Optical tweezers have been used by several groups to manipulate human red blood
cells (see Fig. 2-9), which have no space-ï¬lling cytoskeleton but only a membrane-
associated 2D protein polymer network (spectrin network). The 2D shear modulus
measured for the cell membrane plus spectrin network varies between 2.5 ÂµN/m
(Henon et al., 1999; Lenormand et al., 2001) and 200 ÂµN/m (Sleep et al., 1999), pos-
sibly due to different modeling approaches in estimating the modulus. The nonlinear
part of the response of red blood cells has been explored by using large beads and high
laser power achieving a force of up to 600 pN. The shear modulus of the cells levels
off at intermediate forces before rising again at the highest forces, which was simu-
lated in ï¬nite element models of the cells under tension (Dao et al., 2003; Lim et al.,
32 P. Janmey and C. Schmidt
Fig. 2-9. Stretching of red blood cells by optical tweezers, using a pair of beads attached to diamet-
rically opposed ends of the cell. Forces are given next to the panels. From Henon et al., 1999.
Using magnetic particles has the advantage that large forces (comparable to AFM) can
be exerted, while no open surface is required. One can use magnetic ï¬elds to apply
forces and/or torques to the particles. Ferromagnetic particles are needed to apply
torques; paramagnetic particles are sufï¬cient to apply force only. A disadvantage
of the magnetic force method is that it is difï¬cult to establish homogeneous ï¬eld
gradients (only gradients exert a force on a magnetic dipole), and the dipole moments
of microscopic particles typically scatter strongly. Furthermore, one is often limited
Experimental measurements of intracellular mechanics 33
Fig. 2-10. A magnetic manipulation system to measure viscoelasticity in a single cell. From
Freundlich and Seifriz, 1922.
to video rates for displacement detection when using the force method. Rotations can
be detected by induction for ensembles of particles.
One of the ï¬rst reports of an apparatus to measure intracellular viscoelasticity
was from Freundlich and Seifriz (1922). A diagram of the instrument is shown in
In this instrument, a micromanipulator mounted next to the microscope objective
was used to insert a magnetic particle, made of nickel or magnetite, into a relatively
large cell like a sand dollar egg. Then a magnetic ï¬eld gradient, produced by an
electromagnet placed as close as possible to the cell, was used to impose a force on
the bead, whose displacement was measured by the microscope. The strength of the
force on the bead could be calibrated by measuring the rate of its movement through
a calibration ï¬‚uid of viscosity that could be measured by conventional rheometers.
This magnetic manipulation instrument was the precursor of current magnet-based
microrheology systems, and was further enhanced by work of Crick and Hughes
(1950), who made two important modiï¬cations of the experimental design. One was
to ï¬rst magnetize the particle with a large magnetic ï¬eld, and then use a smaller
probing magnetic ï¬eld directed at a different angle to twist particles on or within the
cell. The second change was to use phagocytic cells that would engulf the magnetic
particle, thereby avoiding possible damage to the cell when magnetic particles were
forced through its membrane. These early studies were done before the cytoskeleton
was visualized by electron or ï¬‚uorescence microscopy and before the phospholipid
bilayer forming the cell membrane was characterized, so a critical evaluation could not
be done of how disruptive either way of introducing the beads was. Further pioneering
work was done on amoebae (Yagi, 1961) and on squid axoplasm (Sato et al., 1984).
34 P. Janmey and C. Schmidt
In principle, the motion of embedded probes will depend on the probe size. Small
particles can diffuse through the meshes, and this has been used to determine effective
mesh sizes in model systems (Jones and Luby-Phelps, 1996; Schmidt et al., 1989;
Schnurr et al., 1997) and in cells (Jones et al., 1997; Luby-Phelps, 1994; Valentine
et al., 2001). On the other hand, the beads might interact with and stick to the cyto-
skeleton, possibly mediated by an enveloping lipid membrane and by motor proteins,
which would cause active motion. How micron-sized beads are coupled to the net-
work in which they are imbedded is still a major issue in evaluating microrheology
measurements, and no optimal method to control or prevent interactions yet exists.
Entry of a particle through phagocytosis certainly places it in a compartment distinct
from the proteins forming the cytoskeleton, and how such phagosomes are bound to
other cytoplasmic structures is unclear. Likewise, both the mechanical and chemical
effects of placing micron-sized metal beads in the cell raises issues about alignment
and reorganization of the cytoskeleton. These issues will be further considered in the
Pulling by magnetic ï¬eld gradients
Magnetic particles can either be inserted into cells or bound â“ possibly via speciï¬c
attachments â“ to cell surfaces. Both superparamagnetic particles (Bausch et al., 1998;
Keller et al., 2001) and ferro- as well as ferrimagnetic particles (Bausch et al., 1999;
Trepat et al., 2003; Valberg and Butler, 1987) have been used. Paramagnetic particles
will experience a translational force in a ï¬eld gradient, but no torque. With sharpened
iron cores reaching close to the cells, forces of up to 10 nN have been generated
(Vonna et al., 2003). Ferromagnetic (as well as ferrimagnetic) particles have larger
magnetic moments and therefore need less-strong gradients, which can be produced
by electromagnetic coils without iron cores and can therefore be much more rapidly
modulated. The particles have to be magnetized initially with a strong homogeneous
ï¬eld. Depending on the directions of the ï¬elds, particles will experience both torque
and translational forces in a ï¬eld gradient (see Fig. 2-11). Forces reported are on the
order of pN (Trepat et al., 2003).
Forces exerted by the cell in response to an imposed particle movement, both
on the membrane and inside the cell, are mostly dominated by the cytoskeleton.
Exceptions are cases where the particle size is smaller than the cytoskeletal mesh
size; specialized cells, such as mammalian red blood cells without a three-dimensional
cytoskeleton; or cells with a disrupted cytoskeleton after treatment with drugs such
as nocodazole or cytochalasin. The interpretation of measured responses needs to
start from a knowledge of the exact geometry of the surroundings of the probe and
it is difï¬cult, even when the particle is inserted deeply into the cell. This is due to
the highly inhomogeneous character of the cytoplasm, consisting of different types of
protein ï¬bers, bundles, organelles, and membranes. Because a living cell is an active
material that is slowly and continuously changing shape, responses are in general
time dependent and contain passive and active components. Passive responses to low
forces are often hidden under the active motions of the cell, while responses to large
forces do not probe linear response parameters, but rather nonlinear behavior and
rupture of the networks. Given all the restrictions mentioned, a window to measure
Experimental measurements of intracellular mechanics 35
Fig. 2-11. Schematic diagram of a magnetic tweezers device using a magnetic ï¬eld gradient. From
Trepat et al., 2003.
the passive mechanical properties of cells appears to be to apply large strains, or to
apply relatively high-frequency oscillatory strain at small amplitudes, while active
responses can best be measured at low frequencies.
A number of experiments performed inside cells have observed the creep response
to the instantaneous application of large or small forces (Bausch et al., 1999; Bausch
et al., 1998; Feneberg et al., 2001). Bulk shear moduli were found to vary from â¼20 Pa
in the cytoplasm of Dictyostelium to â¼300 Pa inside macrophages. At higher forces
and strains, differences in rupture forces were found between mutant Dictyostelium
cells and wild-type controls, highlighting the roles of regulatory proteins for the
properties of the cytoskeleton (Feneberg et al., 2001).
With particles attached to the surface of cells, a shear modulus between 20 and
40 kPa was measured in the cortex of ï¬broblasts (Bausch et al., 1998), qualitative
differences were measured between unstimulated and stimulated (stiffening) vascu-
lar endothelial cells (Bausch et al., 2001), and an absolute value of about 400 Pa
was estimated from subsequent work (Feneberg et al., 2004). Active responses of
macrophages and the formation of cell protrusion under varying forces were also
tested with externally attached magnetic beads (Vonna et al., 2003).
Twisting of magnetized particles on the cell surface and interior
Applying a pure torque to magnetic particles avoids the difï¬culties of constructing
a well-controlled ï¬eld gradient. Homogeneous ï¬elds can be created rather easily.
The method most widely used was pioneered by Valberg and colleagues (Valberg
and Butler, 1987; Valberg and Feldman, 1987; Wang et al., 1993) and consists of
using a strong magnetic ï¬eld pulse to magnetize a large number of ferromagnetic
particles that were previously attached to an ensemble (20,000â“40,000) of cells. A
weaker probe ï¬eld oriented at 90â—¦ to the induced dipoles then causes rotation, which
is measured in a lock-in mode with a magnetometer. In an homogeneous inï¬nite
medium, an effective shear modulus can be determined simply from the angle Î±
rotated in response to an applied torque T : G = T /Î±. On the surface of cells, however,
36 P. Janmey and C. Schmidt
the boundary conditions are highly complex, and a substantial polydispersity within
the bead ensemble is expected (Fabry et al., 1999). Therefore the method has been
mainly used for determining qualitative behavior, for comparative studies of different
cell types, and for studies of relative changes in a given cell population. Frequency
dependence of the viscoelastic response was measured with smooth muscle cells
between 0.05 and 0.4 Hz (Maksym et al., 2000) and with bronchial epithelial cells
up to 16 Hz (Puig-de-Morales et al., 2001) (see also Chapter 3). The shear elastic
modulus was found to be around 50 Pa with a weak frequency dependence in both
Rotation in response to torque can also be detected on individual particles by video
tracking when beads are attached to the outsides of cells. In that case the torque causes
center-of-mass displacement, which can be tracked with nm accuracy (Fabry et al.,
2001). Tracking individual particles makes it possible to study the heterogeneity of
response between different cells and in different locations on cells. In conjunction
with ï¬‚uorescent labeling it is possible to explore the strains caused by locally imposed
stresses; initial studies reveal that the strain ï¬eld is surprisingly long-range (Hu et al.,
2003). Using oscillatory torque and phase-locking techniques, the bandwidth of this
technique was extended to 1 kHz (Fabry et al., 2001). While absolute shear moduli
were still not easy to determine because of unknown geometrical factors such as depth
of embedding, the bandwidth was wide enough to study the scaling behavior of the
complex shear modulus more extensively. The observed weak power-laws (exponent
between 0.1 and 0.3) appear to be rather typical for cells in that frequency window and
were interpreted in terms of a soft glass model. Finite element numerical modeling
has been applied to analyze the deformations of cells when attached magnetic beads
are rotated (Mijailovich et al., 2002) to test the limits of linearity in the response as
well as the effect of ï¬nite cell thickness and surface attachment.
To measure the viscoelastic properties of a system, it is not necessary to apply external
forces when one employs microscopic probes. In a soft-enough medium, thermal ï¬‚uc-
tuations will be measurable and these ï¬‚uctuations precisely report the linear-response
viscoelastic parameters of the medium surrounding the probe. This connection is for-
malized in the ï¬‚uctuation-dissipation (FD) theorem of linear-response theory (Landau
et al., 1980). When possible, that is when the medium is soft enough, this method
even elegantly circumvents the need to extrapolate to zero-force amplitude, which is
usually necessary in active methods to obtain linear response parameters. Particularly
in networks of semiï¬‚exible polymers such as the cytoskeleton, nonlinear response
occurs typically for rather small strains on the order of a few percent (Storm et al.,
A microscopic probe offers both the possibility to study inhomogeneities directly
in the elastic properties of the cytoskeleton, and to measure viscoelasticity at higher
frequencies, above 1 kHz or even up to MHz, because inertia of both probe and em-
bedding medium can be neglected at such small length scales (Levine and Lubensky,
2001; Peterman et al., 2003b). The possibility to observe thermal ï¬‚uctuations instead
of actively applying force or torque in principle exists for all the techniques using
Experimental measurements of intracellular mechanics 37
microscopic probes described above. It has, however, mainly been used in several
related and recently developed techniques, collectively referred to as passive mi-
crorheology, employing beads of micron size embedded in the sample (Addas et al.,
2004; Lau et al., 2003; MacKintosh and Schmidt, 1999; Mason et al., 1997; Schmidt
et al., 2000; Schnurr et al., 1997). Passive microrheology has been used to probe, on
microscopic scales, the material properties of systems ranging from simple polymer
solutions to the interior of living cells.
Optically detected individual probes
The simplest method in terms of instrumentation uses video microscopy to record the
Brownian motion of the embedded particles. Advantages are the use of standard equip-
ment coupled with well-established image processing and particle tracking (Crocker
and Grier, 1996), and the fact that massively parallel processing can be done (100 s of
particles at the same time). Disadvantages are the relatively low spatial and temporal
resolution, although limits can be pushed to nm in spatial displacement resolution and
kHz temporal resolution with specialized cameras. Much higher spatial and temporal
resolution still can be achieved using laser interferometry with laser beams focused
on individual probe particles (Denk and Webb, 1990; Gittes and Schmidt, 1998; Pralle
et al., 1999). Due to high light intensities focused on the particles, high spatial res-
olution (sub-nm) can be reached. Because the detection involves no video imaging,
100 kHz bandwidth can be reached routinely.
Once particle positions as a function of time are recorded by either method, the com-
plex shear modulus G(Ï) of the viscoelastic particle environment has to be calculated.
This can be done by calculating the mean square displacement xÏ of the Brownian
motion by Fourier transformation. The complex compliance Î±(Ï) of the probe particle
with respect to a force exerted on it is deï¬ned by:
xÏ = (Î± + iÎ± ) f Ï (2.12)
The FD theorem relates the imaginary part of the complex compliance to the mean
4k B T Î±
xÏ = .
Using a Kramers-Kronig relation (Landau et al., 1980):
Î¶ Î± (Î¶ )
Î± (Ï) = P dÎ¶ , (2.14)
Ï Î¶ 2 â’ Ï2
where Î¶ is the frequency variable to integrate over and P denotes a principal value
integral, one can then calculate the real part of the compliance. Knowing both real
and imaginary parts of the compliance, one then ï¬nds the complex shear modulus via
38 P. Janmey and C. Schmidt
a generalized Stokes law:
Î±(Ï) = , (2.15)
where R is the probe bead radius. This procedure is explained in detail in Schnurr
et al. (1997).
The shear modulus can also be derived from position ï¬‚uctuation data in a different
way. After ï¬rst calculating the mean square displacement as a function of time, one
can obtain (using equipartition) a viscoelastic memory function by Laplace transfor-
mation, The shear modulus follows by again using the generalized Stokes law (Mason
et al., 1997).
Large discrepancies between macroscopic viscoelastic parameters and those deter-
mined by one-particle microrheology can arise if the presence of the probe particle
itself inï¬‚uences the viscoelastic medium in its vicinity or if active particle movement
occurs and is interpreted as thermal motion. The shear strain ï¬eld coupled to the mo-
tion of a probe particle extends into the medium a distance that is similar to the particle
radius. Any perturbation of the medium caused by the presence of the particle will
decay over a distance that is the shorter of the particle radius or characteristic length
scales in the medium itself, such as mesh size of a network or persistence length of a
polymer. Thus it follows that if any characteristic length scales in the system exceed
the probe size, the simple interpretation of data with the generalized Stokes law is not
valid. This is probably always the case when micron-sized beads are used to study
the cytoskeleton, because the persistence length of actin is already 17 Âµm (Howard,
2001), while that of actin bundles or microtubules is much larger still. A perturbation
of the medium could be caused by a chemical interaction with the probe surface,
which can be prevented by appropriate surface coating. It is unavoidable, though,
that the probe bead locally dilutes the medium by entropic depletion. To circumvent
these pitfalls, two-particle microrheology has been developed (Crocker et al., 2000;
Levine and Lubensky, 2000). In this variant, the cross-correlation of the displacement
ï¬‚uctuations of two particles, located at a given distance from each other, is measured
(Fig. 2-12). The distance between the probes takes over as relevant length scale and
probe size or shape become of secondary importance.
Instead of the one-particle compliance, a mutual compliance is deï¬ned by:
xÏ = Î±imn (Ï) f Ï j ,
with particle indices n, m and coordinate indices i, j = x, y. If the particles are
separated by a distance r along the x-axis, two cases are relevant, namely Î±x x , which
will be denoted as Î±|| , and Î± 12 , which will be denoted as Î±â¥ (the other combinations
are second order). The Fourier transform of the cross-correlation function is related
to the imaginary part of the corresponding compliance:
4k B T 12
Si12 (Ï) = X i1 (Ï)X 2 (Ï)â = Î±i j (Ï), (2.17)
where â denotes the complex conjugate.
Experimental measurements of intracellular mechanics 39
Fig. 2-12. Sketch of 1-particle and 2-particle microrheology using lasers for trapping and detection.
Either one laser beam is focused on one particle at a time, or the two beams, displaced by some
distance, are each focused on seperate particles. In both cases the motion of the particle in the two
directions normal to the laser propagation direction is measured by projecting the laser light onto
quadrant photodiodes downstream from the sample.
A Kramers-Kronig integral can again be used to calculate the real parts, and elastic
moduli can be derived according to Levine and Lubensky (2002) from:
Î±|| (Ï) =
4Ïr Âµ0 (Ï)
Î»0 (Ï) + 3Âµ0 (Ï)
Î±â¥ (Ï) = ,
8Ïr Âµ0 (Ï) Î»0 (Ï) + 2Âµ0 (Ï)
written here (following Levine and Lubensky (2002)) with the LamÂ´ coefï¬cients Î»0 (Ï)
and Âµ0 (Ï), where Âµ0 (Ï) = G(Ï). One can thus measure directly the compressional
modulus and the shear modulus in the sample.
The technique can again be implemented using video recording and particle track-
ing (Crocker et al., 2000) or laser interferometry. In cells, so far only a video-based
variant has been used (Lau et al., 2003), exploring the low-frequency regime of the
cellular dynamics in mouse macrophages and mouse carcinoma cells. It was found
that the low-frequency passive microrheology results were strongly inï¬‚uenced by
active transport in the cells, so that the ï¬‚uctuation-dissipation theorem could not be
used for calculating viscoelastic parameters.
Dynamic light scattering and diffusing wave spectroscopy
A well-established method to study the dynamics of large ensembles of particles
in solutions is dynamic light scattering (DLS) (Berne and Pecora, 1990). To obtain
smooth data and good statistics, it is obviously advantageous to average over a large
number of particles. In DLS, a collimated laser beam is typically sent through a
sample of milliliter volume, and scattered light is collected under a well-deï¬ned
40 P. Janmey and C. Schmidt
angle with a photomultiplier or other sensitive detector. The intensity autocorrelation
I (t)I (t + Ï„ )
= 1 + Î²eâ’q r 2 (Ï„ ) /3
g2 (Ï„ ) = (2.20)
I (t) 2
can be used to calculate the average mean square displacement r 2 (Ï„ ) of particles,
with the scattering vector q = 4Ïn sin(Î¸/2)/Î», wavelength Î», scattering angle Î¸ and
index of the solvent n, and a coherence factor Î². This relationship assumes that
all particles are identical and that the solution is homogeneous across the scattering
volume. It also assumes that the particles dominate the scattering intensity compared to
the scattering from the embedding medium itself. DLS has also been used extensively
to study polymer solutions without added probe particles (Berne and Pecora, 1990).
In that case the medium has to be modeled to extract material properties from the
observed intensity autocorrelation function. This has been done for example for pure
actin networks as models for the cytoskeleton of cells (Isambert et al., 1995; Liverpool
and Maggs, 2001; Schmidt et al., 1989). Unfortunately this technique is not well
applicable to study the interior of cells, because the cellular environment is highly
inhomogeneous and it is not well deï¬ned which structures scatter the light in any
given location. Larger objects dominate the scattered intensity (Berne and Pecora,
1990). DLS has been applied to red blood cells (Peetermans et al., 1987a; Peetermans
et al., 1987b), but results have been qualitative. It is also difï¬cult to introduce external
probe particles that scatter light strongly enough in sufï¬cient concentrations without
harming the cells.
A related light-scattering technique is diffusing wave spectroscopy (DWS) (Pine
et al., 1988; Weitz et al., 1993), which measures again intensity correlation functions
of scattered light, but now in very dense opaque media where light is scattered many
times before it is detected so that the path of a photon becomes a random walk
and resembles diffusion. There is no more scattering-vector dependence in the ï¬eld
correlation function, which is directly related to the average mean square displacement
r 2 (Ï„ ) of the scattering particles (Weitz and Pine, 1993):
r 2 (Ï„ )
g1 (Ï„ ) â P(s) exp â’ ds. (2.21)
P(s) is the probability that the light travels a path length s, k0 = 2Ï/Î» is the wave
vector, and l â— is the transport mean free path. The ï¬nal steps to extract a complex
shear modulus are the same as described above, either using the power spectral den-
sity method (Schnurr et al., 1997) or the Laplace transform method (Mason et al.,
The advantage of the technique is that it is sensitive to very small motions (of less
than nm) because the path of an individual photon reï¬‚ects the sum of the motions of
the all particles by which it is scattered. The bandwidth of the technique is also high
(typically 10 Hz to 1 MHz), so that ensemble-averaged mean-square displacements,
and from that viscoelastic response functions, can be measured over many decades in
frequency. The technique has been used to study polymer solutions, colloidal systems,
and cytoskeletal protein solutions (actin) (Mason et al., 1997; Mason et al., 2000) and
Experimental measurements of intracellular mechanics 41
Fig. 2-13. (a) Experimental set-up for ï¬‚uorescence correlation
spectroscopy. A laser beam is expanded (L1, L2) and focussed
through a microscope objective into a ï¬‚uorescent sample. The
ï¬‚uorescence light is collected through the same objective and
split out with a dichroic mirror toward the confocal pinhole
(P) and then the detector. (b) Magniï¬ed focal volume with the
ï¬‚uorescent particles (spheres) and the diffusive path of one
particle highlighted. From Hess et al., 2002.
results agree with those from conventional methods in the time/frequency regimes
where they overlap. The application to cells is hindered, just as in the case of DLS, by
the inhomogeneity of the cellular environment. Furthermore, typical cells are more
or less transparent, in other words one would need to introduce high concentrations
of scattering particles, which likely would disturb the cellâ™s integrity.
Fluorescence correlation spectroscopy
Many complications can be avoided if speciï¬c particles or molecules of interest in
a cell can be selected from other structures. A way to avoid collecting signals from
unknown cellular structures is to use ï¬‚uorescent labeling of particular molecules or
structures within the cell. This method is extensively used in cell biology to study the
localization of certain proteins in the cell. Fluorescence can also be used to measure
dynamic processes in video microscopy, but due to low emission intensities of ï¬‚uo-
rescent molecules and due to their fast Brownian motion when they are not ï¬xed to
larger structures, it is difï¬cult to use such data to extract diffusion coefï¬cients or vis-
coelastic parameters inside cells. A method that is related to dynamic light scattering
and is a nonimaging method that can reach much faster time scales is ï¬‚uorescence
correlation spectroscopy (Hess et al., 2002; Webb, 2001), where a laser is focused to
a small volume and the ï¬‚uctuating ï¬‚uorescence originating from molecules entering
and leaving this volume is recorded with fast detectors (Fig. 2-13).
42 P. Janmey and C. Schmidt
From the ï¬‚uorescence intensity ï¬‚uctuations Î´F(t) = F(t) â’ F(t) one calculates
the normalized autocorrelation function:
Î´ F(t)Î´ F(t + Ï„ )
G(Ï„ ) = , (2.22)
from which one can calculate in the simplest case, in the absence of chemical reactions
involving the ï¬‚uorescent species, the characteristic time Ï„ D a diffusing molecule
spends in the focal volume:
G D (Ï„ ) = (2.23)
N (1 + Ï„/Ï„ D ) (1 + Ï„/Ï2 Ï„ D )1/2
with an axial-to-lateral-dimension ratio Ï and the mean number of ï¬‚uorescent
molecules in the focus N. Eq. 2.23 is valid for a molecule diffusing in 3D. The
method can also be used in other cases, for example 2D diffusion in a membrane.
With some knowledge of the geometry of the situation â“ for example 2D membrane-
bound diffusion â“ one can again extract diffusion coefï¬cients. This has been done on
cell surfaces and even inside cells (see references in Hess et al., (2002)). Data typically
have been interpreted as diffusion in a purely viscous environment or as diffusion in an
inhomogeneous environment with obstacles. Compared with the rheology methods
described above, ï¬‚uorescence correlation spectroscopy is particularly good when
studying small particles such as single-enzyme molecules. For the motion of such
particles it is not appropriate to model the environment inside a cell as a viscoelastic
continuum, because characteristic length scales of the cytoskeleton are as large or
larger than the particles.
A novel optical method related to optical traps employs two opposing nonfocused
laser beams to both immobilize and stretch a suspended cell (Guck et al., 2001; Guck
et al., 2000). Viscoelastic properties are determined from the time-dependent change
in cell dimensions as a function of optical pressures. This method has the signiï¬cant
advantage over other optical trapping methods that it can be scaled up and automated
to allow measurement, and potentially sorting, of many cells within a complex mixture
for use in diagnosing abnormal cells and sorting cells on the basis of their rigidity
(Lincoln et al., 2004).
Ultrasound transmission and attenuation through cells and biological tissues can also
provide measurements of viscoelasticity, and acoustic microscopy has the potential to
provide high-resolution imaging of live cells in a minimally invasive manner (Viola
and Walker, 2003). Studies of puriï¬ed systems such as F-actin (Wagner et al., 2001;
Wagner et al., 1999), and alginate capsules (Klemenz et al., 2003), suggest that acous-
tic signals can be related to changes in material properties of these biopolymer gels,
but there are numerous challenges related to interpreting the data and relating them
Experimental measurements of intracellular mechanics 43
to viscoelastic parameters before the potential of this method for quantitative high-
resolution elastic imaging on cells is realized.
Outstanding issues and future directions
The survey of methods used to study the rheology of cells presented here shows the
wide range of methods that various groups have designed and employed. At present
there appears to be no one ideal method suitable for most cell types. In many cases,
measurements of similar cell types by different methods have yielded highly different
values for elastic and viscous parameters. For example, micropipette aspiration of
leukocytes can variably be interpreted as showing that these cells are liquid droplets
with a cortical tension or soft viscoelastic ï¬‚uids, while atomic force microscopy mea-
sures elastic moduli on the order of 1000 Pa. In part, differences in measurements stem
from differences in the time scale or frequency and in the strains at which the measure-
ments are done. Also, it is almost certain that cells respond actively to the forces needed
to measure their rheology, and the material properties of the cell often cannot be inter-
preted as those of passive material. Combining rheological measurements with simul-
taneous monitoring or manipulation of intracellular signals and cytoskeletal structures
can go a long way toward resolving such challenges.
Currently a different and equally serious challenge is presented by the ï¬nding that
even when studying puriï¬ed systems like F-actin networks, micro- and macrorheology
methods sometimes give very different results, for reasons that are not completely
clear. In part there are likely to be methodological problems that need to be resolved,
but it also appears that there are interesting physical differences in probing very
small displacements of parts of a network not much larger than the network mesh
size and the macroscopic deformations that occur as the whole network deforms in
macrorheologic measurements. Here a combination of more experimentation and new
theories is likely to be important.
The physical properties of cells have been of great interest to biologists and phys-
iologists from the earliest studies that suggested that cells may be able to convert
from solid to liquid states as they move or perform other functions. More recently,
unraveling the immense complexity of the molecular biology regulating cell biol-
ogy and high-resolution imaging of intracellular structures have provided molecular
models to suggest how the dynamic viscoelasticity of the cell may be achieved. Now
the renewed interest in cell mechanics together with technological advances allowing
unprecedented precision and sensitivity in force application and imaging can com-
bine with molecular information to increase our understanding of the mechanisms by
which cells maintain and change their mechanical properties.
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3 The cytoskeleton as a soft glassy material
Jeffrey Fredberg and Ben Fabry
Using a novel method that was both quantitative and reproducible, Francis Crick
and Arthur Hughes (Crick and Hughes, 1950) were the ï¬rst to measure the mechanical proper-
ties inside single, living cells. They concluded their groundbreaking work with the words: âIf
we were compelled to suggest a model (of cell mechanics) we would propose Motherâ™s Work
Basket â“ a jumble of beads and buttons of all shapes and sizes, with pins and threads for good
measure, all jostling about and held together by colloidal forces.â
Thanks to advances in biochemistry and biophysics, we can now name and to a large de-
gree characterize many of the beads and buttons, pins, and threads. These are the scores of
cytoskeletal proteins, motor proteins, and their regulatory molecules. But the traditional re-
ductionist approach â“ to study one molecule at a time in isolation â“ has so far not led to a
comprehensive understanding of how cells are able to perform such exceptionally complex
mechanical feats as division, locomotion, contraction, spreading, or remodeling. The question
then arises, even if all of the cytoskeletal and signaling molecules were known and fully char-
acterized, would this information be sufï¬cient to understand how the cell orchestrates complex
and highly speciï¬c mechanical functions? Or put another way, do molecular events playing out
at the nanometer scale necessarily add up in a straightforward manner to account for mechanical
events at the micrometer scale?
We argue here that the answer to these questions may be â˜No.â™ We present a point of view
that does not rely on a detailed knowledge of speciï¬c molecular functions and interactions, but
instead focuses attention on dynamics of the microstructural arrangements between cytoskeletal
proteins. Our thinking has been guided by recent advances in the physics of soft glassy materials.
One of the more surprising ï¬ndings that has come out of this approach is the discovery that,
independent of molecular details, a single, measurable quantity (called the â˜noise temperatureâ™)
seems to account for transitions between ï¬‚uid-like and solid-like states of the cytoskeleton.
Although the interpretation and precise meaning of this noise temperature is still emerging,
it appears to give a measure of the â˜jostlingâ™ and the â˜colloidal forcesâ™ that act within the
Measurements of mechanical properties afford a unique window into the dynamics of
protein-protein interactions within the cell, with elastic energy storage reï¬‚ecting num-
bers of molecular interactions, energy dissipation reï¬‚ecting their rate of turnover, and
The cytoskeleton as a soft glassy material 51
remodeling events reï¬‚ecting their spatio-temporal reorganization (Fredberg, Jones
et al., 1996). As discussed in Chapter 2, probes are now available that can measure
each of these features with temporal resolution in the range of milliseconds and spatial
resolution in the range of nanometers. Using such probes, this chapter demonstrates
that a variety of phenomena that have been taken as the signature of condensed systems
in the glassy state are prominently expressed by the cytoskeletal lattice of the living
adherent cell. While highly speciï¬c interactions play out on the molecular scale, and
homogeneous behavior results on the integrative scale, evidence points to metasta-
bility of interactions and nonequilibrium cooperative transitions on the mesoscale
as being central factors linking integrative cellular function to underlying molecular
events. Insofar as such fundamental functions of the cell â“ including embryonic
development, contraction, wound healing, crawling, metastasis, and invasion â“ all
stem from underlying cytoskeletal dynamics, identiï¬cation of those dynamics as
being glassy would appear to set these functions into an interesting context.
The chapter begins with a brief summary of experimental ï¬ndings in living cells.
These ï¬ndings are described in terms of a remarkably simple empirical relationship
that appears to capture the essence of the data with very few parameters. Finally, we
show that this empirical relationship is predicted from the theory of soft glassy rheol-
ogy (SGR). As such, SGR offers an intriguing perspective on mechanical behavior of
the cytoskeleton and its relationship to the dynamics of protein-protein interactions.
Experimental ï¬ndings in living cells
To study the rheology of cytoskeletal polymers requires a probe whose operative
frequency range spans, insofar as possible, the internal molecular time scales of
the rate processes in question. The expectation from such measurements is that the
rheological behavior changes at characteristic relaxation frequencies, which in turn
can be interpreted as the signature of underlying molecular interactions that dominate
the response (Hill, 1965; Kawai and Brandt, 1980). Much of what follows in this
chapter is an attempt to explain the failure to ï¬nd such characteristic relaxation times
in most cell types. The experimental ï¬ndings of our laboratory, summarized below, are
derived from single cell measurements using magnetic twisting cytometry (MTC) with
optical detection of bead motion. Using this method, we were able to apply probing
frequencies ranging from 0.01 Hz to 1 kHz. As shown by supporting evidence, these
ï¬ndings are not peculiar to the method; rather they are consistent with those obtained
using different methods such as atomic force microscopy (Alcaraz, Buscemi et al.,
Magnetic Twisting Cytometry (MTC)
The MTC device is a microrheometer in which the cell is sheared between a plate at
the cell base (the cell culture dish upon which the cell is adherent) and a magnetic
microsphere partially embedded into the cell surface, as shown in Fig. 3-1. We use
ferrimagnetic microbeads (4.5 Âµm diameter) that are coated with a panel of antibody
and nonantibody ligands that allow them to bind to speciï¬c receptors on the cell surface
52 J. Fredberg and B. Fabry
1 Âµm displacement
(a) (b) (d)
Fig. 3-1. (a) Scanning EM of a bead bound to the surface of a human airway smooth muscle cell.
(b) Ferrimagnetic beads coated with an RGD-containing peptide bind avidly to the actin cytoskeleton
(stained with ï¬‚uorescently labeled phalloidin) of HASM cells via cell adhesion molecules (integrins).
(c) A magnetic twisting ï¬eld introduces a torque that causes the bead to rotate and to displace. Large
arrows indicate the direction of the beadâ™s magnetic moment before (black) and after (gray) twisting.
If the twisting ï¬eld is varied sinusoidally in time, then the microbead wobbles to and fro, resulting
in a lateral displacement, (d), that can be measured. From Fabry, Maksym et al., 2003.
(including various integrin subtypes, scavenger receptors, urokinase receptors, and
immune receptors). The beads are magnetized horizontally by a brief and strong
magnetic pulse, and then twisted vertically by an external homogeneous magnetic
ï¬eld that varies sinusoidally in time. This applied ï¬eld creates a torque that causes the
beads to rotate toward alignment with the ï¬eld, like a compass needle aligning with
the earthâ™s magnetic ï¬eld. This rotation is impeded, however, by mechanical forces
that develop within the cell as the bead rotates. Lateral bead displacements during
bead rotation in response to the resulting oscillatory torque are detected by a CCD
camera mounted on an inverted microscope.
Cell elasticity (g ) and friction (g ) can then be deduced from the magnitude and
phase of the lateral bead displacements relative to the torque (Fig. 3-2). Image ac-
quisition with short exposure times of 0.1 ms is phase-locked to the twisting ï¬eld so
that 16 images are acquired during each twisting cycle. Heterodyning (a stroboscopic
technique) is used at twisting frequencies >1 Hz up to frequencies of 1000 Hz. The
images are analyzed using an intensity-weighted center-of-mass algorithm in which
sub-pixel arithmetic allows the determination of bead position with an accuracy of
5 nm (rms).
Measurements of cell mechanics
The mechanical torque of the bead is proportional to the external magnetic ï¬eld
(which was generated using an electromagnet), the beadâ™s magnetic moment (which
was calibrated by measuring the speed of bead rotation in a viscous medium), and the
cosine between the beadâ™s magnetization direction with the direction of the twisting
ï¬eld. Consider the speciï¬c torque of a bead, T , which is the mechanical torque per
bead volume, and has dimensions of stress (Pa). The ratio of the complex-speciï¬c
torque T to the resulting complex bead displacement dË (evaluated at the twisting fre-
quency) then deï¬nes a complex modulus of the cell g = TË/d, and has dimensions of
The cytoskeleton as a soft glassy material 53
0 1 2 3 4
-50 -25 0 25 50
Fig. 3-2. (a) Speciï¬c torque T (solid line) and lateral displacement d (ï¬lled circles connected by
a solid line) vs. time in a representative bead measured at a twisting frequency of 0.75 Hz. Bead
displacement followed the sinusoidal torque with a small phase lag. The ï¬lled circles indicate when
the image and data acquisition was triggered, which was 16 times per twisting cycle. (b) Loops
of maximum lateral bead displacement vs. speciï¬c torque of a representative bead at different
frequencies. With increasing frequency, displacement amplitude decreased. From Fabry, Maksym
et al., 2003.
Pa/nm. These measurements can be transformed into traditional elastic shear (G ) and
loss (G ) moduli by multiplication of g and g with a geometric factor that depends
on the shape and thickness of the cell and the degree of bead embedding. Finite ele-
ment analysis of cell deformation for a representative bead-cell geometry (assuming
homogeneous and isotropic elastic properties with 10 percent of the bead diameter
embedded in a cell 5 Âµm high) sets this geometric factor to 6.8 Âµm (Mijailovich, Kojic
et al., 2002). This geometric factor need serve only as a rough approximation, how-
ever, because it cancels out in the scaling procedure described below, which is model
independent. For each bead we compute the elastic modulus g (the real part of g),
the loss modulus g (the imaginary part of g), and the loss tangent Î· (the ratio g /g )
at a given twisting frequency. These measurements are then repeated over a range of
Because only synchronous bead movements that occur at the twisting frequency
are considered, nonsynchronous noise is suppressed by this analysis. Also suppressed
are higher harmonics of the bead motion that may result from nonlinear material
properties and that â“ if not properly accounted for â“ could distort the frequency
dependence of the measured responses. However, we found no evidence of nonlinear
54 J. Fredberg and B. Fabry
0 50 100 150
Fig. 3-3. g and g vs. speciï¬c torque amplitude T . g and g were measured in 537 HASM cells at
f = 0.75 Hz. Speciï¬c torque amplitudes T varied from 1.8 to 130 Pa. g and g were nearly constant,
implying linear mechanical behavior of the cells in this range. Error bars indicate one standard error.
From Fabry, Maksym et al., 2003.
cell behavior (such as strain hardening or shear thinning) at the level of stresses we
apply with this technique, which ranges from about 1 Pa to about 130 Pa (Fig. 3-3).
Throughout that range, which represents the physiological range, responses were
Frequency dependence of g and g
The relationship of G and G vs. frequency for human airway smooth muscle
(HASM) cells under control conditions is shown in Fig. 3-4, where each data point
represents the median value of 256 cells. Throughout the frequency range studied, G
increased with increasing frequency, f, according to a power law, f xâ’1 (as explained
below, the formula is written in this way because the parameter x takes on a special
meaning, namely, that of an effective temperature). Because the axes in Fig. 3-4 are
logarithmic, a power-law dependency appears as a straight line with slope x â’ 1. The
power-law exponent of G was 0.20 (x = 1.20), indicating only a weak dependency
of G on frequency. G was smaller than G at all frequencies except at 1 kHz. Like
G , G also followed a weak power law with nearly the same exponent at low fre-
quencies. At frequencies larger than 10 Hz, however, G exhibited a progressively
stronger frequency dependence, approaching but never quite attaining a power-law
exponent of 1, which would be characteristic of a Newtonian viscosity.
This behavior was at ï¬rst disappointing because no characteristic time scale was
evident; we were unable to identify a dominating relaxation process. The only charac-
teristic time scale that falls out of the data is that associated with curvilinearity of the
G data that becomes apparent in the neighborhood of 100 Hz (Fig. 3-4). As shown
below, this curvilinearity is attributable to a small additive Newtonian viscosity that is
entirely uncoupled from cytoskeletal dynamics. This additive viscosity is on the order
of 1 Pa s, or about 1000-fold higher than that of water, and contributes to the energy
dissipation (or friction) only above 100 Hz. Below 100 Hz, friction (G ) remained a
The cytoskeleton as a soft glassy material 55
10 -2 10 -1 10 0 10 1 10 2 10 3
Fig. 3-4. G and G (median of 256 human airway smooth muscle cells) under control conditions
measured at frequencies between 0.01 Hz and 1000 Hz. The solid lines were obtained by ï¬tting Eq.
3.2 to the data. G and G (in units of Pa) were computed from the measured values of g and g
(in units of Pa/nm) times a geometric factor Î± of 6.8 Âµm. G increased with increasing frequency,
f , according to a power law, f xâ’1 , with x = 1.20. G was smaller than G at all frequencies except
at 1 kHz. At frequencies below 10 Hz, G also followed a weak power law with nearly the same
exponent as did G ; above 10 Hz the power law exponent increased and approached unity. From
Fabry, Maksym et al., 2003.
constant fraction (about 25 percent) of elasticity (G ). Such frictional behavior cannot
be explained by a viscous dissipation process.
It is intriguing to note the combination of an elastic process (or processes) that
increases with frequency according to a weak power-law over such a wide range
of time scales, and a frictional modulus that, except at very high frequencies, is a
constant, frequency-independent fraction of the elastic modulus. Similar behavior
has been reported for a wide range of materials, biological tissue, and complex man-
made structures such as airplane wings and bridges. Engineers use an empirical
description â“ the structural damping equation (sometimes referred to as hysteretic
damping law, or constant phase model) â“ to describe the mechanical behavior of such
materials (Weber, 1841; Kohlrausch, 1866; Kimball and Lovell, 1927; Hildebrandt,
1969; Crandall, 1970; Fredberg and Stamenovic, 1989), but as regards mechanism,
structural damping remains unexplained.
The structural damping equation
The mechanical properties of such a material can be mathematically expressed either
in the time domain or in the frequency domain. In the time domain, the mechanical
stress response to a unit step change in strain imposed at t = 0 is an instantaneous
component attributable to a pure viscous response together with a component that
rises instantaneously and then decays over time as a power law,
g(t) = ÂµÎ´(t) + g0 (t/t0 )1â’x . (3.1)
g0 is the ratio of stress to the unit strain measured at an arbitrarily chosen time t0 , Âµ is a
Newtonian viscous term, and Î´(t) is the Dirac delta function. The stress response to unit
amplitude sinusoidal deformations can be obtained by taking the Fourier transform of
56 J. Fredberg and B. Fabry
the step response (Eq. 3.1) and multiplying by jÏ, which gives the complex modulus
g (Ï) as:
g(Ï) = g0 (1 â’ i Î·) (2 â’ x) cos (t â’ 1) + iÏÂµ
Ë Â¯ (3.2)
where Î· = tan(x â’ 1)Ï/2 and Ï is the radian frequency 2Ï f (Hildebrandt, 1969).
g0 and 0 are scale factors for stiffness and frequency, respectively, denotes the
Gamma function, and i 2 is â’1. g0 and Âµ depend on bead-cell geometry. Î· has been
called the structural damping coefï¬cient (Fredberg and Stamenovic, 1989). The elastic
modulus g corresponds to the real part of Eq. 3.2, which increases for all Ï according
to the power-law exponent, x â’ 1. The loss modulus g corresponds to the imaginary
part of Eq. 3.2 and includes a component that also increases as a power law with
the same exponent. Therefore, the loss modulus is a frequency-independent fraction
(Î·) of the elastic modulus; such a direct coupling of the loss modulus to the elastic
modulus is the characteristic feature of structural damping behavior (Fredberg and
As mentioned already, the loss modulus includes a Newtonian viscous term, jÏÂµ,
which turns out to be small except at very high frequencies. At low frequencies, the
loss tangent Î· approximates Î·. In the limit that x approaches unity, the power-law slope
approaches zero, g approaches g0 and Î· approaches zero. In the limit that x approaches
2, the power-law slope approaches unity, G approaches Âµ and Î· approaches inï¬nity.
Thus, Eq. 3.2 describes a relationship between changes of the exponent of the power
law and the transition from solid-like (x = 1, Î· = 0) to ï¬‚uid-like (x = 2, Î· = â)
The structural damping equation describes the data in Fig. 3-4 exceedingly well and
with only four free parameters: the scale factors g0 and 0 , the Newtonian viscosity
Âµ, and the power-law exponent x â’ 1. The structural damping coefï¬cient Î· is not an
independent parameter but depends on x only.
We now go on to show that three of the four parameters of the structural damping
equation (g0 , 0 and Âµ) can be considered constant, and that changes in the cellâ™s
mechanical behavior during contraction, relaxation, or other drug-induced challenges
can be accounted for by changes of the parameter x alone.
Reduction of variables
When smooth muscle cells are activated with a contractile agonist such as histamine,
they generate tension and their stiffness (G ) increases, as has been shown in many
studies (Warshaw, Rees et al., 1988; Fredberg, Jones et al., 1996; Hubmayr, Shore
et al., 1996; Fabry, Maksym et al., 2001; Butler, Tolic-Norrelykke et al., 2002; Wang,
Tolic-Norrelykke et al., 2002). Interestingly, in HASM cells this increase in G after
histamine activation (10â“4 M) was more pronounced at lower frequencies. While G
still exhibited a weak power-law dependence on frequency, x fell slightly (Fig. 3-5).
When cells were relaxed with DBcAMP (1 mM), the opposite happened: G de-
creased, and x increased. When the actin cytoskeleton of the cells was disrupted
with cytochalasin D (2 ÂµM), G decreased even more, while x increased further.
The cytoskeleton as a soft glassy material 57
105 (G0, Î¦ 0/2Ï)
10-2 10-1 100 101 102 103 104 105 106 107
Fig. 3-5. G vs. frequency in HASM cells under control conditions ( , n = 256), and after 10
min. treatment with the contractile agonist histamine [10â“4 M] (â™¦, n = 195), the relaxing agonist
DBcAMP [10â“3 M] ( , n = 239) and the actin-disrupting drug cytochalasin D [2 Ã— 10â’6 M] ( ,
n = 171). At all frequencies, treatment with histamine caused G to increase, while treatment with
DBcAMP and cytoD caused G to decrease. Under all treatment conditions, G increased with
increasing frequency, f , according to a power law, f xâ’1 . x varied between 1.17 (histamine) and
1.33 (cytoD). A decreasing G was accompanied by an increasing x, and vice versa. Solid lines
are the ï¬t of Eq. 3.2 to the data. Surprisingly, these lines appeared to cross at a coordinate close to
[G 0 , 0 /2Ï ], well above the experimental frequency range. According to Eq. 3.2, an approximate
crossover implies that in the HASM cell the values of G 0 and 0 were invariant with differing
treatment conditions. From Fabry, Maksym et al., 2003.
Remarkably, the G data deï¬ned a family of curves that, when extrapolated, appeared
to intersect at a single value (G 0 ) at a very high frequency ( 0 ) (Fig. 3-5). Such a
common intersection, or ï¬xed point, of the G vs. frequency curves at a very high
frequency means that G 0 and 0 were invariant with different drug treatments.
With all drug treatments, G and G tended to change in concert. The relationship
between G and frequency remained a weak power law at lower frequencies, and
the power-law exponent of G changed in concert with that of G . At the highest
frequencies, the curves of G vs. frequency for all treatments appeared to merge onto
a single line with a power-law exponent approaching unity (Fig. 3-6).
The ï¬nding of a common intersection of the G vs. f relationship stands up to rig-
orous statistical analysis, meaning that a three-parameter ï¬t of the structural damping
equation (G 0 , 0 and x) to the full set G data (measured over ï¬ve frequency decades
and with different pharmacological interventions) is not statistically different from
a ï¬t with G 0 and 0 being ï¬xed, and with x being the only free parameter (Fabry,
Maksym et al., 2003). The very same set of parameters â“ a ï¬xed value for G 0 and
0 , respectively, and a drug-treatment-dependent x â“ also predicts the G vs. f rela-
tionship at frequencies below 100 Hz. Because the G data appears to merge onto a
single line at higher frequencies, a single Newtonian viscosity Âµ that is common for
all drug treatments can account for the data, although a rigorous statistical analysis
indicates that a negligible but signiï¬cant improvement of the ï¬t can be achieved with
different Âµ-values for different drug treatments (Fabry, Maksym et al., 2003). For all
practical purposes, therefore, the mechanical behavior of HASM cells is restricted
to vary only in a very particular way such that a single parameter, x, is sufï¬cient to
characterize the changes of both cell elasticity and friction.
58 J. Fredberg and B. Fabry
10-2 10-1 100 101 102 103
Fig. 3-6. G vs. frequency in HASM cells under control conditions ( , n = 256), and after 10 min.
treatment with histamine [10â“4 M] (â™¦, n = 195), DBcAMP [10â“3 M] ( , n = 239) and cytochalasin
D [2 Ã— 10â’6 M] ( , n = 171). At all frequencies, treatment with histamine caused G to increase,
while treatment with DBcAMP and cytoD caused G to decrease. Under all treatment conditions,
G increased at frequencies below 10 Hz according to a power law, f xâ’1 , with exponents that
were similar to that of the corresponding G -data (Fig. 3-5). Above 10 Hz the power-law exponents
increased and approached unity for all treatments; the G curves merged onto a single relationship.
From Fabry, Maksym et al., 2003.
This surprising and particular behavior is not restricted to HASM cells. We found the
very same behavior â“ a power-law relationship of G and G vs. frequency, common
intersection of the G vs. f data for different drug treatments at a very high frequency,
and merging of the G vs. f data onto a single line â“ in all other animal and human
cell types we have investigated so far, including macrophages, neutrophils, various
endothelial and epithelial cell types, ï¬broblasts, and various cancer cell lines (Fabry,
Maksym et al., 2001; Fabry, Maksym et al., 2003; Puig-de-Morales, Millet et al.,
Power-law behavior and common intersection were also found with an almost ex-
haustive panel of drugs that target the actin cytoskeleton and the activity of myosin
light chain kinase (including BDM, ML-7, ML-9, W-7, various rho-kinase inhibitors,
latrunculin, jasplakinolide) (Laudadio, Millet et al., 2005). Moreover, power-law be-
havior and common intersection are not peculiar to the details of the coupling between
the bead and the cell, and can be observed in beads coated with different ligands (in-
cluding RGD-peptide, collagen, vitronectin, ï¬bronectin, urokinase, and acetylated
low-density lipoprotein), and antibodies that speciï¬cally bind to various receptors
(activating and nonactivating domains of various integrins and other cell adhesion
molecules) (Puig-de-Morales, Millet et al., 2004). Neither is this behavior peculiar
to the magnetic twisting technique. The power-law dependence of G and G on fre-
quency is consistent with data reported for atrial myocytes, ï¬broblasts, and bronchial
endothelial cells measured with atomic force microscopy (AFM), for pellets of mouse
embryonic carcinoma cells measured with a disk rheometer, for airway smooth mus-
cle cells measured with oscillatory magneto-cytometry, and for kidney epithelial cells
measured by laser tracking of Brownian motion of intracellular granules (Shroff,
Saner et al., 1995; Goldmann and Ezzell, 1996; Mahaffy, Shih et al., 2000; Maksym,
The cytoskeleton as a soft glassy material 59
G â² [Pa]
10-1 100 101 102 103 104 105 106
Fig. 3-7. G vs. frequency in kidney epithelial cells measured with laser tracking microrheology
under control conditions ( ), and after 15 min. treatment with Latrunculin A [1 Ã— 10â’6 M] ( ).
Solid lines are the ï¬t of Eq. 3.2 to the data, with x = 1.36 under control conditions, and x = 1.5 after
Latrunculin A treatment. These lines crossed at G 0 = 5.48 kPa and 0 = 6.64â— 105 Hz. Adapted
from Yamada, Wirtz et al., 2000.
Fabry et al., 2000; Yamada, Wirtz et al., 2000; Alcaraz, Buscemi et al., 2003). Using
laser tracking microrheology, Yamada, Wirtz et al. also measured cell mechanics be-
fore and after microï¬laments were disrupted with Latrunculin A, and obtained two
curves of G vs. f that intersected at a frequency comparable to the value we mea-
sured with our magnetic twisting technique, as shown in Fig. 3-7 (Yamada, Wirtz
et al., 2000).
Finally, power-law behavior in the frequency domain (Eq. 3.2) corresponds to
power-law behavior in the time domain (Eq. 3.1): when we measured the creep mod-
ulus of the cells by applying a step change of the twisting ï¬eld, we did indeed ï¬nd
power-law behavior of the creep modulus vs. time, and a common intersection of the
data at a very short time (Lenormand, Millet et al., 2004).
Scaling the data
Not surprisingly, although structural damping behavior always prevailed, substantial
differences were observed in the absolute values of our G and G measurements
between different cell lines, and even larger (up to two orders of magnitude) differences
when different bead coatings were used (Fabry, Maksym et al., 2001; Puig-de-Morales,
Millet et al., 2004). Still larger (up to three orders of magnitude) differences were
observed between individual cells of the same type even when they were grown within
the same cell well (Fabry, Maksym et al., 2001). It is difï¬cult to specify to what extent
such differences reï¬‚ect true differences in the âmaterialâ properties between different
cells or between different cellular structures to which the beads are attached, vs.
differences in the geometry (cell height, contact area between cell and bead, and so
forth). Theoretically, these geometric details could be measured and then modeled, but
in practice such measurements and models are inevitably quite rough (Mijailovich,
Kojic et al., 2002).
Rather than analyzing âabsoluteâ cell mechanics, it is far more practical (and in-
sightful, as shown below) to focus attention instead upon relative changes in cell
60 J. Fredberg and B. Fabry
mechanics. To ï¬rst order, such relative changes are independent of bead-cell geome-
try. Thus, the measurements need to be normalized or scaled appropriately such that
each cell serves as its own control.
Two such scaled parameters have already been introduced. The ï¬rst one is the slope
of the power-law relationship (x â’ 1), which is the log change in G or G per fre-
quency decade (for example, the ratio of G 10 Hz /G 1 Hz ). The second scaled parameter
is the hysteresivity Î·, which is the ratio between G and G at a single frequency.
Both scaling procedures cause factors (such as bead-cell geometry) that equally
affect the numerator and denominator of those ratios to cancel out. Moreover, both
scaling procedures can be performed on a bead-by-bead basis. The bead-by-bead
variability of both x and Î· are negligible when compared with the huge variability in
G (Fabry, Maksym et al., 2001; Fabry, Maksym et al., 2001).
A third scaling parameter is the normalized stiffness G n , which we deï¬ne as the
ratio between G (measured at a given frequency, say 0.75 Hz) and G 0 (the intersection
of the G vs. f curves from different drug treatments, Fig. 3-5). Here, G 0 serves as
an internal stiffness scale that is characteristic for each cell type and for each bead
coating (receptor-ligand interaction), but that is unaffected by drug treatments. Thus,
the normalized stiffness G n allows us to compare the drug-induced responses of cells
under vastly different settings, such as different bead coating, and so on.
Collapse onto master curves
The data were normalized as follows. We estimated x from the ï¬t of Eq. 3.2 to the
pooled (median over many cells) G and G data. The hysteresivity Î· was estimated
from ratio of G /G measured at 0.75 Hz (an arbitrary choice). The normalized cell
stiffness G n was estimated as G measured at 0.75 Hz divided by g0 . log G n vs. x and
Î· vs. x graphs were then plotted (Fig. 3-8).
In human airway smooth muscle (HASM) cells, we found that drugs that increased
x caused the normalized stiffness G n to decrease (Fig. 3-8a, black symbols). The
relationship between log G n and x appears as nearly linear: ln G n â¼ â’x. The solid
line in Fig. 3-8a is the prediction from the structural damping equation under the
condition of a common intersection of all G data at radian frequency 0 :
ln G n = (x â’ 1) ln (Ï/ 0 ). (3.3)
How close the normalized data fell to this prediction thus indicates how well a
common intersection can account for those data.
Conversely, drugs that increased x caused the normalized frictional parameter Î· to
increase (Fig. 3-8b, black symbols). The solid line in Fig. 3-8b is the prediction from
(and not a ï¬t of) the structural damping equation:
Î· = tan (x â’ 1)Ï/2. (3.4)
How close the normalized data fell to this prediction thus indicates how well the
structural damping equation can account for those data.
Surprisingly, the normalized data for the other cell types collapsed onto the very
same relationships that were found for HASM cells (Fig. 3-8). In all cases, drugs that
increased x caused the normalized stiffness G n to decrease and hysteresivity Î· to
The cytoskeleton as a soft glassy material 61
1.1 1 1.4
1 1.3 1.4 1.1 1.2
Fig. 3-8. Normalized stiffness G n vs. x (left) and hysteresivity Î· (right) vs. x, of HASM cells
(black, n = 256), human bronchial epithelial cells (light gray, n = 142), mouse embryonic carcinoma
cells (F9) cells (dark gray, n = 50), mouse macrophages (J774A.1) (hatched, n = 46) and human
neutrophils (gray, n = 42) under control conditions ( ), treatment with histamine ( ), FMLP ( ),
DBcAMP ( ) and cytochalasin D ( ). x was obtained from the ï¬t of Eq. 3.2 to the pooled (median)
data. Drugs that increased x caused the normalized stiffness G n to decrease and hysteresivity Î· to
increase, and vice versa. The normalized data for all types collapsed onto the same relationships.
The structural damping equation by Eq. 3.2 is depicted by the black solid curves: ln G n = (x â’ 1)
ln(Ï/ 0 ) with 0 = 2.14 Ã— 107 rad/s, and Î· = tan((x â’ 1)Ï/2). Error bars indicate Â± one standard
error. If x is taken to be the noise temperature, then these data suggest that the living cell exists close
to a glass transition and modulates its mechanical properties by moving between glassy states that
are âhot,â melted and liquid-like, and states that are âcold,â frozen and solid-like. In the limit that x
approaches 1 the system behaves as an ideal Hookean elastic solid, and in the limit that x approaches
2 the system behaves as an ideal Newtonian ï¬‚uid (Eq. 3.2). From Fabry, Maksym et al., 2003.
increase. These relationships thus represent universal master curves in that a single
parameter, x, deï¬ned the constitutive elastic and frictional behaviors for a variety of
cytoskeletal manipulations, for ï¬ve frequency decades, and for diverse cell types.
The normalized data from all cell types and drug treatments fell close to the pre-
dictions of the structural damping equation (Fig. 3-8). In the case of the Î· vs. x data
(Fig. 3-8b), this collapse of the data indicates that the coupling between elasticity and
friction, and their power-law frequency dependence, is well described by the structural
damping equation. In the case of the log G n vs. x data (Fig. 3-8a), the collapse of
the data indicates that a common intersection of the G vs. f curves exists for all cell
types, and that this intersection occurs approximately at the same frequency 0 .