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Themechanicalpropertiesofactingels:Elasticmodulusandfilamentmotions

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0 1994 by The American Society for Biochemistry and Molecular Biology, Inc. THE JOURNAL OF B I O ~ I C A L CHEMISTRY Vol. 269, No. 51, Issue of December 23, pp. 32503-32513, 1994

Printed in U.S.A.

The Mechanical Properties of Actin Gels ELASTIC MODULUS AND FILAMENT MOTIONS*

(Received for publication, July 12, 1994, and in revised form, October 7, 1994)

Paul A. JanmeySOnll, SGren Hvidt**, Josef KasSO, Dietmar LercheSS, Anthony MaggsOO, Erich SackmannllII, Manfred Schliwa***, Thomas P. StosselSOn From the Wxperimental Medicine Division, Brigham and Women’s Hospital, Boston, Massachusetts 02115, Departments of §Medicine and TlBiomedical and Biological Sciences, Harvard Medical School, Boston Massachusetts 02115, **Department of Chemistry, Roskilde University, DK-4000 Roskilde, Denmark, $Unstitut fur Medizinische Physik und Biophysik, Medizinische Fakultat (Charite) der Humboldt-Uniuersitat, Berlin, 0-1040 Federal Republic of Germany, QBGroupe de Physico-chimie Thdoretique, Ecole Supdrieure de Physique et de Chimie Zndustrielle de la ViEZe de Paris, Paris Ceder 06 France, I[ IIBiophysik, Physik Department, Technische Universitat Munchen, 0-85748 Garching bei Miinchen, Federal Republic of Germany, and ***Znstitut fur Zellbiologie, Ludwig-Maximilians-Universitat, 0-80336 Munchen 2, Federal Republic of Germany

To address large discrepancies reported in the litera- ture, the viscoelastic properties of gels formed by puri- fied actin filaments have been measured by five differ- ent techniques and five different instruments using actin preparations purified separately in four different laboratories. These measurements consistently showed that the elastic shear modulus of 2 mg/ml F-actin is on the order of several hundred pascals, and depends very strongly on the length of the filaments and on the his- tory of the sample prior to measurement. Shortening of actin filaments with gelsolin and mechanical perturba- tions reduce the shear modulus to low values identical to some reported in the literature, indicating that such perturbations account for low shear moduli and poor responsiveness to filament modifying treatments re- ported previously. The structures of individual actin filaments within gels very similar or identical to those studied rheometrically were also examined by dynamic light scattering and fluorescence microscopy. Dynamic light scattering data were analyzed by a new method to confirm that actin filaments have no stable associations with each other and fluctuate in solution at a rate gov- erned by the filament bending modulus or persistence length, determined to be approximately 10 pm. Fluores- cence microscopy confirmed that applying even small shear stresses to F-actin can orient and rupture the fila- ments, and that in a minimally perturbed viscoelastic gel, long actin filaments are free to diffuse within a limit of constraints formed by their neighbors. These findings confirm that relatively isotropic F-actin networks are sufficiently strong to stabilize cells.

Fluid flow along an arterial wall imparts a shear stress on an adherent leukocyte of at least 10-100 Pa,’ and the shear stress exerted in the periphery of a white blood cell extending protru-

AR38910, HL19429, and HL07680, the Deutsche Forschungsgemein- * This work was supported by National Institutes of Health grants

shaft (SFB 266), the Whitaker Foundation, and NATO. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

icine Division, Brigham and Women’s Hospital, 221 Longwood Ave., 11 To whom correspondence should be addressed: Experimental Med-

Boston MA 02115. Tel.: 617-278-0382; Fax: 617-734-2248; E-mail: janmey@fas.harvard.edu.

‘The abbreviations used are: Pa, pascal(s); DLS, dynamic light scattering.

sions to engulf a yeast has been measured to be 1000 Pa (1). Similar large forces are exerted by the cortex of locomoting keratocytes (2). The intracellular structure widely believed to prevent cell collapse under such stress and to transmit high internal forces is a network of actin polymers (3-6). Therefore, rheologic studies of the viscoelastic properties of actin ought to have direct relevance to its function in vivo (7). But despite the agreement in principle about this function of actin filaments, reported rheologic parameters about actin filaments are widely divergent and confound interpretations concerning actin rheology’s contribution to cell shape and motility.

A fundamental quantity, for example, the absolute value for the rigidity of actin filament networks, varies by orders of mag- nitude, depending on the observers. Some investigators find that purified actin filaments that are presumed to be many micrometers in length at physiologic concentrations (2-10 mg/ ml) have shear elastic moduli of several thousand pascals, as do shorter actin filaments bound together by exogenous cross-link- ing agents (8-12). These studies have also documented high ratios of elastic (G’) t o loss ( G ) moduli from measurements of stress in and out of phase to an oscillating strain and an in- crease in the elastic modulus as the strain increases (from 0 to 20%), a phenomenon known as strain hardening. In exhibiting these properties F-actin obeys rules generally accepted for gels in general (13) and qualitatively resembles the fibrin clot (12, 14, 151, a well characterized protein polymer gel. In most in- stances these investigations also find that higher strains (>20%) and treatments that shorten actin filaments, such as with actin filament-severing proteins, markedly diminish the measured shear moduli. The high elastic moduli measured for F-actin in these experiments provided a base line to document subtle changes in the mechanical properties of the polymers, depending on whether they were polymerized from ATP-con- taining or ADP-containing subunits (16). The findings summa- rized above lend themselves to interpretation by widely ac- cepted theories of polymer rheology (17, 18). According to this view, actin filaments are semiflexible chains that can manifest elastic behavior from impeded rotational diffusion which greatly depends on filament length but does not require cross- links that would prevent translational (reptating) movements. Recent high resolution videomicroscopic studies of F-actin di- rectly demonstrating both entanglement and reptation are con- sistent with this interpretation (19). It follows that F-actin could stiffen a cell either by interpenetration or, at filament lengths close to the transition between impeded and free rota- tional diffusion, through being immobilized by exogenous cross- links.

32503

32504 Elasticity of F-actin and Filament Diffusion

In contrast, other rheologic results that are in approximate agreement with each other differ from those just described in nearly every aspect. In particular, the magnitude of the elastic strength of actin networks is more than a factor of 100 lower than the measurements cited above. The investigators do not observe a large ratio of G' to G or strain hardening at small strains, and at moderate to high strains filament severing pro- teins, mechanical perturbations, or adenine nucleotides have little or no effect on the rheologic values (20-26). If an actin network has an elastic modulus of 1 Pa, as reported by these studies, it would be impossible for it t o be responsible for main- taining the integrity of the cell cortex. Even taking into account the intracellular actin concentration, which may be as high as 10 mg/ml, the shear modulus extrapolated from the data of these studies would be no more than 30 Pa, since the elastic modulus of protein gels scales with the square of the protein concentration (9, 10, 12). Furthermore, if actin rheology does not depend heavily on filament length (24), then models of secretion that invoke actin filament severing to facilitate ves- icle movement (27) are not likely to be relevant. Some investi- gators who found low moduli for purified F-actin have reported higher moduli for F-actin in the presence of cross-linking agents (25,261. The implication of these results is that in order to exert resistance to high shear forces, F-actin must be extra- neously cross-linked. In that case supposedly pure F-actin with a high measured shear modulus is possibly contaminated with cross-linking factors.

To try and resolve these discrepancies, we have performed a series of rheologic measurements on 10 different actin samples made from five different muscle preparations in four independ- ent laboratories, employed five different rheometers, and supplemented our rheologic measurements with new methods for analysis of dynamic light scattering and fluorescence mi- croscopy of single actin filaments. We compare directly the viscoelastic properties of purified F-actin with those of a cyto- plasmic extract gel to put the results in a cellular perspective. We conclude that the low values reported for the rigidity of F-actin are not due to actin purification methods or to differ- ences in instruments used to make measurements but result from the mechanical disruption of actin filaments prior to or during the rheological determinations.

MATERIALS AND METHODS Proteins-Actin was purified from rabbit skeletal muscle indepen-

dently in four different laboratories in Boston, Berkeley, and Munich by the method of Spudich and Watt, and, when indicated, further subjected to gel filtration chromatography using Sephacryl S200 or a Superose 6B FPLC column. The viscoelasticity of purified actin was measured di- rectly after elution from the column, after storage for 1 day at 4 "C, or after freezing in liquid N, and thawing at 37 "C. Gelsolin was purified from human blood plasma by the method of Kurokawa et al. (28). Actin was polymerized by addition of 2 mM MgCl, and 150 m~ KC1 to G-actin in solutions containing 2 mM Tris, 0.2 mM CaCl,, 0.5 m~ ATP, 0.2 mM dithiothreitol, pH 7.6.

Macrophage Extracts-Rabbit alveolar macrophages were obtained as described previously (29). 5 ml of packed cells were suspended on ice in 10 ml of TBS (10 mM Tris, 100 mM NaCl, pH 7.4), and 40 ml of water were added immediately prior to centrifugation at 1000 x g for 10 min at 4 "C. Under these conditions the cells swell but do not rupture. The cell pellet which now had a volume of approximately 7 ml was SUS- pended in 15 ml of cold homogenizing buffer (0.34 M sucrose, 2 mM ATP, 2 mM EDTA, 5 mM dithiothreitol, 10 mM Tris, pH 7.0). The cells were broken by approximately 50 strokes in a Dounce homogenizer until few if any intact cells were visible in a microscope, and the extract was centrifuged at 100,000 x g for 40 min at 4 "C. The supernatant fraction was immediately frozen in liquid nitrogen in 800-pl aliquots. When supplemented with 100 m~ KC1 (delivered from a 3 M KC1 stock) and warmed to 25 "C, such extracts form gels (30), and the Ca2+-sensitive solation of these gels was the basis for the original isolation of cytoplas- mic gelsolin (31).

Rheology-Five different rheometers were used to measure shear viscoelasticity of actin by both free and forced oscillatory methods and by direct determination of shear stress after imposition of a sudden or a slowly increasing shear strain. Free oscillations were measured by torsion pendulums constructed and operated independently in labora- tories in Boston, Roskilde, and Munich. The principles of these instru- ments and their application to actin gels are described elsewhere (9,321. Forced oscillatory measurements were done with both a Rheometrics RFS instrument and a Bohlin VOR rheometer, using cone and plate geometries and either titanium or stainless steel surfaces. Strain- dependent forced oscillatory measurements were also made using a Mettler-Toledo LS40DIN412 instrument and couette geometry (33). Additional measurements were made with a novel highly sensitive magnetically driven oscillator recently described (34, 35). Unless actin was polymerized by addition of 2 m~ MgCl, or 150 mM KC1, a high elastic modulus was never observed ruling out a contribution of surface layers of drying or denatured protein at the edge of the rheometer. Moreover, as shown below, addition of gelsolin to purified F-actin or activation of gelsolin in extracts by Ca" very strongly lowered or elimi- nated the elastic modulus; these maneuvers would not be expected to alter surface denaturation if it occurred. We find that reliable measure- ments are difficult or impossible after many hours because eventually the sample does dry or denature at the outer edge even when the edge of the aqueous gel is covered with a hydrophobic liquid.

Dynamic Light Scattering (DLS)-Light scattering (36) was per- formed on 6 m F-actin using a Brookhaven Instruments BI30ATN apparatus as described previously (8, 16). Light scattering was meas- ured at 23 "C for 30 min at each angle over a range of angles from 30" to 150" using a 4-domain multiple sample time method to extend the range of times in the autocorrelation functions.

Light scattering data were analyzed by the method of Farge and Maggs (37) to determine the stiffness of the filaments. For long poly- mers arranged in an interpenetrating meshwork with mesh size 5 much less than the filament length, measurements of DLS are dominated by bending motions of the filaments (36,3841). For 6 p~ F-actin, the mesh size is approximately 1 pm, and the average filament length is at least 10 times greater (19) as confirmed in Fig. 12. Filament stiffness is most usefully characterized by the length over which correlations in the direction of the tangent are lost, a quantity termed the persistence length. An excellent recent summary of how these quantities can be derived is found in Gittes et al. (42). Analysis of DLS data by this method also can confirm that on a scale of times comparable to those sampled by the light scattering technique (1 ps to 100 ms), there is no significant interaction between filaments. We summarize here the re- sults for the scattering from individual filaments and then discuss the effect of hydrodynamic interactions between filaments on the scattering functions.

DLS from dilute polymer solutions can be described by the intensity autocorrelation function Z(t , q ) = B + I g(q, t ) I ', where B is an exper- imental base line, conventionally set to 0, and g(q, t ) is the dynamic structure factor (43). Farge and Maggs (37) showed that the dynamic structure factor of a single actin filament was given by the following expression

g(q,t) = exp(-q2G(t)) (Eq. 1)

where q = 4m/A sin(W2); n is the refractive index of the solution (1.331, A is the wavelength of light (633 nm), and 8 is the scattering angle.

G(t) is a dynamic correlation function which expresses the propaga- tion of density fluctuations along a semiflexible filament.

The quantity K is the bending modulus of the filament. The function H ( q ) is the Fourier transform of the Oseen tensor (18) and takes account of the hydrodynamic interactions between two parts of the same filament.

H ( q ) = (-ln(Cqd))/4~q (Eq. 3)

where d is the diameter of the actin filament and C = 1.78. The Oseen tensor expresses the fact that movement of the filament causes a hydrodynamic flow which acts back on the filament even at large distances.

The integral (2) is analytically intractable; however we can evaluate the integral approximately as follows: by changing the variable of the

Elasticity of F-actin and Filament Diffusion 32505

integration from q to z = q (t~U47rq)" where 1 is an arbitrary parameter, we find

(Eq. 4) 1 - exp(z41n(z~(t)/l%)/1))

z4 dzl where a( t ) (4mTr7J~t)'~Cd. This is the main result of Farge and Maggs (37). Farge and Maggs continued numerically, but further analytical progress is also possible. This final result is independent of the value of the parameter 1; we choose it in such a way that the integral in Equation 4 is almost independent oft and of K. To do this we set it equal to the typical value of the logarithmic term in the integrand for the values of the physical parameters that interest us. Thus we shall choose 1 = l ( t ) = ln((4~q/3~t)"Cd). The integral should be cut off at a high wave vector corresponding to the inverse of the diameter of the filament. The pa- rameter Z(t ) is a weakly varying function of the rigidity of the filament and the time scales involved in the experiment; a typical value found in our experiments is Z = 3.5. We thus evaluate the integral once and for all for this value of Z and find numerically the results 1 = 1.52. Numerical work shows that a factor of 2 variation in t or K leads to a variation of less than 2% in this integral. In the following analysis of experimental results we shall use the following formula for the analysis of light scattering data.

I

This is our fundamental result for the dynamic scattering due to a single filament or for a solution which is extremely dilute. By plotting ln(g(q, t ) ) as a function of q2(t1(t))3'4 we can hope to deduce the elastic constant K.

Such a scaling plot with our experimental data was, however, only moderately successful. For large momentum transfers (large angles) the results do follow this theoretical law; at small angles however, there are significant deviations from the theoretical result. The long range nature of the hydrodynamic interactions is such that the system is very sen- sitive to the structure of the solution at large length scales. The integral in (Equation 4) contains a logarithmic divergence near z = 0 which is cut off in physical systems by the finite distance between the filaments. The integral should be cut off at an unknown lower wave vector which is independent of the scattering vector, q, and the correlation time, t.

To take into account this hydrodynamic screening we cut off the integral in Equation 2 at a lower, constant value qo (which is only a function of the concentration but not t or q). Expanding 2 in powers of 40 gives

g,,,(q,t) =gt,(q,t)(l + A ( ( q o t ~ z ) / 4 ~ ) ~ ) (Eq. 6)

where A is a constant we needed for further analysis. Data fitted with the analytical form show excellent scaling properties and can be used to measure the persistence length with confidence.

Fluorescence Microscopy-A very dilute solution (actin concentration c = 1 x mg/ml) of rhodamine-phalloidin-labeled F-actin (molar ratio actirdphalloidin, 1:l) was mixed with semidilute solutions of non-la- beled F-actin ( c = 0.1-1.4 mg/ml) by slowly sucking both solutions into a 1-ml pipette. The pipette tip was cut to prevent filament breakage (diameter of the pipette after cutting the tip: 0.7 cm). To prevent bleach- ing of the fluorescent dye, 4 pg/ml catalase, 0.1 mg/ml glucose, 20 pg/ml glucose oxidase, and 0.5 vol % mercaptoethanol were added after the solution was degassed for 1 h. The samples were put between a glass slide and a cover glass separated by about 100 pm and sealed by vac- uum grease. To impede adsorption of F-actin on glass, the slide and the cover glass were coated with G-actin. To equilibrate the sample from strain and shear alignment we waited for 2 h until the sample was investigated. The selective labeling of filaments in an unlabeled F-actin solution permits the observation of the dynamics of a single filament.

To observe whether transfer of F-actin by pipetteting caused filament breakage or alignment, we used a dilute solution of rhodamine-phalloi- din-labeled F-actin alone without mixing it with non-labeled F-actin and allowed the filaments to adsorb to the glass surface to allow a more accurate measurement of the length distribution. Fluorescent filaments were observed either with a confocal microscope (Bio-Rad MR6000) or

A 1000 ~ ' " ' 1 ' ' " 1 " " l " " l " " I " " l ' " ' ~

100

10 m

0 10 20 30 40 50 60

B Time (min)

1

m a Y

Z

0.1

0.01 0 10 20 30 40 50 60 70

Time (min) FIG. 1. Time course of increase in elastic modulus. A, the

buildup of G' during polymerization of a variety of different prepara- tions of actin was measured in three different instruments. The meas- urements denoted by the symbols A, 0, m, A, and 0 were obtained using actin prepared from three different acetone powders and measured in an RFS rheometer at a frequency of 1 rads and a maximal strain amplitude of 1 or 2%. CZosed symbols denote that the samples were gel-filtered. Similar measurements were made in a torsion pen- dulum at resonance frequencies between 1 and 10 rad/s and a maximal strain amplitude less than 2%. The symbols 0 , m, and denote measurements made with a torsion pendulum in Boston using actin prepared in Boston with (El3 ) or without ( 0 modification by biotin or in Berkeley ( 0 , and kindly provided by David Drubin. All samples con- tained 2 mg/ml actin and were polymerized within the rheometers by adding 2 mM MgCl, and 150 r m KC1 to G-actin in buffer A. The symbols S, P, and Z denote results reported in Refs. 21,22, and 44, respectively. B , measurements similar to those shown in Fig. l.4 except that gelsolin was included at a molar ratio of 1:lOOO with respect to actin to produce samples with an average filament length of 2.7 pm. The symbols are as described for A, and N denotes data from Ref. 20.

with an inverted Zeiss microscope (Axiovert) equipped with a filter set for rhodamine fluorescence, a Zeiss Plan Neofluar 63xPh3 objective and a Zeiss HBOlOO light source. For documentation the Axiovert micro- scope was connected to a SIT-camera (SIT68, MTI, Michigan City, IN). The images were recorded on videotape by a SVHS recorder (Panasonic) and analyzed on a Macintosh IIci, supplemented with a fast frame grabber (Pixel Pipeline, Perceptics, Knoxville, TN). Image analysis was carried out with a modified version of the Image processing software (Wayne Rasband, NIH).

RESULTS

Magnitudes of Elastic Moduli-The mechanical properties of polymer networks can be described quantitatively by measur- ing their resistance to deformation in shear, i.e. to stresses (force/area) applied parallel to the surface of the sample. The elastic resistance to such deformations is defined by the shear modulus G ( t ) (stressktrain, where strain is a quantitative

32506 Elasticity of F-actin and Filament Diffusion

t r'

0 1 1 1 1 1 , / , 1 , 1 , , / ~ , , , ~ ,

0 2000 4000 6000 8000 10000

FIG. 2. Effect of filament length on shear modulus of F-actin. Measurements at 1 rads in an RFS instrument were made using a single preparation of gel filtered actin and adding various amounts of gelsolin to vary the average filament length. Experimental conditions are as described above, and measurements were begun 90 min after initiating polymerization.

measure of deformation) or, for oscillatory deformations, by the storage shear modulus, G'. Fig. 1 shows how G' increases dur- ing the polymerization of actin in eleven different measure- ments using eight different actin preparations and four differ- ent instruments. The gel-filtered actin samples (solid symbols) tend to have somewhat higher shear moduli, but this is not always the case, and there is no difference in results using the same actin preparation in different machines (see also Ref. 9). After 60 min, when pyrene-labeled actin fluorescence or light scattering show that greater than 95% of the actin has polymerized (91, the values of G' for different actin samples vary by more than a factor of ten from each other, whereas the reproducibility of multiple measurements from the same sample is within a factor of 2 (32). The average value of these samples was 158 Pa with a range from 20 to 420 Pa. The average value of our measurements using multiple differently treated and measured samples is 100 times greater than the values of G' reported in three recent studies (20,21,44) (points marked S, P , and N in Fig. 1) under what were reported to be identical experimental conditions. The value of 158 Pa is 1000 times greater than that reported in another study (22) (2 in Fig. lA) on a similar sample using the same method as re- ported in ref. (20).

Fig. 1B shows that the enormous difference in elastic moduli measured by different groups largely disappears if we first make filaments short using the filament severing protein gel- solin, which accurately determines the average length of fila- ments because of its ability to nucleate and cap filaments (8, 45). When five of the same actin samples as used for the data in Fig. lA are polymerized with a 1:740 molar ratio of gelsolin to produce an average filament length of 2 pm, the values of G' drop by approximately 2 orders of magnitude, to 0.82 2 0.33 Pa and now are very close to the values reported in Refs. 20-22. The variation among different samples is also much less, and G' now no longer depends on whether the actin was gel-filtered, frozen, or purified from a different acetone powder. This result strongly suggests that both the high shear moduli and the variability between samples are a result of differences in aver- age filament length. Since the average filament length cannot be controlled using only purified actin and depends on the kinetics of polymerization and trace impurities such as CapZ in the preparations which cannot be removed by gel filtration (46), a clear interpretation of the rheology of pure F-actin requires an independent measurement of filament length.

Effects of Filament Length-Fig. 2 shows that the shear mod- ulus of a constant amount of F-actin depends very strongly on the average filament length (L) . Changing the average length of F-actin from 0.5 to 4 pm, by varying the gelso1in:actin ratio changes the elastic shear modulus by more than a factor of 100. This strong dependence of G' on L is consistent with a model of individual interpenetrated filaments that form a viscoelastic network because of steric interactions and impeded rotational diffusion. Only filaments long enough to be immobilized by their neighbors contribute to the elasticity of the sample, and the elastic modulus falls to very low levels when the average filament length approaches the average filament separation which is determined by the protein concentration. Because of the strong dependence of G' on filament length, the low values of G' reported by some groups (20-22) could result from a relatively modest amount of filament breakage or contamina- tion by as little as 0.3% of a capping protein. Since, even after gel filtration, actin contains between 5 and 8 nM CapZ (46) and since some forms of modified actin monomers also act as caps at the barbed end (47), such effects may be significant in many preparations.

Another aspect of actin rheology in which there is a large disparity of results is the frequency dependence of the shear moduli and the relative contribution of elastic and viscous re- sistance to deformation, as measured by G' and G , the in phase and out of phase components of the complex modulus. Fig. 3A shows that F-actin networks with long filaments and a high storage modulus, G' , exhibit two interesting features. G' is very nearly independent of the frequency, over a large range of fre- quencies, and is much larger (>50 times) than G over the entire range of measurements. These features imply that there are few molecular motions with a time constant similar to the reciprocal of these frequencies (0.01-50 s) that can dissipate the mechanical energy in the network, which remains stored in an elastically deformed state. In contrast, Fig. 3B shows that solutions of relatively short (800 nm; 1:300 gelso1in:actin) fila- ments exhibit a much lower G' that is somewhat more frequen- cy-dependent and, most importantly, is not much larger than G. A comparison with the measurements in studies ((21) (Fig. 3C) and (20) (Fig. 3 0 ) ) on actin polymerized without or with a 1:lOOO molar ratio of gelsolin, respectively, suggests that their results are closer to those of short actin filaments, even though their presumed filament lengths are much longer. At the high end of the frequencies they report, G is steeply increasing and nearly equal in magnitude to G', suggesting that G' and G' would cross at only slightly higher frequencies. Previous meas- urements (Fig. 2b of Ref. 23) actually show a crossover at which G > G' at a frequency of 1 rads (23). This result has two important implications. First, a material with G > G' at a frequency near 1 rads does not satisfy the most rudimentary definition of a gel (13). Second, such a material is overdamped when subjected to a transient mechanical displacement, mean- ing that oscillations or traveling waves cannot be sustained in such a material. These two implications are at odds with com- mon observations of the material properties of F-actin. Even at concentrations lower than 34 p ~ , at which most of these ex- periments have been done, F-actin can maintain its own weight when a tube containing it is inverted, and F-actin can also suspend small metal spheres which exert a shearing stress of greater than 10 Pa (48). These discrepancies suggest that the low shear moduli reported for F-actin networks in some studies may not accurately represent the properties of the unperturbed material.

Mechanical Loss of Actin Gels and Their Disruption by Large Deformation-Fig. 4 demonstrates that F-actin can sustain free mechanical oscillations after transient displacement with

Elasticity of F-actin and Filament Diffusion 32507

FIG. 3. Frequency dependence of dynamic shear moduli. The storage (triangles) and loss (circles) shear moduli were measured in an RFS8400 instru- ment at a strain amplitude of 2% and at the frequencies denoted for samples con- taining 34 pv F-actin without (A) or with ( B ) gelsolin at a molar ratio to actin of 1500 and compared to data obtained from similar measurements of the same con- centration of F-actin reported in Fig. 4 of Ref. 21 ( C ) or Fig. 3 of Ref. 20 (Dl .

loo m A

0.01 0.1 1 10 100 Frequency (rad/s)

C

w

u 0.01 0.1 1

Frequency (rad/s)

"

4

3

2

1 = 5.0; G'= 73 Pa; G / G ' = 0.06

0

-1 0 2 4 6 8 10

Time (s)

FIG. 4. Free oscillations of biotin-labeled F-actin. Oscillatory measurements were made in a torsion pendulum using two similar samples of biotin-labeled F-actin 65 min after initiating polymerization. One sample had been measured periodically by similar oscillations dur- ing the course of polymerization (upper curue, and H in Fig. L4 1, and the other sample had been left undisturbed prior to this measurement.

a low mechanical loss near that predicted from the data of Fig. 3A. Two samples of F-actin were polymerized between the plates of a torsion pendulum and momentarily deformed to a strain of approximately 2% by a pulse of air directed at the end of the pendulum arm. In the case of the upper curve, the sample had been measured by such a method every 10 min for one hour prior to the measurement shown, and the lower curve repre- sents a sample that had been left undisturbed for 60 min prior to measurement. In both cases, the measured oscillations are well fit by a theoretical curve for an exponentially damped

B - 10 1:500 gelso1in:actin -

h

k v

0.1 0.1 1 10 100

Frequency (rad/s)

D

7

0.01 0.1 1 10 Frequency (rad/s)

sinusoid, and the frequency, G', and ratio G'/G of the two samples are in close agreement with each other and with the data of Fig. 3 A . If the ratio of G'/G had been as low as those reported in some studies (20-221, no oscillations could have been induced, and previous studies using larger amounts of gelsolin confirm that such materials do not support free oscil- lations (32). Although the magnitude of G' calculated from free oscillations depends on measurements of sample volume, and moment of inertia, and assumes that the apparatus itself does not damp the elastic response, the simple observation of oscil- lations is independent of any possible calibration errors, and therefore impossible to reconcile with the data of Fig. 3, C and D , on the basis of quantitative miscalculations. Furthermore, since the relatively high elastic moduli might conceivably be affected by cross-linking between reactive thiols of actin, the data of Fig. 4 were obtained using nearly 100% biotin iodoac- etamide-labeled actin in which the only surface-exposed cys- teine of actin has been protected (11). Very similar results have also been obtained using native F-actin (9, 32).

A third aspect by which our results and those of some other reports differ radically is the strain dependence of elastic moduli. Newman et al. (20) report very little strain dependence of G' over a range of strains from 1 to 200%, whereas we have previously reported a strong increase in G' a t small strains followed by an abrupt decrease attributed to network rupture at strains greater than approximately 20% (11). Fig. 5A com- pares the relative value of G' measured by oscillations over a range of maximal strain amplitude taken on the same sample at two times, and compares them to the results of Newman et al. (20). The first measurement confirms the strain hardening and rupture we previously reported using different instru- ments and is in contrast to the results of Newman et al. (20) for strains below 30%. However, if we first broke the sample either by large oscillatory strains or by constant shearing to several hundred percent in one direction, the large values of G' previ-

32508 Elasticity of F-actin and Filament Diffusion

A

25 C'' I 4

20 -

15 -

10 -

5 i 1

B

4

2

0

10 100 Strain (Yo)

I 1 10 100

Strain (YO) FIG. 5. Dependence of shear modulus on the degree of defor-

mation. A, the shear modulus was measured at 1 rads in an RFS rheometer by oscillatory measurements at increasing strain amplitudes ranging from 1 to 50% (A). Additional measurements (A) were then made at decreasing strain amplitudes from 50 to 1%. The data are shown relative to the shear modulus measured at the maximal strain of 50% which was 1.4 Pa and are compared to the similar data (0) ob- tained from Fig. 4 of Ref 20. B, stress was measured at increasing (A) strains every 3 s during shear deformation at a rate of 0.02/s up to a maximal strain of 10. The rheometer plates were then returned to their original position and the measurement was immediately repeated (A).

ously observed at small deformations are lost, and we obtain data very similar to those of Newman et al. (20). We confirmed that F-actin networks undergo irreversible damage when sub- jected to large strains by a direct measurement of resisting stress when a sample was strained at a slow shear rate (0.02/s) to increasing extents (Fig. 5B). In close agreement to the re- sults of forced oscillations, and to our previous results, the stresdstrain curve shows significant strain hardening at small strains, followed by an abrupt loss of mechanical resistance. Again, once broken, the network does not rapidly repair itself, although if left for several minutes, and depending on the av- erage filament length, some recovery is observed. The initial increase and abrupt decrease is characteristic of F-actin be- cause it is not observed if the filaments are first made short by gelsolin or when materials that are more resistant to rupture, such as fibrin gels, or vimentin networks are subjected to the same stresses (data not shown).

To verify that strain hardening was not an artifact intro- duced by sample drying at the edges of the sample in conelplate geometry, we also obtained data using a different rheometer equipped with a bob and couette geometry in which nearly all of the mechanical resistance is provided by surfaces far from the edge of the sample exposed to air, which is limited to a very

v k u

50

40

30

20

10

0

0 10 20 30 40 50 Strain (%)

FIG. 6. Strain dependence of F-actin measured in couette ge- ometry. The shear modulus is shown for measurements at increasing maximal strain amplitudes at a frequency of 1 rads using a Mettler- Toledo rheometer.

small area near the axis of rotation at the bob, where the shear stresses are negligible compared to surfaces at larger radii. Under these conditions, the increase in G' at small strains, and the abrupt decrease at strains >lo% shown in Fig. 6 are similar to previous measurements observed in torsion pendulums and in Rheometrics or Bohlin instruments employing either paral- lel plate or cone and plate geometries.

Stress Relaxation of Actin Gels-An independent measure- ment of the viscoelasticity of F-actin samples was obtained from stress relaxation experiments in which the sample is first rapidly deformed to a constant shear strain of 10% and the resulting stress is measured as a function of time that the sample is held in the deformed state. The ratio of stress to strain, the shear modulus G(t), is shown in Fig. 7 for 1.7 mg/ml actin in the absence and presence of a 1500 molar ratio of gelsolin. The initial resisting stress is much lower in the gel- solin-containing sample, and in both samples the resisting stress slowly relaxes after several hours nearly to zero, consist- ent with an absence of permanent cross-links or other irrevers- ible changes in filament structure. The magnitudes of the shear moduli of both samples at times less than 1 s, a value roughly equivalent to G' at a frequency of 1 rads, are very close to the corresponding measurements shown in Fig. 3, A and B.

Macrophage Extract Gels-If actin filaments dominate the viscoelasticity of the cell, then the elastic moduli measured for purified actin samples should be similar in magnitude to that of a cytoplasmic extract gel, to the extent that such gels qualita- tively resemble the intracellular actin gel. Fig. 8 shows the increase in G' when a low ionic strength extract containing 30% macrophage cytosol formed by centrifugation at 4 "C is slowly warmed and forms a gel. Fig. &I shows that such an extract develops a shear modulus near 1000 Pa after several hours. Fig. 8B shows that the cytoplasmic extract gel exhibits an increase in G' at increasing strains (strain hardening) and an abrupt rupture above 7% strain, similar to our previous results obtained with purified F-actin (9, 16). The composition of cell extract gels is complex, containing not only actin filaments but also actin binding proteins, tubulin, and other constituents. However, the composition of proteins associated with the easily sedimentable component of gelled extracts has been analyzed in several studies, and is mostly actin and substoichiometric amounts of actin cross-linking proteins such as ABP, a-actinin, fimbrin, and myosin (49-51). The formation of actomyosin rigor complexes as ATP is depleted may account for the very high elastic modulus observed at later times. 30 p~ cytochalasin B, which inhibits actin polymerization, completely prevents gela-

Elasticity of F-actin and Filament Diffusion 32509 A A

h

k v

c3

h

v k w

0.1 1 10 100 1000

Time ( s ) B

lo R -1

1000

100

10

1 0 50 100 150 200 250 300

Time (min) B

3000 1 t \ j

5001 actin:gelsolin 1

0.1 1 10 100 1000 Time ( s )

F-actin samples (1.7 mg/ml) without (A) or with ( B ) a 1:500 molar ratio FIG. 7. Stress relaxation of F-actin with and without gelsolin.

of gelsolin were polymerized for 1 h and then strained in less than 0.05 s to 10%. The shear modulus (ratio of stress to strain) is shown during the time that the stress relaxes in the samples.

tion, whereas 90 p~ phalloidin, a stabilizer of actin polymers, accelerates the initial increase in G (data not shown). These results confirm that the viscoelasticity of macrophage extract gels is dominated by the actin filaments within them.

Dynamic Light Scattering; Determination of Filament Bend- ing Stiffness-Taken together the viscoelastic properties of in- tact F-actin networks suggest that they are elastic because the filaments are so long that diffusion is severely restricted as predicted by theories of deGennes (17) and Doi and Edwards (181, even when the mass concentration of actin is low, and the average distance between filaments (the mesh size of the net- work) is large, on the order of a micrometer, and longer than the wavelength of visible light. Therefore, as recognized by Schmidt et al. (36), Piekenbrock et al. (39), and Piekenbrock and Sackmann (401, dynamic light scattering should be domi- nated by internal motions of individual filaments due to flex- ing, and therefore the persistence length or equivalently the bending energy of the filament can be obtained from the inten- sity autocorrelation of dynamic light scattering. We have per- formed DLS experiments at a variety of scattering angles and analyzed the results using Equations 5 and 6.

Fig. 9A shows the raw normalized intensity autocorrelation functions of light scattered from 6 p~ F-actin measured at angles from 30" to 150". When these data are replotted as a function of the scaling variable q2(tl(t))3/4, we obtain the curves shown in Fig. 9B. This analysis accounts for hydrodynamic interactions between different parts of the same filament, but not for effects between filaments. For short times and large

h

v

w F 0

0 1 2 3 4 5 6 Strain (%)

FIG. 8. Shear modulus of macrophage extract gel. Oscillatory measurements were made in a torsion pendulum at strains of less than 1% after warming the extract to room temperature (A) or at increasing strains 4 h after the gel had formed (B) .

scattering angles, the scaling of the data is good; however as noted in the theoretical discussion, at longer times and smaller angles, there are large deviations due to hydrodynamic inter- actions between filaments in the solution. We must therefore replot the experimental data to find a reliable value for the elastic constant of an actin filament. In accordance with Equa- tions 5 and 6, we plotted ln(l(t))/(l + A (tl(t))1'4) as a function of q2(tl(t))3" and adjusted the coeficient A so that the scaling of the data was restored. The results are shown in Fig. 9C. All of the curves now superimpose over the entire range of times and angles. The superposition of the curves confirms that the only interactions between filaments are purely hydrodynamic and not, for example, filament-filament cross-links or lateral align- ment of filaments into bundles. From the slope of the curve and from Equation 5 we can deduce a value of the persistence length of the actin filament of 16 pm, with an error of approxi- mately 30%. This value is in approximate agreement with measurements of this quantity by fluorescence microscopy (42, 52-54).

Fluorescence Microscopy of Intertwined Actin Filaments- The formation of an interpenetrating meshwork in which actin filaments are still able to diffuse by reptation was directly visualized by fluorescence microscopy. Fig. 1OA shows a typical solution of actin filaments. Even the small fraction of labeled filaments (ratio of labeled to unlabeled filaments 1:2000) illus- trates the entangled state of an equilibrated F-actin solution. Directly after pipetteting the sample onto the cover slide the solution of actin filaments exhibits a highly aligned state as shown in Fig. 1OB and the network needs 2-3 h to relax into its entangled, entropic state.

Despite the superficial resemblance of the filament intersec- tions to cross-links, these entanglements are not static cross-

32510 Elasticity of F-actin and Filament Diffusion

1 -

0.8 -

0.6 ;

0.4 -

0.2 -

n -

0 1 2 3 4 q' ( t / ~ ( t ) ) ~ ' ~ (l+A((q,t~1)/4q)"~) /1013 (SI units)

FIG. 9. Dynamic light scattering. Normalized intensity autocorre- lation functions were obtained a t angles of 30" to 150' from 6 g~ F-actin (A). Experimental conditions are described in the text. The data inA are replotted against reduced variables to account for hydrodynamic inter- actions within ( B ) and between ( C ) filaments according to Equations 5 and 6 as described in the text.

links, because the filaments can still diffuse. Individual fila- ments diffuse through the entangled mesh of surrounding F-actin by a snakelike sliding motion called reptation (19). The reptation motion of a 7-pm long rhodamine-phalloidin-labeled filament through a matrix of unlabeled filaments of mesh size 0.1 pm is shown in Fig. 11. F-actin of this average filament length, equivalent to a 1:1850 molar ratio of gelsolin, would have a shear modulus above 70 Pa, as shown in Fig. 2. The filament slides a distance which is comparable to its own length through the entangled network. This finding excludes the ex- istence of cross-links between the filaments, because these would limit motion of the filament to less than the mesh size of the network.

Breakage of Actin Filaments by Shear Flow-The conforma- tion of actin filaments is drastically affected by the bulk flow of an F-actin sample caused by macroscopic mechanical pertur- bations. If an F-actin solution is pipetted with a usual 1-ml Eppendorf pipette a large decrease in filament length occurs due to filament breakage. To illustrate this effect, and how easily breakage of actin filaments takes place, we pipetted a very dilute F-actin solution (2.5 nM, stabilized by rhodamine- phalloidin) with an Eppendorfpipette with two different diam- eters, in the first case we cut the top of the pipette tip and in the second case we used a normal 1-ml Eppendorf tip. The uncut

FIG. 10. Fluorescence microscopy of single actin filaments in an interpenetrating F-actin solution. A, Ihodamine-phalloidtn-la- beled actin filaments embedded in an unlabeled F-actin solution of 2.0 mg/ml concentration after 2.5 h of relaxation time. The filaments form a non-cross-linked, entangled network. B, rhodamine-phalloidin-la- beled actin filaments embedded in an unlabeled F-actin solution of 2.0 mg/ml concentration directly after the solution is pipetted onto the cover slide. The filaments are in a highly aligned state due to flow during pipetteting.

FK:. 11. Diffusion of an actin filament within an F-actin mesh- work. Rme sequence of the reptation motion of a i.l-pm long actin filament embedded in a solution of actin filaments of concentration c = 2.0 mg/ml.

pipette tip has a minimal diameter of -1 mm and the cut pipette tip diameter is -7 mm. To quantify the degree of fila- ment breakage we measured the length distribution by video

Elasticity of F-actin and Filament Diffusion 32511

FIG. 12. Effect of filament breakage. The upper panel shows fluorescence mi- crographs of a 2.5 nM rhodamine-phalloi- din-labeled F-actin solution pipetted with a Eppendorf-pipette of 0.7-cm tip diame- ter (left side) or with an Eppendorf pipette of 1-mm tip diameter (right side). The lowerpanel compares the resulting length distributions. In the case of the pipette with the narrow diameter =50% fila- ments breakage occurs as indicated by the change in the length distribution.

l i

microscopy. The use of an uncut pipette tip leads to a 50% reduction in the length distribution of the actin filaments in comparison to a cut pipette tip as shown in Fig. 12. Due to the very dilute F-actin solution few if any entanglements occur and the breakage is caused by friction forces between the polymer and the solvent. At higher concentrations, above the so-called semi-dilute transition, which for such filaments is on the order of 0.05 mg/ml (55) filarnenUfilament contacts would greatly increase the forces applied to each filament and therefore in- crease the probability that they break. This experiment shows the lower limit of filament breakage, and the effect should be larger for an entangled F-actin solution, where shear forces between filaments also contribute.

Reptational Diffusion of F-actin during Oscillatory Shear-If the actin meshwork is composed of individual filaments whose diffusion is retarded but not prevented, then lowering the fila- ment concentration and the rate of deformation should reveal a viscoelastic relaxation due to the center of mass diffusion of the long polymers through the meshwork by the reptation process shown in Fig. 11. Conventional rheometers are not sufficiently sensitive to measure such relaxation under these dilute condi- tions in oscillatory measurements, which are therefore nor- mally measured by creep experiments under constant stresses. However, a recently constructed high sensitivity rheometer specifically designed to measure very small stresses a t low shear rates and which avoids protein denaturation at the air interface by use of a phospholipid monolayer (35) shows that F-actin solutions do exhibit the rheology predicted for semi- flexible filaments. Fig. 13 shows the frequency dependent measurement of the storage modulus G' of an F-actin solution of -7 PM concentration. The measurement was performed over five frequency decades and three regimes are clearly visible. The low frequency region is characterized by the transition to fluid behavior indicated by a drop of G'(o). In this regime the network relaxes from strain by the reptation motion of single filaments. The plateau regime extending from 10"' rads < o < 10' r a d s resembles rubber-like elastic behavior. At higher fre- quencies G' again increases. In this regime the dynamic elas- ticity is determined by the internal dynamics of the actin fila- ments. The observed frequency dependence is in good agreement with the theoretical predictions (18) and was re- cently confirmed by microrheological measurements (56).

DISCUSSION Using actin independently prepared in several different labo-

ratories and measured in various instruments we have repeat- edly documented high elastic moduli, large G ' /G ratios, strain hardening, and large effects of actin fragmenting maneuvers on

7 i <I I / I 2 IS 17 19

filament length (pm)

I : . . I 7 h 7 x 9 1 0 filament length (prn)

;a

b p 0.1 -1 -

0.01 10.' I o-2 1 oo

Frequency (rad/?.)

FIG. 13. Rheology of F-actin at very low concentration. Fre- quency dependence of the storage modulus G' of an F-actin solution a t monomer concentration c = 300 pg/ml. The sample was polymerized a t 20 "C for 4 h, heated to 36 "C for 2 h, and cooled down again to 20 "C. The measurements were started after annealing a t 20 "C for 6 h.

reducing the elastic moduli. Similar rheologic properties were observed in an actin-rich cellular extract. For these reasons and since the actin used in our previous studies has been ex- tensively characterized in many other assays, contamination by cross-linking proteins, failure to purify and maintain native actin or other unspecified artifacts are either extremely preva- lent or unlikely to account for the differences between our find- ings and those of other reports concerning the rheology of F- actin. Errors in calibration are similarly ruled out since consistent results are obtained using free oscillations in torsion pendulums of different design, and several types of forced os- cillators, as well as a stress relaxation measurement. Operat- ing below the accurate range of force transducers can be a significant problem especially for samples with low shear moduli and at low strains and frequencies. Indeed our instru- ments would be incapable of measuring accurately materials with shear moduli of 1 Pa a t < Hz at strain amplitudes of a few percent, as reported in some studies (21, 44, 57).

The discrepant results in the literature appear rather to be due to the presence or absence of irrecoverable changes in the F-actin samples that can occur when the sample is placed in a device for making measurements or when an initial large de- formation is imposed on the sample prior to measurements. In particular, the very low elastic moduli reported by some groups, and the minimal dependence of these parameters on the degree of deformation of actin gels or the actions of filament severing proteins can be reproduced by our experimental methods if the gels are first broken or subjected to high flow rates.

Other findings corroborate the susceptibility of F-actin to breakage by shear forces. The flow of F-actin and of microtu-

32512 Elasticity of F-actin and Filament Diffusion

bules at high shear rates does not occur as predicted for the flow of filamentous polymers in solution, but rather occurs as a critical disruption of some complex structure within the solu- tion characteristic of an indeterminate fluid (9,58). The critical disruption during such flow is likely to be the breaking of in- dividual filaments and their ordering into bundles or filaments aligned with the flow field, as shown in Fig. 11. A transient overshoot in viscosity occurs when a rheometer is first turned on to high shear rates, and some of the lowering of apparent viscosity is recovered as the sample is left unperturbed, attrib- utable to filament reannealing (59). The results of Fig. 5B show a similar overshoot in resistance to flow, but at shear strains more than one order of magnitude smaller than those meas- ured previously in (59). Part of the large resistance to defor- mation that is destroyed by increasing the strain cannot be recovered when the sample is left unperturbed, confirming that once a sample has ruptured or aligned, subsequent measure- ments even at low shear strains do not measure the same material as was present initially.

The effects of shear flow on the structure of F-actin are most clearly seen in the fluorescence micrographs shown in Figs. 10-12. The highly aligned state of filaments immediately after even gentle pipetteting of a dilute actin solution indicates a high sensitivity of F-actin to shear alignment. Such alignment superficially resembles a transition to a liquid crystalline state (60-62), but is fundamentally different since it does not repre- sent a stable phase of the solution. Well defined rheological experiments can be only performed in the case of relaxed, en- tangled F-actin solutions. For the aligned states the measure- ments should result in an anisotropic shear modulus with a low value parallel and high value perpendicular to the aligned fila- ments. A second and possibly more important effect of shear flow on F-actin is the breakage of filaments that results from shear deformations of entangled actin filament samples. The forces needed to break an actin filament in uitro are less than one nanonewton (631, and such forces are achieved even when a dilute solution of F-actin is drawn through a pipette, or strained too far or too fast in a rheometer.'

Since viscoelastic moduli large enough to support physiologi- cally significant mechanical stresses can be measured in puri- fied F-actin solutions under well defined conditions, we con- clude that isotropic actin filament networks are intrinsically strong enough to account for many aspects of cell rheology and that recent reports that actin networks are extremely weak in the absence of cross-linking proteins are the result of filament breakage or alignment in flow resulting from the rheologic measurements themselves. While F-actin itself can have a large viscoelastic modulus in uitro, its properties are certainly modified by biochemical and mechanical factors in uiuo. For example, whether an actin filament is better modeled as a stiff rod or a semi-flexible chain depends on the filament length in comparison to the persistence length. Most fluorescence meas- urements of single filaments seek to maximize contour length for better imaging, and therefore the filaments look flexible because they are longer than their persistence length. How- ever, in cells extracts and in cytoskeletal networks in deter- gent-extracted cells, the average filament length (64) as well as the mesh size (65) are on the order of several hundred nm, and therefore the actin filament may be better modeled as a stiff rod. In addition to the structural alterations caused by many well-described actin-binding proteins, it is likely that the struc- ture of networks formed when actin polymerizes in a moving cell is also in part determined by the flow of the cytoplasm in which the filaments grow. Both the strains and strain-rates of cytoplasm in a living cell (66, 67) are within the range where

J. G s , unpublished results.

actin filaments align or break in uitro, and the formation of anisotropic structures in actin filament solutions can result from shear flow (59, 61, 68) or the presence of appropriate chemical conditions (60, 62).

Acknowledgments-We are grateful to Jennifer Lamb and Maura Shutt for help with some experiments and to Phil Allen and Jean- Francois Leterrier for critical reading of the manuscript.

1. 2. 3. 4. 5. 6. 7. 8.

9.

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11. 12.

13.

14. 15. 16.

17. 18.

19. 20.

21.

22. 23. 24.

25. 26. 27.

28.

29. 30. 31. 32. 33.

34.

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36.

37. 38. 39.

40. 41. 42.

43.

44.

45.

46.

47. 48. 49. 50. 51.

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