes

equation for an impermeable membrane and non-slip boundary

conditions gives the relaxation frequency for a single excited

q-mode:17,29

u(q) ¼ (kq3 + sq)/(4h), (5)

where h presents the mean viscosity of the two fluids separated by

the membrane. An excited wave is hence dissipated by the

medium and, as the system is overdamped, the amplitude decays

exponentially following the autocorrelation function:

hhq(t)hq(0)i ¼ hh2qiexp[�u(q)t]. (6)

To gain access to the power spectral density (PSD), we can use

the Wiener–Khinching theorem30 that identifies the PSD as the

Fourier transform of the autocorrelation function. Furthermore,

we measure the PSD at a single point, hence all modes contribute

and it is necessary to integrate over all the accessible q-modes:

P S D ¼ðqmax

qmin

d2 q

ð2 pÞ2ð�N

N

�hq ðtÞ hq ð0Þ

�expði u tÞ d t (7)

¼ 1

p

ðqmax

qmin

q d qDh2q

E u ðqÞu ðqÞ2þu2

(8)

¼ 4 h kB T

p

ðqmax

qmin

dq

ðkq3 þ sqÞ2þð4 h uÞ2: (9)

where qmin and qmax are the cut-off wave-numbers that are

defined by the perimeter of the liposome (2pR) and the diameter

of the laser focus df, namely qmin ¼ 2 p

2 p R¼ 1

Rand qmax ¼ 2 p

df.

The integral over q cannot be expressed analytically, however the

limiting cases of high and low frequency can be calculated for an

infinite membrane with point-like detection area, qmin ¼ 0 and

qmax ¼ N. This leads to a tension dominated PSDs for the low

frequency limit, and a bending dominated PSDk for the high

frequency limit:

P S Ds ¼u/0

kB T

2 s u¼ kB T

2 s ð2 p f Þ; (10)

P S Dk ¼u/N

kB T

6 p ð2 h2 kÞ1=3 u5=3¼ kB T

6 p ð2 h2 kÞ1=3 ð2 p f Þ5=3:

(11)

It should be noted that in the tension-dominated regime,

the PSD only depends on the tension and not on the viscosity.

This is surprising since the PSD presents a dynamic measure-

ment. It is due to the fact that the expression depends inversely

on the frequency. As the viscosity increases, the frequency for

a given oscillation mode decreases, but simultaneously the

PSD value increases since it presents a fluctuation per frequency

bin. In simple terms, the curve is shifted to the left and simul-

taneously shifted upwards, thus the PSD for the tension-domi-

nated regime is independent of the viscosity in the presented

limiting case.

Soft Matter

3.1.2 Spherical harmonics description. The integrals of eqn

(10) and eqn (11) are divergent when u / 0. This is due to the

fact that a flat membrane has fluctuations divergent for the limit

of infinite wavelength. In practice, however, fluctuation ampli-

tudes are limited by the finite size of the liposome which we take

into account using spherical harmonics. This is an extension of

previous work by Milner and Safran15 who developed the

spherical harmonics membrane theory. This theory assumes

nearly spherical liposomes and expands the deviations from the

sphere by the angle dependent radius r in spherical harmonics:

r ðUÞ ¼ R

1þ

Xl;m

ulm Ylm ðUÞ!; (12)

where U is the solid angle, R is the mean radius of the sphere, ulmis the amplitude associated with the l,m mode and Ylm are

the spherical harmonics.15 This expression combined with the

Helfrich Hamiltonian (eqn (3)) leads to the mean squared

amplitude for each fluctuation mode:Djulmj2

E¼ kB T

k ðl þ 2Þ ðl � 1Þ l ðl þ 1Þ þ s R2 ðl þ 2Þ ðl � 1Þ: (13)

The decay frequency of each mode is calculated by solving the

Navier–Stokes equation in spherical harmonics, and the result is

presented by the Lamb solution.31 Each mode decays with the

relaxation frequency:

ul ¼ k ðl þ 2Þ ðl � 1Þ l ðl þ 1Þ þ s R2 ðl þ 2Þ ðl � 1Þh R3 Z ðlÞ ; (14)

with Z ðlÞ ¼ ð2 l þ 1Þ ð2 l2 þ 2 l � 1Þl ðl þ 1Þ . As before we write the

autocorrelation function:

hulm(t)ul0m0(0)i ¼ dl,l0dm,m0h|ulm|2iexp(�ult), (15)

Similar to the case of plane membranes, the Fourier transform

of eqn (15) yields the PSD as:

P S D ¼ð dt

Xlmax;m¼þl

l¼2;m¼�l

Djulmj2

Eexpð�ul tÞ expði u tÞ (16)

¼Xlmax

l¼2

Djulmj2

E ul

u2l þ u2

2 l þ 1

2 p: (17)

The last term presents the sum over the m-modes confined

between �l and +l. The sum starts at the second mode, as the l ¼0 mode is a compressive mode that is excluded by the incom-

pressibility condition of the medium, and the l¼ 1 mode presents

a displacement of the whole sphere. This expression was used to

fit the measured data in order to get the mechanical and

dynamical parameters k, s and h. The radiusR is the radius of the

liposome which is measured before the fit function is applied.

Since the detection laser has a finite lateral resolution of about

D x ¼ 0:61 l

1:2z0:5 mm, the upper limit for the sum has to be

restricted to a cut-off mode that is calculated by the ratio between

the perimeter of the liposome and the diameter of the laser focus

lmaxz2 p R

D x, hence for most liposomes only the first 100–200

modes need to be included, making the fit procedure very

efficient.

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3.2 Theoretical prediction for the PSD

To better understand the predicted changes of the PSD under

various experimental conditions, we examine the theoretical

expression and give the results in Fig. 1. These predictions have

been confirmed by experimental results as discussed later in the

paper. It does furthermore show the limits of the plane

membrane approximation.

3.2.1 Comparison between plane membrane approximation

and spherical harmonics description. As shown in Fig. 1A,

a difference between spherical harmonics and the plane

membrane approach is expected for the low frequency regime.

Since a high frequency corresponds to a short wavelength it is

expected that both descriptions converge in this regime, as the

curvature of the liposome can be neglected at the scale of a short

wavelength. The general behavior is not dependent on liposome

size, only the frequency at which the difference is important can

be estimated as a function of the liposome size. The frequency

limit at which the difference between the plane membrane

approximation and the spherical harmonic description becomes

important is found to be the relaxation frequency uq (see eqn (5))

at a q value that corresponds to about 10% of the liposome size,

which corresponds in the presented case to ulim ¼ (kq3lim + sqlim)/

(2p � 4h) z 40 Hz with qlim ¼ 2 p

0:1� R. As the tension is mainly

affecting the low frequency regime, this analysis shows that using

the approximative plane membrane description will systemati-

cally produce too high tension values. This is a general remark

that should be considered when using the plane membrane

Fig. 1 Differences of the expected PSD for various conditions. If not mark

bending rigidity k¼ 10kBT, tension s¼ 1� 10�6 Nm�1, viscosity h¼ 1� 10�3

powerspectrum for a spherical membrane and the simplification of a plane me

liposome size in m. (C) PSD for variable laser focus size in m. (D) PSD for varia

for variable viscosity in Pa s.

This journal is ª The Royal Society of Chemistry 2012

approximation in any analysis. To avoid this problem, we

exclusively use the spherical harmonics description to analyse the

measured PSDs.

3.2.2 Geometrical conditions affect the PSD. First we study

the effect of geometric parameters, such as the size of the lipo-

some or the size of the laser focus. To illustrate this, we plot the

expected effect of the liposome size on the PSD (see Fig. 1B),

which is shown to mainly affect the plateau regime in the low

frequencies. This can be explained by the fact that the amplitudes

are apparently limited by the vesicle size. Large scale fluctuations

are only expected on large liposomes and are suppressed in small

liposomes due to the limiting liposome size. Moreover, the size of

the laser focus sets a limit of lateral resolution to our measure-

ments. This means that we cannot detect membrane fluctuations

with a wavelength smaller than the size of the focus. In the

theory, this cut-off is represented by the qmax of the integral in

eqn (9) and the lmax in eqn (17). The effect of this cut-off on the

PSD is illustrated in Fig. 1C, where the size of the laser

focus determines a high frequency crossover at which the

expected�5/3 powerlaw changes to�2. This change is due to the

dominant u�2 term in eqn (9) for high frequencies.

3.2.3 Mechanical conditions affect the PSD. As mentioned,

the tension of the liposome has its main impact on the low

frequency regime, decreasing the plateau with increasing values.

This can be understood by the fact that the increased tension

hinders large amplitude fluctuations dominant at low frequen-

cies. As presented in Fig. 1D, increasing the tension does exclu-

sively lower the PSD in the low frequency regime, while at high

ed differently, the parameters of the calculation are radius R ¼ 10 mm,

Pa s and laser focus size Dx¼ 0.5 mm. (A) Difference between the expected

mbrane of the same size as the spherical membrane. (B) PSD for variable

ble tension in Nm�1. (E) PSD for variable bending modulus in J. (F) PSD

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frequencies, all curves converge to a value that depends on the

bending rigidity. This is further illustrated in Fig. 1E, where only

the membrane bending modulus is varied. This has an effect on

the whole spectrum shifting the amplitudes to lower values with

increasing membrane bending rigidity. An increased bending

modulus directly corresponds to increased energy costs for

a given amplitude and the constant thermal energy can hence

only excite smaller amplitudes, thus shifting the curves to smaller

values. In contrast, an increase in medium viscosity increases the

low frequency PSD while it reduces the high frequency ampli-

tudes (see Fig. 1F). This effect is due to the fact that an integral

over the whole frequency spectrum has to be independent of any

dynamic parameter such as viscosity.

4 Experimental results

4.1 Mechanical parameters of a liposome

As shown in the previous section, each mechanical parameter has

a different effect on the PSD of liposome membrane fluctuations.

We record the PSD of 32 electroformed EPC liposomes and fit

the spherical harmonics theory on the measured datasets. As fit

parameters we use tension s, bending modulus k and viscosity h,

while the size of the liposome is directly measured using

video microscopy and the size of the laser is calculated to be Dx¼0.5 mm. As presented in Fig. 2 the data can be very well explained

by the fit function over the accessible frequency range of more

than 4 orders of magnitude (R2 value 0.93). In this example

liposomes are not held by a micropipette, but are slightly

attached to the substrate to prevent overall diffusive movement.

Such a movement could easily be identified in the PSD by its f�2

dependency, which would dominate the low frequency regime.

The plot also shows the different limiting cases as described in the

theory section. At low frequency, the data are dominated by the

size of the liposome and by the tension, which gives a f�1 regime

as predicted in eqn (10). At higher frequencies the spectrum is

Fig. 2 PSD of an EPC liposome and the best fit function. The param-

eters that are extracted for this particular liposome are: k ¼ 4.5 � 10�20 J

¼ 11.7 kBT, tension s¼ 1.02� 10�6 N m�1, viscosity h¼ 20.7� 10�3 Pa s.

The different powerlaws are presented by the dashed lines to show that

the curve can be decomposed into different regimes, each accounting for

the different effects of tension, bending modulus and the size of the focus.

Soft Matter

dominated by the bending rigidity that follows a f�5/3 behavior as

predicted in eqn (11), while for even higher frequencies the

limiting cut-off frequency is passed and we measure the expected

f�2. This final regime can be explained by eqn (9) for frequencies

higher than fmax z (kq3max + sqmax)/(4h), where the u�2 term

dominates. The cut-off qmax is determined by the size of the laser

as explained above. We performed 320 experiments on 32 EPC

liposomes resulting in an average value of the bending modulus

k ¼ 7.8 � 3.4 � 10�20 J. The average tension under these

experimental conditions was determined to be s ¼ 8.5 �1.0 � 10�6 N m�1.

4.2 Membrane tension

The membrane tension of a liposome is fixed by the experimental

conditions, like the initial osmotic pressure difference or the

attachment of the liposomes to the substrate. To determine the

accuracy of the presented fluctuation spectroscopy measure-

ments, we fixed the tension by micropipette aspiration, and

simultaneously recorded the membrane fluctuations. Fig. 3A

Fig. 3 (A) PSD of a given liposome under different applied tensions.

The increasing tension mainly affects the low frequency part as predicted

by the theory. (B) Rescaling of the data by the crossover point allows the

collapse of the curves on a master curve. (Inset) Distribution of error

between the detected and the applied tension.

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Table 1 Measured values of tension (in 10�6 N m�1), bending modulus(in 10�20 J) and effective viscosity and buffer viscosity (both in 10�3 Pa s)for different lipid compositions. The upper panel describes the results forthe different lipid compositions under the same buffer conditions, whilethe lower panel gives the different results obtained for variable bufferviscosity on SOPC liposomes. The index of the SOPC lipids denotes thatthese are measured under different buffer viscosities (in 10�3 Pa s)

Lipids s k heff hbuf

EPC 8.5 � 1 7.8 � 3.3 20.3 � 1.5 0.9DOPC 1.5 � 0.3 4.3 � 3.6 9.7 � 2.0 0.9SOPC 3 � 1 13.5 � 5.0 23 � 3.5 0.9SOPC(1) 3 � 1 13.5 � 5.0 23 � 3.5 0.9SOPC(2) 10 � 2 13.5 � 5.0 45 � 4 1.1SOPC(3) 8 � 2 13.5 � 5.0 117 � 10 2.7

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shows the resulting PSD for a series of different tensions. The

data confirm the theoretical prediction (see Fig. 1D) that

the overall PSD decreases as a function of tension. Having the

analytical expression for the approximation of a plane

membrane we further checked the theory by collapsing the

curves, as shown in Fig. 3A, onto one master curve that accounts

for the different tensions. This collapse does divide both the

frequency and the PSD by the expected crossover point with

coordinates uc and PSDc. The crossover point describes the

change between the ideal case of tension dominated (f�1) and the

bending modulus dominated (f�5/3) regime. It can be calculated

by:

PSDs ¼ PSDk (18)

kB T

2suc

¼ kB T

6 ð2 h2 kÞ1=3 u5=3c

(19)

uc ¼

s

3 ð2 h2 kÞ1=3!3=2

(20)

P S Dc ¼ kB T

2 s

s

3 ð2 h2 kÞ1=3!�3=2

(21)

As presented in Fig. 3B the rescaled curves all indeed collapse

on a single master curve. For the calculation of the crossover

point we use the tension applied by the micropipette, the viscosity

of water hw ¼ 0.001 Pa s and a bending modulus of 10�19 J, and

hence no fitting or data treatment besides the rescaling is applied.

Note that the point (1,1) lies below the curve, which is due to the

fact that the real data is not fully represented by the plane

membrane approximation (see Fig. 1A) as used in the calculation

for the crossover point. The observed collapse of the curve does

however hint that the predicted behavior for the tension is

confirmed by the data. To further check if the applied model does

allow for a correct prediction of the tension, we apply the fitting

procedure to 5 liposomes while applying a large range of tension

for each liposome (from 1.2 � 10�6 N m�1 to 5.8 � 10�5 N m�1)

and compare the applied tension to the tension extracted from

our fits. The inset in Fig. 3B shows the relative error for 76

measurements of tension and allows the quantification of an

accuracy of 11% (SEM of the error distribution). These results

directly confirm that the presented analysis of the membrane

fluctuation is consistent with the applied membrane tension even

to very low values of z10�6 N m�1.

Fig. 4 (A) The PSD of liposomes made up from different lipid

compositions, DOPC, EPC and SOPC. The differences in the high

frequency part which shows the (�5/3) powerlaw is consistent with

the measured differences in the bending modulus of these particular

liposomes (kDOPC ¼ 2.9 � 10�20 J, kEPC ¼ 7.4 � 10�20 J, kSOPC ¼ 13.2 �10�20 J). The differences in the low frequency regime are due to

different tensions (sDOPC ¼ 1.5 � 10�6 N m�1, sEPC ¼ 6.0 � 10�6 N m�1,

sSOPC ¼ 10.7 � 10�6 N m�1).

4.3 Different lipid compositions

As the mechanical parameters of a liposome depend on the lipids

used, we measured the PSD functions of a variety of lipid

compositions to determine the differences of the PSD for

different lipids. During these experiments the liposomes were not

aspirated by a micropipette, but only sedimented onto the glass

surface. To have a significant effect on the bending modulus we

use DOPC, EPC and SOPC, which are expected to vary in

bending modulus. Our experiments provide values of k, s and h

as shown in Table 1, and the typical PSD of liposomes repre-

senting each lipid is shown in Fig. 4. These values confirm the

This journal is ª The Royal Society of Chemistry 2012

previously measured values of bending elasticity.11,32 As pre-

dicted by the theoretical expression (see Fig. 1E), the bending

modulus affects the prefactor of the high frequency regime that

has the powerlaw dependence of (�5/3), hence the curves do not

converge at high frequencies, but differ while keeping the correct

powerlaw dependence. The differences at low frequencies are

probably dominated by different liposome tensions in these

experiments where the liposomes were slightly adhering to

the glass.

Besides the bending modulus, we also measure the effective

viscosity of the medium. As already seen in the example above

(Fig. 2), the effective viscosity derived by the theory is not

consistent with the applied medium viscosity as predicted by the

theory. Although all liposomes were prepared in buffers with

a viscosity of h ¼ 0.9 Pa s, we measure different effective

viscosities as reported in Table 1. An interesting finding is that

the measured effective viscosity varies as a function of the

measured bending modulus, as presented in the inset of Fig. 4.

This result hints for a possible lipid dependent effect that leads to

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the additional dissipative effect which could potentially explain

the differences in the effective viscosity as found in this study.

4.4 Different buffer viscosities

In contrast to an increasing tension or increasing bending

modulus, a change in viscosity is expected to have opposing

effects on the high and the low frequency regime. As shown in

Fig. 1F, an increased viscosity leads to an increase of the PSD for

low frequencies and a decrease for high frequencies. This was

experimentally verified by applying three different medium

viscosities to SOPC liposomes. Experimental conditions are: (A)

non modified buffers (h ¼ 0.9 mPa s), (B) inside buffer non

modified, outside buffer 35 mg ml�1 (h ¼ 1.3 mPa s), leading to

a mean viscosity of �h ¼ 1.1 mPa s and (C) inside and outside

buffer containing 75 mg Dextran leading to a mean viscosity of

�h¼ 2.7 mPa s. For a change of viscosity, our values as reported in

Table 1 and Fig. 5 demonstrate the trend as predicted for the

relative changes under an increase in buffer viscosity.

Even though the general predicted behavior is reproduced by

the experiments, the viscosity values found by the theoretical fit

function are consistently higher than the medium viscosity. The

medium viscosity was measured using the thermal fluctuation of

microspheres. To further quantify the systematic overestimation

of the effective viscosity, we checked the dependence of the

measured effective viscosities as a function of the applied

viscosities, as shown in the inset in Fig. 5. The presented linear fit

heff ¼ ghappl + h0 provides fit parameters of g ¼ 52 and h0 ¼�0.015 Pa s. The fact that the prefactor g dominates over the

offset h0 hints that the difficulty in the theoretical description

might be found in a modified prefactor for the viscosity.

4.5 Different formation methods

Up to now all experiments were performed on electroformed

liposomes. To check if the measurements do depend on the

Fig. 5 The PSD of SOPC liposomes under different medium viscosities.

As predicted by the theory, the low frequency regime increased while the

high frequency fluctuations were reduced under increasing viscosity.

(Inset) Measured dependence of the effective viscosity from the applied

viscosity. The line shows a linear fit heff ¼ ghappl + h0 with fit parameters

determined to be g ¼ 50, h0 ¼ �0.017 Pa s.

Soft Matter

formation method we further study liposomes created by the

inverted emulsion technique,5 as described in the Materials and

Methods section. The lipid composition is EPC, DOGS-NTA-Ni

and cholesterol, with a ratio of 58 : 5 : 37, as used in the biomi-

metic system that mimics the cell cortex.6 In both methods, the

same buffer solutions are applied. We measure the PSD of

electroformed liposomes (190 measurements on 19 liposomes)

and of liposomes formed by an inverted emulsion (60 measure-

ments on 6 liposomes). For the presented measurements, we use

the same micropipette for both the liposome populations. The

resulting PSDs are shown in Fig. 6. In the high frequency regime

we do not find a significant difference, while the difference in the

low frequency regime is due to the slightly different tensions

applied during the experiments. This difference is a result of the

different liposome size, which enters in the calculation of the

tension. The applied suction pressure for both liposomes is

constant. The average values of the membrane tension as

determined by the fit of sfitef ¼ 1.1 � 0.3 � 10�5 N m�1 and sfit

ie ¼ 7

� 1.3 � 10�6 N m�1 that correspond to the known tensions as

fixed by the micropipette, which were sapplef ¼ 1.5 � 10�5 N m�1

and sapplie ¼ 6.3 � 10�6 N m�1. The average lower tension for the

liposomes formed by the inverted emulsion is the reason for the

higher PSD in the low frequency regime. However, it should be

pointed out that this difference is purely explained by the

externally fixed tension and does not imply differences due to the

formation method. In contrast, differences in the bending

modulus would hint at an effect of the formation method on the

liposomes. In detail, the bending moduli for electroformation

and the inverted emulsion are kef ¼ 7.5 � 5.6 � 10�20 J (n ¼ 19)

and a kie ¼ 5.7 � 2.0 � 10�20 (n ¼ 6). While not equal, these

values are not significantly different when evaluated by the

students t-test (p-value of 0.17) and they are consistent with the

values of the EPC liposomes reported above.

Fig. 6 The PSD of liposomes formed with different methods. Both, the

electroformed and inverted emulsion formed EPC mixture liposomes

show the same distributions within the error of the distribution. The

slight difference in the low frequency part is due to a slightly different

tension during the measurement. The missing points in the intermediate

regime were left out, because they show strong peaks resulting from

oscillations of the micropipette tip used on the day of the measurement.

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5 Discussion

5.1 Measurement of bending rigidity and tension

Our results show that the time resolved membrane spectroscopy

provides a fast and direct access to the mechanical parameters of

phospholipid liposomes. Compared to classical fluctuation

spectroscopy based on video microscopy, it has the advantage of

a high time resolution of 100 ms and a high spatial resolution of

<1 nm. While these spatial and temporal resolutions are well

below methods such as X-ray photon correlation spectroscopy

and neutron spin echo,24,25 NMR or X-ray scattering,26 our

method can be applied to a single bilayer liposome directly in the

buffer solution, whereas the former methods generally only apply

to stacked multilayered membranes.

The PSD of a 10 s measurement can be used to directly fit

classical membrane theory, showing that the model does describe

the data very accurately over timescales ranging from 100 ms up

to 10 s. By applying a controlled tension, we confirm that the

PSD can be used to extract membrane tension within 11% of the

expected value. This is an important advantage over video

microscopy, which is limited to very floppy vesicles as fluctuation

amplitudes smaller than 10 nm can not be resolved. However,

relevant tensions of more that 10�7 N m�1 usually constrain the

fluctuation amplitudes below that limit. In the presented method,

the measurement of tension is also limited by the fluctuation

amplitude, which decreases with increasing tension. However,

due to the high resolution of fluctuation amplitudes we success-

fully recovered applied tensions of up to 10�4 N m�1, and the

calculated limit of the setup is z10�3 N m�1.

Furthermore, we confirm literature values of the bending

modulus using the presented analysis of the PSD. A critical

discussion of the resulting errors shows that the measurement of

the bending modulus is in principle less reliable than the

measured tension. The reason for this is found in the way the

tension and the bending modulus influence the PSD, and is most

obviously seen in the limit case of tension or bending dominated

PSDs shown in eqn (10) and eqn (11). The tension is inversely

proportional to the PSD, therefore a 2-fold decrease in the PSD

corresponds to a 2-fold increase in tension. Hence, any inac-

curacy of the PSD only linearly influences the estimation of the

tension. However, in the case of the bending modulus, eqn (11)

shows that the PSD depends on k�1/3. Hence, a 2-fold decrease of

the PSD corresponds to an 8-fold increase in the calculated

bending modulus.

5.2 Systematic overestimation of viscosity

Besides the very accurate description of the data by the theory we

detect an effective viscosity which is systematically higher than

that of the known medium viscosity. This leads to a fit routine

where we leave the medium viscosity as a fit parameter. Imposing

known buffer viscosities on the fit is found to result in a failure of

the fit. In this case, the best fitted curve does not overlay the

measured data. To investigate if this discrepancy between the

buffer viscosity and the measured effective viscosity is a system-

atic measurement error we rule out the following hypothetical

error sources:

(A) The most simple hypothesis for an experimental error

would be a wrong calibration factor that is used to determine the

This journal is ª The Royal Society of Chemistry 2012

fluctuation amplitude from the QPD data. Errors in this linear

prefactor on the amplitudes would propagate through the

calculation of the PSD, and effectively shift the full curve. Hence

it would have an effect on both the low and the high frequency

regime. Using the found difference between the effective and real

buffer viscosity allows the estimation of the calibration error

required to understand this difference. This hypothesis would

mean that the calibration has to be off by a factor of z3. Since

the calibration curves enter the calculation of the PSD as a power

of 2, the effect of the PSD, and hence on the measured tension

would be almost a factor of 10. However, we verify that the

extracted tension is very accurate, and hence an overall offset can

not explain the differences between the effective viscosity and the

buffer viscosity. We furthermore check the calibration routine by

applying a defined movement of the vesicle using the piezo-stage.

For this, we displace the stage by 50 nm during the measurement

of the membrane position, and successfully find this displace-

ment in the measured edge movement. Again, an error in the

calibration would directly lead to a conflict between the applied

displacement and the measured movement of the membrane

position.

(B) Another hypothesis is that the presented measurements

include a systematic error that only affects the high frequency

part, which corresponds to the small wavelength regime.

However, when we apply a controlled tension using the

micropipette we test the same frequency range for many

different tensions. Considering the extreme cases of high

tension (see the black curve in Fig. 3A) we see that the tension

dominated (f�1) regime extends up to frequencies of 1 kHz. If

any systematic error would only affect the high frequency

range, it should also directly affect the measured tension, which

is not observed. Hence, the fact that we do not observe such

a problem at the high frequency range disfavors the hypoth-

esis of a systematic error only acting on the high frequency

regime. Another finding that disfavors the hypothesis of

frequency dependent measurement error is the fact that the

theory works extraordinarily well over the large frequency

range. Any frequency dependent error should in principle

influence the found powerlaw, thus making the fit function

inadequate.

As a final conclusion we examine possible problems of the

theoretical expression. As the overall fit works well, we can

narrow possible problems down to the influence of the parameter

h introduced by the relaxation frequency. Our systematic change

of the external viscosity (inset Fig. 5) points out that a prefactor

in the viscosity would most probably explain the changes in the

effective viscosity. Phenomenologically, the data would be

explained by an effective viscosity heff ¼ g � hbuffer. However,

this hypothesis will have to be tested on a direct measurement of

the relaxation rate for each mode in a similar situation. This

measurement is currently beyond the possibilities of the setup.

Finally, the very interesting finding that the effective viscosity

seems to depend on the lipid composition might lead the way to

possible molecular effects that would have to be taken into

account. It should be noted that the clear correlation between the

membrane bending rigidity and the observed effective viscosity

(inset Fig. 4) hints for a fundamental mechanical relation

between membrane mechanics and the observed effective

viscosity.

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Finally, we investigate the possible effect of the recently

observed stretched exponential. For this we numerically generate

a stretched exponential and integrate eqn (7) to determine the

PSD that results from such a modified autocorrelation function.

We find that the results are not in agreement with the experi-

mentally found �5/3 powerlaw, and can hence not explain the

found differences in viscosity.

Therefore our data call for an unproved theoretical description

of membrane dynamics. As mentioned, other experiments show

a discrepancy between membrane theory and measured

membrane dynamics.20,22,23 One effect that might be taken into

account is that the water viscosity close to a surface is signifi-

cantly increased.33 This might provide an additional dissipative

mechanism that is yet not included in the model. Such an

increased viscosity could in fact also depend on the molecular

details of the membrane and might therefore explain the found

differences of effective viscosity for the different lipids.

6 Conclusion

The present paper reports a new method to quantify the

mechanical parameters of liposomes. This method allows the

measurement of membrane dynamics at a yet unexplored

frequency and amplitude regime, giving the correct membrane

tension and membrane bending rigidity. However, we find

significant differences in the viscosity estimates that are not

covered by the theory. After careful exclusion of possible

systematic error, there is a possibility that the theoretical

description at high frequencies and for small amplitudes might

need to be modified. This conclusion leads to the question why

such a difference has not yet been observed previously. In fact,

the focus during the last 20 years was mainly on static

measurements of membrane mechanics, which did not investi-

gate dynamic variables such as the viscosity. The few reports that

checked membrane dynamics in the form of the autocorrelation

data already found a discrepancy between the classical model

and their measurements which they explained by additional

dissipative processes.20,22,23 This hints that the dissipative

processes in membrane fluctuations are still to be fully under-

stood. It should be noted that these studies were based on video

microscopy and hence limited to a temporal resolution of >40 ms

and a spatial resolution of 10 nm, and therefore did not measure

the same regimes as presented in this work. A important further

difference between our measurement and previous investigations

is that we generally probe the regime of small fluctuation

amplitudes (z50 nm) while the measurements using video

microscopy generally investigate very floppy liposomes with

fluctuation amplitudes z500 nm. Furthermore, a recent work

directly measured membrane dynamics using video microscopy

for very floppy liposomes.21 In this work the found decay times

are well in accordance with the expected viscosity of water. It has

to be tested to see if these differences to the presented work are

due to time or spatial resolution, or depend on the vesicle

conditions used in the experiments.

Overall, our measurements combined with the previous

investigations of membrane dynamics suggest that the theoretical

Soft Matter

description needs further improvement to fully describe

membrane dynamics.

Acknowledgements

The authors would like to thank Pietro Cicuta, Jean Francois

Joanny and Martin Lenz for their helpful discussions. TB was

supported by an EMBO long term fellowship. This work was

supported by the ANR SYSCOM (ANR-08-SYSC-013-03).

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