| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
Biophys J, February 1999, p. 1063-1071, Vol. 76, No. 2
*Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218, and #Department of Biomedical Engineering, The Johns Hopkins University, Baltimore, Maryland 21205 USA
 |
ABSTRACT |
|---|
 Top
 Abstract
 Introduction
Experimental techniques
Results
Discussion
References
|
|---|
Filamentous actin (F-actin), one of the constituents of the cytoskeleton, is believed to be the most important participant in the motion and mechanical integrity of eukaryotic cells. Traditionally, the viscoelastic moduli of F-actin networks have been measured by imposing a small mechanical strain and quantifying the resulting stress. The magnitude of the viscoelastic moduli, their concentration dependence and strain dependence, as well as the viscoelastic nature (solid-like or liquid-like) of networks of uncross-linked F-actin, have been the subjects of debate. Although this paper helps to resolve the debate and establishes the extent of the linear regime of F-actin networks' rheology, we report novel measurements of the high-frequency behavior of networks of F-actin, using a noninvasive light-scattering based technique, diffusing wave spectroscopy (DWS). Because no external strain is applied, our optical assay generates measurements of the mechanical properties of F-actin networks that avoid many ambiguities inherent in mechanical measurements. We observe that the elastic modulus has a small magnitude, no strain dependence, and a weak concentration dependence. Therefore, F-actin alone is not sufficient to generate the elastic modulus necessary to sustain the structural rigidity of most cells or support new cellular protrusions. Unlike previous studies, our measurements show that the mechanical properties of F-actin are highly dependent on the frequency content of the deformation. We show that the loss modulus unexpectedly dominates the elastic modulus at high frequencies, which are key for fast transitions. Finally, the measured mean square displacement of the optical probes, which is also generated by DWS measurements, offers new insight into the local bending fluctuations of the individual actin filaments and shows how they generate enhanced dissipation at short time scales.
| |
INTRODUCTION |
|---|
|
Top
Abstract
Introduction
Experimental techniques
Results
Discussion
References
|
|---|
Filamentous actin (F-actin), one of the constituents of the cytoskeleton, is believed to be one of the most important participants in the mechanical integrity of eukaryotic cells. The mechanism by which F-actin networks provide cells with structural rigidity is attributed to their viscoelastic nature, characterized by viscoelastic moduli.
However, major discrepancies between reported values of the magnitude
of the viscoelastic moduli of uncross-linked F-actin networks, their
concentration dependence, strain dependence, and frequency dependence
have appeared in the literature. These vastly different rheological
data have yielded different conclusions regarding the separate role of
F-actin in the cell's structural rigidity. Traditionally, the linear
viscoelastic moduli of F-actin networks in vitro have been investigated
by imposing a mechanical strain and quantifying the resulting stress.
Reported elastic moduli of uncross-linked F-actin networks, measured
using mechanical deformation, differ by more than two orders of
magnitude, between 10 and 40 dynes/cm2 (Pollard et al.,
1986
; Sato et al., 1987; Wachstock et al., 1993) and 1000-5000
dynes/cm2 (Hvidt and Heller, 1990; Janmey et al., 1990,
1991, 1994; Hvidt and Janmey, 1990). Therefore, concentrated F-actin
solutions have been described as forming networks that are either weak
or stiff (note: we use the terms "network" and "solution"
interchangeably; both terms describe a semidilute solution of highly
purified, uncross-linked F-actin). Furthermore, some researchers have
reported that networks of uncross-linked F-actin harden for strains
between 1% and 10% (Hvidt and Heller, 1990; Janmey et al., 1990,
1994), which corresponds to elastic moduli increasing with strain, and yield beyond 10%. In contrast, other researchers have not observed any
strain-hardening effect, only the onset of yielding when the strain
reaches 10% (Pollard et al., 1986; this work). Finally, vastly
different values of the relative magnitudes of elastic and loss moduli,
which characterize the nature of the viscoelasticity, have helped
researchers to characterize F-actin networks as either viscoelastic
solid or viscoelastic liquids.
The actual magnitude of the elastic modulus is important because a high
value would support the hypothesis that isotropic F-actin networks
alone are strong enough to stabilize cells (Janmey et al., 1994). A
high value of the elastic modulus also provides a baseline against
which to monitor subtle changes in the mechanical properties of F-actin
networks due to regulating proteins such as capping and severing
proteins, as well as small changes due to polymerization of actin
filaments from ATP-containing or ADP-containing subunits (Newman et
al., 1993; Pollard et al., 1986; Janmey et al., 1991). Finally, it
supports the hypothesis that F-actin alone can effectively provide
structural rigidity for the reinforcement of a new cellular protrusion
(Condeelis, 1992).
The strain dependence of F-actin networks' rheology is also important: if F-actin networks strain-harden, they can potentially stabilize the cytoskeleton, even when it is subject to large deformations. Moreover, the strain dependence of the viscoelastic moduli establishes the extent of the rheological linear regime (for which moduli are independent of strain), which is important because it allows for meaningful comparison with theoretical models and other mechanical measurements.
Given the controversy surrounding traditional methods, we have
developed a novel, light-scattering-based technique, diffusing wave
spectroscopy (DWS), which probes the linear rheological properties of
biopolymer networks noninvasively (Palmer et al., 1998; Petka et al.,
1998). This technique and associated analysis are based on the
measurement of the autocorrelation function of the light multiply
scattered from microspheres imbedded in the network. From the measured
autocorrelation function and using a generalized Stokes-Einstein
equation, we extract the mean square displacement of the probing
microspheres. From these mean square displacements, we calculate the
viscoelastic moduli of networks of highly purified, uncross-linked
F-actin. This technique is somewhat similar to single particle tracking
microrheology (SPTM), recently developed by Wirtz and co-workers (Mason
et al., 1997a,b; Xu et al., 1998c; Ganesan et al., unpublished data)
and Schmidt and co-workers (Schnurr et al., 1997a,b). Unlike SPTM,
which monitors the displacement of a single particle at a time,
DWS monitors many thousands of microspheres simultaneously, which
allows for superior statistics. However, unlike DWS, SPTM allows us to
probe the mechanical properties of an anisotropic F-actin
specimen and the cytoskeleton of a living cell in situ (Ganesan et al.,
unpublished data).
Our optical rheometry assay generates measurements of the viscoelastic moduli of F-actin networks that are less ambiguous than mechanical measurements. In particular, this instrument exploits the small, random, thermally induced force generated by imbedded microspheres and therefore avoids possible flow-induced orientation and bundling of actin filaments. In the absence of externally applied forces, optical rheometry probes the viscoelastic properties of networks in the linear small-strain regime almost by definition. Moreover, this optical instrument greatly increases the frequency range of the probed viscoelastic moduli: we routinely probe frequency-dependent loss and elastic moduli at frequencies up to 1 MHz, which is four orders of magnitude larger than possible with mechanical rheometry. This extended range of frequency allows us to probe the internal dynamics of actin filaments in concentrated solutions as well as the effect of fast cellular transitions on F-actin rheology. Finally, the measured mean square displacement of the optical probes, which is also generated by DWS measurements, offers new insight into the local bending fluctuations of the individual actin filaments and shows how these generate large dissipation at short time scales.
| |
EXPERIMENTAL TECHNIQUES |
|---|
|
Top
Abstract
Introduction
Experimental techniques
Results
Discussion
References
|
|---|
Actin preparation
Actin was purified from rabbit skeletal acetone powder by the
method of Spudich and Watt (1971). The resulting actin was then gel
filtered on Sephacryl S-300 HR instead of Sephadex G-150
(MacLean-Fletcher and Pollard, 1980). The purified actin was stored as
Ca2+-actin in continuous dialysis at 4°C against buffer G
(0.2 mM ATP, 0.5 mM dithiothreitol, 0.1 mM CaCl2, 1 mM
sodium azide, and 2 mM Tris-Cl, pH 8.0). The final actin concentration
was determined by ultraviolet absorbance at 290 nm, using an extinction
coefficient of 2.66 × 104 M
1 cm1 and
a cell path length of 1 cm. Mg2+ actin filaments were
generated by adding 0.1 volume of 10× KME (500 mM KCl, 10 mM
MgCl2, 10 mM EGTA, 100 mM imidazole, pH 7.0) polymerizing
salt buffer solution to 0.9 volume of G-actin in buffer G. We did not
observe large variations in the mechanical properties of F-actin
networks measured by optical and mechanical rheometry on actin
extracted and purified by the method of MacLean-Fletcher and Pollard
(1980) and by the new method of Casella et al. (Casella and Torres,
1994; Casella et al., 1995). This new actin purification method
includes an additional cycle of polymerization/depolymerization followed by an additional gel filtration step. We verified that G-actin, which is filtrated once, contained negligible amounts of
capping proteins by monitoring the polymerization of actin, using
time-resolved fluorescence spectroscopy and pyrene-labeled actin (not
shown here). We also note that our actin samples were never frozen for
storage purpose.
For this study, six actin concentrations ranging from 10 µM (0.42 mg/ml) to 164 µM (6.89 mg/ml) are reported. While uncross-linked F-actin solutions of concentrations smaller than ~48 µM are
isotropic liquids, F-actin solutions of concentrations greater than 48 µM form a highly ordered liquid-crystalline phase (Coppin and Leavis, 1993): we report optically measured viscoelastic moduli of F-actin solutions in the liquid-crystalline phase and in the isotropic phase.
Diffusing wave spectroscopy
Instrument
The beam from an Ar+ ion laser operating in the single-line frequency mode at a wavelength of 514 nm is focused and incident upon a flat scattering cell that contains the F-actin network and spherical optical probes. The light multiply scattered from the solution is collected by two photomultiplier tubes (PMTs) via a single-mode optical fiber with a collimator lens of very narrow angle of acceptance at its front end and a beamsplitter at its back end (Weitz and Pine, 1993). The outputs of the PMTs are directed to a
correlator working in the pseudo-cross-correlation mode to generate the
autocorrelation function g2(t) 
1, from which
quiescent rheological properties of the F-actin solutions can be calculated.
Actin is polymerized in situ for 12 h before measurement by
loading the scattering cell with a solution of G-actin mixed with the
polymerizing salt solution and a dilute suspension of monodisperse latex microspheres (Duke Scientific Corp.) of radius 0.48 µm at a
volume fraction of 1%. The scattering cell is then immediately tightly
capped. Using time-resolved mechanical rheology, we verified that
G-actin was fully polymerized into F-actin and that F-actin had formed
an equilibrium network at all concentrations presented in this paper
before 12 h. Using mechanical rheology, we also verified that the
rheology of F-actin is identical in the presence and in the absence of
the added latex beads. Using static light scattering, we verified that
more than 98% of the scattering intensity in the transmission geometry
was due to the microspheres, and less than 2% was due to the F-actin
filament network itself. We also verified that the measured mean square
displacement scaled inversely with the radius of the probing bead for
diameters larger than 0.55 µm and smaller than 1.95 µm (see Eq. 1
below). Smaller particles did not display saturation of their
displacement at large times (they percolated through the network), and
larger particles sedimented too rapidly (data not shown). All DWS
measurements are conducted at a temperature of 23°C.
Analysis
The general method used to extract G'(
) and
G"() from g2(t) 1 via the
calculated values of the mean square displacement of the optical probes
is described in Mason and Weitz (1995) and Palmer et al. (1998). The
method is based on a relation between the mean square displacement of a
microsphere imbedded in a viscoelastic medium and the viscoelastic
modulus of that medium. By assuming that Stokes' law, which holds for
a purely viscous fluid, can be generalized to all frequencies, one
obtains
|
(1) |
(s) 
L[X(t)] 
0
X(t)exp(st)dt. Equation 1, which neglects
inertial effects, relates the unilateral Laplace transforms of the
stress relaxation modulus Gr(t),
(s) sr(s), and of the
mean square displacement 
r2(t)
. We use
the analytic continuation between the real function (s) and the complex function G*() G'() + iG"() to extract elastic and loss moduli by
extracting real and imaginary parts from the imaginary function
(s = i).
Mechanical rheometry
To compare our optical measurements with classical mechanical
measurements (Ferry, 1980), we employ a strain-controlled mechanical rheometer (ARES-100 Rheometrics) equipped with a 50-mm-diameter cone-and-plate geometry. To prevent possible evaporation of the buffer,
the cone-and-plate tools are enclosed in a custom-made vapor trap. The
temperature of the sample is fixed at 23°C to within 0.1°C. The
G-actin solution is placed between the cone-and-plate tools and allowed
to polymerize in the presence of polymerizing salt for 12 h before
the measurements. The viscoelastic properties of F-actin networks were
found independently of time after 12 h at all actin concentrations
used in this work. The linear equilibrium values G'() and
G"() of the actin solutions are measured by setting the
amplitude of the oscillatory strain at 
= 1% and sweeping from low
to high frequency. The strain-dependent viscoelastic moduli were
measured by subjecting the F-actin solution to a 1-rad/s oscillatory
deformation of increasing amplitude; G' and G"
are computed from the maximum magnitude of the measured stress (Palmer et al., 1998; Mason et al., 1998).
| |
RESULTS |
|---|
|
Top
Abstract
Introduction
Experimental techniques
Results
Discussion
References
|
|---|
Computation of the mechanical properties of F-actin networks from
multiple light scattering measurements proceeds via several steps. We
report the autocorrelation function g2(t) 1
of the multiply scattered light intensity, measured with a fast
intensity correlator. From g2(t) 1, we
compute the mean square displacement r2(t) of the microspheres (for further
details about how r2(t) is calculated
from g2(t) 1, see Weitz and Pine, 1993;
Palmer et al., 1998), from which we extract the time-dependent
diffusion constant D(t). From
r2(t), we calculate the viscoelastic
modulus (s), and finally the frequency-dependent
viscoelastic moduli G'() and G"().
Correlation function, mean square displacement, and time-dependent diffusion coefficient
The light of an Ar+ ion laser is incident upon the
F-actin network and multiply scattered by the microbeads imbedded
inside the network. We measure the instantaneous intensity of the light intensity multiply scattered by the microspheres, transmitted through
the network, detected by the PMTs, and then autocorrelated to generate
the autocorrelation function g2(t) 1. Fig.
1 shows g2(t) 1
over a large temporal range corresponding to ~10 time decades,
between 107 s and 103 s. By collecting at
least four DWS runs for each actin concentration, we confirmed the
reproducibility of our optical measurements of g2(t) 1. Sample aging effects are negligible; no significant optical
and rheological changes occurred within 3-5 days after the preparation
of G-actin and (immediate) subsequent polymerization for 12 h. At
large actin concentration, problems from lack of ergodicity can arise
because of the onset of inhomogeneities in F-actin networks. To confirm
ergodicity, we conducted runs on the same F-actin samples with the
laser beam incident upon five different points of the face of the
scattering cell. Local variations of g2(t) 1
were found to be negligible for actin concentration smaller than 48 µM but observed to slightly increase for increasing actin
concentration (data not shown).
|
From g2(t) 1 data, we extract the
time-dependent mean-square displacement
r2(t) of the spherical probes imbedded
in the mesh formed by the overlapping actin filaments. The result of
this operation is displayed in Fig. 2,
which shows that the motion of the probing spheres is rapid at short
times and dramatically slowed past a characteristic time. The time at
which probe motion becomes hindered is more rapidly attained for
increasing actin concentrations, and, as expected, the maximum
displacements reached by the diffusing microspheres decrease with actin
concentration. Fig. 3 displays
r2(t) as a function of actin
concentration and time scale. All graphs display a similar weak
dependence on actin concentration. From
r2(t), we can also calculate the
time-dependent diffusion coefficient D(t) r2(t)/6t, which is shown in Fig.
4. As discussed below, this figure shows
that the transport of microspheres with a diameter larger than the
average mesh size is not purely diffusive and therefore is not
characterized by a single constant diffusion coefficient, but by a
time-dependent diffusion coefficient.
|
|
|
Complex modulus
To extract the viscoelastic moduli of actin filament networks, we
adapt the analytical framework developed in Mason and Weitz (1995). In
particular, the macroscopic viscoelastic modulus
|(s = i)| = |G*()| = 
, which can
be computed directly by Laplace transformation of the discrete
r2(t) data and is displayed in Fig.
5. Each curve displays a
characteristic plateau at small frequencies up to a
concentration-dependent characteristic crossover frequency and a rapid
increase for increasing frequency after that crossover frequency.
|
We can also extract the concentration dependence of the high-frequency
viscoelastic modulus. Fig. 6 displays
|G*| measured at = 105 rad/s as a
function of actin concentration; a power-law fit of the data yields
|
(2) |
. At short time
scales, the bead moves relative to the background of immobile
filaments. The hydrodynamic flow created by the moving bead is screened
into the network and is nonnegligible up to a distance 
into the
sample from the bead. Thus dissipation per unit volume is
kBT
2/2. Therefore, the total
dissipation is on the order of
~(2/2)(R2).
But c1/2; hence the friction is
proportional to c1/2R2, in
qualitative agreement with our observations.
|
Elastic and loss moduli
Elastic and loss moduli are straightforwardly extracted from
(s) because G'() = |G*()|cos(
()/2) and G"() = |G*()|sin(()/2), where G*() = (s = i). The frequency-dependent phase shift () determines the nature of the transport of the microspheres in
the F-actin network. If = 0, the probed polymer network is purely
elastic; if = 1, the network is purely dissipative. Fig. 7 shows the frequency-dependent elastic
and loss moduli of a 24 µM F-actin network. Fig. 7 also shows
G'() and G"() measured by mechanical
rheometry. Excellent agreement is observed between optical and
mechanical measurements over the limited frequency range probed by our
mechanical rheometer. The elastic modulus G' is observed to
dominate at low frequencies, and the loss modulus G"
dominates at large frequencies. We measure
|
(3) |
) as well as the dominance of
G"() at high frequency are unusual for a polymeric
system.
|
In Fig. 8, we plot the low-frequency
elastic plateau modulus G'p, which is
evaluated at = 1 s1 (i.e.,
G'p G'( = 1 s1)), as a function of actin concentration
c. For the sake of comparison, we also plot
G'p(c) as measured by
mechanical rheometry. For concentration c 
60 µM,
the data are fit by a power law of c; we find
|
(4) |
|
| |
DISCUSSION |
|---|
|
Top
Abstract
Introduction
Experimental techniques
Results
Discussion
References
|
|---|
Fast bending modes at short times and hindered diffusion at long times
Our high-bandwidth measurements of the mean square displacement
are sensitive to both the fast bending fluctuations of single actin
filaments at short times and the macroscopic viscoelasticity of F-actin
networks at long times. Morse's model (Morse, manuscript submitted for
publication) predicts that the viscoelastic modulus of a concentrated
network of F-actin is G*() (i)3/4 at
large frequencies. Therefore, according to Eq. 1,
r2(t) t3/4 at short
times, in excellent agreement with our observations. This good
agreement suggests that the exponent 3/4 describes the fast
bending fluctuations of the filaments at wavelengths shorter than the
entanglement length of the network. This exponent is therefore directly
related to the finite rigidity of individual actin filaments. At long
times, the microspheres become elastically trapped by the actin
filaments. The associated diffusion coefficient D of the
probing 0.48-µm radius microsphere decreases from D 0.3 µm2/s at short times to D 106 µm2/s for large actin
concentrations and long times. The latter value corresponds to
near-arrest of the microsphere by the elastic F-actin network: the
microsphere probes the long-time elasticity of the F-actin mesh. The
former is close but smaller than the diffusion coefficient of the same
microsphere in water of viscosity 1 cP, which is equal to
D0 = kBT/6a 0.44 µm2/s.
The values of D that we measured by DWS can be compared with
values obtained by Newman and co-workers using classical dynamic light
scattering (Newman et al., 1989a,b). These authors measured the
diffusion coefficient of latex spheres imbedded in F-actin networks and
obtained values ranging from D/D0 1 to 0.2 for 270-nm-radius spheres in F-actin networks of concentrations ranging from 1 to 22 µM. The magnitudes of these values agree with our optical measurements, at least if we use D optically
measured at very short times and small actin concentrations. But, as
shown in Fig. 4, the diffusion of latex spheres in concentrated F-actin networks is not well described by a single, constant diffusion coefficient, because the spheres' transport becomes subdiffusive at
times as short as 104 s. Indeed, we obtain values that
are 105 to 106 smaller than
D0 at long times and large actin concentrations. The interpretation of Newman et al. (1989a) only incorporates the
dissipative nature of the F-actin network (i.e., the viscosity) and
does not include the elastic contribution to the hindered diffusion of
the microsphere in the F-actin network.
Agreement between optical and mechanical measurements
Over the limited frequency range probed by mechanical rheometry,
our mechanical and optical measurements agree to within 10-15% (see
Fig. 7). This agreement between optical and mechanical measurements helps clarify the behavior of F-actin under deformation. Researchers have reported conflicting observations regarding the low-strain behavior of F-actin networks. In particular, Janmey et al. (1994) have
reported strain-hardening. We have repeated these measurements with our
instruments, and instead observe that G' and G"
are strain-independent for strains up to 10% (see Fig.
9). Because we obtain the same magnitude
for G' and G" with mechanical rheometry (which involves shear deformation) and optical rheometry (which involves no
applied shear deformation), we conclude that F-actin networks do not
display strain-hardening and therefore do not offer a reserve of
resistance to deformation.
|
Frequency dependence of viscoelastic moduli
The viscoelastic nature of F-actin networks strongly depends on the strain frequency
The description of F-actin rheology has been oversimplified. We find that the frequency content of mechanical deformation governs the viscoelastic response of an F-actin network. Because of the large variations in the reported values of their elasticity, uncross-linked F-actin networks have been characterized as either viscoelastic solids or viscoelastic liquids (Janmey et al., 1988, 1990, 1991, 1994; Sato et
al., 1985, 1987). The extremely large temporal resolution of DWS allows
us to show that the rheological behavior of F-actin networks can be
characterized as both solid-like and liquid-like. At small frequencies,
actin filament networks can be characterized as viscoelastic gels
because they display some of the characteristic rheological properties
of gels. These characteristics include an elastic modulus that is
nearly frequency-independent over a wide frequency range and is much
larger than the loss modulus (solid-like behavior). However, at small
frequencies, uncross-linked F-actin networks are weak: the magnitude of
the plateau modulus is small. Instead, at high frequencies, both
elastic and loss moduli become large. Moreover, at high frequencies,
F-actin networks are liquid-like: the loss modulus systematically
dominates the elastic modulus and F-actin networks lose their
solid-like properties.
Comparison with reported values and theoretical models
Rheological measurements of the frequency dependence of the elastic and loss moduli for > 100 rad/s are not available in the
literature; therefore direct comparison with published data is
impossible. Moreover, like stress-controlled rheometers (Janmey et al.,
1994), magnetic microrheometers (Schmidt et al., 1996; Muller et al.,
1991), and torsion pendulums (Zaner, 1986; Janmey et al., 1994), our
strain-controlled rheometer can measure G'() and
G"() only up to 80-100 rad/s, so a direct comparison
between DWS measurements and mechanical measurements at higher
frequencies is also precluded.
However, recent theoretical models predict the high-frequency
dependence of both G'() and G"().
Measurement of the high-frequency regime offers new insight into the
local dynamics of actin filaments inside their tube formed by the
surrounding filaments. For instance, the frequency dependence of the
elastic modulus can be explained within the framework of the model of
Isambert, Maggs, and Morse (Isambert et al., 1995; Morse, manuscript
submitted for publication). This model, which predicts G*() (i)
with = 3/4, assumes that at
times smaller than a characteristic time 
e, the effects
of entanglements as well as the rigidity of F-actin only allow for
actin filaments to fluctuate laterally. In this model, the exponent
3/4, which would become 1/2 for flexible polymers,
directly reflects the finite rigidity of actin (Isambert et al., 1995;
Gittes et al., 1993). Because the ratio of loss and storage moduli is
related to by G"/G' = tan(/2) 2.41, Morse's
model correctly predicts that dissipation effects dominate the
viscoelastic response of F-actin networks at high frequencies (Morse,
manuscript submitted for publication).
The elastic modulus has a small magnitude and a weak concentration dependence
Comparison with published data and theoretical models
Another direct effect of the finite rigidity of F-actin is the concentration dependence of the plateau modulus. Few systematic studies of the concentration dependence of the elastic plateau modulus are available. Janmey and co-workers have reported G'p(c) c2.2
for 9.6 µM < c < 24 µM with a magnitude
between G'p 150 dynes/cm2
and G'p 1000-5000
dynes/cm2, respectively (Janmey et al., 1988, 1990, 1991,
1994; Hvidt and Janmey, 1990). Similar magnitude and concentration
dependence have been reported by other groups (Hvidt and Heller, 1990;
Muller et al., 1991; Newman et al., 1993). Our mechanical and optical measurements of the amplitude of the elastic modulus, which agree with
one another to within 20%, yield values that are between 15 and 200 times smaller. Our measurements are, however, in good agreement with
those published by Sato et al. (1985, 1987), Wachstock et al. (1993,
1994), and Haskell et al. (1994).
Two models that specifically describe the rheology of F-actin networks
have recently appeared in the literature. Isambert and Maggs (1996) and
Morse (manuscript submitted for publication) assume that the
small-frequency plateau modulus stems from the slow motion of the actin
filaments, hindered by the surrounding cage formed by the other
filaments. MacKintosh et al. (1995) postulate instead that the plateau
modulus originates from the fast, hindered lateral motion of subsection
of actin filaments between entanglements. These models present
different predictions for the magnitude and the concentration
dependence of the plateau modulus. We first test the model of Isambert,
Maggs, and Morse by comparing the measured and predicted magnitudes of
the elastic modulus for c = 24 µm. This model
predicts that the plateau modulus is given by
G'p 7kBT
/5le, where
le is the entanglement length,
lp is the persistence length of F-actin, and is the contour length of F-actin per unit volume. Direct measurement of
the entanglement length is difficult; we estimate it by using a
geometric argument given in Morse (manuscript submitted for
publication), le 
lp(lp2)2/5 0.32-0.41 µm with a persistence length lp = 5-17 µm (Gittes et al., 1993; Isambert et al., 1995). Here, is
given by = clANA 38.5 µm2 when c = 24 µM;
NA is Avogadro's number; and
lA = 2.75 nm is the curvilinear length of a
G-actin monomer. We find G'p 7kBT/5le 6 dynes/cm2, which is slightly smaller than our measured
plateau modulus. The model of Isambert, Maggs, and Morse also correctly
predicts the concentration dependence of the low-frequency plateau
modulus: we find G'p c1.2±0.2, whereas the model predicts
G'p c7/5.
We now compare the measured concentration dependence and magnitude of
the plateau modulus with the predictions of the model of MacKintosh,
Käs, and Janmey (MacKintosh et al., 1995). That model predicts
that the plateau modulus is given by G'p
kBTlp2/2le3,
where is the mesh size of the F-actin network. We find
G'p 7600 dynes/cm2, using
1/2 = 0.15 µm, lp = 17 µm, and le 0.41 µm, which is
much larger than our measured value of the plateau. However, the
amplitude of G'p predicted by the model
of MacKintosh, Kas, and Janmey is highly dependent on the evaluation of
le and lp: using the
estimate of le based on a heuristic argument
that yields an entanglement length that is about twice the tube
diameter De 0.5 µm (Morse, 1997),
le 1 µm, and a smaller persistence length
lp = 5 µm, we obtain
G'p 45 dynes/cm2, which
is still too large. Therefore, unlike the Isambert-Maggs-Morse estimate
of G'p, small variations in the estimates
of le and lp can yield to
large variations in G'p. This
disagreement worsens at actin concentrations larger than c > 24 µM because, according to the model of MacKintosh, Kas, and
Janmey, G'p depends strongly on
concentration, G'p c2.2.
The semiquantitative agreement between our DWS measurements and the
predictions of the model of Isambert, Maggs, and Morse suggests that
the origin of the viscoelasticity in an F-actin network at small
frequencies is not due to the lateral fluctuations of the constitutive
filaments, a mechanism evoked by the model of MacKintosh, Kas, and
Janmey, but is due to the reptation-like motion of each actin filament,
which is restricted by the surrounding filaments of the network.
Resolution of the discrepancy between reported values of the elastic modulus
Our measurements help resolve the large discrepancy between values of the elastic shear modulus reported in the literature. Many explanations have been offered to rationalize this discrepancy. Recently, Janmey et al. (1994) observed the extreme sensitivity of both
the F-actin length and shear modulus to even small shear stresses.
Therefore, they attributed the discrepancy to the effect of
irreversible changes in the F-actin samples that could occur during
sample loading into the rheometer or initial deformation imposed to the
sample before mechanical measurements. Because, in our experiments,
G-actin is polymerized in situ inside the scattering cell and because
our optical measurements are conducted in the linear rheological regime
by definition, our measurements reject these explanations for the
origin of the discrepancy.
Janmey et al. (1994) also pointed out that layers of proteins dried and
denaturated by buffer evaporation at the periphery of the tools of the
rheometer can also affect measurements of elastic moduli. Most of the
stress generated in a fluid placed between cone and plate (or parallel
plates) is contributed by the periphery; therefore, even superficial
evaporation at the edge of the tools can indeed have important effects
on measured F-actin rheological properties. Again, our optical
measurements avoid any possible spurious effect of buffer evaporation,
because the scattering cell is tightly capped after G-actin solution
loading, and the laser beam is incident upon the cell at a point far
away from the air-solution interface.
Implications in cell biology
The actual magnitude of the elastic shear modulus is important
because a high value supports the hypothesis (Janmey et al., 1998,
1990, 1991, 1994) that isotropic F-actin networks alone are
sufficiently strong to stabilize cells and provides a baseline against
which to monitor subtle changes in the mechanical properties of F-actin
networks due to regulating proteins, as well as small changes due to
polymerization of actin filaments from ATP-containing or ADP-containing
subunits (Janmey et al., 1990; Pollard et al., 1986). Our new
measurements directly reject a high value for the elastic modulus and
therefore the possibility that F-actin networks alone can generate the
rigidity necessary for cell activities as important as cell locomotion,
cell protrusion, and prevention of cell collapse. This conclusion is
further supported by the observation (Janmey et al., 1994) that a mild
shear flow can dramatically reduce the elastic modulus of an F-actin
network, which is confirmed by our own rheological measurements in the
nonlinear regime (see Fig. 9). Indeed, because cells are constantly
subjected to large external and internal strains, which can be larger
than 10%, F-actin networks alone cannot provide the necessary strength
to prevent cell collapse. Hence, the cytoskeleton absolutely requires
the combined action of F-actin and either other cytoskeletal
filamentous proteins (microtubules and intermediate filaments) or
actin-cross-linking proteins (-actinin, tensin, vinculin, etc.), a
conclusion reached by other authors (Wachstock et al., 1993; Condeelis,
1993, and references therein).
| |
ACKNOWLEDGMENTS |
|---|
The authors acknowledge D. Morse, A. Maggs, and T. D. Pollard for insightful discussions. DW and SCK acknowledge financial support from the Whitaker Foundation (DW, SCK) and the National Science Foundation, grants DMR 9623972 (DW), CTS 9502810 (DW), and CTS 9625468 (DW).
| |
FOOTNOTES |
|---|
Received for publication 30 April 1998 and in final form 19 October 1998.
Address reprint requests to Dr. Denis Wirtz, Department of Chemical Engineering, Johns Hopkins University, Maryland Hall, Room 221, 3400 North Charles Street, Baltimore, MD 21217. Tel.: 410-516-7006; Fax: 410-516-5510; E-mail: wirtz{at}jhu.edu.
| |
REFERENCES |
|---|
|
Top
Abstract
Introduction
Experimental techniques
Results
Discussion
References
|
|---|
-actinin gels on deformation rate.
Nature.
325:828-830[Medline].-actinin for actin determines the structure and mechanical properties of actin filament gels.
Biophys. J.
65:205-214[Abstract].
Biophys J, February 1999, p. 1063-1071, Vol. 76, No. 2
© 1999 by the Biophysical Society 0006-3495/99/02/1063/09 $2.00
This article has been cited by other articles:
 |
|
 |
|
  M. Jonas, H. Huang, R. D. Kamm, and P. T. C. So Fast Fluorescence Laser Tracking Microrheometry, II: Quantitative Studies of Cytoskeletal Mechanotransduction Biophys. J., July 15, 2008; 95(2): 895 - 909. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
S. S. Rogers, T. A. Waigh, and J. R. Lu Intracellular Microrheology of Motile Amoeba proteus Biophys. J., April 15, 2008; 94(8): 3313 - 3322. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Tassieri, R. M. L. Evans, L. Barbu-Tudoran, J. Trinick, and T. A. Waigh The Self-Assembly, Elasticity, and Dynamics of Cardiac Thin Filaments Biophys. J., March 15, 2008; 94(6): 2170 - 2178. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
M. Jonas, H. Huang, R. D. Kamm, and P. T. C. So Fast Fluorescence Laser Tracking Microrheometry, I: Instrument Development Biophys. J., February 15, 2008; 94(4): 1459 - 1469. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
|
as a function of the
frequency obtained from 
between loss and storage modulus is a strong function of strain. At
small strains,