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PHYSICA L R EVIEW LET T ERS
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Microrheology of Entangled F-Actin Solutions
M. L. Gardel,1 M. T. Valentine,1 J. C. Crocker,2 A. R. Bausch,3 and D. A. Weitz1
1
Department of Physics & DEAS, Harvard University, Cambridge, Massachusetts 02138, USA
2
Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
3
¨ ¨ ¨
Lehrstuhl fu r Biophysik–E22, Technische Universitat Munchen, Garching, Germany
(Received 28 March 2003; published 7 October 2003)
We measure the viscoelasticity of entangled F-actin over length scales between 1 and 100 m using
one- and two-particle microrheology, and directly identify two distinct microscopic contributions to the
elasticity. Filament entanglements lead to a frequency-independent elastic modulus over an extended
frequency range of 0:01?30 rad= sec; this is probed with one-particle microrheology. Longitudinal
fluctuations of the filaments increase the elastic modulus between 0.1 and 30 rad= sec at length scales up
to the filament persistence length; this is probed by two-particle microrheology.
DOI: 10.1103/PhysRevLett.91.158302 PACS numbers: 83.10.Mj, 83.80.Lz, 87.15.La, 87.16.Ka
Filamentous actin (F-actin) is a semiflexible polymer approaches a plateau at even lower frequencies. The ex-
and an essential building block in the cytoskeletal net- cess elasticity is believed to arise from longitudinal fluc-
works that determine cellular mechanics and function tuations of the filaments which are coherent over their
[1,2]. In vitro, globular actin (G-actin) polymerizes in persistence lengths; such relaxations are unique to semi-
the presence of K or Mg2 to form polymers whose flexible polymers [5,6]. Thus, the microscopic origins of
lengths are polydisperse and of order 10 to 20 m. the rheology canbe ascertained only by determining both
These filaments have a persistence length of lp the frequency and length scale dependencies.
15 m, 3 orders of magnitude larger than their diameter, Microrheological techniques have the potential of
d 7 nm [3,4]. Thus, solutions of entangled F-actin are probing the response at length scales that are important
ideal for the study of the dynamics of semiflexible poly- for actin networks [9?14]. Significant discrepancies are
mer networks. The rheology of semiflexible polymer observed between microrheological and bulk measure-
networks has important contributions from numerous ments of the viscoelasticity of actin networks [13].
different characteristic length scales and concomitant However, to date, there have been no attempts to exploit
frequency scales, and is qualitatively different than that microrheology to elucidate the microscopic origins of the
of flexible-polymer networks [5?14]. The large persis- rheology at different length scales and to resolve these
tence length leads to important dynamics of individual discrepancies.
filaments that profoundly affect the viscoelasticity of the In this Letter, we use one-particle (1P) and two-
network. For example, semiflexible polymers become en- particle (2P) microrheology to probe the length scale
tangled at extremely low volume fractions to form net- dependence of the rheology and determine the contribu-
works with a significant elastic modulus and long viscous tions of both the longitudinal fluctuations of the filaments
relaxation time in comparison to flexible polymers at the and entanglements to the bulk viscoelasticity of F-actin
same volume fractions. These networks are characterized networks between 0:01?30 rad= sec. We use 2P micro-
p
by an average mesh size, 0:3= cA , where cA is the rheology to probe the contributions of large length scale
actin concentration in mg=ml and is measured in m fluctuations and 1P microrheology to isolate the contri-
[15]. However, for semiflexible networks, individual fila- butions of fluctuations at short length scales. We show that
ments are sterically hindered due to the presence of other the elasticity approaches a frequency-independent plateau
1=5
filaments at the entanglement length, le 4=5 lp , rather when the longitudinal fluctuations have diffusively de-
cayed at the length scale of our probe; by contrast, this
than the mesh size, as is the case for flexible polymers
time scale is determined by diffusive decay over lp for
[6,16]. These steric entanglements should result in an
elastic modulus that remains constant over an extended bulk measurements. However, the magnitude of this pla-
range of frequencies. The lower bound in frequency is set teau is independent of length scale, allowing both 1P and
by the reptation time, r 104 sec, the time it takes a 2P microrheology to be used to probe the concentration
filament to diffusely relax laterally along its tube formed dependent plateau elasticity of F-actin networks.
by neighboring filament entanglements. The upper bound G-actin solutions are prepared by dissolving lyophi-
is set by the entanglement time, e 0:1 sec, resulting lized G-actin in deionized water and dialyzing against
from transverse fluctuations at the largest possible length fresh G-buffer (2 mM Tris HCl, 0.2 mM ATP, 0.2 mM
CaCl2 , 0.2 mM DTT, 0.005% NaN3 , pH 8:0) at 4 C for
scale, le . However, bulk rheological measurements of F-
24 h. Solutions of G-actin are kept at 4 C and used within
actin networks do not exhibit this extended plateau [8,13],
but instead G0
! increases monotonically with ! at 7 days of preparation. Carboyxlate modified colloidal
frequencies above 0:1 rad=sec, and only asymptotically spheres (Molecular Probes) [14] are mixed with G-actin
? 2003 The American Physical Society
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to a volume fraction of 0:4%. Actin polymerization is To examine dynamics at length scales much larger
initiated by adding 1=10 of the final sample volume of than a, we determine the correlated displacements of
10 F-buffer (20 mM Tris HCl, 20 mM MgCl2 , 1 M KCl, pairs of particles, i and j, separated by a distance,
10 m < Rij < 100 m. We average over time, t, and
2 mM DTT, 2mM CaCl2 , 5 mM ATP, pH 7:5) and mixing
gently for 10 sec. The sample is loaded into a 5 mm over all distinct pairs of particles (i Þ j) to calculate
10 mm 1 mm glass sample chamber, sealed with high- the two-particle displacement correlation tensor,
vacuum grease and equilibrated for 1 h at about 25 C. We
D
r; hri
t; ri
t; r ÿ Rij
tiiÞj;t ; (1)
image approximately 100 spheres with an inverted micro-
scope in bright field (40 ; N:A: 0:85, air) and record
where ri is the displacement of the ith particle in
their dynamics at 30 frames= sec using a CCD camera
component of the direction. At length scales, r, where the
with a shutter speed of 0.5 msec [17]. Several thousand
material behaves as a bulk viscoelastic material, the
frames are captured, ensuring good statistical accuracy
correlated motion will decay as 1=r [17,20,21]. From
for time scales up to 100 times the frame rate. Particle
the Rij where we observe a 1=r decay, we extrapolate
centers are detected in each frame to an accuracy of
the correlated motion to a, and define the two-particle
20 nm and the time evolution of the position of each
mean-squared displacement, hr2
iD 2r=aDrr
r;
particle is determined [18]. To avoid wall effects, we
(2P MSD). Physically, the 2P MSD reflects the one-
image 100 m into the sample.
particle motion expected from long-wavelength modes
We calculate the one-dimensional ensemble averaged
in the material; thus, by examining correlations up to
mean-squared displacement hx2
i (1P MSD) and scale
100 m, we are able to use 0:5-m particles to measure
the results by the particle radius, a, to reflect the size-
mechanical response in entangled F-actin at length scales
dependent viscous drag. In a solution of 0:9 mg=ml F-
up to an order of magnitude longer than individual fila-
actin, where 0:3 m, the scaled MSD of particles
ments. The scaling factor of 2=a is obtained by assuming
with a 0:42 m (a= 1:5) exhibits very little time
the surrounding medium is incompressible, thus having a
evolution, reaching a constant value for > 0:1 sec, as
Poisson ratio of 1=2 [20]. We can directly test this
shown by the solid circles in Fig. 1. The magnitude and
assumption by comparing the transverse, D and D , to
frequency dependence of the MSD remain unchanged for
the longitudinal, Drr , components of the two-particle
smaller particles with a 0:32 m (a= 1), as shown
correlation tensor from Eq. (1), and calculating
by the solid triangles in Fig. 1. By contrast, the scaled
MSD of even smaller particles, with a 0:23 m 3 ÿ 4
D
D
(2)
:
(a= 0:6) is dramatically different, evolving as 0:4 , 4
1 ÿ
Drr Drr
as shown by the solid squares in Fig. 1. By examining
This ratio results from calculating the strain field of a
individual particle trajectories, we find that all the
point stress in an elastic medium [17,20,21]. We find that
0:32-m and 0:42-m particles remain caged, whereas
F-actin is incompressible for 0:03 < ! < 30 rad= sec and
some of the 0:23-m particles can permeate through the
1=2 as shown in Fig. 2.
network [19]. Since the 1P MSD probes the local micro-
environment at length scales of a, small changes in The frequency dependence and magnitude of the 2P
MSD of 0:42-m spheres in 0:9 mg=ml actin is qualita-
particle size have dramatic effects on the 1P MSDs.
tively different than the 1P MSD. At 0:1 sec, the 2P
MSD is an order of magnitude smaller than the 1P MSD;
moreover, it scales as 0:5 whereas the 1P MSD is essen-
tially constant in time, as shown in Fig. 2. However, at
FIG. 1 (color online). Comparison of 1P (symbols) and 2P
(open symbols with lines) MSDs in 0:9 mg=ml F-actin ( FIG. 2 (color online). The Poisson ratio of 0:9 mg=ml F-actin
probed with 0:32-m particles.
0:3 m).
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10 sec, the 1P and 2P MSDs converge to similar we infer Go 0:01 Pa. While the magnitudes and fre-
values. Similar behavior is observed with the 0:32-m quency dependence of 2P moduli are robust, crossover
spheres. Remarkably, the 2P MSD of the 0:23-m par- frequencies cannot be precisely determined as they are
ticles also overlays well with the 2P MSDs obtained with very sensitive to variations in the smoothing of the
the larger probes. Although the 0:23-m spheres can 2P MSD required to determine the viscoelastic moduli
permeate through the network, these motions are not [22]. Thus, examining the pairwise correlated motion of
correlated at long distances; the two-particle analysis micron-sized particles separated between 10 and 100 m
reflects only that portion of the tracer motion due to successfully probes the properties observed at macro-
advection by strain fluctuations of the network. scopic length scales with traditional rheology.
We measure the 2P MSD of 0:42-m particles in a To elucidate the microscopic origins of this viscoelas-
1 mg=ml F-actin solution and use the generalized Stokes- ticity, we use the 1P MSD, interpreted with the general-
Einstein relation [22] to obtain a good approximation of ized Stokes-Einstein relation, to examine length scale
the frequency-dependent bulk elastic modulus, G0
!, and dependence of the viscoelastic response. For 1 mg=ml
viscous modulus, G”?! as shown by the closed and open F-actin, the 1P microrheology using 0:42-m particles
(a= 1:5) exhibits a well-defined plateau over the ex-
circles, respectively, in Fig. 3(a). These are in excellent
accord with the bulk values, shown by the closed tended frequency range of 0.03 to 30 rad= sec, as shown
and open triangles, obtained using a home-built stress- in Fig. 3(a) by the solid and open squares. By contrast, the
controlled rheometer with a parallel plate geometry dynamics of 0:5-m particles in 0:3 mg=ml F-actin yield
[8,13]; bulk properties are also correctly measured with 1P microrheology results that significantly underestimate
the 2P MSDs of the 0.32 and 0:23-m particles. In the bulk viscoelasticity, by nearly an order of magnitude
1 mg=ml F-actin, between 0.1 and 30 rad= sec, 2P micro- over the entire frequency range, as shown in Fig. 3(b). In
this case, a= 0:6, and particle motion is again domi-
rheology measures a viscoelastic response with the elastic
and loss moduli similar in magnitude, and proportional nated by permeation through the network; any similarity
to 0.5 as shown by the solid line in Fig. 3(a). At the lowest in the frequency dependence between the microrheology
frequencies, below 0:1 rad= sec, the elastic modulus be- and the bulk measurements is a pure coincidence [19]. We
gins to dominate and we infer a plateau modulus, Go find that when a= 1, 1P microrheology measures a
0:2 Pa. Similarly, we find the 2P MSD of 0:5-m par- frequency-independent elastic modulus between 0.01 and
ticles yields results in good accord with bulk measure- 30 rad= sec consistent with the plateau modulus observed
ments of the frequency-dependent viscoelasticity for a in a 2P or bulk measurement of entangled actin at fre-
0:3 mg=ml F-actin solution, as shown in Fig. 3(b), and quencies below 0:1 rad= sec.
The observed independence of the 1P viscoelasticity in
both frequency and particle size, and its convergence with
the two-particle results at the lowest frequency strongly
suggest that the discrepancies between 1P and 2P micro-
rheology for particle sizes a arise from the nature of
the coupling between the particles and the excitations
responsible for the elasticity of the network [23]. At
intermediate frequencies, between 0.1 and 30 rad= sec,
longitudinal density fluctuations of the filaments signi?-
cantly contribute to the bulk rheological response; these
relax by diffusing along the filament [5,6]. Thus, the
lowest frequency of these excitations that affects both
2P microrheology and bulk rheology is determined by the
time taken for the density fluctuation to diffuse a persis-
tence length, l e
lp =le 10 sec, where e
l4 =lp kB T 0:1 sec and where is the effective friction
e
coefficient of the filament in solution, kB is Boltzmann’s
constant, and T is the temperature. However, 1P motion
will sense only fluctuations on length scales of a, and
these p much more quickly, on time scales of 0
relax l
FIG. 3 (color online). Comparison between the elastic modu-
e
a= le lp 2 0:1 sec; thus 1P microrheology will not
lus, G0
! (closed symbols), and loss modulus, G00
! (open
probe elasticity due to the longitudinal fluctuations for
symbols) obtained from 1P (squares) and 2P (circles) micro-
! < !l
0 ÿ1 [23]. By contrast, because of its larger
rheology and from a conventional rheometer (triangles) for (a) l
effective length scale, 2P microrheology samples these
1:0 mg=ml F-actin probed with 0:42-m beads and (b)
excitations in the same fashion as bulk measurements,
0:3 mg=ml F-actin with 0:5-m beads. The solid line in both
(a) and (b) shows !0:5 . and thus yields the bulk response in the frequency regime
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for the application of 1P microrheology to in vitro net-
works that more closely mimic the cytoskeleton through
the addition of crosslinking proteins. Moreover, it also
suggests that 1P microrheology may also be adequate for
in vivo measurements of cells.
We thank Tony Maggs and Fred MacKintosh for valu-
able discussions and A. Popp and B. Hinner for bulk
rheology data. This work was supported by the NSF
(No. DMR-9971432 and No. DMR-0243715). A. B. ac-
knowledges support from the Emmy Noether Program of
the DFG. M. L. G. was supported by Lucent GRPW.
FIG. 4 (color online). The concentration dependence of the
elastic modulus at 0:05 rad= sec, Go , obtained from 1P (trian-
gles) and 2P (squares) microrheology. The two techniques are
similar in magnitude and both scale as Go c1:80:4 . The
A
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A
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